## Someone is wrong on the internet

The self-described "grumpy old coot" who writes the blog  Right Wing Nation has recently put up a generally admirable post ("On An Entirely Different Note", 2/19/2009) on the relations among music, mathematics, physics. Unfortunately, his account of the Pythagorean comma is psychoacoustically, historically and mathematically wrong.

He gets off to a good start by introducing the Circle of Fifths, showing that 12 fifths bring you back to the name nominal pitch class:

Tune a starting note, say middle C (or if you’re tuning a piano, A-440). The human ear can hear, and tune, a perfect fifth, so we would next tune G, because G is a fifth above C. We would then tune a fifth up from G (D), then a fifth above D (A), and so forth, until it led us finally back to our starting note, C (and that brings us back to do!) It looks like this (the acoustic space between each adjacent pair of tones is a fifth):

C-G-D-A-E-B-F#-C#-G#-D#-A#-F-C

But then he goes off the rails:

Acoustically, what the human ear hears as a perfect fifth is just slightly more than a fifth (about 2 cents, and there are 100 cents in a semi-tone). But as we tune from fifth to fifth, that tiny difference adds up, one after another, so when we get back to C, it is WAY out of tune with our starting C.

No, no, no.  The problem is that a perfect fifth is a ratio of 3/2, while an octave is a ratio of 2/1; and (3/2)12 — the result of 12 successive intervals of a fifth–  is not the same as 27 — the result of 7 successive intervals of an octave.

The results are close — 27 is 128, while (3/2)12 is around 129.7463. Another way to see the relationship is to note that

(3/2)12/27 = 312/219 = 531441/524288

But they're not the same, and the difference is what is known as the "Pythagorean Comma". This is just mathematics, and has nothing to do with the properties of the human auditory system.

As discussed here a couple of years ago ("Pavarotti and the crack to chaos", 9/9/2007), the Pythagoreans saw this difference as one of three crucial flaws in the fundamental mathematical fabric of reality that challenged their belief in its coherence.  So when the G.O.C. writes that "the ear is imperfect, so we have the Pythagorean Comma, a serious problem identified by Pythagoras that was not solved until the 18th Century", he's wrong.  And it's historically as well as physically and mathematically important that this issue is not merely a matter of some imperfection in the human sensory apparatus.

I shouldn't be writing this, I should be working on the announcements for the new journal that I'm helping to start,  or one of the papers that I owe, or the problem set that I need to create.   But as Randall Munroe (or his cartoon character, anyhow) put it, "What do you want me to do?  LEAVE?  Then they'll keep being wrong!"

And I know that the G.O.C. appreciates mathematical rigor and historical accuracy.

1. ### JLR said,

February 20, 2009 @ 1:42 pm

Leaving aside the historical and psychoacoustic issues, I don't think he's mathematically wrong. He is simply comparing a perfect fifth (3/2) with an equal-tempered fifth 2^(7/12) (~1.498) and noting that (3/2)^12 is not equal to (2^(7/12))^12 (= 2^7), which is exactly what you said.

2. ### Stephen said,

February 20, 2009 @ 1:48 pm

I think the author is trying to get at the notion of temperament, which you discuss in your previous post. He poses the problem backwards, saying "what the human ear hears as a perfect fifth is just slightly more than a fifth" when he should be saying "the mathematical and historical perfect fifth is larger than the equal-tempered fifth". Acoustics are only important in that the human ear, at least a human ear unsullied by equal temperament, will find a perfect fifth to be more in tune.

3. ### Craig Daniel said,

February 20, 2009 @ 1:50 pm

But what we hear as a perfect fifth is. An equal-tempered fifth is an approximation that isn't quite perfect, designed to make every part of the scale equally wrong rather than (as in older systems) one part of the scale that you can avoid in your music very wrong and the rest just right or (as in well temperament) everything differently wrong in ways that are slight but give each key its own character.

