Maggie, a little more for you on the 1.23 cents difference. If you go back in this listserver to October 26, 2021, "My aural P12ths tuning sequence (including a video demonstration)" and listen to/watch Patrick Wingren's video at https://vimeo.com/395337359, he is setting a P12 ET temperament from D3 to D4. A very nice video. This time I was interested in the width of the octave. If you look around 6:30 and again around 7:00 you will see/hear him test the D3-D4 octave using a 4:2 (M3 & 10th) test. The stretch of the octave is clear.
Elsewhere in our discussion in that thread you will see a refinement of the description of the octave as being "clean" versus "pure." Bottom line, there is a difference in the octave in 12-TET versus 19-TET based on the twelfth (P12 ET). If the mathematics is not the appropriate measure, which is very specific on the difference, then we can listen. This is not to be negative on P12 ET. I agree with what Patrick Draine wrote there: "You are correct that a pure 12th temperament won't yield absolutely pure 2:1 or 2:4 octaves, but it is "close enough", and has other desirable aspects."
The Cybertuner might not at first glance show you a difference between the two temperaments, but they are there.
Original Message:
Sent: 10-03-2022 16:21
From: Maggie Jusiel
Subject: Pure Twelfth tuning
OMG...from brain fart to light bulb in .5 seconds. Haha!
Kent wrote: When I specify an amount of temper in cents, I am almost always speaking exclusively of the (zero inharmonicity) math model.
JUST now registered. Thanks to everyone who tried to say the same thing. Haha!
Kent wrote:Multiple pairs of coincident partials are blended to achieve an objective compromise. I cannot help but think this is analogous to Virgil's whole sound tuning, and is a superior method of dealing with inharmonicity. And with a smart ETD like Verituner, this multi-partial approach automatically adapts to any inharmonicity condition, from beautiful scales like a Steinway D to impossible scales like the Yamaha CP-70.
That is so cool. Thanks, Kent! Sorry I'm a little slow on the uptake sometimes. lol
------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
Original Message:
Sent: 10-03-2022 15:59
From: Kent Swafford
Subject: Pure Twelfth tuning
Maggie wrote:
"This difference between this 2:1 "unison" is 1.23 cents."
When I specify an amount of temper in cents, I am almost always speaking exclusively of the (zero inharmonicity) math model. Applying that model to real pianos with random inharmonicity is something quite different. But, I believe in being very familiar with equal temperament unaffected by inharmonicity. (By the way, the target temper of the octave is always 1.23 cents, be it 2:1, 4:2, 6:3, etc.)
Braid-White published beat rate tables a century ago of the ideal beat rates of pure octave equal temperament. He did this to calculate the idealized beat rate relationships of equal temperament. Keep in mind that the modern piano already existed before inharmonicity was generally known to exist, let alone how to deal with it. So the idealized beat rate relationships were what tuners had to work with.
Fast forward 50 years or so and Virgil Smith came along and again stressed the importance of relative beat rate relationships.
Pure12 ET uses a different math model and slightly different beat rate relationships distinct from pure8 ET. So when Verituner came along, with its multi-partial approach, I took the the idealized beat rate relationships of p12 ET, and programmed them into Verituner with my Style Files.
So, my Verituner styles throw out the window most all specified interval tempering (of individual pairs of coincident partials) and try to strike a compromise between the various pairs of coincident partials. (That is, in this tuning, the ideal temper of 1.23 cents in the octave exists only as a target, not an absolute.) Multiple pairs of coincident partials are blended to achieve an objective compromise. I cannot help but think this is analogous to Virgil's whole sound tuning, and is a superior method of dealing with inharmonicity.
And with a smart ETD like Verituner, this multi-partial approach automatically adapts to any inharmonicity condition, from beautiful scales like a Steinway D to impossible scales like the Yamaha CP-70.
