The modern major/minor scale is a compromise. Maybe it is better understood if put as the following. For brevity, denote K
as \sqrt[12]{2}=1.05946
, the twelfth root of 2 (sorry but no LaTeX support here).
Verbally, to divide an octave as 12 equal-sized intervals.
And use scientific notation, that is the middlemost octave is C4 to B4, with C5 to B5 the higher, and C3 to B3 the lower, and so on.
Than, starting with C2
, denote its frequency to be f_0
. The harmonic series, those human most easily perceive, with a particular note (here C2
) given.
We see:
C3 =2f_0
G3 =3f_0
C4 =4f_0
E4 =5f_0
G4 =6f_0
B♭4 =7f_0
C5 =8f_0
D5 =9f_0
And so on. The introduction of constant K
seems to be a good approximation of
K^7 ~ 3/2
K^4 ~ 5/4
K^3 ~ 7/6
K^2 ~ 8/7
And so on. The rest story is that the major/minor scale is gradually accepted within European music community. You may refer the relevant history by looking up in any serviceable music history book, the Renaissance section.
Why aren't there other scales around the world? Because there are! When you listen to a traditional Japanese folksong, or Chinese folksong, they each use a scale different from the major/minor scale.
The exoticness gives us the flavor of Chinese-ness or Japanese-ness, but while the difference is true, the relation is artificial.
Indeed, a scale is not intrinsic Western European, or Japanese, or Chinese, but because people there use it, and we are informed so. In the end, the classical conditioning results, I suppose.
The view that a modern piano always use K
as the freq. for adjacent keys --- the so-called equal temperament --- seems not to be true.
A tuning is often a compromise between equal temperament and just temperament (a strict adherence of the proportions you said and I repeated above).
Different tuners apply different methods too.
You has urged me to check relevant source or ask a question myself. Correction welcomed.