Original Message:
Sent: 02-17-2025 12:17
From: Tim Foster
Subject: Pure 12th calculations
Kent,
Aha, I think you got through my thick skull. Thank you for bearing with me, I think I see the light at the end of this tunnel. This is mind bending stuff when you're used to thinking inside an octave.
I came to the calculation of half of 1.23 so that I could conceive of the octaves without having to extend each octave outside of a temperament octave, if that makes sense. I think I see where this thinking went wrong, and I appreciate your time.
Best,
Tim
------------------------------
Tim Foster RPT
New Oxford PA
(470) 231-6074
Original Message:
Sent: 02-17-2025 09:50
From: Kent Swafford
Subject: Pure 12th calculations
Tim wrote:
"The Pythagorean comma is normally thought about as being distributed between fifths. P12ET distributed between fifths and octaves. Fifths are narrowed by 1.23 cents. 1.23x12 =14.76. Octaves are stretched 1.23 cents."
Correct, so far.
"Since octaves extend over the normal parameters of the temperament octave, we can distribute half of 1.23 cents between all 12 notes to keep within an octave."
No. Specifically, how are you deriving the figure of "one half"?
There are 12 fifths, but only 7 octaves.
So, for the fifths, 1.2347x12 =14.816 , and for the octaves, 1.2347x7 = 8.643
14.82 + 8.64 = 23.46
"why is my number just short of the Pythagorean comma?"
The tempering isn't split equally between 5ths and octaves, but 7/19 of the comma is made up from the 7 octaves and 12/19 by the 12 fifths.
No?
Original Message:
Sent: 2/17/2025 7:40:00 AM
From: Tim Foster
Subject: RE: Pure 12th calculations
Anthony,
Thank you for this, that really helps me visualize what P12ET is doing.
Kent,
After thinking about it some more over the weekend, I think I am starting to wrap my head around what you're saying, particularly in relation to distributing part of the Pythagorean comma in octaves. I was getting hung up on this since the Pythagorean comma is normally only a discussion of fifths within a pure octave.
To visualize, I needed to figure out the math within the octave. Please let me know if the following is on the right track.
The Pythagorean comma is normally thought about as being distributed between fifths. P12ET distributed between fifths and octaves. Fifths are narrowed by 1.23 cents. 1.23x12 =14.76. Octaves are stretched 1.23 cents. Since octaves extend over the normal parameters of the temperament octave, we can distribute half of 1.23 cents between all 12 notes to keep within an octave. 1.23/2=0.615. 0.615x12=7.38. 7.38+14.76=22.14. To complete the octave, we would need to add another 0.615 to "close" the octave. 22.14+0.615=22.755.
The 22.755 does not quite equal the Pythagorean comma, but it's really close. Am I on the right track? If so, why is my number just short of the Pythagorean comma?
Thanks again for your help!
------------------------------
Tim Foster RPT
New Oxford PA
(470) 231-6074
Original Message:
Sent: 02-17-2025 02:19
From: Anthony Willey
Subject: Pure 12th calculations
I'm a bit late to this, but I'd like to take a stab at it. Setting aside the discussion about comma, which confuses me too, I'll see if I can explain the "pure 12th" idea at the theoretical (no inharmonicity) level and the real piano (with inharmonicity) level.
First: the theoretical zero inharmonicity case:
With zero inharmonicity, everything magically lines up in equal temperament. For regular twelve-tone equal temperament, we double the frequency every octave, and each octave is divided into twelve equal pieces of 100 cents each. On this magical piano, it doesn't matter if you tune 2:1 or 4:2 or 6:3 octaves because they're all exactly in tune. Let's look at some numbers:
If A4 is at 440 Hz, then A3 is at exactly 220 Hz. And the 2nd harmonic of A3 is 440. Perfect 2:1 octave. The 4:2 octave is at 880 Hz, also where A5 is, so you've also got perfect double octaves, triple octaves, etc. Yay.
Now let's look at what we call pure 12ths equal temperament. Instead of doubling the frequency every 12 notes, we are tripling the frequency every 19 notes. So if A4 is at exactly 440 Hz, then E6 would be at exactly (440 x 3) 1320 Hz. Which is also conveniently the 3rd harmonic of A4. So you have a beatless twelfth where the 3rd harmonic/partial of the lower note interacts with the 1st harmonic of the upper note. And of course the 6:2 and 9:3 twelfths also come out pure.
But the octaves don't come out pure anymore. They are all stretched slightly wide of pure. The fifths, which were all a bit narrow in regular 12-tone equal temperament, are a bit less narrow. The fourths would be a bit more wide.
