Pianotech

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

Kent! Thank you for speaking "my language". Everything you wrote makes perfect sense. Regarding your last three paragraphs, I didn't realize there were ETD's measuring more than one partial at a time. Good to know! I'm curious as to how they compromise. I figured they were getting there but I didn't realize how close they were getting. With the last part, thank you for saying that. It's very validating for me. To me, the arguments between those who primarily use fast beating intervals and those who primarily use slow beating intervals are as silly as those who argue ETD vs aural (even though I used to be one of those people...lol). I often add a caveat to help pad that, or to help others understand where I might be coming from with a particular topic as I did this time. You didn't need that and seemed to understand me completely. 

Thanks so much. I appreciate your input. :-) 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net

Original Message:
Sent: 04-01-2024 14:29
From: Kent Swafford
Subject: P12, ETD's, stretch, & the target (warning: long post)

Original Message:
Sent: 4/1/2024 12:08:00 PM
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Greetings, Maggie

You went: "To me, the arguments between those who primarily use fast beating intervals and those who primarily use slow beating intervals are as silly as those who argue ETD vs aural (even though I used to be one of those people...lol)"

I hope you're not referring to the conversation between Peter Grey and myself. Yes, an "argument" over whose choice of temperament is better, would be silly and a waste of time. But it was not an argument, but a sharing of ideas, in particular on the point in a temperament forces us to choose between even 4ths & 5ths or for even 3rds. We were of the same page there. I did mention that the 6ths were a bridge between the RBIs and the SBIs. Having started off with the "William Braid White" (essentially 4ths&5ths), I've spent the last 40 years enjoying what M3ds can do in mapping out the temperament compass at the outset.

No, I wouldn't dream of telling fellow NH Chapter member Peter that he's using the wrong temperament. Here we are, fellow townspeople in the village square. sharing ideas and experiences. Maybe though, you were referring to somebody else.

------------------------------
William Ballard RPT
WBPS
Saxtons River VT
802-869-9107

"Our lives contain a thousand springs
and dies if one be gone
Strange that a harp of a thousand strings
should keep in tune so long."
...........Dr. Watts, "The Continental Harmony,1774

Original Message:
Sent: 04-01-2024 19:26
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Kent! Thank you for speaking "my language". Everything you wrote makes perfect sense. Regarding your last three paragraphs, I didn't realize there were ETD's measuring more than one partial at a time. Good to know! I'm curious as to how they compromise. I figured they were getting there but I didn't realize how close they were getting. With the last part, thank you for saying that. It's very validating for me. To me, the arguments between those who primarily use fast beating intervals and those who primarily use slow beating intervals are as silly as those who argue ETD vs aural (even though I used to be one of those people...lol). I often add a caveat to help pad that, or to help others understand where I might be coming from with a particular topic as I did this time. You didn't need that and seemed to understand me completely. 

Thanks so much. I appreciate your input. :-) 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net

Original Message:
Sent: 04-01-2024 14:29
From: Kent Swafford
Subject: P12, ETD's, stretch, & the target (warning: long post)

Original Message:
Sent: 4/1/2024 12:08:00 PM
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Hi Bill - No need for paranoia. I haven't had time to be on the forum in many months and have never seen this conversation you speak of. I said what I did because I have heard people argue about this and have, myself, been criticized for using fast beating intervals when I tune. I don't understand why it's such a big deal. We each use whatever works for us. I'm just reacting to 1) those critics, and 2) those who don't understand how some others use fast beating intervals and see them as a very separate skill. As Kent pointed out: "I don't distinguish between slow interval checks and fast interval checks; they all have target beat rates that should be taken into consideration." That is a wise statement. 😉 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net

Original Message:
Sent: 04-01-2024 20:45
From: Bill Ballard
Subject: P12, ETD's, stretch, & the target (warning: long post)

Greetings, Maggie

You went: "To me, the arguments between those who primarily use fast beating intervals and those who primarily use slow beating intervals are as silly as those who argue ETD vs aural (even though I used to be one of those people...lol)"

I hope you're not referring to the conversation between Peter Grey and myself. Yes, an "argument" over whose choice of temperament is better, would be silly and a waste of time. But it was not an argument, but a sharing of ideas, in particular on the point in a temperament forces us to choose between even 4ths & 5ths or for even 3rds. We were of the same page there. I did mention that the 6ths were a bridge between the RBIs and the SBIs. Having started off with the "William Braid White" (essentially 4ths&5ths), I've spent the last 40 years enjoying what M3ds can do in mapping out the temperament compass at the outset.

No, I wouldn't dream of telling fellow NH Chapter member Peter that he's using the wrong temperament. Here we are, fellow townspeople in the village square. sharing ideas and experiences. Maybe though, you were referring to somebody else.

------------------------------
William Ballard RPT
WBPS
Saxtons River VT
802-869-9107

"Our lives contain a thousand springs
and dies if one be gone
Strange that a harp of a thousand strings
should keep in tune so long."
...........Dr. Watts, "The Continental Harmony,1774

Original Message:
Sent: 04-01-2024 19:26
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Kent! Thank you for speaking "my language". Everything you wrote makes perfect sense. Regarding your last three paragraphs, I didn't realize there were ETD's measuring more than one partial at a time. Good to know! I'm curious as to how they compromise. I figured they were getting there but I didn't realize how close they were getting. With the last part, thank you for saying that. It's very validating for me. To me, the arguments between those who primarily use fast beating intervals and those who primarily use slow beating intervals are as silly as those who argue ETD vs aural (even though I used to be one of those people...lol). I often add a caveat to help pad that, or to help others understand where I might be coming from with a particular topic as I did this time. You didn't need that and seemed to understand me completely. 

Thanks so much. I appreciate your input. :-) 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net

Original Message:
Sent: 04-01-2024 14:29
From: Kent Swafford
Subject: P12, ETD's, stretch, & the target (warning: long post)

Original Message:
Sent: 4/1/2024 12:08:00 PM
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

I'll reply to this little bit about ETDs "This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). "

Yes, it is usually up to the tech to investigate available stretch options or create their own custom stretches to best match the piano. What can happen, at least with PiaTune since it measures all of the A0-C7 notes for inharmonicity constants is that in a blink of an eye, the projected beat rates of important interval matches can be calculated and then scored to provide information to the tech as to what stretch is likely to have the best results (or least amount of problems) I believe this is an industry first. I paid $100 for the pro version and have no financial connection to the developer, though I am an active beta tester/feature suggester.

Ron Koval

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

Ron - Thank you for enlightening me about more than one partial being read at a time. You and Kent have made me aware of that which is fascinating. As far as stretch, I get what you are saying. I think it's amazing what these new ETD's can do. What I'm curious about is if programs like the one you describe can compensate for shifts in inharmonicity. I know it's coming. I just haven't seen one yet that does an exceptional job (to my tastes, because there are always several choices depending on the personal goal of balance one is looking for). I haven't tried your program but have tried several others. I may look into it, but if I have to measure every note, I may skip it. By the time I've measured every note, I could have a temperament done & be working on the midrange. I'm not curious enough to spend that much time on it....yet. I'm sure I will be eventually. When I hear that they really can compensate for inharmonicity shifts, I'll probably make time to play with some. ;-)  Thanks! :-) 

I'll reply to this little bit about ETDs "This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). "

Yes, it is usually up to the tech to investigate available stretch options or create their own custom stretches to best match the piano. What can happen, at least with PiaTune since it measures all of the A0-C7 notes for inharmonicity constants is that in a blink of an eye, the projected beat rates of important interval matches can be calculated and then scored to provide information to the tech as to what stretch is likely to have the best results (or least amount of problems) I believe this is an industry first. I paid $100 for the pro version and have no financial connection to the developer, though I am an active beta tester/feature suggester.

Ron Koval

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

Maggie - there are a couple of different aspects to the term 'multiple partial' ... Yes, ever since the Verituner decades ago, it has been possible for an ETD to directly respond and plan around shifts in inharmonicity with varying degrees of success. The current new crop (PianoMeter, PiaTune and PianoScope) use a quicker measuring by calculating inharmonicity constants for each note instead of direct measurement of the 'spaces' between the individual partials. That is one way the term multiple partials is used. Older ETDs also used inharmonicity constants, but had to make educated guesses when filling in the blanks between the notes measured. Sometimes with great success, often not...

The other possible use of the term 'multiple partials' is by generating targets for many partials instead of just one to be used for tuning - almost like having many spinners active that then get 'boiled down' to driving the main spinner. When the data is accurate and the calculations proper, all of the spinners should agree. But since things aren't ever perfect, the different decay rate of partials can lead to display instability when all of those spinners don't agree and the main display gets confusing data.

So yes, all of the current crop of apps are responsive to inharmonicity breaks and blips, and it is best to measure the notes between A0-C7 first. This usually takes under 5 minutes the first time, and then can be used for repeat visits to the same piano. (Although there is an aural approach option in PiaTune which allows you to begin tuning right away building the temperament between F3-A4. Then that data can be used to work out towards the bass and treble. The analyzing/suggestion of the "best" stretch won't work until all the notes have been tuned, so I've never used this method) I've been regularly pushing for better use of the data not only for good scales, but felt this should be most helpful on challenging pianos. It is still a work in progress!