4. ### Dmitri said,

February 20, 2009 @ 1:54 pm

I'm going to dock you half a point for saying that 12 fifths brings you back to the same "nominal pitch class." This is pretty much equivalent to the statement that you're in a 12 note temperament.

In some temperaments, like 19 tone equal, 12 fifths brings you somewhere else:

Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#

With Bb and A# represented by *different keys* on the keyboard. If memory serves, there are actually 17th-c. instruments built this way.

5. ### Eric P. said,

February 20, 2009 @ 2:02 pm

Also, it's probably worth pointing out that, even if you're tuning a piano, "Middle C" is not the same thing as "A-440." A-440 occurs a major sixth above Middle C.

6. ### Andy Hollandbeck said,

February 20, 2009 @ 2:18 pm

The first flutes had no keys, only holes, and were therefore not completely chromatic. When Quantz and his ilk began experimenting with adding keys to get more notes, the first key added produced an Eb. Further experimentation produced a second key — which produced a D# in the same octave.

Although this is great fodder for flautist jokes (which are the band equivalent of dumb blonde jokes), there is a legitimate argument that these two keys produced two acoustically different notes, even though by today's standards they are harmonic equivalents.

7. ### rpsms said,

February 20, 2009 @ 2:18 pm

Reminds me of the arguments regarding dozenal vs. decimal.

I suppose the fact that you can't efficiently box up 10 eggs in a cube shaped box might be construed as a problem with space-time.

Or something.

8. ### Andy Hollandbeck said,

February 20, 2009 @ 2:20 pm

@Eric: GOC indicated that A-440 is "a starting note" for tuning a piano, not as a description of middle C.

9. ### Bloix said,

February 20, 2009 @ 2:26 pm

I would love to have these concepts illustrated with actual tones. Is anyone aware of a website, or a recording, that does so?

10. ### Andy Hollandbeck said,

February 20, 2009 @ 2:33 pm

There's also a great, more involved explanation of the whole thing here:
http://www.amarilli.co.uk/piano/theory/pythcrcl.asp

(I'm not affiliated with either site — this isn't self-promotion.)

11. ### Bruce Balden said,

February 20, 2009 @ 2:48 pm

GOC is referring to two different concepts of a 'perfect fifth', an equal tempered one (as evidenced by the tuning of an ideal piano) and a 'natural one' (Pythagorean, or judged by ear).

For purposes of discussing these things in detail, each equal-tempered 'semi-tone' is divided into 100 cents, thereby dividing an octave into 1200 'cents'. In mathematical terms cents(r) = 1200*(log(r)/log(2)) where r = pitch ratio and logarithms are in your favorite base.

Having established the frame of reference and noting (as Mark does) that for a Pythagorean (natural) perfect fifth r=3/2, we find that cents(3/2)=701.955… thereby establishing the author's point that a piano's equal-tempered 'perfect fifth' is about 2 cents less than an ideal one.

12. ### Sili said,

February 20, 2009 @ 2:52 pm

This is and has always been my 'problem' with music.

I get the maths. No problem whatsoever. Makes perfect sense.

But … I don't get what it means. Or sounds like, I guess.

Any suggestions for how to 'learn'? That doesn't involve investing a piano - can't afford that on the dole, I fear.

13. ### Cameron said,

February 20, 2009 @ 2:53 pm

Craig Daniel wrote: "An equal-tempered fifth is an approximation that isn't quite perfect, designed to make every part of the scale equally wrong rather than (as in older systems) one part of the scale that you can avoid in your music very wrong and the rest just right or (as in well temperament) everything differently wrong in ways that are slight but give each key its own character."

But that's not quite right. Equal temperment doesn't make each part of the scale equally wrong. In fact, the fifths are almost the same as perfect fifths, but the thirds and sixths are quite a bit off. What equal temperment buys you is that all keys are off in the same ways, allowing a composer to modulate freely using all keys.

