In western music an octave is devided into 12 equal steps.
The frequencies form a geometric sequence with factor
2 1/12 . This multiplicative structure comes from the fact
that the human ear listens logarithmically. Rational Frequency relations are
considered "harmonic" . A geometric sequence is translational
invariant but can only approximate rational numbers. With 12 steps, this
can be achieved well, better for example than in Stockhausen's
5 1/5 scale.
Euler's music theory assigns to a frequency ratio
p/q a number G(p,q)=G(p/q) called "gradus suavitatis" which could
be translated as "degree of pleasure". G(p/q)-1 is defined as
the product (p i-1), where p i are the prime
factors (with multiplicity) of the least common multiple of
p and q.
For example, the little decime 12/5 has the degree of pleasure
(1+ (2-1)(2-1)(3-1)(5-1)) = 9. The geometric scale with 12 steps
interpolating the frequency doubling 1:2 allows to approximate
some rational numbers "with pleasure". For example,
2 2/12 = 1.122462048 ... is close to 9/8=1.125.
The "antique doric tune" is obtained by
taking from c-d and d-e the frequency relation 9/8 and
from e-f the relation 128/117, (c-f has then a relation 4/3)
then again 9/8 from f-g, g-a and a-h and again 128/117 from h to c.
Other tunes are the "diatonic tune" , 4/3=(9/8)(8/7)(28/27) the
"chromatic tune" 4/3 = (32/27)(36/35)(28/27), or the
"enharmonic tune" using
4/3=(5/4)(36/35)(28/27). The function G(n,m) can be determined with the
online-calculator to the left.
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