Ron Koval

Ron - Thank you for enlightening me about more than one partial being read at a time. You and Kent have made me aware of that which is fascinating. As far as stretch, I get what you are saying. I think it's amazing what these new ETD's can do. What I'm curious about is if programs like the one you describe can compensate for shifts in inharmonicity. I know it's coming. I just haven't seen one yet that does an exceptional job (to my tastes, because there are always several choices depending on the personal goal of balance one is looking for). I haven't tried your program but have tried several others. I may look into it, but if I have to measure every note, I may skip it. By the time I've measured every note, I could have a temperament done & be working on the midrange. I'm not curious enough to spend that much time on it....yet. I'm sure I will be eventually. When I hear that they really can compensate for inharmonicity shifts, I'll probably make time to play with some. ;-)  Thanks! :-) 

I'll reply to this little bit about ETDs "This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). "

Yes, it is usually up to the tech to investigate available stretch options or create their own custom stretches to best match the piano. What can happen, at least with PiaTune since it measures all of the A0-C7 notes for inharmonicity constants is that in a blink of an eye, the projected beat rates of important interval matches can be calculated and then scored to provide information to the tech as to what stretch is likely to have the best results (or least amount of problems) I believe this is an industry first. I paid $100 for the pro version and have no financial connection to the developer, though I am an active beta tester/feature suggester.

Ron Koval

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

Maggie - there are a couple of different aspects to the term 'multiple partial' ... Yes, ever since the Verituner decades ago, it has been possible for an ETD to directly respond and plan around shifts in inharmonicity with varying degrees of success. The current new crop (PianoMeter, PiaTune and PianoScope) use a quicker measuring by calculating inharmonicity constants for each note instead of direct measurement of the 'spaces' between the individual partials. That is one way the term multiple partials is used. Older ETDs also used inharmonicity constants, but had to make educated guesses when filling in the blanks between the notes measured. Sometimes with great success, often not...

The other possible use of the term 'multiple partials' is by generating targets for many partials instead of just one to be used for tuning - almost like having many spinners active that then get 'boiled down' to driving the main spinner. When the data is accurate and the calculations proper, all of the spinners should agree. But since things aren't ever perfect, the different decay rate of partials can lead to display instability when all of those spinners don't agree and the main display gets confusing data.

So yes, all of the current crop of apps are responsive to inharmonicity breaks and blips, and it is best to measure the notes between A0-C7 first. This usually takes under 5 minutes the first time, and then can be used for repeat visits to the same piano. (Although there is an aural approach option in PiaTune which allows you to begin tuning right away building the temperament between F3-A4. Then that data can be used to work out towards the bass and treble. The analyzing/suggestion of the "best" stretch won't work until all the notes have been tuned, so I've never used this method) I've been regularly pushing for better use of the data not only for good scales, but felt this should be most helpful on challenging pianos. It is still a work in progress!

Ron Koval

Ron - Thank you for enlightening me about more than one partial being read at a time. You and Kent have made me aware of that which is fascinating. As far as stretch, I get what you are saying. I think it's amazing what these new ETD's can do. What I'm curious about is if programs like the one you describe can compensate for shifts in inharmonicity. I know it's coming. I just haven't seen one yet that does an exceptional job (to my tastes, because there are always several choices depending on the personal goal of balance one is looking for). I haven't tried your program but have tried several others. I may look into it, but if I have to measure every note, I may skip it. By the time I've measured every note, I could have a temperament done & be working on the midrange. I'm not curious enough to spend that much time on it....yet. I'm sure I will be eventually. When I hear that they really can compensate for inharmonicity shifts, I'll probably make time to play with some. ;-)  Thanks! :-) 

I'll reply to this little bit about ETDs "This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). "

Yes, it is usually up to the tech to investigate available stretch options or create their own custom stretches to best match the piano. What can happen, at least with PiaTune since it measures all of the A0-C7 notes for inharmonicity constants is that in a blink of an eye, the projected beat rates of important interval matches can be calculated and then scored to provide information to the tech as to what stretch is likely to have the best results (or least amount of problems) I believe this is an industry first. I paid $100 for the pro version and have no financial connection to the developer, though I am an active beta tester/feature suggester.

Ron Koval

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Thanks for the well-thought out and interesting post.  It's gotten me rethinking some of my own assumptions about how I've been tuning...  But let me jump straight to the italicized bit (If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears) so I don't get onto too many tangents.

First off, let me restate the problem in my own words, to make sure we're talking about the same thing.

Assumptions:

• • We're tuning from bottom to top, for simplicity
  • The low bass has perfectly smooth inharmonicity, again for simplicity 
  • The fundamental frequencies of notes F2 through C3 are already tuned to the notes below, so they follow a perfectly smooth progression.
  • There's a break in the scaling between F2 and C3, where F2 has very low inharmonicity, and C3 has very high inharmonicity
  • The inharmonicity above C3 is smooth.

We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks.  So

Tune C4 to F2, which had low inharmonicity.  C4 will come out relatively flat.

Tune G4 to C3, which had high inharmonicity.  G4 will come out relatively sharp.

If the inharmonicity difference is great enough, C4-G4 will be an inverted (wide) 5th.

You could also go the other way.  I.e. if C4-G4 fundamentals were instead tuned smoothly, and you tuned P12's down to F2-C3 you'd end up with an inverted 5th there instead.  I suppose in practice we ought to split the difference some kind of way, but I don't know that P12ET really has a prescription for how to do that (Kent or anybody else please correct me if I'm wrong).  Edit: regardless, there's gonna be a breaking point where at least one of the P5's will be inverted now matter how you split the difference.

Are inharmonicity breaks on some pianos bad enough to cause this to happen?  Yes they are.  ET P5's are only contracted by -1.96c to begin with.  I have a tuning file from a particularly nasty little Howard spinet where the generated tuning offset for a 3:1 midrange goes from -6.63c at G3 to -2.82c at D4.  So there's your expanded 5th.

But here's the thing I'm still kinda chewing on... The traditional 4:2 octave can totally suffer from the same phenomenon.  i.e.

(same assumptions)

Tune F3 to F2 dead pure at 4:2.  F3 comes out relatively flat.

Tune C4 to C3 dead pure at 4:2.  C4 comes out relatively sharp.

Again, we have the potential for an inverted (wide) 5th at F3-C4, assuming the break in inharmonicity is enough.  That begs the question, is the 4:2 octave more or less reactive to inharmonicity than the 3:1 P12?  This is harder to calculate than it seems at first blush, because these first few partials don't exactly follow the "expected" mathematical formula for pitch offsets -- apparently the motion at the bridge causes them to be flatter (i.e. less inharmonious) than they would be if completely rigidly anchored at both ends.  So it's probably going to take some empirical data to answer that question, assuming it hasn't already been answered.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-01-2024 12:08
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Thanks for the well-thought out and interesting post.  It's gotten me rethinking some of my own assumptions about how I've been tuning...  But let me jump straight to the italicized bit (If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears) so I don't get onto too many tangents.

First off, let me restate the problem in my own words, to make sure we're talking about the same thing.

Assumptions:

• • We're tuning from bottom to top, for simplicity
  • The low bass has perfectly smooth inharmonicity, again for simplicity 
  • The fundamental frequencies of notes F2 through C3 are already tuned to the notes below, so they follow a perfectly smooth progression.
  • There's a break in the scaling between F2 and C3, where F2 has very low inharmonicity, and C3 has very high inharmonicity
  • The inharmonicity above C3 is smooth.

We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks.  So

Tune C4 to F2, which had low inharmonicity.  C4 will come out relatively flat.

Tune G4 to C3, which had high inharmonicity.  G4 will come out relatively sharp.

If the inharmonicity difference is great enough, C4-G4 will be an inverted (wide) 5th.

You could also go the other way.  I.e. if C4-G4 fundamentals were instead tuned smoothly, and you tuned P12's down to F2-C3 you'd end up with an inverted 5th there instead.  I suppose in practice we ought to split the difference some kind of way, but I don't know that P12ET really has a prescription for how to do that (Kent or anybody else please correct me if I'm wrong).  Edit: regardless, there's gonna be a breaking point where at least one of the P5's will be inverted now matter how you split the difference.

Are inharmonicity breaks on some pianos bad enough to cause this to happen?  Yes they are.  ET P5's are only contracted by -1.96c to begin with.  I have a tuning file from a particularly nasty little Howard spinet where the generated tuning offset for a 3:1 midrange goes from -6.63c at G3 to -2.82c at D4.  So there's your expanded 5th.

But here's the thing I'm still kinda chewing on... The traditional 4:2 octave can totally suffer from the same phenomenon.  i.e.

(same assumptions)

Tune F3 to F2 dead pure at 4:2.  F3 comes out relatively flat.

Tune C4 to C3 dead pure at 4:2.  C4 comes out relatively sharp.