There's a good intro to historical tunings here: http://www.kylegann.com/histune.html

14. ### Jim said,

February 20, 2009 @ 3:15 pm

Let me second Cameron's recommendation to check out Kyle Gann's excellent website. He is currently writing a wonderful textbook on this topic. For two different (and equally contentious) takes on the historical aspect of this stuff, check out the books by Ross W. Duffin's and Stuart Isacoff.

If you'd like to hear what some of these intervals sound like, try out my (very beta) software, Interval Calculator, at http://tweeg.net/software.html

I would like to go ahead and chime in though, that the original author is correct in saying that what we hear as a perfect fifth (a 3/2 fifth) is larger than an equal tempered fifth. And also, where you say he is correct, he is actually incorrect, because as other commenters have pointed out, if you move 12 fifths away from C, you'll end up at B#.

15. ### Mark Liberman said,

February 20, 2009 @ 3:22 pm

Jim: [T]he original author is correct in saying that what we hear as a perfect fifth (a 3/2 fifth) is larger than an equal tempered fifth.

Reference? But in any case, the point is that the Pythagorean Comma is not a fact about psychoacoustics, it's a fact about mathematics, namely that the 12th power of 3/2 and the 7th power of 2 are close but not identical.

Jim: where you say he is correct, he is actually incorrect, because as other commenters have pointed out, if you move 12 fifths away from C, you'll end up at B#

Right, but if the universe were competently constructed, by the lights of the Pythagoreans, this wouldn't be true, and we wouldn't have to detune all our intervals in order to modulate freely.

16. ### Theodore said,

February 20, 2009 @ 3:27 pm

@Sili: Lucky for you, a piano is the wrong instrument to hear these differences since it's tuned more or less to equal temperament and hard to change on the fly for listening purposes. What you need is a cheap guitar, where you can tune the intervals between strings pure and use strings stopped at the frets (more or less set to equal temperament) for comparison.

Another good site, about one particular temperament that's neither equal nor pythagorean, is Scott Lehman's Larips.com. Lehman claims to have deciphered J.S. Bach's tuning from a figure drawn on the cover of the Well-Tempered Clavier. Background and links to audio are there too.

17. ### Andy Hollandbeck said,

February 20, 2009 @ 3:57 pm

There is a perspective issue here, too. We must remember that math is a language created by humans to describe the world. Just because we use a base-10 system of counting doesn't mean that the universe runs in groups of ten. The relationship of pi to the circumference of a circle is a natural fact. Pi isn't an irrational number because of nature; it's irrational because that's the only way we can fit the concept into our mathematical constructs. Reality isn't the numbers, the numbers just describe reality.

What Pythagoras saw as a flaw in the fabric of reality is just a problem of perspective. What he actually saw was a flaw in his human-created mathematics to describe something in nature. Perhaps if we had been born with six fingers per hand, these mathematical descriptions would fall neatly into place.

[(myl) I don't see how changing the base of the number system will change anything about this problem. If you translate it all into base 12, for example, the issue becomes the fact that 3/2 raised to the 10th power (i.e. the 12th power in base 12) is not the same as 2 raised to the 7th power. In binary, it's the fact that 11/10 raised to the 1100th power is not the same as 10 raised to the 111th power. Expressed in any base, it's the same fact. Or am I missing the point you're trying to make? ]

18. ### Robert Coren said,

February 20, 2009 @ 5:37 pm

Ever since I've understood (what little I do understand) about wavelength ratios and the circle of fifths, it's struck me as a remarkable coincidence that 2^7 and (3/2)^12 are so close to each other, without which our whole tonal system wouldn't work at all (or would have developed in some other way, I guess).

19. ### Forrest said,

February 20, 2009 @ 5:50 pm

Do you ever read a web comic called Cyanide and Happiness? For a moment I thought you had named this post after one of their cartoons … I'll try to dig up a url.

