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I've been reading an excellent book on the mathematics and history of scales called "The Arithmetic of Listening". Based on what I have read, I am thinking that both the 4th and 5th have an optimal size (namely 4/3 and 3/2 respectively). I assume that these ratios are the most pleasing to the ear.

Now, the text says that the major and minor 3rds generated in the Pythagorean scale are not very optimal - "the minor 3rds are bitter and unsatisfying, its major 3rds harsh..."

This left me wondering whether this is an optimal major and minor 3rd? Ones that are the most pleasing to the ear? If so, how would they be defined?

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    Oof, this is complicated. It's easy to talk about the mathematics involved; it's less easy to make absolute statements (as that text does) about perception. Mankind has been dividing the octave in probably literally thousands of ways across time and place, so "pleasing" or unpleasing is construed based on context, culture, and goal. Dec 14, 2022 at 0:23
  • Isn’t a just major third 5/4? Is that the same as a Pythagorean major third? Dec 14, 2022 at 0:28
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    5/4 doesn't exist on the Pythagorean scale since it is a 3-limit scale (constructed from only the numbers 2 and 3). 5/4 is a major 3rd in a suitable 5-limit scale. Perhaps it is optimal in the sense that it is a simple ratio of small integers. But maybe that is an oversimplification.
    – Darren
    Dec 14, 2022 at 1:10
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    You might have to define “optimal” in your question. “Most pleasing to the ear” is subjective. A just third (5/4) has essentially the same harmonic character relative to intonation as just fifths (3/2) and fourths (4/3) do. So if you like just fourths and fifths, you might like just thirds in the same way. By “harmonic character” I mean all the partials that overlap between the two notes of the interval are (perfectly, as much as possible) in tune. This is only true of just intonation. Dec 14, 2022 at 1:50
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    @ToddWilcox a Pythagorean major third is 81:64.
    – phoog
    Dec 14, 2022 at 4:29

2 Answers 2

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the text says that the major and minor 3rds generated in the Pythagorean scale are not very optimal - "the minor 3rds are bitter and unsatisfying, its major 3rds harsh..."

In my opinion, a minor triad with a Pythagorean third of 32:27 is perfectly acceptable even though it is definitely different from a minor triad with a third of 6:5. I don't perceive a strong functional difference, and I certainly don't find the Pythagorean interval "bitter and unsatisfying."

By contrast, I do find the 81:64 major third harsh and the 5:4 major third restful and pleasant. Still, the 81:64 third may be acceptable or even useful in some contexts, notably in the dominant triad, which may benefit from the tension of the "harsh" third increasing the Listener's desire for resolution to the tonic. The tonic, by contrast, should be more settled, so my preference for a 5:4 major third there is stronger. This is why I mentioned the relative lack of "functional difference" with the minor triads.

I suppose that this difference is because of the relative complexity of minor triads with respect to the overtone series. The pitches of a 4:5:6 major triad are all overtones of an implied pitch two octaves below the root, and the three difference tones are 1, 1, and 2, two of which are this implicit bass pitch and the other is its first overtone. With a 64:81:96 Pythagorean major triad, the implied bass tone is a full six octaves below the root, and the difference tones are 15, 17, and 32: instead of having two difference tones reinforcing each other two octaves below the root, one is roughly a half step lower than that and the other roughly a half step higher. With a just minor triad of 10:12:15, the implied common bass is three octaves plus a major third below the root, which isn't a chord tone, and the difference tones, 2, 3, and 5, define a different (but closely related) triad, the major triad a major third below, the ♭VI, if you will. For a Pythagorean minor triad of 54:64:81, the implied fundamental is five octaves and a major sixth below the root, and the difference tones are 10, 17, and 27, or, in the C minor example, a rather flat G1, a somewhat sharp E♮2, and C3. Instead of a well tuned ♭VI, they define a poorly tuned major triad on the same root.

In other words, the difference between the just and Pythagorean major triads is the difference between perfect acoustical alignment and poor acoustical alignment, or between high consonance and moderate dissonance, while the difference between the minor triads is the difference between fair acoustical alignment and poor acoustical alignment, or between moderate consonance and moderate dissonance.

This left me wondering whether this is an optimal major and minor 3rd? Ones that are the most pleasing to the ear? If so, how would they be defined?

Not only is this subjective, it also depends on the harmonic context. As noted, the Pythagorean major third might be more acceptable in a dominant chord than a tonic chord; you may find a similar preference for one tuning or the other of minor triads depending on the context, too.