Again, we have the potential for an inverted (wide) 5th at F3-C4, assuming the break in inharmonicity is enough.  That begs the question, is the 4:2 octave more or less reactive to inharmonicity than the 3:1 P12?  This is harder to calculate than it seems at first blush, because these first few partials don't exactly follow the "expected" mathematical formula for pitch offsets -- apparently the motion at the bridge causes them to be flatter (i.e. less inharmonious) than they would be if completely rigidly anchored at both ends.  So it's probably going to take some empirical data to answer that question, assuming it hasn't already been answered.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-01-2024 12:08
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Ron - Can you elaborate on this: "calculating inharmonicity constants for each note instead of direct measurement of the 'spaces' between the individual partials"

Nathan - That was a really thoughtful reply, and a REALLY good question about octaves! The way I tune, I see slow beating intervals as flexible to compensate for these things, so focus on the other intervals just as much. I understand your "assumptions" sections is for the sake of discussion. However, the sentence right after that, "We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks." is not something I would do. I suppose that's why I'm so interested in this, because my other interval checks tell me when I can't push a 12th or an octave too far. I do listen to larger intervals, but I fix most things with many intervals in mind. I care an awful lot about not inverting fifths & fourths. I should care as much about octaves, I suppose. I don't think I tend to invert them, but I do keep them flexible. For example, If I am tuning into the bass and the fifth, fourth, and fast beating intervals put the note in a particular spot, but the octave comes out closer to a 2:1 for whatever reason (let's say there is a drop in inharmonicity that is affecting upper partial matches), as long as it's not inverted, I'll leave it even if the octaves around it are around a 6:3. If the other intervals work, and it sounds good when played in a harmonic context, it stays. 

I'm curious to see what others think. Hopefully I'll have time to keep up! LOL

Thanks for such a thoughtful post!!! 

Thanks for the well-thought out and interesting post.  It's gotten me rethinking some of my own assumptions about how I've been tuning...  But let me jump straight to the italicized bit (If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears) so I don't get onto too many tangents.

First off, let me restate the problem in my own words, to make sure we're talking about the same thing.

Assumptions:

• • We're tuning from bottom to top, for simplicity
  • The low bass has perfectly smooth inharmonicity, again for simplicity 
  • The fundamental frequencies of notes F2 through C3 are already tuned to the notes below, so they follow a perfectly smooth progression.
  • There's a break in the scaling between F2 and C3, where F2 has very low inharmonicity, and C3 has very high inharmonicity
  • The inharmonicity above C3 is smooth.

We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks.  So

Tune C4 to F2, which had low inharmonicity.  C4 will come out relatively flat.

Tune G4 to C3, which had high inharmonicity.  G4 will come out relatively sharp.

If the inharmonicity difference is great enough, C4-G4 will be an inverted (wide) 5th.

You could also go the other way.  I.e. if C4-G4 fundamentals were instead tuned smoothly, and you tuned P12's down to F2-C3 you'd end up with an inverted 5th there instead.  I suppose in practice we ought to split the difference some kind of way, but I don't know that P12ET really has a prescription for how to do that (Kent or anybody else please correct me if I'm wrong).  Edit: regardless, there's gonna be a breaking point where at least one of the P5's will be inverted now matter how you split the difference.

Are inharmonicity breaks on some pianos bad enough to cause this to happen?  Yes they are.  ET P5's are only contracted by -1.96c to begin with.  I have a tuning file from a particularly nasty little Howard spinet where the generated tuning offset for a 3:1 midrange goes from -6.63c at G3 to -2.82c at D4.  So there's your expanded 5th.

But here's the thing I'm still kinda chewing on... The traditional 4:2 octave can totally suffer from the same phenomenon.  i.e.

(same assumptions)

Tune F3 to F2 dead pure at 4:2.  F3 comes out relatively flat.

Tune C4 to C3 dead pure at 4:2.  C4 comes out relatively sharp.

Again, we have the potential for an inverted (wide) 5th at F3-C4, assuming the break in inharmonicity is enough.  That begs the question, is the 4:2 octave more or less reactive to inharmonicity than the 3:1 P12?  This is harder to calculate than it seems at first blush, because these first few partials don't exactly follow the "expected" mathematical formula for pitch offsets -- apparently the motion at the bridge causes them to be flatter (i.e. less inharmonious) than they would be if completely rigidly anchored at both ends.  So it's probably going to take some empirical data to answer that question, assuming it hasn't already been answered.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-01-2024 12:08
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Hi Maggie, I'll give it a try. 

I'm sure you are familiar with partials and how inharmonicity causes the sounded pitch of higher partials to be higher than a note without inharmonicity.

If you graph how many cents higher than expected for each partial and then pass a straight trend line through for the slope, I believe you get something like an inharmonicity constant. That would be a way using math to predict the location of any partial from another. (hope this comes through from pianoworld)  

This is note F2 from a Kawai grand. The slope would make the inharmonicity constant for this note 0.097

This information is then the basis for calculating beatspeeds, interval matches, stretch etc.  

If the graph came through, you should be able to see that some of the partial offsets don't actually touch the line... that's what Verituner tried to avoid by actually using the measurements for each partial and then the 'space' or difference between the offsets for each partial to make it's calculations. It was a way to try to improve on the inharmonicity constant approach. Slower measuring is one of the drawbacks.

The topic not really talked about much is the "flip side" to any of the apps that use all of the measurements to make a more customized tuning. What if one, more or many of the measurements taken at the piano aren't really accurate. I've measured differences from multiple measuring passes that will make up to around 0.3 cents difference in some of the notes calculated. If the measuring pass was sloppy, that can increase to over a cent. It should give us pause when we talk about tuning to within 0.1 cents or closer ---- closer to what?(not such an issue for unison tuning) That's why the graphing capabilities of some of the newer apps are important to use so that we can catch errors before the piano gets tuned. 

Ron Koval

------------------------------
Ron Koval
CHICAGO IL

Original Message:
Sent: 04-06-2024 12:12
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Ron - Can you elaborate on this: "calculating inharmonicity constants for each note instead of direct measurement of the 'spaces' between the individual partials"

Nathan - That was a really thoughtful reply, and a REALLY good question about octaves! The way I tune, I see slow beating intervals as flexible to compensate for these things, so focus on the other intervals just as much. I understand your "assumptions" sections is for the sake of discussion. However, the sentence right after that, "We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks." is not something I would do. I suppose that's why I'm so interested in this, because my other interval checks tell me when I can't push a 12th or an octave too far. I do listen to larger intervals, but I fix most things with many intervals in mind. I care an awful lot about not inverting fifths & fourths. I should care as much about octaves, I suppose. I don't think I tend to invert them, but I do keep them flexible. For example, If I am tuning into the bass and the fifth, fourth, and fast beating intervals put the note in a particular spot, but the octave comes out closer to a 2:1 for whatever reason (let's say there is a drop in inharmonicity that is affecting upper partial matches), as long as it's not inverted, I'll leave it even if the octaves around it are around a 6:3. If the other intervals work, and it sounds good when played in a harmonic context, it stays. 

I'm curious to see what others think. Hopefully I'll have time to keep up! LOL

Thanks for such a thoughtful post!!! 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net

Original Message:
Sent: 04-02-2024 10:48
From: Nathan Monteleone
Subject: P12, ETD's, stretch, & the target (warning: long post)

Thanks for the well-thought out and interesting post.  It's gotten me rethinking some of my own assumptions about how I've been tuning...  But let me jump straight to the italicized bit (If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears) so I don't get onto too many tangents.

First off, let me restate the problem in my own words, to make sure we're talking about the same thing.

Assumptions:

• • We're tuning from bottom to top, for simplicity
  • The low bass has perfectly smooth inharmonicity, again for simplicity 
  • The fundamental frequencies of notes F2 through C3 are already tuned to the notes below, so they follow a perfectly smooth progression.
  • There's a break in the scaling between F2 and C3, where F2 has very low inharmonicity, and C3 has very high inharmonicity
  • The inharmonicity above C3 is smooth.

We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks.  So

Tune C4 to F2, which had low inharmonicity.  C4 will come out relatively flat.

Tune G4 to C3, which had high inharmonicity.  G4 will come out relatively sharp.

If the inharmonicity difference is great enough, C4-G4 will be an inverted (wide) 5th.

You could also go the other way.  I.e. if C4-G4 fundamentals were instead tuned smoothly, and you tuned P12's down to F2-C3 you'd end up with an inverted 5th there instead.  I suppose in practice we ought to split the difference some kind of way, but I don't know that P12ET really has a prescription for how to do that (Kent or anybody else please correct me if I'm wrong).  Edit: regardless, there's gonna be a breaking point where at least one of the P5's will be inverted now matter how you split the difference.

Are inharmonicity breaks on some pianos bad enough to cause this to happen?  Yes they are.  ET P5's are only contracted by -1.96c to begin with.  I have a tuning file from a particularly nasty little Howard spinet where the generated tuning offset for a 3:1 midrange goes from -6.63c at G3 to -2.82c at D4.  So there's your expanded 5th.

But here's the thing I'm still kinda chewing on... The traditional 4:2 octave can totally suffer from the same phenomenon.  i.e.

(same assumptions)

Tune F3 to F2 dead pure at 4:2.  F3 comes out relatively flat.