20. ### Forrest said,

February 20, 2009 @ 5:55 pm

Sorry - I was completely wrong. This was xkcd, and it looks like you did title the post after the cartoon.

http://xkcd.com/386/

[(myl) Yes, I even linked to the cartoon (or to my earlier posting on it), and quoted the mouseover text, attributint it to xkcd's author, Randall Munroe. ]

21. ### Craig Daniel said,

February 20, 2009 @ 6:06 pm

Cameron said: "But that's not quite right. Equal temperment doesn't make each part of the scale equally wrong. In fact, the fifths are almost the same as perfect fifths, but the thirds and sixths are quite a bit off. What equal temperment buys you is that all keys are off in the same ways, allowing a composer to modulate freely using all keys."

What I meant but didn't say entirely clearly was basically exactly what you said - in contrast to the original "equal temperament" (which we now call "well temperament"), the system which Bach wrote The Well-Tempered Clavier to show off. In well temperament, you can modulate freely between keys - but because the semitones aren't all exactly equal, the different keys sound different. (I've not had the chance to hear The Well-Tempered Clavier played on a well tempered piano or organ, but am told that each piece therein sounds "off" if played in the wrong key - because part of Bach's genius was to demonstrate how one tuning gave a different euphonious result no matter what you were playing.)

This is also distinct from just intonation (the kind of tuning Pythagoras used), where there is a "wolf interval" at some point and chords that cross it sound unpleasant, or from the meantone tuning (used in most Rennaisance music) where the wolf interval was broken up into several pieces spread across the scale so you could play one key and have it sound reasonable almost no matter what you played but sound very wrong if you tried to transpose. Well temperament freed musicians to play in any key without re-tuning their instruments, and equal temperament frees composers to write in any key regardless of what they're writing - but some people argue that a certain richness is lost in equalizing your temperament, and what little early music I've heard played in meantone tunings has born that idea out.

22. ### dr pepper said,

February 20, 2009 @ 6:28 pm

Math is a construct, but numbers are real. Musical intervals are a construct but harmonics is an observal fact. Still, most instruments produce a mix of sounds, not simple frequncies so the difference between tempers is probably inside the bandwidth, at least for untrained ears like mine.

Otoh, now that i know about this, i've got this image of a hellenistic Doctor Who. He has dual pan pipes, whose notes span the comma. When he plays them together, a crack appears in the air in front of him, which he can walk through and be elsewhere. Or when.

23. ### Mark Liberman said,

February 20, 2009 @ 8:26 pm

dr pepper: Musical intervals are a construct but harmonics is an observal fact. Still, most instruments produce a mix of sounds, not simple frequncies so the difference between tempers is probably inside the bandwidth, at least for untrained ears like mine.

Except for sine-wave generators (and some exceptionally pure whistles), all pitched instruments produce an overtone series. In most cases, this is very close to a perfect series of 1*f, 2*f, 3*f, … , i.e. integer multiples of the fundamental frequency f — with the steady-state timbre depending on the relative amplitude of the different partials.

Due to the stiffness of the strings, the overtone series of a piano note is unusually and characteristically inharmonic; this is a crucial part of the perception of piano-ness. (See e.g. this paper and the references it cites.)

I've always wondered whether one of the reasons for the popularity of pianos, in the era of equal temperment, has been the fact that the inharmonic overtones mean that mistuned intervals don't sound quite as bad on a piano as they do (for example) on a harpsichord. (Obviously the ability to vary dynamics and to play really loud was more important, but still…)

Anyhow, there's no question that many tempered intervals sound pretty bad on most instruments. Try (for example) playing tenths in the neighborhood of middle C, on a harpsichord that's properly tuned so that all semitones are 2(1/12). Doing this on the piano doesn't make me cringe the way it does on the harpsichord.

24. ### Nick Lamb said,

February 20, 2009 @ 8:49 pm

“The relationship of pi to the circumference of a circle is a natural fact. Pi isn't an irrational number because of nature; it's irrational because that's the only way we can fit the concept into our mathematical constructs. Reality isn't the numbers, the numbers just describe reality.”

Yes / No / Sort of - but mostly No.