Original Message:
Sent: 10/3/2022 2:03:00 PM
From: Maggie Jusiel
Subject: RE: Pure Twelfth tuning
Hi All - I typed all this out yesterday but it never sent. I'll try again. In addition, I will reply to Kent: I'll be at MRCO and am hoping to catch your class if I can sneak away from the exam floor. ;-)
I thought I understood Pat but let me double check. In answering my question, Pat was saying that the 1.23 cents is the difference between the fundamental of the upper note in an octave as compared to the second partial from the lower note. This difference between this 2:1 "unison" is 1.23 cents. Therefore, if the octave is 1.23 cents wide at the first CP, then it is less wide at each CP as you move up, just comparing the CP's of that one octave. The 4:2 measurement would be less than 1.23 cents wide, the 3:2 less, and the 8:4 less again, assumingly narrow at that point. Right?
Anthony - A lot of what you write make sense to me, so I'm assuming we think in a similar way. Whether that's good or bad I can't say for sure. Haha! I do appreciate your input. It is helpful to me. Thanks.
Larry & Norman - Perhaps I understand a P12 tuning even less than I think. I don't have that much stretch moving up. I would never leave a piano with noisy octaves. A roll is fine, but not noisy. A customer is going to play their piano after the tuner is gone. One can't just play what sounds good and expect the customer won't notice anything odd later. Having said that, the P12 setting on Cybertuner doesn't leave noisy octaves and is very similar to what I would do naturally with the occasional exception of me wanting to push both the bass & treble a hair lower. *shrug*
------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
Original Message:
Sent: 10-02-2022 13:40
From: Norman Brickman
Subject: Pure Twelfth tuning
Thanks to all for the interesting discussion.
I don't have a formula for the Railsback (inharmonicity) curve, but it looks like there is very little inharmonicity in the middle of the scale. So maybe in the middle of the scale the theoretical 2:1 octave for 12-TET really does have a natural stretch of near-zero. Anyone have the actual numbers?
Larry - my experience is that the 2.5 cents for the double octave in P12 ET is going to be significant. That is why I suggested not to test the octaves in a P12 ET tuning in front of one's customer – the tuner can find more appropriate intervals and chords that demonstrate the quality of their tuning. In 12-TET you can hear the fourths having significant beating as you get up in the scale to around C4, and that is only 1.96 cents.
Or put in terms of whether the 1.23 cents artificial stretch of the octave in P12 ET is relevant – I think it is very relevant - if you play an octave or multiples of an octave. I wonder if a 1.23 cents artificial stretch after one octave would get a pass on the PTG 12-TET tuning exam?
Regards, Norman
------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321
Original Message:
Sent: 10-02-2022 13:01
From: Anthony Willey
Subject: Pure Twelfth tuning
Of course :-) I realize the terminology is provocative, and 1:1 is, as you say, a unison. Think of it this way: I'm defining an octave as a 1:1 unison that is tuned 1200 or 1201.23 cents wide. That's the best analogy I can think of to demonstrate the "1.23 cents" number's irrelevance to piano tuning.
------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com
Original Message:
Sent: 10-02-2022 12:43
From: Patrick Draine
Subject: Pure Twelfth tuning
Nomenclature, as already noted, is very important. Anthony, 1:1 is not an octave. 1:1 is a unison. 2:1 is an octave relationship.
------------------------------
Patrick Draine RPT
Billerica MA
(978) 663-9690
Original Message:
Sent: 10-02-2022 12:02
From: Anthony Willey
Subject: Pure Twelfth tuning
"1.23 cents between which partials?"
Exactly! Outside the framework of theoretical equal temperament widths with zero inharmonicity, the 1.23 cents loses all relevance.
Here's how I understand it: if you ignore all harmonics and only look at the fundamental frequencies, regular 12-note equal temperament makes the fundamental frequency double each octave. A3=220, A4=440, A5=880. In a perfect-twelfth (19-note) equal temperament, the fundamental frequency triples every 19 notes, and when you do the math that makes the fundamental frequency slightly more than double for each 12-note octave. Each octave is about 1.23 cents wider than in normal equal temperament, again measured from the fundamental. So A3=219.84, A4=440, A5=880.63. But this is irrelevant for piano tuners because we don't tune based on the fundamentals. We tune based on the harmonics/partials, and that also stretches the octave (as measured from the fundamental frequencies). So the difference between a 4:2 stretched octave and an octave resulting from a perfect-twelfths tuning is much smaller than 1.23 cents because of the inharmonic partials we used to tune the octaves or twelfths.