Tune C4 to C3 dead pure at 4:2.  C4 comes out relatively sharp.

Again, we have the potential for an inverted (wide) 5th at F3-C4, assuming the break in inharmonicity is enough.  That begs the question, is the 4:2 octave more or less reactive to inharmonicity than the 3:1 P12?  This is harder to calculate than it seems at first blush, because these first few partials don't exactly follow the "expected" mathematical formula for pitch offsets -- apparently the motion at the bridge causes them to be flatter (i.e. less inharmonious) than they would be if completely rigidly anchored at both ends.  So it's probably going to take some empirical data to answer that question, assuming it hasn't already been answered.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-01-2024 12:08
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Maggie,

I too have been beta testing Piatune and I'm liking many of it's features. One in particular is that when I don't agree with a decision it makes I can quickly bend it to my will with a custom offset. The feature is quick and easy (as opposed to other ETD's) and I move on. 

Also, it listens only to the note it is set to listen to which allows me to use all of my analog tests to confirm/deny/adjust anything it tries to do. I am in control (sounds ominous but it's necessary). I refuse to relegate all thinking to the machine. It is my tool and helper...and that's the end of it. 

The developer is also quick to accept I put and suggestions and implement what he agrees with. 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 04-06-2024 21:32
From: Ron Koval
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi Maggie, I'll give it a try. 

I'm sure you are familiar with partials and how inharmonicity causes the sounded pitch of higher partials to be higher than a note without inharmonicity.

If you graph how many cents higher than expected for each partial and then pass a straight trend line through for the slope, I believe you get something like an inharmonicity constant. That would be a way using math to predict the location of any partial from another. (hope this comes through from pianoworld)  

This is note F2 from a Kawai grand. The slope would make the inharmonicity constant for this note 0.097

This information is then the basis for calculating beatspeeds, interval matches, stretch etc.  

If the graph came through, you should be able to see that some of the partial offsets don't actually touch the line... that's what Verituner tried to avoid by actually using the measurements for each partial and then the 'space' or difference between the offsets for each partial to make it's calculations. It was a way to try to improve on the inharmonicity constant approach. Slower measuring is one of the drawbacks.

The topic not really talked about much is the "flip side" to any of the apps that use all of the measurements to make a more customized tuning. What if one, more or many of the measurements taken at the piano aren't really accurate. I've measured differences from multiple measuring passes that will make up to around 0.3 cents difference in some of the notes calculated. If the measuring pass was sloppy, that can increase to over a cent. It should give us pause when we talk about tuning to within 0.1 cents or closer ---- closer to what?(not such an issue for unison tuning) That's why the graphing capabilities of some of the newer apps are important to use so that we can catch errors before the piano gets tuned. 

Ron Koval

------------------------------
Ron Koval
CHICAGO IL

Original Message:
Sent: 04-06-2024 12:12
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Ron - Can you elaborate on this: "calculating inharmonicity constants for each note instead of direct measurement of the 'spaces' between the individual partials"

Nathan - That was a really thoughtful reply, and a REALLY good question about octaves! The way I tune, I see slow beating intervals as flexible to compensate for these things, so focus on the other intervals just as much. I understand your "assumptions" sections is for the sake of discussion. However, the sentence right after that, "We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks." is not something I would do. I suppose that's why I'm so interested in this, because my other interval checks tell me when I can't push a 12th or an octave too far. I do listen to larger intervals, but I fix most things with many intervals in mind. I care an awful lot about not inverting fifths & fourths. I should care as much about octaves, I suppose. I don't think I tend to invert them, but I do keep them flexible. For example, If I am tuning into the bass and the fifth, fourth, and fast beating intervals put the note in a particular spot, but the octave comes out closer to a 2:1 for whatever reason (let's say there is a drop in inharmonicity that is affecting upper partial matches), as long as it's not inverted, I'll leave it even if the octaves around it are around a 6:3. If the other intervals work, and it sounds good when played in a harmonic context, it stays. 

I'm curious to see what others think. Hopefully I'll have time to keep up! LOL

Thanks for such a thoughtful post!!! 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net

Original Message:
Sent: 04-02-2024 10:48
From: Nathan Monteleone
Subject: P12, ETD's, stretch, & the target (warning: long post)

Thanks for the well-thought out and interesting post.  It's gotten me rethinking some of my own assumptions about how I've been tuning...  But let me jump straight to the italicized bit (If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears) so I don't get onto too many tangents.

First off, let me restate the problem in my own words, to make sure we're talking about the same thing.

Assumptions:

• • We're tuning from bottom to top, for simplicity
  • The low bass has perfectly smooth inharmonicity, again for simplicity 
  • The fundamental frequencies of notes F2 through C3 are already tuned to the notes below, so they follow a perfectly smooth progression.
  • There's a break in the scaling between F2 and C3, where F2 has very low inharmonicity, and C3 has very high inharmonicity
  • The inharmonicity above C3 is smooth.

We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks.  So

Tune C4 to F2, which had low inharmonicity.  C4 will come out relatively flat.

Tune G4 to C3, which had high inharmonicity.  G4 will come out relatively sharp.

If the inharmonicity difference is great enough, C4-G4 will be an inverted (wide) 5th.

You could also go the other way.  I.e. if C4-G4 fundamentals were instead tuned smoothly, and you tuned P12's down to F2-C3 you'd end up with an inverted 5th there instead.  I suppose in practice we ought to split the difference some kind of way, but I don't know that P12ET really has a prescription for how to do that (Kent or anybody else please correct me if I'm wrong).  Edit: regardless, there's gonna be a breaking point where at least one of the P5's will be inverted now matter how you split the difference.

Are inharmonicity breaks on some pianos bad enough to cause this to happen?  Yes they are.  ET P5's are only contracted by -1.96c to begin with.  I have a tuning file from a particularly nasty little Howard spinet where the generated tuning offset for a 3:1 midrange goes from -6.63c at G3 to -2.82c at D4.  So there's your expanded 5th.

But here's the thing I'm still kinda chewing on... The traditional 4:2 octave can totally suffer from the same phenomenon.  i.e.

(same assumptions)

Tune F3 to F2 dead pure at 4:2.  F3 comes out relatively flat.

Tune C4 to C3 dead pure at 4:2.  C4 comes out relatively sharp.

Again, we have the potential for an inverted (wide) 5th at F3-C4, assuming the break in inharmonicity is enough.  That begs the question, is the 4:2 octave more or less reactive to inharmonicity than the 3:1 P12?  This is harder to calculate than it seems at first blush, because these first few partials don't exactly follow the "expected" mathematical formula for pitch offsets -- apparently the motion at the bridge causes them to be flatter (i.e. less inharmonious) than they would be if completely rigidly anchored at both ends.  So it's probably going to take some empirical data to answer that question, assuming it hasn't already been answered.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-01-2024 12:08
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Ron - Can you elaborate on this: "calculating inharmonicity constants for each note instead of direct measurement of the 'spaces' between the individual partials"

Nathan - That was a really thoughtful reply, and a REALLY good question about octaves! The way I tune, I see slow beating intervals as flexible to compensate for these things, so focus on the other intervals just as much. I understand your "assumptions" sections is for the sake of discussion. However, the sentence right after that, "We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks." is not something I would do. I suppose that's why I'm so interested in this, because my other interval checks tell me when I can't push a 12th or an octave too far. I do listen to larger intervals, but I fix most things with many intervals in mind. I care an awful lot about not inverting fifths & fourths. I should care as much about octaves, I suppose. I don't think I tend to invert them, but I do keep them flexible. For example, If I am tuning into the bass and the fifth, fourth, and fast beating intervals put the note in a particular spot, but the octave comes out closer to a 2:1 for whatever reason (let's say there is a drop in inharmonicity that is affecting upper partial matches), as long as it's not inverted, I'll leave it even if the octaves around it are around a 6:3. If the other intervals work, and it sounds good when played in a harmonic context, it stays. 

I'm curious to see what others think. Hopefully I'll have time to keep up! LOL

Thanks for such a thoughtful post!!! 

Thanks for the well-thought out and interesting post.  It's gotten me rethinking some of my own assumptions about how I've been tuning...  But let me jump straight to the italicized bit (If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears) so I don't get onto too many tangents.

First off, let me restate the problem in my own words, to make sure we're talking about the same thing.

Assumptions:

• • We're tuning from bottom to top, for simplicity
  • The low bass has perfectly smooth inharmonicity, again for simplicity 
  • The fundamental frequencies of notes F2 through C3 are already tuned to the notes below, so they follow a perfectly smooth progression.
  • There's a break in the scaling between F2 and C3, where F2 has very low inharmonicity, and C3 has very high inharmonicity
  • The inharmonicity above C3 is smooth.

We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks.  So

Tune C4 to F2, which had low inharmonicity.  C4 will come out relatively flat.

Tune G4 to C3, which had high inharmonicity.  G4 will come out relatively sharp.

If the inharmonicity difference is great enough, C4-G4 will be an inverted (wide) 5th.