Pi isn't quite what you think it is. The ratio shows up in several places in mathematics, you can find it without needing any circles or geometry at all. In Euclidean geometry the ratio between a circle's diameter and its circumference is Pi, but our universe isn't Euclidean (though it's a convincing approximation) and in any case contains no perfect circles or instruments suitable to measure their circumference to exacting standards.

Being irrational or transcendental isn't extraordinary, very nearly all real numbers are transcendental (and thus irrational), it's just that we tend to work with rational numbers for purely pragmatic reasons. The Pythagoreans were startled to come across a tree, but if they'd had our broader view they'd have realised it's not so extraordinary, they were stood in a forest.

And as Mark pointed out, the base used (fingers on a hand) has nothing whatsoever to do with these ratios. The pleasing ratio between the lengths of the sides of A series sheets of paper isn't nice because of the fingers we count with or some accident of our eyes, but because of a fundamental mathematical relationship. If an octopus can find the ratio between things pleasing, it would be just as pleased with the A series ratios as we are.

25. ### JLR said,

February 20, 2009 @ 9:19 pm

Nick Lamb said: "The pleasing ratio between the lengths of the sides of A series sheets of paper isn't nice because of the fingers we count with or some accident of our eyes, but because of a fundamental mathematical relationship."

While it's nice that the whole sqrt(2) thing works out so well, I would hesitate to make any sweeping generalizations about its universal pleasingness. Being used to letter sized paper, I find A4 paper irritatingly long and narrow, and I suspect others do to. I'm fairly sure that any aesthetic appreciation of A series paper is learned and not innate.

26. ### TB said,

February 20, 2009 @ 9:22 pm

Well, speak for yourself. I hate A4 paper. And I love those "wolf intervals", though I did grow up listening to Indonesian music which intentionally has a lot of that beating (or whatever it's called).

27. ### TB said,

February 20, 2009 @ 9:23 pm

(high five to JLR)

28. ### Randy said,

February 21, 2009 @ 2:01 am

MYL: "Reference?"

One good book that I recently read on the matter of music and the brain is
"This is your brain on music", by Daniel J. Levitin.

Website here: http://www.yourbrainonmusic.com/

It's written for a popular audience, but it's quite scholarly.

[(myl) I own this book, and I don't see anything in it to support your assertion. On the contrary, on pp. 74-75, there's a discussion that briefly explains that 12 perfect fifths doesn't quite equal 7 octaves, and suggests that human interval perception hears fifths correctly, in the context of explaining the origins of perceptions of consonance and dissonance. So I'm going to assume that in fact, you don't have any basis for your assertion. I don't know for sure that it's wrong, but I suspect that it is — and in any case it has nothing to do with the origins of systems of temperament.]

I've returned the book to the person who lent it to me, but I recall that there is a discussion on the ability of the human ear/brain to correct for slight variations in pitch. As well, there is a discussion on equal temperament vs well temperament.

As for paper sizes, I have one copy of a paper that was given to me in A4 size. I didn't notice that it was different until I tried to stack it with the rest of my papers, and it didn't match up. This, rather than the aspect ratio, was the primary irritation. I'd be happy to use either size, provided I only have to use one of them.

The advantage of A series paper is apparently that upon folding so that the short sides of the paper meet, the aspect ratio of the folded paper is the same as the original, although the size has changed. I do not see what is pleasing about this.

29. ### Andy said,

February 21, 2009 @ 2:39 am

This being a Language Blog, I thought it was interesting that you didn't pick up on this assertion for correction:

"For those of you who find the math-music correlation studies dubious, let me fill you in. There’s a reason for that correlation: Music is math. Or physics, if you prefer."

No, it is not. Music certainly has a mathematical component (as you note), but this is mainly due to music's formation from sound (and hence the physics of sound waves). But music is not just sound- the "rules" of music theory have more to do with grammar and rhetoric than with the laws of physics.

For example, in studying Bach chorale writing techniques, musicians are taught to avoid certain interval progressions (such as parallel fifths and octaves, especially in outer voices). But this is not because this violates any acoustical principles; rather, this is because violating those rules creates a musical sound which was not considered stylistically appropriate to that period in history. (Later composers- especially in the 20th century- chose to ignore those rules as they no longer felt constrained by them.)

