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Commas in 12-eq: Pythagorean and Others
The musical term “comma” derives from the Greek word komma, to cut off. A comma breaks the flow of written words, and a musical comma does the same in a harmonic progression. In just intonation, when you travel through a series of intervals, sometimes you come to a place where two intervals are very close. Twelve perfect fifths, for example, take you to a pitch 23.46 cents above where you land when you ascend by seven octaves. This is the Pythagorean comma, the first in history to be discovered in the Western world. In pure just intonation, this would be a “pause” or a “comma” in the broken circle of fifths. The Pythagorean system of twelve perfect fifths evolved into our modern chromatic keyboard with twelve notes. Every comma also has a ratio, which is the mathematical difference between the two comma-pitches. The ratio for the Pythagorean comma is the incredibly clumsy 531441/524288, the ratio of the B# of twelve perfect fifths vs. the C of seven octaves. In 12-eq, each note on the keyboard ends up being around two cents sharp of a pure just fifth, so the comma becomes almost inaudible as it is evenly divided across the twelve notes, around two cents per note times twelve notes. A piano tuner does not tune for perfect harmony; instead, s/he tunes by listening to the “beats” until each fifth is two cents out of tune. Those “beats” are the sound of wave forms clashing against one another, canceling each other out and strengthening each other a few times a second. This is why all of Western music has a certain nervously unresolved quality to it, even if you are playing a nice C Major chord. Pure just fifths, which Pythagoras employed, give us this unclosed circle of fifths (I go up a fifth and then down a fourth to keep the pitches within a couple octaves, or it would take up the whole range of the piano):
Figure 11-14 circle of fifths
C – G –D –A – E –B –F# – C# – G# – D# – A# – E# – B# (with B# 23.64 cents sharper than C)
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