You could also go the other way.  I.e. if C4-G4 fundamentals were instead tuned smoothly, and you tuned P12's down to F2-C3 you'd end up with an inverted 5th there instead.  I suppose in practice we ought to split the difference some kind of way, but I don't know that P12ET really has a prescription for how to do that (Kent or anybody else please correct me if I'm wrong).  Edit: regardless, there's gonna be a breaking point where at least one of the P5's will be inverted now matter how you split the difference.

Are inharmonicity breaks on some pianos bad enough to cause this to happen?  Yes they are.  ET P5's are only contracted by -1.96c to begin with.  I have a tuning file from a particularly nasty little Howard spinet where the generated tuning offset for a 3:1 midrange goes from -6.63c at G3 to -2.82c at D4.  So there's your expanded 5th.

But here's the thing I'm still kinda chewing on... The traditional 4:2 octave can totally suffer from the same phenomenon.  i.e.

(same assumptions)

Tune F3 to F2 dead pure at 4:2.  F3 comes out relatively flat.

Tune C4 to C3 dead pure at 4:2.  C4 comes out relatively sharp.

Again, we have the potential for an inverted (wide) 5th at F3-C4, assuming the break in inharmonicity is enough.  That begs the question, is the 4:2 octave more or less reactive to inharmonicity than the 3:1 P12?  This is harder to calculate than it seems at first blush, because these first few partials don't exactly follow the "expected" mathematical formula for pitch offsets -- apparently the motion at the bridge causes them to be flatter (i.e. less inharmonious) than they would be if completely rigidly anchored at both ends.  So it's probably going to take some empirical data to answer that question, assuming it hasn't already been answered.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-01-2024 12:08
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Thanks Maggie :)

Just to make sure it didn't come across wrong: I did not think you were saying that you tune the way I described with pure 3:1's. Rather the opposite, I understand you find it de-optimizes the 5ths (and likely other relevant intervals) too much when large changes in inharmonicity are involved. Just wanted to make sure we were thinking of the same situation, and it sounds like we are.

On the subject of octaves... I suppose one difference between P12ET and traditional octave temperaments, is that we're rather accustomed to "feathering" the octave sizes as we progress through the piano, i.e. 6:3 (or wider) in the bass, 4:2 for the temperament octave, and something a bit narrower as we get up to the top. So perhaps it's always been natural for aural tuners to do as you say, and narrow down to something more like a 2:1 if it helps reduce nastiness at the break.

(You probably know all of this, I'm just restating for clarity:) One of the appeals of P12ET -- on a piano with a smooth international curve anyway -- is that it sort of avoids having to do this ad-hoc feathering and instead hopes to come up with a good compromise by just making the 12ths as pure-sounding at possible. There are still issues with choosing between 3:1, 6:2, and even 9:3 coincident partials when doing this, but I would give 12ths the edge here in that for most of the piano, you can achieve this by simply targeting the 3:1 level. In theory it's part of what gives wide chords and dampers-up sympathetic vibration such a nice character -- at least in the opinion of us P12ET nerds ;)

But your original question hits on something else entirely of course -- what do we do about inharmonicity breaks when they happen? Based on the advantage I mentioned in the previous paragraph, I think maybe we tend to assume, incorrectly, that P12ET somehow holds an edge here as well. But from the discussion do far, I don't think P12ET is dramatically better or worse than the traditional progression of octave sizes at dealing with this problem. With inharmonicity breaks, either one can mess up P5s and other important intervals if you follow their prescription legalistically. Does P12ET lose its advantages if you compromise a few of the 12ths to smooth out a break? In my experience no, simply because the sound I like from it is fairly resilient to normal tuning errors. IOW I think an attentive aural tuner -- or clever ETD algorithm -- can make improvements around an inharmonicity break without doing harm to a P12ET tuning. 

This is, more or less, supposed to be the same thing Kent said, just with a wordier justification.

Ron - Can you elaborate on this: "calculating inharmonicity constants for each note instead of direct measurement of the 'spaces' between the individual partials"

Nathan - That was a really thoughtful reply, and a REALLY good question about octaves! The way I tune, I see slow beating intervals as flexible to compensate for these things, so focus on the other intervals just as much. I understand your "assumptions" sections is for the sake of discussion. However, the sentence right after that, "We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks." is not something I would do. I suppose that's why I'm so interested in this, because my other interval checks tell me when I can't push a 12th or an octave too far. I do listen to larger intervals, but I fix most things with many intervals in mind. I care an awful lot about not inverting fifths & fourths. I should care as much about octaves, I suppose. I don't think I tend to invert them, but I do keep them flexible. For example, If I am tuning into the bass and the fifth, fourth, and fast beating intervals put the note in a particular spot, but the octave comes out closer to a 2:1 for whatever reason (let's say there is a drop in inharmonicity that is affecting upper partial matches), as long as it's not inverted, I'll leave it even if the octaves around it are around a 6:3. If the other intervals work, and it sounds good when played in a harmonic context, it stays. 

I'm curious to see what others think. Hopefully I'll have time to keep up! LOL

Thanks for such a thoughtful post!!! 

Thanks for the well-thought out and interesting post.  It's gotten me rethinking some of my own assumptions about how I've been tuning...  But let me jump straight to the italicized bit (If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears) so I don't get onto too many tangents.

First off, let me restate the problem in my own words, to make sure we're talking about the same thing.

Assumptions:

• • We're tuning from bottom to top, for simplicity
  • The low bass has perfectly smooth inharmonicity, again for simplicity 
  • The fundamental frequencies of notes F2 through C3 are already tuned to the notes below, so they follow a perfectly smooth progression.
  • There's a break in the scaling between F2 and C3, where F2 has very low inharmonicity, and C3 has very high inharmonicity
  • The inharmonicity above C3 is smooth.

We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks.  So

Tune C4 to F2, which had low inharmonicity.  C4 will come out relatively flat.

Tune G4 to C3, which had high inharmonicity.  G4 will come out relatively sharp.

If the inharmonicity difference is great enough, C4-G4 will be an inverted (wide) 5th.

You could also go the other way.  I.e. if C4-G4 fundamentals were instead tuned smoothly, and you tuned P12's down to F2-C3 you'd end up with an inverted 5th there instead.  I suppose in practice we ought to split the difference some kind of way, but I don't know that P12ET really has a prescription for how to do that (Kent or anybody else please correct me if I'm wrong).  Edit: regardless, there's gonna be a breaking point where at least one of the P5's will be inverted now matter how you split the difference.

Are inharmonicity breaks on some pianos bad enough to cause this to happen?  Yes they are.  ET P5's are only contracted by -1.96c to begin with.  I have a tuning file from a particularly nasty little Howard spinet where the generated tuning offset for a 3:1 midrange goes from -6.63c at G3 to -2.82c at D4.  So there's your expanded 5th.

But here's the thing I'm still kinda chewing on... The traditional 4:2 octave can totally suffer from the same phenomenon.  i.e.

(same assumptions)

Tune F3 to F2 dead pure at 4:2.  F3 comes out relatively flat.

Tune C4 to C3 dead pure at 4:2.  C4 comes out relatively sharp.

Again, we have the potential for an inverted (wide) 5th at F3-C4, assuming the break in inharmonicity is enough.  That begs the question, is the 4:2 octave more or less reactive to inharmonicity than the 3:1 P12?  This is harder to calculate than it seems at first blush, because these first few partials don't exactly follow the "expected" mathematical formula for pitch offsets -- apparently the motion at the bridge causes them to be flatter (i.e. less inharmonious) than they would be if completely rigidly anchored at both ends.  So it's probably going to take some empirical data to answer that question, assuming it hasn't already been answered.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-01-2024 12:08
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Nathan,

It's exactly stuff like this that repeatedly tells us analog tuners that its not safe to put all our eggs into the ETD basket. (This is a generalized opinion from me alone). We must keep our eyes and ears open, using everything we know (or think we know) to ensure that the DA (Digital Assistant) is cooperating as intended. Eventually as one sees how the specific DA handles oddities and inconsistencies (or that we learn how to make it do things well) we can put more confidence in it and relax a bit more. 

Remember that I am speaking from a relatively adolescent viewpoint (ETD speaking), as my confidence in them was shattered years ago, but I'm giving it a second chance now with more up to date equipment and algorithms, etc.

This may not be relevant to this specific thread but I thought I'd just throw it in.

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 04-07-2024 13:10
From: Nathan Monteleone
Subject: P12, ETD's, stretch, & the target (warning: long post)

Thanks Maggie :)

Just to make sure it didn't come across wrong: I did not think you were saying that you tune the way I described with pure 3:1's. Rather the opposite, I understand you find it de-optimizes the 5ths (and likely other relevant intervals) too much when large changes in inharmonicity are involved. Just wanted to make sure we were thinking of the same situation, and it sounds like we are.

On the subject of octaves... I suppose one difference between P12ET and traditional octave temperaments, is that we're rather accustomed to "feathering" the octave sizes as we progress through the piano, i.e. 6:3 (or wider) in the bass, 4:2 for the temperament octave, and something a bit narrower as we get up to the top. So perhaps it's always been natural for aural tuners to do as you say, and narrow down to something more like a 2:1 if it helps reduce nastiness at the break.