Music is no more "math" than language is biology (since glottal, mouth, and tongue movements cause the formation of different vowels and consonants).

30. ### rightwingprof said,

February 21, 2009 @ 6:16 am

"He is simply comparing a perfect fifth (3/2) with an equal-tempered fifth"

Indeed.

As for the post here, there is nothing I do not know. I'm a piano technician. I could have posted a far more technical article, and as a result, bored everyone.

[(myl) I'm sure that you have an excellent ear and can tune pianos well. But your post contains one straightforwardly false claim — that the difference between perfect and tempered fifths exists because people don't hear perfect fifths correctly — and it omits one straightforward mathematical truth — that twelve perfect fifths are almost, but not exactly, the same as seven equally-perfect octaves.

The truth, in this case as often, is not boring at all. ]

31. ### Nick Lamb said,

February 21, 2009 @ 9:12 am

What? How do you not see it? Compare the alternative (it wouldn't matter if you picked another one, but there basically aren't any in common use) US Letter. What we can we do with that ratio? Nothing. No-one even really seems to know how the ratio came about.

Some posters seem to have assumed I meant that the shape had a mystical aesthetic charm. Not at all, a variety of rectangles are sort of vaguely appealing to humans, there's no reason to pick a specific ratio just for that (enthusiasm for golden rectangles in architecture not withstanding) and no reason to think an octopus would have the same aesthetic. But if the octopus can appreciate the ratio itself it will see how ingeniously chosen it is.

In practical terms its pleasing to anyone who has to actually print things. Europeans are used to it not mattering how you print something, if you print something intended for A5 at A4 it will simply be larger, and if printed at A5 when intended for A4 it will be smaller, the proportions are unaltered. It's also trivial to automatically "two up" a document for printing two pages per side to save paper. European photocopiers almost universally offer to shrink or enlarge documents for you. With other ratios, including US Letter such ordinary everyday practices become cumbersome and difficult.

In purely mathematical terms you'd have as good a claim for the golden ratio, but it turns out that nobody's everyday business requirements involve cutting squares out of rectangles.

32. ### Randy said,

February 21, 2009 @ 10:34 am

MYL: "and I don't see anything in it to support your assertion."

It wasn't my assertion, it was someone else's. Although, I suppose I implicitly approved of it by attempting to provide a reference for it. In any case, I thought I remembered reading something about that. Isn't there also a chapter on the ability to recognize tunes, regardless of what pitch they're played at? I'm almost certain of it. Perhaps I only thought to myself while reading this chapter, that you could also slightly alter the size of various intervals and the tune would still be recognizable. That is what I took correction by the brain to mean.

[(myl) Though it was someone else's assertion, you're so far the only one who has offered any evidence for it. And I believe that your suggested evidence is based on a misunderstanding of Levitan's book. You may be remembering the discussion (p. 133 and following) of experiments by Benjamin White, showing that people could recognize tunes not only in transposition (where intervals are exactly retained), but also "in almost every case … more often than chance could account for" when the melodic contour is retained in a more qualitative way, with interval sizes shrunk or stretched in various ways. The modified tunes always still stayed within the standard tempered chromatic scale, so that the question of how people hear slightly mis-tuned intervals didn't come up. And Levitan doesn't put this experiment forward as evidence that the brain hears intervals in general in a non-veridical way, or more specifically that fifths are heard as tempered rather than exact intervals, but rather as evidence for a gestalt sense of overall melodic contour, an "abstract generalization" independent of particular intervallic sequences. (The citation is Benjamin White, "Recognition of Distorted Melodies", American Journal of Psychology, 73(1): 100-107, 1960. Levitan's conclusions are probably true, but there are some problems with White's experiment, for example the fact that the procedure involved forced choice among ten tunes that were identified to the subjects at the beginning, and heard repeatedly in various modified versions during the course of the experiment.)

