The Pythagorean Symmetry That Never Was Quite Right

While many of the superstitious mathematical beliefs of the Pythagoreans seem silly today, like ascribing philosophical/religious meaning to certain numbers, they were in some ways ahead of their time. Check out their stance on animal rights and women in philosophy:

The Pythagoreans advanced the argument that unless an animal posed a threat to a human, it was not justifiable to kill an animal and that doing so would diminish the moral status of a human (…) Pythagoreans believed that human beings were animals, but with an advanced intellect (…) Pythagoreans reasoned that the logic of this argument could not be avoided by killing an animal painlessly. The Pythagoreans also thought that animals were sentient and minimally rational.
(…)
The biographical tradition on Pythagoras holds that his mother, wife and daughters were part of his inner circle. Women were given equal opportunity to study as Pythagoreans. (…) Theano of Croton, the wife of Pythagoras, is considered a major figure in early-Pythagoreanism. She was noted as distinguished philosopher and in the lore that surrounds her, is said to have taken over the leadership of the school after his death.
https://en.wikipedia.org/wiki/Pythagoreanism

But one area where we still feel the influence of the Pythagoreans is in their mathematical theory of musical harmony. Essentially, their theory was that notes with frequencies which are rational-with-small-denominator multiples one another sound harmonious. For example, the octave corresponds to multiplying the frequency by 2, a perfect fifth (in Pythagorean tuning) corresponds to multiplying the frequency by 3/2, and a perfect fourth to multiplying the frequency by 3/4.

To understand this, first let’s consider the frequencies note in equal temperament (the modern replacement to Pythagorean temperament) defined by the equation $f = 440 \cdot 2^{n/12}$ . Notes $n=0,1,2,3,4,5,6,7,8,9,10,11$ correspond to the note letters A,A#,B,C,C#,D,D#,E,E#,F,G,G#, respectively, $n=12$ corresponds to the $A$ an octave higher, and so on. Generally, if $n_1 \equiv n_2$ (mod 12) these notes have frequencies which differ by a power of $2$ , and we give them the same note letter. If we graph this equation with the radius $r=f$ and angle $\theta = n \pi/6$ (allowing $n$ to take on non-integer values), we obtain a logarithmic spiral. The points on this spiral where $n$ is an integer are all on one of twelve radial lines, spaced equally with angle $\pi/6$ .

(I know, this drawing isn’t perfect!)

The interesting thing about this spiral is that it is an accurate representation of the relationship between frequency a perceived pitch. If you move continuously along the curve from one point to another, the perceived change in pitch corresponds to the angular distance, while the measured change in frequency corresponds to the radial distance. Points on the same radial line sound “similar”, and are in fact some number of octaves apart. The entire picture has a symmetry by rotating by $2\pi$ and scaling every point radially by a factor of $2$ .

Now, you might wonder, why twelve? We could divide this graph by any number of equally spaced radial lines, and still have this same symmetry. Is there some kind of extra geometric structure when we divide it into twelve pieces?

Well, it certainly seems so! Let’s compare $n_1=0$ (corresponding to A) and $n_2=7$ (corresponding to the E a perfect fifth above). If we compute the ratio of their frequences $\frac{f_2}{f_1} = \frac{440 \cdot 2^{7/12}}{440 \cdot 2^{0/12}} = 2^{7/12}$ and round it to two decimal places, we get $1.50$ . This certainly sounds familiar doesn’t it? Remember, the Pythagorean idea of a perfect fifth was that the ratio of the frequencies should $3/2 = 1.50000\cdots$ . Alas, it’s too good to be true. $2^{7/12}$ is not rational. If we compute more digits, we find $2^{7/12} = 1.498307\cdots$ . So it’s very close, but not quite equal to $\frac{3}{2}$ .

What we’ve stumbled on is an almost symmetry. It’s almost the case that if you rotate the entire picture by $7\pi/6$ , and scale radially by a factor of $3/2$ you get the same picture back. If this were so, we would have a very beautiful description of why we should have twelve radial lines (and thus twelve note letters): the symmetries $\sigma: (r,\theta) \mapsto (\frac{3}{2}r, \theta + \frac{7\pi}{6})$ and $\tau: (r,\theta) \mapsto (2r, 2\pi \theta)$ would generate all the notes of the piano starting with $(440,0)$ (corresponding to the note A), and every note thus generated would be harmonically related: their frequencies would both be integer multiples of some common frequency (meaning they can coexist as vibrational energies of the same vibrating string). Any different collection of notes would either not be closed under these symmetries, or else would have notes that are not harmonically related.

Our Western system of music, with twelve notes, it based on this almost symmetry. The well-known “circle of fifths” arranged notes in a circle, where adjacent notes differ by a perfect fifth, or seven semitones. A measure of “consonance” (or harmony) between notes is given by how many steps along the circle of fifths you need to go to get from one note to another. This corresponds to how many times you need to apply $\sigma$ or $\sigma^{-1}$ . Notes which are separated by one or two applications of $\sigma$ will be almost-harmonics of a common fundamental frequency which is not much lower than the frequencies of those notes, and so will sound nice together.

This has been studied rigorously by psycho-acoustic researchers. You can experiment for yourself, if you want! Pick two notes on the piano. Now try to get from one note to the other by only going up/down a certain number of fifths and octaves. The fewer steps you need to take, the better your brain is able to hear the connection between those pitches, and the more consonant they sound. If you play A and Eb (a “tritone” interval), which are opposite on the circle of fifths, they will sound dissonant.

Now, you might be asking, does this mean we need to throw away all of our music theory? Not quite. Let’s talk a bit about equal temperament, it’s main advantage, and how classical musicians cheat.

First, there is an actual symmetry of the equal tempered pitches which is really close to $\sigma$ . Let’s call this $\hat{\sigma} : (r, \theta) \mapsto (2^\frac{7}{12} r, \theta + \frac{7\pi}{6})$ . This actual symmetry corresponds to the fact that it’s possible to transpose a song into any key, and the ratios of all the notes will remain the same. This is a huge advantage! Other temperaments, such as the Pythagorean tuning, would make a song take on a different character if it were played in a different key. Some keys would sound sour. But equal temperament has the disadvantage that none of the notes (except those which are some number of octaves apart) are exactly harmonically related. This makes it impossible to tune to equal temperament by ear or using harmonics. It also means that harmonies played on equal temperament sound a bit off because of a “beating” sensation. Take a look at this video to see what I mean. In practice, classical musicians will typically cheat by dynamically adjusting their intonation to sound as harmonious as possible with the other musicians. This means, usually, playing Pythagorean intervals, rather than equal tempered intervals.

The Pythagorean Symmetry That Never Was Quite Right

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