the sound of music

Another year, another illusion shattered.

regale me with a story!

The thing that fascinated me about learning music, as a proto-geek (I gave up because I didn't have the discipline for practice) was the elegant way the scale fitted together. Playing in C major, G was clearly the next-best note - and G had its own scale, differing by just one note. D was the best note in that scale, and continuing in perfect 5ths took you all the way around - C G D A E B F# C#/Db Ab Eb Bb F C.

the radical antagonist

The pitches formed a nice pattern too - C₄ (middle C) was half the frequency of C₅, and double the frequency of C₃. Semitones, the indivisible atoms of notes, were equally spaced with 12 to an octave. Multiplying rather than adding seemed to be the way to go, so I figured out that to get a semitone higher you multiplied by ¹²√2. I then wrote a cute little program to play Mary Had a Little Lamb, as if to prove that math really can't teach you about music.

discord

I didn't really wonder why C made that G sound so good. That was an Art question, I didn't own a beret, and I was still cutting my hair at that point. But it was closely related to why all Cs sound the same.

Notes forming a 'perfect' fifth have frequencies in the ratio 3/2. A fourth is the ratio 4/3 (so a fourth plus a fifth is an octave, 3/2 x 4/3 = 2). A third is 5/4 (major) or 6/5 (minor). Problem is, these fractions aren't what you get when you glue a bunch of your ¹²√2 semitones together. 7 semitones is 2^7/12 ≅ 1.4983 - 2% away from perfect harmony. The circle of fifths raises the pitch by (3/2)¹² ≅ 129.74 times - 2% more than 7 octaves.

So this music thing less of a deep truth about the universe and more of a weird and meaningless approximate coincidence.

You can tune your piano by

• Setting the notes to nice fractions of C (but music in other keys will be out of tune)
• Finding notes with a circle of perfect fifths, but the fifth that joins the circle up will be too small, leaving (typically) G# and Eb badly out of tune
• Spreading the discrepancy equally across all notes, which is what we generally do now. It's regular and you can modulate betwteen keys freely.

the thing about coincidences

You can find more if you look hard enough. There's no way to find an equal division of the octave that will give you a perfect fifth - or any nice harmonic. n 'semitones' in an m-division scale is 2^n/m which is irrational.

The next equal division size after 12 that can measure a fifth pretty accurately is 29 (3.6% error). 41 does better than 12 (1.6% error) and 53 is very good indeed (0.03% error). Music has been composed in these. I'm curious though, whether our familiar time signatures of 2, 3, 4, and 6 notes work as well in other divisions, or whether dividing evenly into the scale is an important attribute.

And of course you can divide intervals other than an octave to form a scale. In particular I'm curious about an octave + fifth such as C₄ - G₅ - a 3:1 frequency ratio.

Time to write some code, I guess.