Again, I'm very skeptical of the claim that human interval perception hears tempered rather than exact fifths — and I'm somewhat skeptical of the claim that our judgment of fifths is systematically different from the mathematical 3/2 ratio, though I don't know the literature on this subject well enough to be certain about it. If anyone has any credible evidence one way or the other, I'd be interesting in seeing it.]

Nick: Explains A series

Thanks for the explanation.

I simply took pleasing to mean "easy on the eyes" or something like that. I don't typically want to print things two-up, or have need to. There are too many subscripts in my work to make that an attractive option.

33. ### TB said,

February 21, 2009 @ 3:19 pm

Nick Lamb:

I see, I see. I guess I was thrown off because I would never use the word "pleasing" in that context.

On the other hand it sounds sort of like the arguments for equal temperment: it may be ugly but you can change it to any size (or key in the case of equal temperment.)

34. ### JLR said,

February 21, 2009 @ 4:19 pm

Nick Lamb said:"If an octopus can find the ratio between things pleasing, it would be just as pleased with the A series ratios as we are"

That's a particularly un-apt sentiment given the long and tragic history of cephalopods and writing.

February 21, 2009 @ 5:31 pm

Is it possible that this is all a misunderstanding of the original post on Right Wing Nation? Perhaps rightwingprof was referring not to the problems of equal temperament - which is what Mark assumes - but to the phenomenon of scale stretching in piano tuning? (For an explanation see e.g. this reference or several of the other links that you get if you google the phrase {octave stretching}.) I can well imagine that - as he seems to be suggesting in his response to Mark - rightwingprof might have thought that a detailed explanation of scale stretching might have come across as boringly technical.

36. ### Mark Liberman said,

February 21, 2009 @ 8:35 pm

Bob Ladd: Is it possible that this is all a misunderstanding of the original post on Right Wing Nation? Perhaps rightwingprof was referring not to the problems of equal temperament - which is what Mark assumes - but to the phenomenon of scale stretching in piano tuning?

In a word, no. In his own words:

However, the ear is imperfect, so we have the Pythagorean Comma, a serious problem identified by Pythagoras that was not solved until the 18th Century. In ancient Greece (until the 18th Century) perfect fifths were used to tune instruments, and this led to a problem that limited what could be played.

Aside from his explicit claim that the Pythagorean Comma is due to the ear being imperfect rather than to the fact that (3/2)^12 != 2^7, there's also the fact that pianos were thin on the ground in ancient Greece.

February 22, 2009 @ 4:43 am

Mark Liberman: Aside from his explicit claim that the Pythagorean Comma is due to the ear being imperfect rather than to the fact that (3/2)^12 != 2^7, there's also the fact that pianos were thin on the ground in ancient Greece.

Fair enough, but the Ancient Greeks did have harps and lyres, and if I understand the physical basis of scale stretching on a piano correctly - the fact that the overtone series of a stiff vibrating string runs slightly sharp - then they ought to have run into the same problem. At the very least, the practical problems of tuning a harp or a lyre may have helped confuse Pythagoras's thinking about the issue.

That said, however, I've now read the original posting on Right Wing Nation, and I agree that scale stretching doesn't seem to be what the writer is talking about.

38. ### Mark Liberman said,

February 22, 2009 @ 8:27 am

Bob Ladd: the Ancient Greeks did have harps and lyres, and if I understand the physical basis of scale stretching on a piano correctly - the fact that the overtone series of a stiff vibrating string runs slightly sharp - then they ought to have run into the same problem.

The amount of inharmonicity in a plucked string depends on its stiffness — at least that's what I've always read — and steel piano wires are a lot stiffer than gut harpstrings. In the case of the guitar, for example, Järveläinen et al., "Perceptibility of Inharmonicity in the Acoustic Guitar", Acta Acustica 92(5), 2006, found that in listening tests conducted using "test tones resynthesized from real recordings using high-resolution parametric modeling", listeners could easily detect inharmonicity in the lowest notes for steel strings, but "mean thresholds were close to, though above, typical amounts of inharmonicity in the [nylon-stringed] guitar".