(You probably know all of this, I'm just restating for clarity:) One of the appeals of P12ET -- on a piano with a smooth international curve anyway -- is that it sort of avoids having to do this ad-hoc feathering and instead hopes to come up with a good compromise by just making the 12ths as pure-sounding at possible. There are still issues with choosing between 3:1, 6:2, and even 9:3 coincident partials when doing this, but I would give 12ths the edge here in that for most of the piano, you can achieve this by simply targeting the 3:1 level. In theory it's part of what gives wide chords and dampers-up sympathetic vibration such a nice character -- at least in the opinion of us P12ET nerds ;)

But your original question hits on something else entirely of course -- what do we do about inharmonicity breaks when they happen? Based on the advantage I mentioned in the previous paragraph, I think maybe we tend to assume, incorrectly, that P12ET somehow holds an edge here as well. But from the discussion do far, I don't think P12ET is dramatically better or worse than the traditional progression of octave sizes at dealing with this problem. With inharmonicity breaks, either one can mess up P5s and other important intervals if you follow their prescription legalistically. Does P12ET lose its advantages if you compromise a few of the 12ths to smooth out a break? In my experience no, simply because the sound I like from it is fairly resilient to normal tuning errors. IOW I think an attentive aural tuner -- or clever ETD algorithm -- can make improvements around an inharmonicity break without doing harm to a P12ET tuning. 

This is, more or less, supposed to be the same thing Kent said, just with a wordier justification.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-06-2024 12:12
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Ron - Can you elaborate on this: "calculating inharmonicity constants for each note instead of direct measurement of the 'spaces' between the individual partials"

Nathan - That was a really thoughtful reply, and a REALLY good question about octaves! The way I tune, I see slow beating intervals as flexible to compensate for these things, so focus on the other intervals just as much. I understand your "assumptions" sections is for the sake of discussion. However, the sentence right after that, "We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks." is not something I would do. I suppose that's why I'm so interested in this, because my other interval checks tell me when I can't push a 12th or an octave too far. I do listen to larger intervals, but I fix most things with many intervals in mind. I care an awful lot about not inverting fifths & fourths. I should care as much about octaves, I suppose. I don't think I tend to invert them, but I do keep them flexible. For example, If I am tuning into the bass and the fifth, fourth, and fast beating intervals put the note in a particular spot, but the octave comes out closer to a 2:1 for whatever reason (let's say there is a drop in inharmonicity that is affecting upper partial matches), as long as it's not inverted, I'll leave it even if the octaves around it are around a 6:3. If the other intervals work, and it sounds good when played in a harmonic context, it stays. 

I'm curious to see what others think. Hopefully I'll have time to keep up! LOL

Thanks for such a thoughtful post!!! 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net

Original Message:
Sent: 04-02-2024 10:48
From: Nathan Monteleone
Subject: P12, ETD's, stretch, & the target (warning: long post)

Thanks for the well-thought out and interesting post.  It's gotten me rethinking some of my own assumptions about how I've been tuning...  But let me jump straight to the italicized bit (If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears) so I don't get onto too many tangents.

First off, let me restate the problem in my own words, to make sure we're talking about the same thing.

Assumptions:

• • We're tuning from bottom to top, for simplicity
  • The low bass has perfectly smooth inharmonicity, again for simplicity 
  • The fundamental frequencies of notes F2 through C3 are already tuned to the notes below, so they follow a perfectly smooth progression.
  • There's a break in the scaling between F2 and C3, where F2 has very low inharmonicity, and C3 has very high inharmonicity
  • The inharmonicity above C3 is smooth.

We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks.  So

Tune C4 to F2, which had low inharmonicity.  C4 will come out relatively flat.

Tune G4 to C3, which had high inharmonicity.  G4 will come out relatively sharp.

If the inharmonicity difference is great enough, C4-G4 will be an inverted (wide) 5th.

You could also go the other way.  I.e. if C4-G4 fundamentals were instead tuned smoothly, and you tuned P12's down to F2-C3 you'd end up with an inverted 5th there instead.  I suppose in practice we ought to split the difference some kind of way, but I don't know that P12ET really has a prescription for how to do that (Kent or anybody else please correct me if I'm wrong).  Edit: regardless, there's gonna be a breaking point where at least one of the P5's will be inverted now matter how you split the difference.

Are inharmonicity breaks on some pianos bad enough to cause this to happen?  Yes they are.  ET P5's are only contracted by -1.96c to begin with.  I have a tuning file from a particularly nasty little Howard spinet where the generated tuning offset for a 3:1 midrange goes from -6.63c at G3 to -2.82c at D4.  So there's your expanded 5th.

But here's the thing I'm still kinda chewing on... The traditional 4:2 octave can totally suffer from the same phenomenon.  i.e.

(same assumptions)

Tune F3 to F2 dead pure at 4:2.  F3 comes out relatively flat.

Tune C4 to C3 dead pure at 4:2.  C4 comes out relatively sharp.

Again, we have the potential for an inverted (wide) 5th at F3-C4, assuming the break in inharmonicity is enough.  That begs the question, is the 4:2 octave more or less reactive to inharmonicity than the 3:1 P12?  This is harder to calculate than it seems at first blush, because these first few partials don't exactly follow the "expected" mathematical formula for pitch offsets -- apparently the motion at the bridge causes them to be flatter (i.e. less inharmonious) than they would be if completely rigidly anchored at both ends.  So it's probably going to take some empirical data to answer that question, assuming it hasn't already been answered.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-01-2024 12:08
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Peter,

I agree, I think there's a lot of value in understanding exactly what our ETDs are telling us, and what they are/aren't taking into consideration.  Granted that opinion is shaped by having mostly used TuneLab, which in some ways I find analogous to driving a car with a manual transmission...

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-07-2024 14:15
From: Peter Grey
Subject: P12, ETD's, stretch, & the target (warning: long post)

Nathan,

It's exactly stuff like this that repeatedly tells us analog tuners that its not safe to put all our eggs into the ETD basket. (This is a generalized opinion from me alone). We must keep our eyes and ears open, using everything we know (or think we know) to ensure that the DA (Digital Assistant) is cooperating as intended. Eventually as one sees how the specific DA handles oddities and inconsistencies (or that we learn how to make it do things well) we can put more confidence in it and relax a bit more. 

Remember that I am speaking from a relatively adolescent viewpoint (ETD speaking), as my confidence in them was shattered years ago, but I'm giving it a second chance now with more up to date equipment and algorithms, etc.

This may not be relevant to this specific thread but I thought I'd just throw it in.

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 04-07-2024 13:10
From: Nathan Monteleone
Subject: P12, ETD's, stretch, & the target (warning: long post)

Thanks Maggie :)

Just to make sure it didn't come across wrong: I did not think you were saying that you tune the way I described with pure 3:1's. Rather the opposite, I understand you find it de-optimizes the 5ths (and likely other relevant intervals) too much when large changes in inharmonicity are involved. Just wanted to make sure we were thinking of the same situation, and it sounds like we are.

On the subject of octaves... I suppose one difference between P12ET and traditional octave temperaments, is that we're rather accustomed to "feathering" the octave sizes as we progress through the piano, i.e. 6:3 (or wider) in the bass, 4:2 for the temperament octave, and something a bit narrower as we get up to the top. So perhaps it's always been natural for aural tuners to do as you say, and narrow down to something more like a 2:1 if it helps reduce nastiness at the break.

(You probably know all of this, I'm just restating for clarity:) One of the appeals of P12ET -- on a piano with a smooth international curve anyway -- is that it sort of avoids having to do this ad-hoc feathering and instead hopes to come up with a good compromise by just making the 12ths as pure-sounding at possible. There are still issues with choosing between 3:1, 6:2, and even 9:3 coincident partials when doing this, but I would give 12ths the edge here in that for most of the piano, you can achieve this by simply targeting the 3:1 level. In theory it's part of what gives wide chords and dampers-up sympathetic vibration such a nice character -- at least in the opinion of us P12ET nerds ;)

But your original question hits on something else entirely of course -- what do we do about inharmonicity breaks when they happen? Based on the advantage I mentioned in the previous paragraph, I think maybe we tend to assume, incorrectly, that P12ET somehow holds an edge here as well. But from the discussion do far, I don't think P12ET is dramatically better or worse than the traditional progression of octave sizes at dealing with this problem. With inharmonicity breaks, either one can mess up P5s and other important intervals if you follow their prescription legalistically. Does P12ET lose its advantages if you compromise a few of the 12ths to smooth out a break? In my experience no, simply because the sound I like from it is fairly resilient to normal tuning errors. IOW I think an attentive aural tuner -- or clever ETD algorithm -- can make improvements around an inharmonicity break without doing harm to a P12ET tuning. 