Edit: Responding to what Larry wrote: to be clear the 1.23 cents is not related to the width of the 2:1 octave, or of any octave measured from higher harmonics. It is a measure of the distance between the fundamental frequencies. Stated a different way: In regular equal temperament, the 1:1 octave is 1200 cents wide, and in perfect-twelfth equal temperament, the 1:1 octave is 1201.23 cents wide.
------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com
Original Message:
Sent: 10-02-2022 10:45
From: Maggie Jusiel
Subject: Pure Twelfth tuning
This is part of my problem: 1.23 cents between which partials?
------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
Original Message:
Sent: 10-02-2022 10:15
From: Norman Brickman
Subject: Pure Twelfth tuning
Anthony, thanks for clarifying on your question, and I am happy to clarify my terminology.
I think that "natural stretch" is a good phrase that Kent used for the context of this thread, and the equivalent that I usually use is "pure octave." You are interested in which partials are used for the natural stretch in a 12-TET tuning. There is probably not one correct answer. I tune aurally and, after establishing the temperament octave, go up and down the scale by first setting a note and then using various tests and progressions to find my mistakes. Going up the scale I usually initially set with 4:2 (M3 & 10th), then switch to 4:1 (M3 & 17th), also 4:1 with double octaves. And other intervals. I then do the detailed corrections. But someone using a good ETD only needs one step. For example purposes only, the ETD going up from the temperament octave might start with 4.4:2.0, maybe later move to 4.3:1.0, etc. It is hard to estimate for an ETD without knowing the computer's algorithm for choosing the right proportion of coincident partials for the particular piano's scale.
The "artificial stretch" that Kent refers to is the resultant octave in a P12 ET tuning. The difference to the octave in a 12-TET tuning is 1.23 cents.
You personally might choose different intervals and sets of coincident partials to concentrate on in your 12-TET tuning. Even if we have slight differences in your and my results, I think we can both be correct is saying that we are tuning pure octaves. For a 12-TET tuning I suggest, for one test, to use the octave test that I mentioned in my April Journal article. This shows the customer the quality of your tuning and helps assure return business. Play different simultaneous multiple-note octaves like C3+C4+C6+C7 and demonstrate how you can pretend you are only playing one note. For a 19-TET tuning based on the twelfth (P12 ET), I suggest that you NOT do the octave test. Instead, you will have other intervals and chords that will demonstrate to your customer the quality of your tuning.
Regards, Norman
------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321
Original Message:
Sent: 10-02-2022 00:57
From: Bill Ballard
Subject: Pure Twelfth tuning
Maggie wrote:
Bill wrote, "<snip> But for most pianos, the 6:3's narrowing of the 2:1 below it and widening of the 4:2 (and other single octaves audible above that) is not objectionable ("noisy" as Peter would say)."
I'm not understanding. Are you saying that you are using a 6:3 octave and allowing the octave below to be more narrow by default?
It'll make more sense to you if I correct my error. Referring to Levitan's chart of expansion units, if the single octave is tuned 6:3 (expansion unit of 27), that width results in a widened (NOT NARROWED) 2:1. The same correction applies to the coincidental partials above the 6:3 (say, the 8:4, with an expansion unit of 48): they'll be pulled flat by being to the 6:3's lower expansion unit, not widened sharp. My bad
This is a natural consequence of selecting one coincidental partial over the others, for tuning.
Bill wrote, "So, for my own work, moving from a 6:3 to a 3:1 as the basis for the temperament would actually make my 5ths busier. <snip>"
I definitely don't understand this one.
I'd be narrowing the "octave relationship" to use for the temperament octave from one of 27 expansion units (the 6:3) to one of 8 (the 3:1 "octave"). Pick a narrower octave relationship and your narrow intervals speed up; same-same, going from the 3:1 to the 2:1.
Don't lose any sleep over it. <G>
P.S. Does anyone know how to change font size with the editor. The various heading sizes don't work reliably.
------------------------------
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: 10-01-2022 19:42
From: Maggie Jusiel
Subject: Pure Twelfth tuning
Anthony! You reminded me of what I forgot! It was basically what you said about the term, "pure". I'm never sure what people mean by that. When you write about something being pure, you say which CP you are considering the target. I find this to be very helpful. Thanks.
