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I recently watched Adam Neely's video about comma pump. He quotes from "Fundamentals of Musical Composition" by Arnold Schoenberg. (Chapter XI, page 99):

...the natural semi-tones differ in size from the tempered (a fact which causes choirs to get off pitch)

Then Adam Neely says it's common for choirs singing a cappella to end in a different pitch standard from they began. This is quite surprising to me, for I had believed that I can sing a cappella in 12-tone equal temperament.

This arises the following questions about a cappella:

  1. As we unconsciously tune intervals to simplest just ratios, does that mean 7/4 will more frequently appear than 9/5, or 13/7 will do so more than 15/8?

  2. In case question #1 holds, will neutral intervals like 7/6 or 11/6 have a chance to appear?

  3. Should we avoid comma pumps? If so, would that mean we should occasionally detune some intervals to let the tones stay in the position? Say, if we got 2 syntonic commas and thus the tonic is 43.01 cents acute, we cannot have another syntonic comma to the tonic, so the preceding interval(s) should be detuned. For example, in the "Benedetti's puzzle" in the video, for the third A-C interval (ignoring the E; if it were present, the C-E-A chord will be rather dissonant), instead of 3/5, I could use 7/12, to let the C stay in the position.

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    I think what he means is choirs (occasionally) drift off key, and end slightly in a different key from the one they started in. Due mainly to the human propensity to sing not using 12tet. Banks of violins have also been known to do similar. – Tim May 10 at 7:16
  • That exact same quote is mentioned here: music.stackexchange.com/q/11898/9426 – Brian THOMAS May 10 at 11:07
  • @BrianTHOMAS - does that make it a dupe? – Tim May 10 at 12:49
  • There's another solution to comma pumps aside from detuning individual chords, which is removing the constraint that notes common to adjacent chords must be sung at the same pitch. – phoog May 10 at 22:51
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    It seems you're title question is very different from the questions in the post. Xenharmonic, to me, suggests tuning that is explicitly non-western, so by definition, I think the acapella you have in mind would not be xenharmonic, even if it's technically microtonal. Occidental music could be considered to have had several different tuning systems besides 12edo, such as meantone and just intonation, and I don't think any of those would be considered xenharmonic. – awe lotta May 11 at 5:38
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I admire a lot of Adam's work, but I think he's exaggerating a bit about the reasons why choirs get off pitch. (Though he's stating a commonly held belief -- or perhaps common excuse.) Yes, most choirs tend to drift in pitch when singing a cappella, but I guarantee you that at least 99% of the time, it's not due to "comma drift." Instead, non-professional choirs drift because of poor technique and because their intonation is poor overall.

Explaining that in detail is a bit off-topic from this answer. Suffice it to say that the research actually measuring intonation in choirs is inconclusive, with most studies showing that it approximates equal temperament (though drifting around quite a bit), with the potential exception of major 3rds in some triads, which trend a bit closer to 5:4 JI thirds in some circumstances (not as often in leading tones). Minor thirds and sixths tend to be closer to ET, and the rest of the intervals either don't differ enough between JI and ET to make a difference (e.g., fifths, fourths, major seconds) or are dissonances that tend to vary a lot more in intonation. There's no evidence that I've ever seen that non-professional vocal ensembles tend to "lock in" to JI intervals over ET or some other approximation, with the exception of groups that specifically train for this type of lock-in (like barbershop warm-ups).

(Most studies on this stuff are behind paywalls. Here's one that's not and shows the high variation in intonation for a quintet of highly trained singers. I chose this article too because it had a decent summary of other experiments in the literature at the beginning. It does suffer a bit from stimulus design problems, as they deliberately avoided leading tones and introduced all sorts on nonsense into an "arrangement" of Bach chorale the breaks voice-leading rules on almost every chord. Even by creating a biased stimulus that seems designed to try to get evidence of JI thirds, they barely did, which shows how much ET listening influences most singers.)

With all that in mind, let's discuss your actual questions.

  1. As we unconsciously tune intervals to simplest just ratios, does that mean 7/4 will more frequently appear than 9/5, or 13/7 will do so than 15/8?