This is, more or less, supposed to be the same thing Kent said, just with a wordier justification.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-06-2024 12:12
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Ron - Can you elaborate on this: "calculating inharmonicity constants for each note instead of direct measurement of the 'spaces' between the individual partials"

Nathan - That was a really thoughtful reply, and a REALLY good question about octaves! The way I tune, I see slow beating intervals as flexible to compensate for these things, so focus on the other intervals just as much. I understand your "assumptions" sections is for the sake of discussion. However, the sentence right after that, "We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks." is not something I would do. I suppose that's why I'm so interested in this, because my other interval checks tell me when I can't push a 12th or an octave too far. I do listen to larger intervals, but I fix most things with many intervals in mind. I care an awful lot about not inverting fifths & fourths. I should care as much about octaves, I suppose. I don't think I tend to invert them, but I do keep them flexible. For example, If I am tuning into the bass and the fifth, fourth, and fast beating intervals put the note in a particular spot, but the octave comes out closer to a 2:1 for whatever reason (let's say there is a drop in inharmonicity that is affecting upper partial matches), as long as it's not inverted, I'll leave it even if the octaves around it are around a 6:3. If the other intervals work, and it sounds good when played in a harmonic context, it stays. 

I'm curious to see what others think. Hopefully I'll have time to keep up! LOL

Thanks for such a thoughtful post!!! 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net

Original Message:
Sent: 04-02-2024 10:48
From: Nathan Monteleone
Subject: P12, ETD's, stretch, & the target (warning: long post)

Thanks for the well-thought out and interesting post.  It's gotten me rethinking some of my own assumptions about how I've been tuning...  But let me jump straight to the italicized bit (If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears) so I don't get onto too many tangents.

First off, let me restate the problem in my own words, to make sure we're talking about the same thing.

Assumptions:

• • We're tuning from bottom to top, for simplicity
  • The low bass has perfectly smooth inharmonicity, again for simplicity 
  • The fundamental frequencies of notes F2 through C3 are already tuned to the notes below, so they follow a perfectly smooth progression.
  • There's a break in the scaling between F2 and C3, where F2 has very low inharmonicity, and C3 has very high inharmonicity
  • The inharmonicity above C3 is smooth.

We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks.  So

Tune C4 to F2, which had low inharmonicity.  C4 will come out relatively flat.

Tune G4 to C3, which had high inharmonicity.  G4 will come out relatively sharp.

If the inharmonicity difference is great enough, C4-G4 will be an inverted (wide) 5th.

You could also go the other way.  I.e. if C4-G4 fundamentals were instead tuned smoothly, and you tuned P12's down to F2-C3 you'd end up with an inverted 5th there instead.  I suppose in practice we ought to split the difference some kind of way, but I don't know that P12ET really has a prescription for how to do that (Kent or anybody else please correct me if I'm wrong).  Edit: regardless, there's gonna be a breaking point where at least one of the P5's will be inverted now matter how you split the difference.

Are inharmonicity breaks on some pianos bad enough to cause this to happen?  Yes they are.  ET P5's are only contracted by -1.96c to begin with.  I have a tuning file from a particularly nasty little Howard spinet where the generated tuning offset for a 3:1 midrange goes from -6.63c at G3 to -2.82c at D4.  So there's your expanded 5th.

But here's the thing I'm still kinda chewing on... The traditional 4:2 octave can totally suffer from the same phenomenon.  i.e.

(same assumptions)

Tune F3 to F2 dead pure at 4:2.  F3 comes out relatively flat.

Tune C4 to C3 dead pure at 4:2.  C4 comes out relatively sharp.

Again, we have the potential for an inverted (wide) 5th at F3-C4, assuming the break in inharmonicity is enough.  That begs the question, is the 4:2 octave more or less reactive to inharmonicity than the 3:1 P12?  This is harder to calculate than it seems at first blush, because these first few partials don't exactly follow the "expected" mathematical formula for pitch offsets -- apparently the motion at the bridge causes them to be flatter (i.e. less inharmonious) than they would be if completely rigidly anchored at both ends.  So it's probably going to take some empirical data to answer that question, assuming it hasn't already been answered.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-01-2024 12:08
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------

Incidentally, from what I have experienced thus far, if you add a sensor to TL your tranny will smooth out quite a bit. It seriously stabilized my Android TL program.

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 04-07-2024 16:44
From: Nathan Monteleone
Subject: P12, ETD's, stretch, & the target (warning: long post)

Peter,

I agree, I think there's a lot of value in understanding exactly what our ETDs are telling us, and what they are/aren't taking into consideration.  Granted that opinion is shaped by having mostly used TuneLab, which in some ways I find analogous to driving a car with a manual transmission...

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-07-2024 14:15
From: Peter Grey
Subject: P12, ETD's, stretch, & the target (warning: long post)

Nathan,

It's exactly stuff like this that repeatedly tells us analog tuners that its not safe to put all our eggs into the ETD basket. (This is a generalized opinion from me alone). We must keep our eyes and ears open, using everything we know (or think we know) to ensure that the DA (Digital Assistant) is cooperating as intended. Eventually as one sees how the specific DA handles oddities and inconsistencies (or that we learn how to make it do things well) we can put more confidence in it and relax a bit more. 

Remember that I am speaking from a relatively adolescent viewpoint (ETD speaking), as my confidence in them was shattered years ago, but I'm giving it a second chance now with more up to date equipment and algorithms, etc.

This may not be relevant to this specific thread but I thought I'd just throw it in.

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 04-07-2024 13:10
From: Nathan Monteleone
Subject: P12, ETD's, stretch, & the target (warning: long post)

Thanks Maggie :)

Just to make sure it didn't come across wrong: I did not think you were saying that you tune the way I described with pure 3:1's. Rather the opposite, I understand you find it de-optimizes the 5ths (and likely other relevant intervals) too much when large changes in inharmonicity are involved. Just wanted to make sure we were thinking of the same situation, and it sounds like we are.

On the subject of octaves... I suppose one difference between P12ET and traditional octave temperaments, is that we're rather accustomed to "feathering" the octave sizes as we progress through the piano, i.e. 6:3 (or wider) in the bass, 4:2 for the temperament octave, and something a bit narrower as we get up to the top. So perhaps it's always been natural for aural tuners to do as you say, and narrow down to something more like a 2:1 if it helps reduce nastiness at the break.

(You probably know all of this, I'm just restating for clarity:) One of the appeals of P12ET -- on a piano with a smooth international curve anyway -- is that it sort of avoids having to do this ad-hoc feathering and instead hopes to come up with a good compromise by just making the 12ths as pure-sounding at possible. There are still issues with choosing between 3:1, 6:2, and even 9:3 coincident partials when doing this, but I would give 12ths the edge here in that for most of the piano, you can achieve this by simply targeting the 3:1 level. In theory it's part of what gives wide chords and dampers-up sympathetic vibration such a nice character -- at least in the opinion of us P12ET nerds ;)

But your original question hits on something else entirely of course -- what do we do about inharmonicity breaks when they happen? Based on the advantage I mentioned in the previous paragraph, I think maybe we tend to assume, incorrectly, that P12ET somehow holds an edge here as well. But from the discussion do far, I don't think P12ET is dramatically better or worse than the traditional progression of octave sizes at dealing with this problem. With inharmonicity breaks, either one can mess up P5s and other important intervals if you follow their prescription legalistically. Does P12ET lose its advantages if you compromise a few of the 12ths to smooth out a break? In my experience no, simply because the sound I like from it is fairly resilient to normal tuning errors. IOW I think an attentive aural tuner -- or clever ETD algorithm -- can make improvements around an inharmonicity break without doing harm to a P12ET tuning. 

This is, more or less, supposed to be the same thing Kent said, just with a wordier justification.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-06-2024 12:12
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Ron - Can you elaborate on this: "calculating inharmonicity constants for each note instead of direct measurement of the 'spaces' between the individual partials"

Nathan - That was a really thoughtful reply, and a REALLY good question about octaves! The way I tune, I see slow beating intervals as flexible to compensate for these things, so focus on the other intervals just as much. I understand your "assumptions" sections is for the sake of discussion. However, the sentence right after that, "We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks." is not something I would do. I suppose that's why I'm so interested in this, because my other interval checks tell me when I can't push a 12th or an octave too far. I do listen to larger intervals, but I fix most things with many intervals in mind. I care an awful lot about not inverting fifths & fourths. I should care as much about octaves, I suppose. I don't think I tend to invert them, but I do keep them flexible. For example, If I am tuning into the bass and the fifth, fourth, and fast beating intervals put the note in a particular spot, but the octave comes out closer to a 2:1 for whatever reason (let's say there is a drop in inharmonicity that is affecting upper partial matches), as long as it's not inverted, I'll leave it even if the octaves around it are around a 6:3. If the other intervals work, and it sounds good when played in a harmonic context, it stays. 

I'm curious to see what others think. Hopefully I'll have time to keep up! LOL

Thanks for such a thoughtful post!!! 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net

Original Message:
Sent: 04-02-2024 10:48
From: Nathan Monteleone
Subject: P12, ETD's, stretch, & the target (warning: long post)

Thanks for the well-thought out and interesting post.  It's gotten me rethinking some of my own assumptions about how I've been tuning...  But let me jump straight to the italicized bit (If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears) so I don't get onto too many tangents.

First off, let me restate the problem in my own words, to make sure we're talking about the same thing.