Anthony said, "A pure 3:1 tuning, besides being impractical, wouldn't put enough stretch in the bass to satisfy other intervals, including the 6:2 twelfths."
OMG... I need to think about this. I can't quite wrap my head around this yet, but it might explain some issues I was having. Thanks!
Anthony said, "Perhaps my view is overly simplistic, but I see "perfect 12ths" tuning as a slightly different means to the same desired end. A good pure 12ths tuning is basically the same thing as a good tuning that stretches octaves."
I have had a similar point of view but focus on 5ths instead of octaves. Either way, you can get the same result. I think Bill may have answered my question about that. If I understand him correctly, he and others are wanting to use a temperament that prioritizes the 12th first with the 5ths and octaves as secondary to avoid having to tweak too much in more general tuning styles. Other than that, as long as the end results are the same, we're just talking about technique, personal style, and time management...I think. I will have to learn more.
Anthony said, "Meanwhile the 3:1 twelfth is fairly pure in the midsection and up into the treble, but is forced wide in the tenor and bass where it competes with the 6:2 twelfth which is narrow in the middle, passes through pure in the upper bass, then forced wide in the low bass by the 9:3 twelfth. This is essentially true for all good tunings, and is mandated by the piano's inharmonicity."
The 6:2 twelfth being narrow in the middle helps me wrap my head around this. The 9:3 twelfth wasn't even on my radar so thank you for helping this make more sense to me.
Anthony said, "Is there a difference between a tuning that favors the twelfths vs. one that favors octaves? It depends on which octaves the tuner is favoring. I suspect a "prefers twelfths" tuning would stretch the 4:2 octaves wider than is traditional in the lower midsection/high tenor, favoring the 6:3 octaves. So when Peter Grey claims, "I tune my octaves as wide as the piano will let me 'without making noise'" and that his sample tuning was indistinguishable from a "pure twelfths" tuning, that makes perfect sense to me."
YES!!! This makes sense to me, too. If I push my fifths, it then pushes the octaves and 12ths. It takes very little tweaking...I think. Perhaps I am not experienced enough to really know yet. lol
Bill wrote, "Anything which expands the temperament compass will slow the narrow intervals (5ths, m3rds) and speed up the wide ones... <snip> ...I think the late and sorely missed David Anderson used the term "happier 4ths). You are absolutely correct in your understand."
Thanks! I am glad I'm not misunderstanding that. My brain need not explode now.
Bill wrote, "But (a minor correction), the top half of that octave bisected by C4 is actually a 4th."
Yes, I am aware. It was a typo. *facepalm*
Bill wrote, "<snip> That's why I was interested in how a structure for a 3:1-based temperament could be consistent arrived at, early on.<snip>"
OOHH! I think I understand now. Thanks! You can check my interpretation of what you said in a response above to Anthony if you want to. I put your name in bold so you could find it easily.
Bill wrote, "<snip> But for most pianos, the 6:3's narrowing of the 2:1 below it and widening of the 4:2 (and other single octaves audible above that) is not objectionable ("noisy" as Peter would say)."
I'm not understanding. Are you saying that you are using a 6:3 octave and allowing the octave below to be more narrow by default? Meaning you are not moving the lowest note of the lower octave to add comparable stretch? Would you not have to stretch downward as you would upward, if not more so?
Bill wrote, "So, for my own work, moving from a 6:3 to a 3:1 as the basis for the temperament would actually make my 5ths busier. <snip>"
I definitely don't understand this one. I will have to ponder it. If I am being a bother, no worries. I don't want to waste your time if I haven't done my homework. I do appreciate you responding to me, and I love your "<snip>" idea. ;-)
I apologize to everyone if I'm interrupting the discussion with uninformed questions, but I do appreciate the responses. Thanks!
------------------------------
Maggie Jusiel, RPT
Athens, WV
(304)952-8615
mags@timandmaggie.net
Original Message:
Sent: 10-01-2022 18:53
From: Anthony Willey
Subject: Pure Twelfth tuning
Perhaps my view is overly simplistic, but I see "perfect 12ths" tuning as a slightly different means to the same desired end. A good pure 12ths tuning is basically the same thing as a good tuning that stretches octaves.
