Music and Math

a single Note, such as C4, as frequency of a string (length unit as L=1), its 2x frequency C5 (1 octave higher note, length=1/2); its half frequency C3 (1 octave lower note, length=1*2 =2);

since short length string making higher frequency sound, long string making lower frequency sound

we can define: 2/3 length of that C4 string is G4, (which is a note in-between C4 and C5, L=2/3)

then, start from G4, as frequency of a new string, 2/3 of that length, we get D5 (L=2/3 * 2/3 = 4/9), so if we start with C3 as 2, then D4 = 2*4/9 = 8/9

then, same circle of C4 (1), G4 (2/3), D5 (4/9) → D4 (8/9), A5 (8/27) → A4 (16/27), E6 (16/81) → E4 (64/81) continues

that way of push making C5 (2^?)/(3^12), a factorial never perfectly matching 1/2

however: the frequency difference between each step, like C to C#/Db, is around 105.3% to 106.8%