A fraction, specifically
531441/524288, which is the
interval (or:
ratio of
frequencies) between a
note generated by 12 successive multiplications of a frequency by 3/2 (going 12 turns around the
'cycle of fifths') and one generated by seven doublings of the original frequency (by "going up 7
octaves")
You can arrive at this ratio as follows: starting with 1, get the next term by multiplying the previous term by 3/2 and then again by 1/2 if the result is greater than 1 (except at the very end.) This gives you:
c 1
g 3/2
'g 3/4
d 9/8
'd 9/16
a 27/32
e 81/64
'e 81/128
b 243/256
f# 729/512
'f# 729/1024
c# 2187/2048
'c# 2187/4096
g# 6561/8192
d# 19683/16384
'd# 19683/32768
a# 59049/65536
f 177147/131072
'f 177147/262144
c 531441/524288
Once you have done this little sum, you can enter the pythagorean comma (
531441/524288) into a search engine like
google and find out all kinds of wonderful things about
music theory!
Essentially, the value represents the amount of disharmony to be distributed around the 12-tone musical scale. It amounts to about 1/55 of an octave, or, in the modern nomenclature, about 23 cents (0.23 of a semitone in equal temperament).
Different tuning systems or intonations may be characterised by where they put this extra 23 cents: the pythagorean tuning (or temperament) hides it all in one fifth (the so-called "wolf fifth" or "wolf tone"), which has the effect of making keys harmonically distant from that particular fifth sound very concordant and harmonically close ones discordant.
Other systems, like just intonation and mean temperament distribute the disharmony using an uneven sequence of rational intervals (ie ratios composed of two integers) between successive semitones, while the modern equal temperament distributes it evenly over all 12 notes, so that each successive note (going up in semitones) has the same ratio to its successor as its predecessor has to it, namely 1::21/12, making all keys equally discordant.