Not likely. 7:4 is a horribly flat minor seventh to most ears that are used to equal temperament. No choir singer is going to naturally gravitate toward that interval unless they are in a blues choir, barbershop chorus, or some other ensemble that specifically highlights that interval. To be clear, 7:4 is perfectly appropriate in those contexts and will "sound right" there, but no choir singing Bach or Mozart is going to drift toward 7:4 tuning for the sevenths in dominant seventh chords. They might simply go flat due to poor technique, and might push a seventh down naturally just a bit (for the same reason that leading tones tend to be sung sharp -- tendency tones get pushed around quite a bit by trained singers toward where they are going to resolve). But 7:4 is only an interval you'll hear a choir sing (1) in specific genres, (2) for specific effect, or (3) by accident because they're really poor at singing "in-tune" (and for most choirs, "in-tune" is generally an approximation close to equal temperament).

For similar reasons, choirs are unlikely to precisely tune a major seventh at a 15/8 or 13/7 interval. Instead, the context of that major seventh is more likely to influence its tuning -- whether as leading tone or descending note or whatever.

  1. In case question #1 holds, will neutral intervals like 7/6 or 11/6 have a chance to appear?

Highly doubtful. Even professional singers often have great difficulty tuning "neutral intervals" precisely when they are well outside the standard ET scale. They're certainly not going to naturally gravitate toward them.

  1. Should we avoid comma pumps? If so, would that mean we should occasionally detune some intervals to let the tones stay in the position?

"Comma pumps" in choral singing are practically almost a myth. If you happen to be in that elite less than 1% of professional vocal ensembles that can perform with very precise tuning, they can become an important worry in pitch drift. But the vast majority of choirs have plenty of other intonation problems to solve first.

And in case you think I'm just being contrary -- no, this is empirical fact. From the introduction of the study linked above: "Devaney et al. found no evidence of pitch drift in an exercise written by Benedetti in the sixteenth century to illustrate potential pitch drift associated with 'pure tuning,' when performed by four expert 3-part ensembles." Yep, there was a literal experiment done using Adam Neely's chosen example of a "comma pump," and choirs didn't pitch drift. Why not? Lots of reasons, but in such a short progression, pitch memory is important. That's actually a good reason why "comma pumps" that are typically used for examples don't function that way in practice -- good singers have good pitch memory and will return to notes very close to what they just have sung rather than reorienting completely with every new chord. (Exceptions tend to be when chords are sustained for long enough for "lock-in" to become desirable in skilled ensembles, or when ensembles specifically rehearse precise intonation.)

If you happen to be in one of those incredibly rare ensembles that can sing so precisely in tune that comma drift actually is a primary reason for ensemble intonation drift, then yes, there might be the need for some planning. I have friends who are professional singers in early music ensembles and spend a great deal of time tuning individual sonorities. Even there, though, the natural variation of notes in various situations will alleviate a lot of comma drift. As noted above, leading tones (for example) tend to be pushed higher. Very few ensembles are going to naturally tune the third of a dominant chord as a just 5:4, when it can have a more vibrant ET or Pythagorean third that wants to resolve upward. This ebb and flow of natural variance in tuning (along with recent pitch memory) is often enough to keep the overall intonation static for professional ensembles, even without extensive planning.