All of this has nothing to do with real pianos until you notice that when you're tuning a real piano you have to stretch the octaves a bit (looking only at the fundamental frequency) and that if you look at how much the stretch is in the midsection, it's close to what you'd get if you were just using the frequencies from the theoretical perfect 12th equal temperament. It's not a lot of stretch, and this relationship breaks down very quickly in the treble and bass where we need a lot more stretch on real pianos.
If you'll forgive a too-long table, here's a table of the frequencies for the theoretical pure octave and pure twelfth equal temperaments. I've highlighted A=440 and the frequencies where that reference frequency doubles/halves (for octaves) or triples/thirds (for twelfths).
If you squint you will see that these two sets of frequencies are really close to each other for any given note. They diverge by 1.23 cents per octave, so at the top and bottom of the piano the notes are 4 and 5 cents apart respectively. Much less than the 20-50ish cents of stretch we often use in the bass and treble.
So that's just the theoretical case. Let's change gears and look at the real piano case.
For real pianos, pure 12th tuning simply means that you are tuning in such a way that you tune the twelfth interval pure, usually with the 3rd partial of the lower note tuned to the 1st partial of the upper note. But in the bass of a piano, pure 12th can refer to tuning a 6:2 twelfth instead of a 3:1.
Of course you can't tune both the 6:2 and the 3:1 pure in the bass. To see why, let's look at some real physical pianos. Here are some images showing the calculated beat rates of all the octaves and twelfths on 3 pianos: a Baldwin Acrosonic, a Yamaha U1, and a Steinway D. The graph is based off the lower note of each interval, so if you care about the A3-A4 octave, look at the point labeled A3.
On all these pianos the 2:1 octave is wide. It must be. But on the D it isn't nearly as wide. The big difference comes when you look at the 4:2 and 6:3 (and higher number) octaves in the tenor and bass. On the Steinway they are all quite close together with near zero beat rates. On the Yamaha the 8:4 is impossible to tune pure without messing up the 6:3; and the 6:3 and 4:2 never quite converge the way they did in the Steinway. The Baldwin has the same issue as the Yamaha except it's worse, with the 8:4 beating at 2 Hz throughout the tenor and bass instead of 1 Hz on the Yamaha.
The twelfths behave similarly. The 3:1 twelfth is surprisingly pure on all 3 pianos, but more so on the Steinway. The 6:2 and the 9:3 on the Steinway converge in the bass, but they remain spread on the Yamaha and Baldwin. In other words, you can have the 3:1 and the 6:2 both sounding pure in the Steinway's low tenor and high bass, but you're going to have to compromise on the other pianos.
By the way, since we're looking at beat rates, these graphs are a bit misleading because they blow up on the right side (because the frequencies are doubling each octave and are really high in the treble) and the difference between intervals looks really small on the left side because the frequencies there are so small down there in the bass. It can be helpful to look at the same graphs plotted in Cents instead of Hertz or Beats per second.
Now the graphs seem to blow up in the bass, where even a small frequency difference equals a lot of cents, and they don't blow up as much in the treble.
Looking at the cents graphs, you can see that the 2:1 octave is generally tuned about 2 cents wide for most of the range where it matters. The 4:2 octave is tuned fairly pure in the midsection, and the 6:3 octave takes over in the tenor. 8:4 and 10:5 must always be narrow.
For the twelfths, 3:1 is pure throughout the piano except for the bass where it's forced wide by other intervals including the 6:2 and eventually 9:3. Except on the Steinway. Long pianos really are nice to tune :-)
Getting back to the original topic, what do these images have to do with theoretical pure twelfth equal temperament? Nothing really I guess. These are real tunings for real pianos and as such they're a big compromise between many competing intervals. But the 12th, especially the 3:1, is a good interval for most of the piano. If you had to pick one interval to tune pure from the tenor to the top of the treble, the 3:1 twelfth is the best choice. And it is given a lot of weight in the ETD (PianoMeter) that calculated these tunings. Other ETDs calculate the tunings with different methods, but the results are similar I believe.
Another takeaway from these graphs that I'd like to emphasize again is that the 2:1 octave is never tuned pure in the treble. It should always be wide. You can do that with the double octaves or the twelfths, and I think it will also just happen naturally if you're just working your way upward aurally with octaves, fourths, and fifths. Deliberately tuning pure 2:1 octaves up through the treble will probably sound fine, but you'll be forcing literally every other interval narrow, and you'll end up with around 10 cents less stretch by the time you get to C8.