In general: The 2:1 octave (what most non-piano-tuners think of when you say octave) is always stretched wide. The 4:2 octave is fairly pure in the midsection and then compromises with the 6:3 octave in the lower midsection and upper tenor, and it then goes wide in the bass. The 6:3 octave goes from narrow in the midsection, to pure in the tenor, and then wide in the bass. The 8:4 octave is basically always narrow, but approaches pure in the bass where it compromises with the 6:3. (On large pianos you can sometimes get away with tuning it pure.) Meanwhile the 3:1 twelfth is fairly pure in the midsection and up into the treble, but is forced wide in the tenor and bass where it competes with the 6:2 twelfth which is narrow in the middle, passes through pure in the upper bass, then forced wide in the low bass by the 9:3 twelfth. This is essentially true for all good tunings, and is mandated by the piano's inharmonicity.
If someone came to me and claimed to have invented a tuning with "pure octaves" I would ask, "Which octaves?" A pure octave tuning is physically impossible. You've got to compromise somewhere. So when people talk about "pure 12ths" tunings, I always wonder "which twelfths?" A pure 3:1 tuning, besides being impractical, wouldn't put enough stretch in the bass to satisfy other intervals, including the 6:2 twelfths.
Is there a difference between a tuning that favors the twelfths vs. one that favors octaves? It depends on which octaves the tuner is favoring. I suspect a "prefers twelfths" tuning would stretch the 4:2 octaves wider than is traditional in the lower midsection/high tenor, favoring the 6:3 octaves. So when Peter Grey claims, "I tune my octaves as wide as the piano will let me 'without making noise'" and that his sample tuning was indistinguishable from a "pure twelfths" tuning, that makes perfect sense to me.
I'm coming at this from more of an Entropy perspective, so I don't pay much attention to the theoretical frequencies of pure octave ET versus pure twelfth ET and how the latter stretches the theoretical octave 1.23 cents wider or whatever. In the context of piano tuning, the standard "1/12th root of 2" ET basically functions as a ruler that I can use to measure my "stretched" tuning. And its usefulness basically ends there. Measuring a tuning with a "1/19th root of 3" ET ruler doesn't change the tuning, it just makes it so the lines on the ruler line up better with what I'm measuring.
Just to be clear, I'm not saying anything against perfect 12th tunings. I think they're great, and I've enjoyed reading and re-reading Kent's articles about them.
------------------------------
Anthony Willey, RPT
http://willeypianotuning.com
http://pianometer.com
Original Message:
Sent: 09-30-2022 23:21
From: Norman Brickman
Subject: Pure Twelfth tuning
In case anyone wants some background on Kent's last post, try the Wikipedia posting on Equal Temperament: https://en.wikipedia.org/wiki/Equal_temperament
Kent, good point in your last post. I usually prefer to expand on the terminology to be more precise, as I did in my April Journal article where I wrote "19-TET based on the 12th" and as I did for my first post in this thread where I said "19-TET based on the P12" in referring to what you call "P12 ET." Then I got lazy and just started saying "19-TET" – sorry. I could edit them. At least the long version is unambiguous, even if "normal" usage of TET is based on the octave. Let me know if there are still issues. Thanks.
BTW, I tune to what you call a "pure, untempered octave," or what you earlier called "natural stretch" which is simply "correction for inharmonicity." So in my opinion and from what I have read 12-TET, or 12-TET based on the octave, or 12-EDO expresses that correctly. An example is in the Wikipedia article I noted above on ET. Is "P12 ET" a standard terminology?
David – your example of 19-TET based on the octave, or 19-EDO, is very interesting. If you are not already familiar with it, you might be interested in the Bohlen-Pierce Scale. It was developed in the 1970's and is an ET based on the 12th, what they call a "tritave." Rather than the 19 notes of Kent's P12 ET, it only has 13 notes (A 13-TET based on the 12th). https://en.wikipedia.org/wiki/Bohlen-Pierce_scale
Regards, Norman
------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321
Original Message:
Sent: 09-30-2022 21:01
From: David Pinnegar
Subject: Pure Twelfth tuning
Kent - thanks so much for clearing this up. TET is a term one's seen and easily confusable.
https://academo.org/demos/19-tet-keyboard/ is rather a fun 19-TET keyboard to play with. Can we tune a 12 note piano to any of those notes in a meaningful way? Would a choice of some approximate for instance to meantone?