"Pleasing to the ear" does not necessarily mean the same thing when listening to a single sonority in isolation as when listening to a complete piece of music. Discussions of tuning often equate consonance and "pleasing to the ear." But even if we eliminate the subjective judgement about individual sounds by calculating or measuring actual acoustical phenomena, we're still left with context: a major seventh or a minor second is objectively "harsher" acoustically, whatever its tuning, than a perfect fifth, but it may be a most beautiful sound nonetheless.

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    Complicating the subjective questions still further, "unpleasant" is a viable aesthetic. I've heard (don't have a source) about baroque (or post, maybe CPE?) works that intentionally modulate to key that is "unsuited" for the temperament, to exploit the disturbing affect of the "wolf." The opening tritone of Saint-Sean's Danse Macabre is far from "pleasing to the ear," especially if done properly by detuning the E and whaling on it fiddle-style. "Pleasant" and "optimal" aren't always the same thing. Dec 14, 2022 at 16:02
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There are no "optimal" intervals in this sense, except for the octave at 2/1. It's worthwhile to check out the Wikipedia articles on temperament, meantone, equal temperament, just scales, Pythagorean temperament, etc.

I will try to describe a couple of approaches. There is an insoluble mathematical problem here. In number theory, it goes by the name of "Catalan's Conjecture" (from 1844) and was proved in 2002 (number theory has lots of simple-to-state theorems with fiendishly difficult proofs.) The main point is to combine musical intervals so that the new intervals formed are "consonant" (which I'll describe later.)

The difficulty comes from there being no power of 3 which is close to a power of 2 (except for the power 0 and the numbers 8 and 9 proved around 1744). Thus there is no combination of fifths that is close to an octave (musical fifths, enough fifths of Laphroaig makes most of these problems moot.) Pythagorean tuning makes use of fifths (ratio 3/2) and octaves (2/1) to make a useful set of notes. If we start with some arbitrary note and call its frequency ratio (with itself) 1, we can apply fifths to see what happens. 1/1 Unison, C (I'll start with C for really obscure historical reasons.) 3/2 Fifth, G 9/8 Second D (3/2 * 3/2 = 9/8; this is converted to a ratio between 1 and 2 for convenience.
27/16 Sixth, A 81/ 64 Third, E 243/ 128 Seventh, B 729/512 Fourth F 2187/1024 Octave C

Now, Houston, we have a problem. The octave ought to be 2048/1024. Some adjustments could be made by having inverse intervals be the reciprocal of ordinary intervals. Then we get the following: C Unison 1:1 D Major Second 9/8 C Major Third 81/64 (audible problem) F Fourth 4/3 (inverse of the Fifth) G Fifth (3/2) A Minor Sixth (27/16) B Major Seventh (243/128)

The "Just Temperament" adds the interval of 5/4 as a major third (rather than 81/64 which is close to 5/4) and gives a "supposedly more satisfactory" set of ratios. It has been though over the centuries (and somewhat confirmed by Helmholz and the moderns) that ratios of notes with "small" numbers in their fractions are "nicer" than those with larger numbers. One now gets the "Just" scale. C 1/1 D 9/8 E 5/4 F 4/3 G 3/2 A 5/3 B 15/8 C 2/1

Chords formed from these ratios are pretty good. C major is C-E-G or 4-5-6 (4/4-5/4-3/2). F major is 4/3-5/3-6/3 or 4-5-6. A minor is 10-12-15 (after clearing denominators), and G major is 4-5-6 again. However, D minor (also a very important chord) is in the ratio 27-32-40 (arithmetic left to the reader) which does not equal 10-12-15 as in A major. The scale is self-inconsistent (again because powers two different primes are not ever close.)

Also, the two whole steps between C-D and D-E are not the same; C to D is a ratio of 9/8 but D to E is a ratio of 10/9; the system has two whole steps. One good method of using this tuning is to change the "base" of the system on-the-fly. One can look ahead the next few notes and just which intervals to use. For keyed or fretted instruments, such adjustments are not possible so some type of temperamenet is needed, but that's another lecture.

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    2187/1024 is 113 cents sharp of an octave rather than the 23 cents of a Pythagorean comma. This is because you have calculated ratios for F sharp and C sharp but labeled them F and C. Of course C sharp isn't C! You should keep going until you get to B sharp, which is a lot closer to C than C sharp is.
    – phoog
    Dec 14, 2022 at 4:43
  • Good point, you are right. I'll still leave the actual computations to the readers.
    – ttw
    Dec 14, 2022 at 4:51
  • Why give this the dv?
    – Tim
    Dec 14, 2022 at 7:38
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    @Tim because the description of the Pythagorean comma is wrong.
    – phoog
    Dec 14, 2022 at 7:39
  • @phoog - ah, now we know which part of a fairly complicated answer is faulty. As in fact verified by yourself. Thanks!
    – Tim
    Dec 14, 2022 at 7:43

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