Well, I was aiming for something easy to understand, but I think I went too long. If you read this far, I hope it was helpful.
------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com
Original Message:
Sent: 02-15-2025 18:46
From: Bill Ballard
Subject: Pure 12th calculations
As an aural tuner, I'll bring up the caboose on this discussion. Which someone mentions extrapolating the temperament into the outer octaves, using pure octaves, I have to ask which octave are they going to make pure. The single octave (2:1) is the most prominent, but in that single octave, there are at least of any practical use: the 2:1, the 4:2, and the 6:3. (My hat is off to anyone who can tune an 8:4, heading north from the temperament.) Then there are the double octaves (4:1, and the 8:2). These can be tuned aurally, as direct unisons between the coinciding partials. So, back to the opening question: which octave relationship are you going to make the "pure one". Inharmonicity is at work, making sure that the ones NOT chosen to be the "pure" one will be beating.
Dan Levitan, in his book "The Craft of Piano Tuning", expresses this nicely in the term "expansion units". Pick, say, a pure 4:2 octave and the octave relationship below it (ie., of lower expansions units, namely the 2:1) will be pulled sharp/wide by the 4:2's expansion units. The same warpage happens to the octave relationships above the 4:2 (the 6:3, 8:4 single, and the double octave 4:1). These relationships will get pulled flat/narrow. I am a fan of the 3:1 in laying out the treble octaves: the 3:1's 8EU is quite mild compared to the 4:2 or the 6:3. I run the 3:1 right up through the 6th octave, at which point I downshift to pure 2:1s. (Full disclosure: my 7th octave gets done by ETD (TuneLab), taking a 2:1 from the 6th octave.)
I hope there's no confusion about how the "octave" choice of the temperament's compass interacts with the choice of interval (single/double octave vs. P5th) to carry that outwards. They are each separate decisions, although inharmonicity's warpage is the main consideration in both.
------------------------------
William Ballard RPT
WBPS
Saxtons River VT
802-869-3161
"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: 02-15-2025 17:07
From: Tim Foster
Subject: Pure 12th calculations
Kent, I didn't see your last post (16) until after submitting post 18. It seems there should be a less confusing out-of-order way for replying to threads on this forum. But that's another issue.
Hopefully my last post clarifies where I think my confusion lies. (And I just saw another reply come in from you.)
I do understand that on paper, a 2:1 and 6:3 octaves are the same math model-- I was speaking about tuning real pianos, and therefore including IH. Especially in a case with high IH, the difference in "cent size" could be notably different.
As for the red herring point about faster 10ths and 17ths, while I agree this is less noticeable on a piano with very little IH, I believe it is noticeable on many "normal" pianos. I've measured the beat rates of thirds in ET with no stretch, 4:2 and 3:1 and I agree, the beat rate increase was negligible and insignificant. My point about thirds has more to do with the sound of 10ths and 17ths, which are very relevant to real music. As a pianist myself, I definitely hear this difference when playing.
In agreement with your last post, the 1/10 cent difference is really insignificant when it comes to the beat rates of thirds, as mentioned above. I mostly use an ETD for tuning, so I'm not trying to make a case for aural tuning superiority/inferiority.
Thanks again for your help. I'll chew on the information you provided.
------------------------------
Tim Foster RPT
New Oxford PA
(470) 231-6074
Original Message:
Sent: 02-15-2025 16:29
From: Tim Foster
Subject: Pure 12th calculations
Kent,
One more follow-up from my last post (#15). The more I think about the cents in both tuning methods not being equal, the more I believe this is the point that is throwing me off. If one cent in P12ET = more than one cent in P8ET in Hz, the 1.23 cents in question would be equivalent to more than 1.23 cents in P8ET. For example, if I set my ETD to tune P12ET and tuned C3 and C4, then reset my ETD to tune a 2:1 octave and rechecked C3 and C4, I would find C3 to be flat and C4 to be sharp, their difference being substantially more than 1.23 cents wide. Is this correct?
This would solve my conceptual problem with the system. As a practical matter, the comparison of thirds in both systems (13.69c vs. 14.1c) is slightly misleading, since the cents in both numbers are not equal.
I understand why we use cents (logarithmic functions). I also think that often when comparing the two systems of pure octave or 12th ET, the deviations are compared without fully acknowledging that the unit of measurement (i.e. cents) is not the same size between numbers. Perhaps I've just been slow in comprehending, it wouldn't be the first time. :)
Have I missed the mark?
