Why are the frequency ratios of notes in the Pythagorean scales 9/8 and 256/243?

Question

In Pythagorean tuning, for every 7th semitones the frequency would increase by a factor of 3/2 (to get that harmonious perfect fifth). If the frequency of C4 is set to 256Hz, the frequency of G4 can be calculated by 256 * 3/2 = 384Hz.

If the 3/2 factor spans over 7 semitones, the ratio between each neighboring semitone has to be x^7 = 3/2 or x = (3/2)^(1/7). If C4 is 256Hz, one semitone up to C4# should reach 256Hz * (3/2)^(1/7) = 271.266Hz. Two semitones up to D4 = 256 * (3/2)^(2/7) = 287.44Hz. So on and so forth until (3/2)^(7/7) = 3/2 where you reach the next octave. The ratio between notes, say D and C, should be (3/2)^(2/7)=1.123.

But the book (Science of Sound - Rossing) states the ratio between the notes to be 9/8=1.125 or 256/243=1.053. Would someone help explain why in my math that I'm not getting these ratios?

Answer 1

If the 3/2 factor spans over 7 semitones, the ratio between each neighboring semitone has to be x^7 = 3/2 or x = (3/2)^(1/7).

That is true if all the semitones are the same size, but they can't be if you want to use pure fifths and also close the circle of fifths on a standard 12-tone keyboard.¹

Pythagorean tuning is not an irrational system. The semitones in Pythagorean tuning are not the (geometric) seventh part of 1.5. Rather, every tone is found with the 3/2 ratio, adjusting for the octave where necessary, around the circle of fifths. So C to G is 3:2, G to D is 3:2, and C to D is 9:4. That's a major ninth, however, which is more than an octave, so dividing by 2 to get the major second gives 9:8. A is found by multiplying yet again by 3:2, so 27:16, and E is found by multiplying again by 3:4, so 81:64. Keep going, and you'll eventually reach B♯, which isn't the same as C, with a value of 3¹²:2¹⁷.

Along the way, you'll hit C♯ at 3⁷:2¹¹, and if you square that, you get C-double-sharp at 3¹⁴:2²², which isn't the same as D at 9:8. The pure Pythagorean system does not close the circle of fifths; it is rather a spiral.

If you use your approach of dividing the fifth from C to G into seven equal semitones, then the fifth between G and D doesn't have a ratio of 3:2 but of 1.497. That is a hair smaller (about 3.35 cents) than a Pythagorean fifth.

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Footnote 1: you can have all the semitones equal if you divide the octave into 12 equal parts, of course, which is twelve-tone equal temperament, but then the fifths are a tiny bit smaller than 3:2.

Answer 2

There are a couple different issues here. As phoog notes, the Pythagoreans loved rational numbers. To them, "irrational numbers" (like various roots of 2 or 3/2) were, well... irrational.

Phoog also notes that you run into a problem when you add a bunch of perfect fifths together. If you use a 3:2 ratio and go through 12 perfect fifths, you theoretically need to add up to seven 2:1 octave ratios for the "circle of fifths" to close. But you don't. No power of 3:2 is going to give you a power of 2:1.

The question presumes that this 3:2 ratio is absolute rather than the 2:1 octave, but that still creates problems in tuning scales.

As to the reason why the Pythagorean scales typically have the particular ratios brought up in the question (9:8 and 256:243), well, it has to do with the ancient Greek method of deriving scales, which was based around perfect fourths, with a 4:3 ratio. The Greeks had many ways of dividing up a fourth into different notes. But one "diatonic" way was to use "whole tones," which were typically tuned to a 9:8 ratio. Why 9:8? Because 9:8 is the difference between a 3:2 perfect fifth and a 4:3 perfect fourth. (Divide 3/2 by 4/3 and you can see this.)

So, when dividing up the perfect fourth, one way of doing it is to use two 9:8 whole tones. But then the last interval you end up with is roughly a semitone. The actual ratio required to complete the 4:3 perfect fourth can be found by taking two whole tones away. So, 4/3 divided by 9/8 divided by 9/8 leaves 256/243. That's where the other ratio comes from.

The question assumes all semitones are of equal size. This was not an assumption in ancient Greece, though it is an assumption in modern equal temperament. Instead, the Greeks would start by tuning the important intervals like a 2:1 octave, 3:2 fifth, and 4:3 fourth. Then they'd fill in the "gaps" as discussed above.

To find the (irrational) ratios for a modern 12-tone equal tempered scale, you'd need to divide the octave (2:1) ratio in 12 parts, hence 2^1/12:1. That will give you the correct equal tempered semitone ratio, though note that seven of those won't give you an exact 3:2 ratio either, for reasons stated in my first couple paragraphs.

Answer 3

There's an important mathematical theorem underlying the problems with tuning. The upshot is that there is no "perfect" tuning. The theorem is that there is no power of 2 that equals a power of 3 (except for the zero power which is 1 in both cases.) Actually, there are no powers of primes that equal each other except for P^0=1. In fact, there are no powers of primes that are close to each other except for 8 and 9.

This implies that no stack of fifths will equal any stack of octaves (replace octaves or fifths by thirds, sixths, etc.)

Pythagorean tuning works with stacks of fifths with ratio 3/2. Two fifths stacked give 9/8 (reducing intervals to between 1 and 2 by octave equivalence when appropriate) giving a major second (C-G and G-D giving C-D). Stacking 2 major seconds gives 81/64 for a Pythagorean major third; however, the "just" major third should be 5/4. The difference is audible and gets worse with other intervals. Even throwing in a 5/4 ratio to the "basic" ratios, doesn't help exactly. A guitar is tuned E-A-D-G-B-E (3 fourths, a third, and a fourth which should yield 2 octaves). The fourth is the complementary interval to the fifth with a ratio of 4/3 (the inverse of 3/2 reduced to be between 1 and 2). By ear (listening to beats) one can easily tune each fourth and the third, then the two Es are out of tune. Mathematically, 4/34/34/3* 5/4*4/3=320/81 but two octaves should 4/1 (or 320/80). (As an aside, one of my guitarists who had perfect pitch found this annoying when trying to tune.)

So compromises are made yielding the entire field of temperament. Equal temperament solves the problems of treating intervals equally but it may not yield as nice intervals.

Answer 4

If one uses the following sequence of interval ratios for a major scale, the product of the ratios does, in fact, add up to 2:1 - 9:8, 10:9, 16:15, 9:8, 9:8, 10:9, 16:15. These major and minor seconds can in fact be derived from either Pythagoras or overtones. Using a C scale, 9:8 is D:C, 10:9 is E:D 16:15 is the semitone C:B and you have the lower tetrachord. Connect it to the upper tetrachord with another 9:8 and, voila! intervals that add up to a perfect octave.