Positive and negative temperaments

The formula given relates the Pythagorean comma to the step size.
If 50 equal is a doubly negative temperament, this isn't such a
useful method of classification. Better is to state the interval
that's defined out of existence.

For generalised keyboard mappings, you need to define two small
intervals for two directions. The way these add up to give
5-limit harmony is determined by which temperament class you
choose. The nthly positive definition is useless here. It is
also undefined for unequal temperaments.

79, 91 and 103 equal are singly negative, but not meantone.

There are two different kinds of triply positive temperaments.
With 15, 27 and 39 equal, the enharmonic diesis (7 0 -3)H is
zeroed. With 51, 63, 75 and 87 equal, zero 3*(13 -4 -1)H, or
set 2 Pythagorean equivalent to 3 syntonic commas.

Now, look at this:

Triply negative 9, 21, 33, 45, 57, 69, 81, 93, 105, 117
Doubly negative 2, 14, 26, 38, 50, 62, 74, 86, 98, 110
Singly negative 7, 19, 31, 43, 55, 67, 79, 91, 103, 115
Zeroly positive 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
Singly positive 5, 17, 29, 41, 53, 65, 77, 89, 101, 113
Doubly positive 10, 22, 34, 46, 58, 70, 82, 94, 106, 118
Triply positive 3, 15, 27, 39, 51, 63, 75, 87, 99, 111

The difference between each scale size in a class is 12. This
follows from the definition of the Pythagorean comma. A better
nomenclature for the same classes would simply be to take the
remainder of the scale size when divided by 12. Defining the
sizes of the octave(T) and Pythagorean comma(X) uniquely defines
the size of the fifth(P5)

12 * P5 - 7 T = X P5 = (X + 7T)/12

T 2 3 5 7 9 10 12
X -2 3 1 -1 -3 2 0
P5 1 2 3 4 5 6 7

1*ln(2)/ln(1.5)=1.7 2*ln(2)/ln(1.5)=3.4
3*ln(2)/ln(1.5)=5.1 4*ln(2)/ln(1.5)=6.8
5*ln(2)/ln(1.5)=8.5 6*ln(2)/ln(1.5)=10.3
7*ln(2)/ln(1.5)=12.0

When you add 12 to the octave size, you add 7 to the fifth size.
So, this is a complete set of consistent scale classes. If you
add the octave sizes of any two scales in this superset, you also
add the fifth sizes. That means:

T_3 = T_1 + T_2 P5_3 = P5_1 + P5_2

For perfect octaves, the fifth is P5/T-ln(3/2)/ln(2) octaves
sharp. The mistuning in terms of scale steps is then:

d = P5 - T*ln(3/2)/ln(2)

When adding scales,

d_3 = P5_3 - T_3*ln(3/2)/ln(2)
d_3 = (P5_1+P5_2) - (T_1+T_2)*ln(3/2)/ln(2)
d_3 = P5_1 - T_1*ln(3/2)/ln(2) + P5_2 - T_2*ln(3/2)/ln(2)
d_3 = d_1 + d_2

So, adding scale sizes means adding the amount of mistuning of
fifths in terms of scale steps. Paul Erlich stated this on the
tuning list last month, but didn't prove it.


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From: gbreed@cix.compulink.co.uk (Graham Breed)
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