🚀 go-pugleaf

SAF Marc <safmarc@aol.com> wrote:
>"Pythagorean comma"?

Take a note. Tune another note a perfect fifth higher (i.e. exactly 3/2
the frequency of the original note). Take that note and give us a perfect
fifth above that (i.e. 3/2 time 3/2 of the original frequency) do that to
derive each of the notes in a 12-tone scale. Of course, you can divide any
of the resulting frequencies by 2 to drop them an octave. You'd get something
like this (assuming you start from a=440)

a=440
e=3/2*440
b=3/2*(3/2*440)
f#=3/2*(3/2*(3/2*440))
c#=3/2*....

If you do this long enough, you eventually arrive via the circle of fifths
back at a (up from d). But you've got a problem - when you finish dividing
by 2 to get the frequencies back into the range of an octave, the frequency
you get for a by tuning up by perfect fifths is *higher* than you'd get if
you simply took 440 and doubled it to 880. I'll let you sit down and figure
out the exact number - it's good practice. The distance between the number
you get by multiplying by 3/2 over and over (that is, by tuning your 
instrument by a series of perfect fifths) and the number you get by simply
doubling 440 is a Pythagorean comma.





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