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question on Scala scale

🔗Joseph Pehrson <josephpehrson@compuserve.com>

5/23/2000 9:19:21 PM

Question on Scala scale...

I have a question concerning the Scala scale 19-31ji.scl:

A septimal interpretation of 19 out of 31 tones, after Wilson, XH7+8
0: 1/1 0.000 unison, perfect prime
1: 25/24 70.672 classic chromatic semitone
2: 16/15 111.731 minor diatonic semitone
3: 9/8 203.910 major whole tone
4: 7/6 266.871 septimal minor third
5: 6/5 315.641 minor third
6: 5/4 386.314 major third
7: 9/7 435.084 septimal major third
8: 4/3 498.045 perfect fourth
9: 7/5 582.512 septimal tritone
10: 10/7 617.488 Euler's tritone
11: 3/2 701.955 perfect fifth
12: 14/9 764.916 septimal minor sixth
13: 8/5 813.686 minor sixth
14: 5/3 884.359 major sixth
15: 7/4 968.826 harmonic seventh
16: 16/9 996.090 Pythagorean minor seventh
17: 15/8 1088.269 classic major seventh
18: 27/14 1137.039 septimal major seventh
19: 2/1 1200.000 octave

Number of notes : 19
Smallest interval : 64/63, 27.2641 cents
Average interval (divided octave) : 63.158 cents
Average / Smallest interval : 2.316523
Largest interval of one step : 135/128, 92.1787 cents
Largest / Average interval : 1.459496
Largest / Smallest interval : 3.380957
Least squares average interval : 63.3253 cents
Median interval of one step : 28/27, 62.9609 cents
Interval standard deviation : 18.6593 cents
Interval skew : 0.0004 cents
Scale is not proper
Number of different intervals : 112 = 6.22222 / class
Lumma stability : 0.258112
Impropriety factor : 0.014055
Prime limit : 7
Odd number limit : 27 (O: 27 U: 15)
Average absolute harmonicity : 0.147808
Specific harmonicity : 0.117219
Fundamental : 1/2520,-11.299 octaves, 0.104 Hz.
Guide tone : 75600, 16.206 octaves,
19778892.74
Hz.
Exponens Consonantiae : 1.90512E+08, 27.5053 octaves
Euler's gradus suavitatis : 42
Wille's k value : 2362
Vogel's harmonic complexity : 2.21053E+01
Wilson's harmonic complexity : 44
Rectangular lattice dimensions : 10
Triangular lattice dimensions : 5
Prime exponents' range and average:
2: -3 .. 4 0.05263
3: -2 .. 3 0.05263
5: -1 .. 2 0.10526
7: -1 .. 1 0.05263
Average exponent except of 2 : 0.21053
Average absolute exponent except of 2: 36 / 19 = 1.89474
Average distance from equal tempered : 9.6189 cents, 0.152299 steps
Standard deviation from equal tempered : 2.6472 cents, 0.041913 steps
Maximum distance from equal tempered : 21.4575 cents, 0.339743 steps
Geometric average of pitches 0..n : 602.170 cents
Arithmetic average of pitches 0..n : 24319/16800, 640.3472 cents
Harmonic average of pitches 0..n : 563.994 cents
Geometric average of pitches 1..n : 633.864 cents
Arithmetic average of pitches 1..n : 23479/15960, 668.2924 cents
Harmonic average of pitches 1..n : 599.435 cents
Geometric average of pitches 1..n-1: 602.412 cents
Arithmetic average of pitches 1..n-1: 21799/15120, 633.3646 cents
Harmonic average of pitches 1..n-1: 571.459 cents

This scale, to me, seems quite resonant and nice in harmonic usage...
but
how is it constructed? It is seven-limit just intonation, but what
does
XH7+8 mean? Is this some kind of notation by Erv Wilson?

Any help would be appreciated...

___________ _____ ____ __
Joseph Pehrson