This comes to something really interesting in so far as building a piano and piano tone to give power, raw power without resonance, and what resonance we can build into the piano tone from resonance.
True 19 TET as heard on that link appears to be unrelated to harmonic series and therefore requires a piano capable of raw power even without resonance.
I posit that fortepianos achieved power through resonance through unequal tunings and, applying those principles to modern pianos I'm achieving greater power available to pianists, and for those wanting extreme power, without breaking strings.
http://www.sbfisica.org.br/rbef/pdf/342301.pdf quoted in this thread is very interesting giving a minimum entropy solution for pianos and the Railsback curve with deviations. However that paper is flawed in terms of minimum entropy:
1. the experimental measurements put harmonics into mathematical bins (or boxes) 1 cent wide. Analysis by cents is fundamentally flawed as it's beats which the ear hears as either still and smooth or fast and rough, independent of cents
2. It calculates the solution assuming equal temperament, and then labels this minimum entropy and then demonstrates that there have to be variations from the equal temperament line to achieve minimum disorder. It doesn't consider whether some forms of unequal temperament might lower entropy more. My numerical analysis with spreadsheets which I published a few years ago here suggest that is possible, and demonstrated experimentally with the analysis of a chromatic harp in equal and unequal tunings.
I believe that there are two harmonic series that pianos can exploit and conventional tuning only follows one - and of which Pure 12th tuning is a good solution to one.
What is really interesting is that where a string has mass in the middle as explained on
https://newt.phys.unsw.edu.au/jw/harmonics.html a plucked string or hit string have odd numbered harmonics with different inharmonicity to the even ones. The natural tendency is for inharmonics to be sharp whilst the strings with mass have odd harmonics to be flatter - so that the tendency to sharpness might be mollified flatter for the odd harmonics. This is why tuning with pure 12ths with focus on the 3rd partial can sound particularly good, and clean.
Perhaps those who are responsible for where PTG is going might consider whether someone who tunes like me who don't tune to ET (and don't want to).should fail PTG exams.
Tuning for festivals and competitions who like my tunings as I do, as well as crème de la crème of performers, it doesn't matter to me whether I'm debarred from registered PTG or other professional status, but to my students in the future it's another matter.
Best wishes,
David P
--
- - - - - - - - - - - - - - - - - - - - - - - -
David Pinnegar, B.Sc., A.R.C.S.
- - - - - - - - - - - - - - - - - - - - - - - -
+44 1342 850594
Original Message:
Sent: 9/30/2022 6:20:00 PM
From: Kent Swafford
Subject: RE: Pure Twelfth tuning
Norman,
Please note that the terms 12-TET and 19-TET are used by the micro-tuning community (not the piano tuning community) and both assume a pure, untempered octave. Therefore, 19-TET means 19 equally spaced tones to the pure octave; this is definitely not a piano tuning and not of much use to us here.
I try to avoid the confusion by specifying the interval being equally divided: for example, 12 tone to the pure octave ET, or 19 tone to the pure 12th ET, or, in the case of the factory stretch tuning for the Rhodes piano, 12 tone to the 2-cent expanded octave ET.
Sent from my iPad
Original Message:
Sent: 9/29/2022 11:21:00 PM
From: Norman Brickman
Subject: RE: Pure Twelfth tuning
Bill, let me try to summarize what you wrote (using Kent's language). And then I'll add my $0.02.
I think you are saying that you like the "stretched octave" of the P12 ET (19-TET) tuning, with its 1.23 cents widening of the "natural octave" of 12-TET, better than you like the "natural octave" of 12-TET. And further, based on Dan Levitan's writings, the octave in 19-TET "messes up the overall tuning" the least.
