In 1998 I experienced a geometric conversion. We students in Douglas Hofstadter's cognitive science course '"Circles and Triangles: Diamonds of Geometry (Cat:DoG)" at Indiana University were encouraged to explain complex concepts in simple ways, as well as simple concepts in complex ways. One of Hofstadter's favorite devices was to write a geometric proof in the form of a sonnet. Whittle the concept down to 14 rhymed lines, with a punchline at the end, and you've got a slam-dunk in the elegance department. Geometric proofs became objets d'art.
Why Six Tones Make an Octave and a Comma
And Why it is the Same Comma as between Three Tones and a Fifth
A Sonnet after Boethius
A fourth makes two tones and a semitone,
Aristoxenus told us this much.
Three tones and semitone renders in stone A fifth. . .
Our own logic tells us as such.
A tone to splice, and our being precise,
Two semitones make not quite a full one.
A semitone twice and a comma suffice
To fill it, but we’ve only just begun!
Now as we look ’twixt a fourth and a fifth,
Three tones make semitone, comma and fourth.
Two fourths and a tone make an octave ’s width,
And six tones are double of three, of courthe!
Algebra decrees how six tones exceed
Of an octave. Indeed it ’s QED’d.
In the spirit of elegance, calcuations with large orders of magnitude are unnecessary. Armed with the knowledge that two semitones fall short of a whole tone, the following proof unfolds easily, and hopefully, obviously as well:
|1 octave||=?||6 tones|
|(1 fourth) + (1 fifth)||=?||6 tones|
|(2 tones + 1 semitone) + (3 tones + 1 semitone)||=?||6 tones|
|5 tones + 2 semitones||=?||6 tones|
|2 semitones||≠||1 tone|
And why to two semitones fall short of a whole tone? Well, we'll need a bit of calculation just to prove this chunk of information. Using the elegant Pythagorean ratios, a fourth is 4/3 and a full tone is 9/8. The difference between two tones and a fourth is a diatonic semitone, which is a very inelegant number, 256/243
|1 fourth||=||2 tones + 1 semitone|
|4/3||=||9/8 * 9/8 * 1 semitone|
|4/3||=||81/64 * 1 semitone|
From here it is obvious that two of these monstrosities are not going to add up to the lovely 9/8, and in fact they come up short:
|2 semitones + 1 comma||=||1 tone|
|256/243 * 1 comma||=||9/8|
Now that we know the comma exists, no more heavy calculations are needed, and we can get back to poetry. Eat your heart out, Quadrivium.