------------------------------
Tim Foster RPT
New Oxford PA
(470) 231-6074
Original Message:
Sent: 02-15-2025 15:10
From: Kent Swafford
Subject: Pure 12th calculations
David wrote:
"What is the relationship between the "pure" 3:1 19th note and the inharmonic 3rd partial of the lower note? Tuned by ear if the partial is aligned then won't the real interval be generally sharper than the 3:1 ratio if the inharmonics are sharp? Won't this lead to wider stretched octaves and even wider major thirds?"
First, it is important to carefully differentiate between discussion of the math model of equal temperament on the one hand and tuning real-world inharmonic instruments on the other hand. Two different things.
The model of pure 12th equal temperament is based upon the math value of the 19th root of 3: so in pure 12th ET, the fundamental frequency of each consecutive ascending half-step increases by a factor of the 19th root of 3 which is about 1.059526.
(The math model is extremely useful in keeping our tuning objectives in mind, and can be used to calculate all of the theoretical interval relationships. My basis for emphasizing the math model is and has always been the simple fact that the modern piano existed before our understanding of inharmonicity existed. Therefore, "we", meaning piano techs of the 19th century, must have used the math model to develop tunings for the inharmonic piano. They didn't specifically know that beat rates had to be adapted for the real piano, let alone knowing precisely _how_ to adapt the beat rates.)
There can be confusion in referring to the ratios of intervals. In math, we may speak of the ratio between fundamental frequencies, but when discussing the tuning of musical instruments, ratios nearly always refer only to coincident partial pairs.
So "coincidentally" a fifth may have a 3:2 ratio of fundamental frequencies but it is formed as a tuning interval by the relationship between the 3rd partial of the lower note with the 2nd partial of the upper note. It is the partial relationship stated as a ratio that is important in tuning, not the ratio between fundamental frequencies. I admit this may be poor terminology and a source of confusion.
Now to answer your question:
"What is the relationship between the "pure" 3:1 19th note and the inharmonic 3rd partial of the lower note?"
There is no difference. The oft-used phrase in tuning is "Beatless is stretched." A pure 12th is defined as a 12th in which the third partial of the lower note is tuned just with the 1st partial of the upper note. The interval, then, as measured at fundamentals will be "stretched", if stretched means that the frequency of the fundamental of the upper note is more than 3 times that of the fundamental of the lower note. So? Only the partial relationship is audible.
"Won't this lead to wider stretched octaves and even wider major thirds?" As Dr. Al Sanderson repeatedly pointed out, pianos are scaled so that the beat rates can be similar to that of the math model. The effect of inharmonicity upon the beat rates of equal temperament is complex and not straight forward. But the bottom line is that the beat rates of the piano approximate that of the math model. So much so that modern tuning was invented without detailed knowledge of inharmonicity.
Piano tunings should be stretched (partly due to inharmonicity); pure 12th ET can help accomplish that stretch but that is a separate subject.
Original Message:
Sent: 2/15/2025 1:18:00 PM
From: David Pinnegar
Subject: RE: Pure 12th calculations
Kent - thanks
What is the relationship between the "pure" 3:1 19th note and the inharmonic 3rd partial of the lower note? Tuned by ear if the partial is aligned then won't the real interval be generally sharper than the 3:1 ratio if the inharmonics are sharp? Won't this lead to wider stretched octaves and even wider major thirds?
Best wishes
David P
- - - - - - - - - - - - - - - - - - - - - - - -
David Pinnegar, B.Sc., A.R.C.S.
- - - - - - - - - - - - - - - - - - - - - - - -
+44 1342 850594
Original Message:
Sent: 2/15/2025 11:10:00 AM
From: Kent Swafford
Subject: RE: Pure 12th calculations
Glad you are interested!
Instead of dividing the comma into 12 equal parts as in pure octave ET, in pure 12th ET the comma is divided into 19 equal parts: 12 5ths plus 7 octaves.
So, 23.46 cents of the comma divided by 19 half-steps = 1.23 cents.
Think of tuning a series of fifths on an instrument. We are generally unable to tune 12 5ths in one direction only. At some point we transfer octave; that is, instead of just tuning G4 to D5, we tune D4 instead. As tuners we may be clever enough to have learned how to tune D4 directly from G4, but think of tuning G4-D5, then down an octave D5-D4. In pure octave equal temperament we carefully tune that octave pure, but in pure 12th ET we stretch that octave by 1.23 cents. This is where the rest of the comma is made up.
