ATON is a hopefully evolving classification theory. It aspires to unify knowledge around numbers and prefers naive methods. Some of the older posts are wrong but I'll keep them for the sake of continuity.

Sunday, March 26, 2006

Equal Tempered Tuning

E0/H0= /{}
SE0/SH0=0!/2^0=1/1=1
The root note, say C.

E1/H1={}/{,{}}
SE1/SH1=1!0!/2^1=1/2
The octave of the root, 2C.

E2/H2={,{}}/{,{,{}}}
SE2/SH2=2!1!0!/2^2=2/4=1/2
The octave of the root, 2C.

E3/H3={,{},{,{}}}/{,{,{,{}}}}
SE3/SH3 = 3!2!1!0!/2^3 = 12/8 = 3/2
3/2 is the frequency ratio of the 5th to it's root, like G/C.
12 is the number of semi-tones in an octave.
(C,#C,D,#D,E,#E,F,G,#G,A,#A,B,C)
The frequency ratio between any two adjacent notes is the 12th root of 2.
When you iterate the same ratio 12 times you reach the octave,
(Root[2,12])^12=2
This homogeneous division gives transposability.
8 is the number of notes in a diatonic scale,
(C,D,E,F,G,A,B,2C)
In fact there are only 7 different notes but only when the 8th is played it gives the ending feeling.
It can be viewed as a HFractal of depth 3;
(((C,D),(E,F)),((G,A),(B,2C)))
Like modular group.
The grouping becomes clearer when the corresponding rational numbers are considered;
(((1,9/8),(5/4,4/3)),((3/2,5/3),(15/8,2)))
G leads the second group on the way forward.
F leads the second group on the way backward.
When two scales are played, the octave 2C leads the second group;
(((C,D),(E,F)),((G,A),(B,(((2C))),2D),(2E,2F)),((2G,2A),(2B,4C)))
The two scales overlap at 2C and become linked.
There is a wonderful modularity in the scales.
This can be heard clearly when one plays the scale (specially over and over again!).

ETT compares the power taking iteration of 5th to 8th and makes them meet at some real number a/b;
(3/2)^a = 2^b
a*log(3/2) = b*log(2)
a/b = log(2)/(log(3)-log(2)) = 1.7095
12/7 = 1.7143 has a very close match.
((3/2)^12)/2^7 = 1.0136
If you take power of 3/2 12 times you get the 7th octave, almost.
If a/b is set exactly to 12/7, then all other ratios have to be adjusted.
12 is the number of semi-tones in an octave, and 7 is the number of octaves it takes for the 5th to make a complete cycle and also the number of notes in CM scale.
Thus harmony is distorted slightly to get transposability.
The miracle of ETT is that harmony and equal tempering meet with a surprisingly small distortion of harmony.
This 7 octave travel of the 5th suggests M6.

I referred to Michael Rubinstein's page for the algebra;
http://www.math.uwaterloo.ca/~mrubinst/index.html

M6 suggests further division of the octave, like some Arabic scales which divide a tone into 9 commas.
12/7 equal tempering seems to correspond to the outer shape of M6, since it is centered around E3/H3.

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