I think it is clear that the "natural stretch" (again using Kent's language) of the octave will always sound best, and P12 ET is not meant to produce the best natural stretch of the octave. P12 ET octaves are 1.23 cents wide. The best (natural stretch) octave is obtained by trusting the piano tech (and their ETD if used), who is tuning in 12-TET, to real-time decide on 2:1 or 4:2 or 8:4 etc depending on the various factors appropriate for the scale of a particular piano. Upon completion of a 12-TET tuning, listen to the quality of the (pure or natural stretch) octaves and then also listen to the narrowed fifths (and twelfths). A 19-TET tuning has different characteristics than 12-TET and is found to be more desirable by some due to that.
Regards, Norman.
------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321
Original Message:
Sent: 09-29-2022 21:52
From: Bill Ballard
Subject: Pure Twelfth tuning
David Pinnegar wrote:
"Please excuse me but 'As can be seen, the 3:1 "octave" is far more immune to the consequences of the prevailing inharmonicity than the 4:2 and 6:3 single octaves.' "
Don't worry, David. You're not being thick-headed; I'm being obtuse.
Let me rephrase that:
The 3:1 octave messes up the overall tuning less than any of the single octaves. The 2:1 drags everything above it flat, an the 4:2 pushes everything below it (the 2:1 and the 3:1) sharp. As does the 4:2, by a bigger amount.
This is all based on Dan Levitan's "interval inharmonicity" (the inharmonicity created by combining the individual inharmonicities of the interval's two pitches).
Many people in this thread have said that the 3:1 is the best compromise among the octaves for carrying out the temperament to each end of the piano. I agree.
You wrote:
"In the bass on shorter instruments I find that P12 tuning results in the bass not being low enough."
That's why I run my 6:3 all the way down to the bottom. Violinists need plenty of expansion in those lower octaves so that overtones from them agree with their intonation above the treble staff. I once (for shorts and gaggles) tuned the single string at the bottom with an 10: 5. I heard about it after rehearsal. The one piano piece on the program was the Ravel P3, and low octaves of the slow movement sounded like indigestion.
------------------------------
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: 09-29-2022 19:07
From: David Pinnegar
Subject: Pure Twelfth tuning
Please excuse me but "As can be seen, the 3:1 "octave" is far more immune to the consequences of the prevailing inharmonicity than the 4:2 and 6:3 single octaves. "
I'm clearly being thick but don't see what is being said here other than tuning to the P12 harmonic being less stretched.
In the bass on shorter instruments I find that P12 tuning results in the bass not being low enough.
Very interestingly there is a difference between inharmonicity of strings with mass in the middle of the string - the odd partials are flatter than the even ones and as a result the P12 will give less stretch compared with 4:2 tuning.
Best wishes
David P
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David Pinnegar BSc ARCS
Hammerwood Park, East Grinstead, Sussex, UK
+44 1342 850594
"High Definition" Tuning
Original Message:
Sent: 09-29-2022 12:18
From: Bill Ballard
Subject: Pure Twelfth tuning
It bears mentioning that the P12 temperament is a creature of the ETD. For years I tried to figure out an aural temperament based on a P12 from A4 to D3. However what the next next would be was always a brick wall. Of course the 3ds/6ths temperament does a beautiful job of dividing the temperament octave into three equal parts (read, contiguous M3ds, the ratio of whose beat rates is constant even after factoring in individual inharmonicity).
But there was nothing available as a 3d step (from D3 onward) which would systematically specify the remaining 11 notes temperament notes. (One of those notes, keep in mind, would be A3, which would remain to be determined during the equal spacing of semitones {inharmonicity included} in that temperament compass.)
I checked with Dan Levitan, who had reached a similar conclusion.
That does not preclude laying out the top half of the tuning in P12s once the temperament is set, as I do. And it doesn't not preclude a well-experienced educated guess as to the width of whatever the next interval following the initial P12 might be. Peter Grey gave an ear-opening demonstration to our Chapter of a temperament beginning on with an F3 fork, using only two intervals: up a 4th then down a M3d, and onwards.
At regular intervals here, I post my favorite page from Dan Levitan's "Craft of Piano Tuning". (The comparisons are based on an inharmonicity of 1.0.) As can be seen, the 3:1 "octave" is far more immune to the consequences of the prevailing inharmonicity than the 4:2 and 6:3 single octaves. The 2:1 simply has its head in the sand, as far as these consequences.
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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
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