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  • Was Benedetti's example supposed to make pitch drift obvious though? Since it's so short. – awe lotta May 11 at 6:05
  • Have to +1 for a very sober and evidence-based answer, but that doesn't mean I agree with the general conclusion. I look at it more this way: our ears are crippled by life-long experience to tempered music, so going back to the correct just intonation requires an awful lot of mending the bad habits. But that doesn't mean composers should just shrug off 12-edo as the status quo. Composition software should include better utilities for consciously taking just intonation into account, then it would also be straightforward to give out MIDI-renderings as practice guides for singing intonation. – leftaroundabout May 11 at 12:29
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    @leftaroundabout: To be clear, I don't think we should just accept ET as the default tuning system and move on. But I also don't think our ears are "crippled." It's just what we're used to. Those raised with traditional Javanese scales (not a lot in-common with 3-limit or 5-limit JI at all, likely due to the emphasis on gong-like instruments with inharmonic partials) are used to those scales, too. There's nothing magical or "correct" about 5-limit JI; it's just another tuning system. That said, I wish more people learned about tuning and spent time listening to variations in tuning systems. – Athanasius May 11 at 16:54
  • @awelotta: No, I don't think it was. In fact, I think Benedetti himself wouldn't have agreed with Neely's conclusion about choral drift, as he was using this to prove why temperament was necessary, not that "comma pumps" occurred in practice. I just skimmed through his original Latin letter, and I'm not sure he addresses practical singing. But subsequent commenters on Benedetti certainly thought that comma drift wasn't a practical problem, or that if it did happen, it would sound offensive. See this article for some discussion. – Athanasius May 11 at 17:41
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    @phoog: If this is true, please make recordings and submit to a study in a journal. From my own personal experience and from everything I've read in studies on this stuff, accuracy in choral intonation is generally significantly less precise than people think or claim. Inevitably in every choral group I've sung in or worked with, people go flat because (aside from rare cases like specific singers that pull things sharp) most choirs inevitably go flat. Comma pump or not. It's often natural pitch drift due to inadequate compensation for vocal strain, etc. But I'd be happy to see data for this. – Athanasius Sep 15 at 18:06
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Well, you'd need to have a really, really excellent choir for this effect to be observable. For most choirs, yes, they drift. A lot. Unless the choir is accompanied, they will end every piece on a different tuning than they started with. The better the choir, the smaller the drift. But even the best choir may end a piece a bit flat one day, and the very next day end that same piece a bit sharp. Even with two performances on the same day, the drift may be different.

This normal pitch drift of choirs is all down to psychology. If the members are not feeling comfortable for some reason, they will tend to drift down. If they feel one with the world and the music they are producing, they will tend to drift up. If they sing in a place with a grand acoustic, they will tend to drift up. If they sing in a place where the room does not give them any positive feedback, they will tend to drift down.

That said, they are musicians, and they are using an instrument that is not chained to an equal temperament. And if they are good, they will tend to correct the errors of equal temperament within the chords. They are aiming for the perfect sound, after all. Nevertheless, singers in a choir do remember where the pitches of recent notes were supposed to be, so any correction of temperament will tend to cancel out in the long run.

In the end, the goal of any good choir singer is to sing the pitch that sounds best at every place. Best in the context of what the other voices are singing, and, to a lesser degree, best in the context of their own melody. And that best may include temperament corrections along with corrections to stay in tune. But if they have a bad day, all bets are off anyways.

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  1. It will depend on the specific piece of music, since the drift in tuning of one harmonic progression might be cancelled by another harmonic progression. It also depends on the abilities of the singers, since most will be unable to sing that precisely. If the singers have practised their part with a piano, and most of us have the mistuning of equal temperament so ingrained, it is not guaranteed the intervals will cause pitch drift.
  2. We do not write music (except for composers like Harry Partch) using harmonics 7, 11, 13... (although there may be a case for claiming the 7th harmonic is used in the Blues). Intervals using those ratios are therfore not going to occur.
  3. Equal temperament is an example of detuned just intonation, invented for that very purpose, though it was not the only one (e.g. 1/6 comma tunings such as Vallotti-Silbermann).
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  • "the drift in tuning of one harmonic progression might be cancelled by another harmonic progression": this is theoretically possible, but it tends not to happen in practice because the circle-of-fifths progressions so common in European music push the pitch down. Progressions pushing the pitch up are rare. The septimal seventh is quite commonly used in performances of 19th and early 20th century harmonies such as in barbershop quartets. – phoog May 10 at 22:54
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As we unconsciously tune intervals to simplest just ratios, does that mean 7/4 will more frequently appear than 9/5, or 13/7 will do so more than 15/8?

