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, its pattern maintaining 2/3 ratio

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

string length ratio can determine the frequency ratio, so simple ratio makes string simple to make.

a perfect ratio like 1x,2x, 1/2 is simplest.

well temperament: 1/2 maintain ratio, a set of defined frequency of each note

equal temperament: 1/2 maintain ratio, half step is x, solving x^12 = 1/2

different music to math definition means different music tuning system

different time, different culture, different history, different instrument means their default sounding may be slightly different, based on their choice of taste of sound.

modern equal temperament tuning is a balance of all for flexibility of sound adapting