Assumptions:

• • We're tuning from bottom to top, for simplicity
  • The low bass has perfectly smooth inharmonicity, again for simplicity 
  • The fundamental frequencies of notes F2 through C3 are already tuned to the notes below, so they follow a perfectly smooth progression.
  • There's a break in the scaling between F2 and C3, where F2 has very low inharmonicity, and C3 has very high inharmonicity
  • The inharmonicity above C3 is smooth.

We then tune P12's above that, dead pure at the 3:1 level, ignoring all other interval checks.  So

Tune C4 to F2, which had low inharmonicity.  C4 will come out relatively flat.

Tune G4 to C3, which had high inharmonicity.  G4 will come out relatively sharp.

If the inharmonicity difference is great enough, C4-G4 will be an inverted (wide) 5th.

You could also go the other way.  I.e. if C4-G4 fundamentals were instead tuned smoothly, and you tuned P12's down to F2-C3 you'd end up with an inverted 5th there instead.  I suppose in practice we ought to split the difference some kind of way, but I don't know that P12ET really has a prescription for how to do that (Kent or anybody else please correct me if I'm wrong).  Edit: regardless, there's gonna be a breaking point where at least one of the P5's will be inverted now matter how you split the difference.

Are inharmonicity breaks on some pianos bad enough to cause this to happen?  Yes they are.  ET P5's are only contracted by -1.96c to begin with.  I have a tuning file from a particularly nasty little Howard spinet where the generated tuning offset for a 3:1 midrange goes from -6.63c at G3 to -2.82c at D4.  So there's your expanded 5th.

But here's the thing I'm still kinda chewing on... The traditional 4:2 octave can totally suffer from the same phenomenon.  i.e.

(same assumptions)

Tune F3 to F2 dead pure at 4:2.  F3 comes out relatively flat.

Tune C4 to C3 dead pure at 4:2.  C4 comes out relatively sharp.

Again, we have the potential for an inverted (wide) 5th at F3-C4, assuming the break in inharmonicity is enough.  That begs the question, is the 4:2 octave more or less reactive to inharmonicity than the 3:1 P12?  This is harder to calculate than it seems at first blush, because these first few partials don't exactly follow the "expected" mathematical formula for pitch offsets -- apparently the motion at the bridge causes them to be flatter (i.e. less inharmonious) than they would be if completely rigidly anchored at both ends.  So it's probably going to take some empirical data to answer that question, assuming it hasn't already been answered.

------------------------------
Nathan Monteleone RPT
Fort Worth TX
(817) 675-9494
nbmont@gmail.com

Original Message:
Sent: 04-01-2024 12:08
From: Maggie Jusiel
Subject: P12, ETD's, stretch, & the target (warning: long post)

Hi All,

I didn't want to reply to a couple threads because I was worried about taking things off topic, so I'm starting a new thread with this multifaceted topic. Not sure if anyone will be interested, but if anyone is, they might be able to fill in a couple gaps in my understanding as well as possibly take something away. Apologies for the length; if that scares you away, no worries. I'll italicize my obvious missing puzzle piece, but I'm sure I have more missing puzzle pieces than that one.

P12 tuning

Regarding P12 tuning, let's look inside a P12 from C3 to G4 (with the P12 pure at the first CP). G4 is at the third partial of C3. If there is a drop in inharmonicity at C3, it should work quite well. If there is an evenness of inharmonicity, as with a high end piano, it should still work ok, but there is this issue I have in my head of the third partial of C3 possibly being sharp enough to pull G4 up enough to make things so wide you could end up with inverted fifths if all else is tuned as equally as possible. If the inharmonicity of C3 shifts upwards, as it may in some poorly scaled pianos, this amplifies that effect. 

I once asked Dan Levitan how it was possible to tune a pure 12th without inverting fifths. He said it actually wasn't possible and that most P12 tunings have a slightly narrow 12th to compensate for that, leaving the inner fifths virtually pure instead of inverted. I can't remember how narrow the 12th was, but it was small. It didn't take much, possibly as little as 3 cents. Other people have told me that no, this isn't correct and that P12 tunings are supposed to be pure at the first CP. So, I don't understand how that could possibly work well all the time with inharmonicity present. The greater the inharmonicity, the more likely you will have to invert fifths for everything to be equal. If someone can explain how a PURE 12th tuning is possible when inharmonicity places the third partial of the lower note of a 12th so relatively sharp, possibly forcing fifths to be inverted, I'm all ears. 

ETD's and P12 tuning

I do know that some ETD's, as mentioned in another thread, do quite a poor job on poorly scaled pianos when attempting a P12 tuning. I had two that not only inverted the fifths in the tenor and across the break, but also inverted the fourths which I definitely don't understand. My instinct says the fourths should be extra wide, which has happened more than not, so I can't explain those two outliers. 

I also know that if I attempt a P12 by ear, I worry less about the P12 and work to have as pure fifths as the piano will allow. These lead to very quiet 12ths without inverting anything. The 12ths aren't always exactly pure, but almost, and it sounds quite good. If I were to start out with a pure 12th and work from there, I'd end up narrowing it half the time anyway in order to keep my fifths from being too wide. 

Stretch 

There is a limit to how much (or how little) stretch you can put in a piano. That limit can be examined with narrow intervals such as a fifth, minor third, and minor sixth. As one moves downward, the fast beating narrow intervals are easier to hear. They must also progress, and balancing the wide and narrow intervals gives you a lowest & highest point of placing any note. Using fast beating intervals is not exactly the same as using slow ones, but they are related: If you are tuning a bottom note from both a wide and narrow interval above it, (such as C3 from F3 and G3, or C3 from E3 and Eb3), and lower it, the wide interval beats will speed up while the narrow ones will slow down (unless you invert them). With a fourth and fifth, the fifth gives a very definite bottom limit. With fast beating intervals, BOTH wide and narrow intervals must slow down when moving downward (& vice versa). When moving downward, the point at which the beats of a wide interval (such as a M3 or M6 or their multiples) start speeding up (compared to the same interval above it) gives a bottom limit, while the point at which the beats of a narrow interval (such as a m3 or m6 or their multiples) start speeding up (compared to the same interval above it) gives an upper limit. When moving upward, it's the opposite. If the stretch is so great that these intervals don't progress, the piano can't handle that much stretch and the tuning won't sound as good as it could. 

Inharmonicity shifts & stretch

This brings me back to ETD's. ETD's listen to a single partial at a time. ETD's don't do what humans can do: jump around and listen to several CP's of several intervals. (This isn't an argument to say aural is better than ETD; please don't go there.) My issue is that ETD's don't deal well with inharmonicity shifts, and the lesser the piano, the more likely it is an ETD will have issues (just like humans, but humans can deal with it better). 

For example, let's say we have a C3 to G3 fifth. If there is a drop in inharmonicity at C3, and the fifth is tuned pure at the first CP, then it will be wide at the second CP. If there is a jump in inharmonicity at C3, and the fifth is tuned pure at the second CP, it will be wide at the first. What do ETD's do with this? Most of the time, my experience is that when there is a sudden drop in inharmonicity, the M3's & M6's suddenly speed up, and vice versa.  Sometimes the M3's speed up but the M6's don't, and vice versa. The ETD isn't compensating for inharmonicity shifts, so most intervals end up extra wide or extra narrow around these shifts, which most often happen around the break between the wound an unwound strings. Sometimes fourths end up pure while fifths end up extra narrow. Sometimes it's the opposite. Sometimes both fourths and fifths end up pure. It all depends on how many shifts there are and how extreme they are. 

This applies to all intervals, not just fifths, including octaves and 12ths. If too much weight is put on 12ths in lesser pianos without compensating for inharmonicity shifts, you can get gobbledygook. 

Side note: "Traditional pure octave tuning" is a bit of an oxymoron. It's impossible to have a pure octave except at one coincident partial unless the inharmonicity shifts in odd ways. A pure octave must be defined by which CP's are tuned pure. 

Allowing intervals other than an octave or twelfth to work well together will tell you what kind of stretch range a piano can handle. Too much or too little and you are out of its best sounding range. I believe one should not force stretch. Line up partials and the stretch will fall into place. Forced stretch can put partials too far out of alignment and you end up with the problems I've detailed above. 

Slow beating intervals have a range of correct. Fast beating intervals have a much smaller range of correct. Use them all to find balance. 

The target

SO, here we are, trying to get the best tuning our ETD's can give with choices of stretch styles, microphone quality, microphone placement, and even a sensor that can directly sense the vibration of the strings, while our ETD's are still faulty. It's like trying to hit the second ring around a bulls eye as accurately as possible when the target is in the center. 

Another aside: Please don't take this as an attack on ETD's. ETD's are amazing and do a great job. Good aural tuners can do a good job as well. Combine them, and you double their strengths. Rick Butler says that an ETD in the hands of an aural tuner is a very powerful tool.

Summary

All this is to say that we are splitting hairs over a target that remains fuzzy to ETD's in the midrange of the piano, particularly in the low tenor. I'm not saying we shouldn't continue to discuss and improve all the above, but we should be aware that our ETD's have not yet caught up with the resolution of our sensors. I believe they will get there, but I don't see that happening soon. Just food for thought. 

PS: Once again, this is not about ETD vs. aural, although the combination of the two is still on topic. Thanks. 

------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
------------------------------