Think of tuning the 5ths upwards and then tuning the octaves downwards. To eradicate the comma we must lose 23.46 cents in 19 half-steps. We tune the higher note of each upward fifth 1.23 cents flat. Then we tune the lower note of each octave 1.23 cents flat. After 19 intervals we will have taken up the entire comma.
Original Message:
Sent: 2/15/2025 10:35:00 AM
From: Tim Foster
Subject: RE: Pure 12th calculations
Thank you everyone for your replies!
Kent, I appreciate your expertise on this. I'm still getting a little caught on the 1.23 cents narrow of fifths and wide octaves. I know that "normal" ET fifths are tempered by 1.96 cents, 1.96 x 12 = Pythagorean comma (23.46 cents). If fifths are tempered by 1.23 cents and 1.23 x 12 = 14.76, the difference between tempered fifths in both systems would be 8.7 cents. If P12 tuning expands the octave by 1.23 cents, where is the additional 7.47 cents (8.7 - 1.23 = 7.47)?
I understand that the tempering I'm referring to regards the fundamental frequencies of mathematically perfect ET which does not practically correlate to the "real world" with inharmonicity and stretch. A 4:2 octave stretch brings the fundamental frequencies of ET fifths closer to pure (widening), but due to IH, not every piano will place the fundamental frequencies at the same place as we match the higher coincident partials. If part of the 7.47 cents discrepancy is due to "normal" octave stretch (e.g. 4:2, 6:3, etc.), where do we get the precise 1.23 cents narrow 5th and wide octave in P12 tuning? Additionally, the stretch that we add to the piano is not exactly the same from register to register on the piano. Does the 1.23 refer to one specific register, and if so, is it an average?
I know I'm missing something here, I'm just trying to figure out what I'm missing. Hopefully my rambling above helps you understand my confusion.
Thanks again!
------------------------------
Tim Foster RPT
New Oxford PA
(470) 231-6074
Original Message:
Sent: 02-14-2025 10:53
From: Paul Larudee
Subject: Pure 12th calculations
Pure 12ths are simply a measure of the width of the intervals ("stretch") in equal temperament. If you tune pure octaves, you get narrow 12ths. If you tune perfect triple octaves, you will get wide 12ths. It's all equal temperament, just a question of how wide to make the equal intervals. In a piano, it's unavoidable to stretch the intervals because the sound is generated on steel strings that have stiffness, making the 1st partial shorter than half the fundamental. It's not a pipe organ or flute, where the vibrating medium is perfectly fluid air. Stretch is a matter of taste, and pure 12ths are pretty much at the limit of octave stretch, but rather good in the double octaves and other intervals. All of piano tuning involves tempering the intervals in one way or another, but a lot of us think that pure 12ths are pretty close to the best compromise for equal temperament.
Original Message:
Sent: 2/14/2025 9:47:00 AM
From: Ron Koval
Subject: RE: Pure 12th calculations
Hi Tim
It may be helpful to realize there are a few intersecting uses of the term "P 12th tuning". Farther back in time using tunelab and other apps, techs found a helpful stretch compromise by widening the midrange to create a 3:1 D3-A4 12th and then using a strict interval matching to complete the tuning. As far as I know, this was the original use of the term. I've never been a fan of this approach.
Stopper tuning introduced the math model that I believe is at the basis of Kent's explanation of actually using octave matches, but expanding them by 1.23 cents. This is what most techs talk about now when they reference a P1 25th tuning, though how each app applies this model may be different. Also, some may still be using the older model of setting a central temperament and then using 3:1/6:2 intervals to extend that to the ends of the piano. Clear as mud?
Both of these shared the ability to come up with a decent tuning approximation without using complete inharmonicity data from the piano. (Stopper app actually took no pre-measuring from the piano)
Hope this helps.
Ron Koval
------------------------------
Ron Koval
CHICAGO IL
Original Message:
Sent: 02-13-2025 17:38
From: Tim Foster
Subject: Pure 12th calculations
Hello,
I'm having trouble understanding pure 12th tuning. When this is calculated in an ETD, is it actually a beatless 3:1, or is it finding an average between the octave and the 12th? Is it possible to tune a beatless 3:1 in any piano (concert grands?) or is there always a little compromise/averaging? Do ETDs calculate an even progression of thirds and add this into the compromise, or is the priority solely on the 12ths and octaves?
I'm curious, and pure 12th tuning is still a little mysterious to me. I've skimmed some of the threads and articles, but not finding definitive answers to these questions.
Thank you!
------------------------------
Tim Foster RPT
New Oxford PA
(470) 231-6074
------------------------------