The thresholds involved, as I understand the paper, are for telling the difference between synthetic sounds with inharmonicity and those without. I don't know what results you'd get if you looked at the effects of tuning octaves by listening for beats — that might well be a more sensitive test. But maybe not — the amount of inharmonicity in piano notes can be pretty large: Fletcher and Rossing (The Physics of Musical Instruments, p. 363) say that "a typical value for the inharmonicity coefficient in the middle register (B=.0004)" will "shift the 17th partial one 'partial position' to the frequency of the 18th partial of an ideal string without stiffness", or a factor of about 1.014. This is in the same range as the the Pythagorean Comma of (3/2)^12/2^7, or about 1.013643. I believe that the inharmonicity coefficient for nylon strings is typically an order of magnitude lower, though this also depends on length and tension.

But anyhow, the Pythagorean Comma is a mathematical fact, and the inharmonicity of partials in plucked strings is a physical fact. Neither one has anything to do with the imperfections of human hearing.

39. ### Chris said,

February 23, 2009 @ 9:35 am

Right, but if the universe were competently constructed, by the lights of the Pythagoreans, this wouldn't be true, and we wouldn't have to detune all our intervals in order to modulate freely.

Now I'm imagining the Pythagoreans coming up with Kripke semantics and trying to invent a possible world without the Pythagorean Comma…

It's not just a matter of the 12-tone scale - the equation (3/2)^j = 2^k has *no* positive integer solutions j,k. (Or equivalently, 3^j = 2^(j+k).) The fundamental problem is that 2 and 3 are relatively prime, which means a power of one can't be written as a power of the other. And they're relatively prime because they *are* prime. So to avoid the PC and have a perfect cycle of fifths, you have to live in a universe where 3 is a power of 2.

40. ### Mark Liberman said,

February 23, 2009 @ 9:41 am

Chris: So to avoid the PC and have a perfect cycle of fifths, you have to live in a universe where 3 is a power of 2.

Shh. Remember the legendary fate of Hippasus.

41. ### Aaron Davies said,

February 25, 2009 @ 9:18 am

on a completely unrelated note (so to speak), does anyone here know anything about the specific mistunings that make ragtime sound better? i'm thinking of something like the piano in black & white rag. i actually found a piano once that was too out of tune for ragtime, so it's possible for random amounts of flatness to go too far, if fairly rare. (the odds that i'll ever have a piano of my own and the free time to tune it specifically for ragtime are extremely low, all knowledge is worthwhile. :)

42. ### Andy Hollandbeck said,

February 25, 2009 @ 11:54 am

@Nick Lamb: You said while back that "our universe contains no perfect circles." That's a pretty big and wholly unproveable statement. I've always found it astounding that no circle can have both its radius and circumference be real numbers. I've always seen it as a shortcoming of our mathematical system, not of our universe.

Why is it impossible for the universe to have created a perfect circle (or a perfect sphere)? Is it because pi is an irrational number? That reasoning is backward.

As for my statement that math would be totally different if we had been born with 12 fingers and ended up with a base-12 mathematical system: The physical world is what it is, but the math we use to describe it evolved over time (remember when there was no zero?). My contention (wholly unproveable though it is) is that math would have evolved in a completely different way if we had started at 12, or any other number.

In the same way that you can't accurately translate between languages by simple word replacement with a bilingual dictionary, the evolved base-12 system wouldn't simply look like a direct base-12 translation of today's base-10 system.

Maybe I'm just underinformed, though. Admittedly, I haven't the slightest idea how to convert 13.5 into base 12, much less what pi looks like in base 12.

43. ### Peter McAndrew said,

January 6, 2010 @ 9:32 pm

I wonder how much different music would be if the perfect fifth was used instead of the octave in determining the tuning for equal temperament. I might have to try an experiment at some stage.

Incidentally, 13.5(base 10) = 11.6(base 12)