This is an extension to my previous question when I tried to do the math to prove the following statement:
going up 12 perfect fifths takes one up 7 octaves plus one-fourth of a semitone extra
Going up 12 perfect fifths is equivalent to one cycle in the circle of fifth (or precisely the circle of
C1-G1-D2-A2-E3-B4-F#4-C#5-G#5-D#6-A#7-E#7-C8). Since each fifth has a frequency ratio of 3/2, and with 12 perfect fifths in the circle, the end frequency should be
(3/2)^12 or 129.75, roughly
1.75 more than that of the 7th octave in equal temperament, which is
2^7 = 128.
If my math is done correctly, the multiplication factor of 1.75 should be equivalent to one-fourth of a semitone extra, but it isn't! I tried comparing the
1.75 extra factor to a semitone in equal temperament and that in Pythagorean scale, and neither matches that statement.
Comparing to a Equal Temperament semitone
The ratio between neighboring semitones is always
2^(1/12) or 1.059.
One-fourth of that is. One-fourth of that is
1.059/4 = 0.265. Well...not even close to
2^[1/(4*12)] = 1.0145.
Comparing to a Pythagorean semitone
If I choose a semitone between C and its closest/neighboring C#, their ratio should be
(3/2)^7 / 2^4 = 2187 / 2048 = 1.068.
Somewhat closer to the value ofstill not quite there.
How is this one-fourth of a semitone extra derived?