Well, no. Simple in this case means having smaller prime factors. Under that definition, 9/4 is simpler than 7/5, because the largest prime factor in the first ratio is 3 while that of the second is 7. Similarly, 15/8 is simpler than 13/7, because the largest prime factor of the first ratio is 5, while that of the second is 13. This preference exists because intervals don't appear in isolation. For example, consider the ratios with some other pitches in the major scale if the seventh degree is tuned to 13/7:

do - ti, major seventh: 13/7
re - ti, major sixth:   13/7 / (9/8) = 104/63
mi - ti, perfect fifth: 13/7 / (5/4) =  52/35
sol - ti, major third:  13/7 / (3/2) =  26/21

Now look at 15/8:

do - ti, major seventh: 15/8
re - ti, major sixth:   15/8 / (9/8) =   5/3
mi - ti, perfect fifth: 15/8 / (5/4) =   3/2
sol - ti, major third:  15/8 / (3/2) =   5/4

For example, in the "Benedetti's puzzle" in the video, for the third A-C interval (ignoring the E; if it were present, the C-E-A chord will be rather dissonant), instead of 3/5, I could use 7/12, to let the C stay in the position.

No, you couldn't use 7/12 to let the C stay in position. The ratio of A to the low G is 9/4. Instead of 3/5 for the C, which results in a ration with the low G of 27/20, you want to use 7/12, because you think that would put it "in position." But that puts C at 63/48. The position that eliminates pitch drift is 4/3. So instead of the syntonic comma of 81/80 demonstrated in the video, all you've done is establish a different comma of 63/64. (Wikipedia reports that this is the septimal comma or Archytas' comma.) Instead of going up 21.5 cents with each iteration, with your system the pitch will drop 27.3 cents. The only interval from the A to the C that avoids pitch drift is 27/16.

(In practice, this progression is quite simple to tune in just intonation simply by allowing a Pythagorean major third of 81/64 between the C and the E instead of requiring a 5-limit just major third of 5/4. Then the E can be a perfect fourth, and all the chords are quite tolerable. The C-E-A chord is a bit buzzy, but that's ok because its function is transient, so it does not require the sense of "repose" associated with the just major third.)

Now barbershop groups (and others performing late-19th century music) will indeed lower their minor sevenths to get a septimal dominant seventh chord, so, for example, a V-I cadence might be like this:

Tenor 1: F4(21/8)  E4(5/2)
Tenor 2: D4(9/4)   C4(2)
Bass 1:  B3(15/8)  G3(3/2)
Bass 2:  G2(3/4)   C3(1)

That dominant seventh chord will have a very satisfying ring, but it comes with some problems: it yields a very small descending half step for the first tenor of 84 cents, and of course you can't use the F of 21/8 for hardly anything else: it's 29 cents flat from equal and 27 cents flat from a perfect fifth below C.

To test the 13/7 interval, I tried this Phrygian cadence in both traditional just intonation and a septimal tuning:

X: 1
M: 4/4
L: 1/4
K: Emin
%% score (V1 V2)
V:V1
V:V2 merge
% 1
[V:V1]c-|cB2A|B4
[V:V2]E|D2C2|B,4

For the traditional tuning, I used these pitches:

B3=247.5; C4=264; D4=293+1/3; E4=330; A4=440; B4=495; C5=528

This yields the following ratios:

8/5  9/5  27/16  15/8  5/3  2/1

For the septimal tuning, I used these pitches:

B3=238+1/3; C4=256+2/3; D4=293+1/3; E4=330; A4=440; B4=476+2/3; C5=513+1/3

This yields the following ratios:

14/9  7/4  13/8  13/7  12/7  2/1

Now I spend a lot of time listening to and singing music in just intonation. The septimal progression sounded so out of tune to me that I could not pay any attention to how the 13/7 interval sounded. But then I modified the example to stress that interval, and it did indeed sound ok:

X: 1
M: 4/4
L: 1/4
K: Emin
%% score (V1 V2)
V:V1
V:V2 merge
% 1
[V:V1]c-|cB2A|B4|B4|B4
[V:V2]E|D2C2|B,2C2|B,2C2|B,4

But whether any singing group would actually use it seems quite doubtful. As Athanasius notes in his answer, tendency notes are generally tuned more closely to the following note, creating a wider interval, but this interval is far narrower. Perhaps there is some other context in which it might appear. I'll think on it a bit and perhaps edit this answer if I come up with anything.

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