Scala analysis: xen15-gilson-generalized-pythagorean-13-8-3
Generalized Pythagorean Scale, 13/8 stacked 3=2+1 times
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 169/128 481.055324 2: 13/8 840.527662 tridecimal neutral sixth 3: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/13 359.4723 tridecimal neutral third 1: 169/128 481.0553 2: 16/13 359.4723 tridecimal neutral third 3: 16/13 359.4723 tridecimal neutral third
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 16/13 359.472 cents tridecimal neutral third 1: 1 169/128 481.055 cents 2: 1 256/169 718.945 cents two (tridecimal neutral third) 2: 2 13/8 840.528 cents tridecimal neutral sixth Highest number of different intervals for one interval class: 2 Average number of different intervals per interval class: 2.00000 = 2
SHOW/LINE/CENTS INTERVALS
1 2 3 0.0 : 481.1 840.5 1200.0 481.1 : 359.5 718.9 1200.0 840.5 : 359.5 840.5 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 359.4723 .. 481.0553 cents 2197/2048, 121.5830 cents 16/13, 359.4723 cents 2: 718.9447 .. 840.5277 cents 2197/2048, 121.5830 cents 32768/28561, 237.8894 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 16/13, 359.4723 cents, class 1 Average step (divided formal octave): 400.0000 cents Largest one step interval : 169/128, 481.0553 cents Average / Smallest step : 1.112742 Largest / Average step : 1.202638 Largest / Smallest step : 1.338226 Median interval of one step : 16/13, 359.4723 cents, amount: 2 Most common interval of one step : 16/13, 359.4723 cents, amount: 2 Least squares average step : 411.57933 cents, oct.: 1234.73800 cents Scale is strictly proper Scale has Myhill's property chromas: 2197/2048, 121.5830 cents and 32768/28561, 237.8894 cents, ratio: 1.956601 generators: 359.47234 cents and 2 of 840.52766 cents ET pair of linear temperament : 10&227 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 227-tone equal temperament with octave 2/1 degrees: 91 159 227 generators: 68 and 159 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABB Step pattern size order : LSS Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 237.88935 cents Scale diversity : 1.301663 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.797362 Rothenberg efficiency : 0.888889 redundancy: 0.111111 Efficiency x scale size : 2.666667 Number of different interval sizes : 4 = 2.00000 / class Number of one step interval sizes : 2 Highest interval variety : 2 Mean interval variety : 2.00000 = 2 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 2197/2048, 121.5830 cents Most common intervals : 16/13, 359.4723 cents & inv., amount: 2 Number of recognisable fifths : 1, average 718.9447 cents Number of appreciable fifths : 0 Best fifths form a closed circle Best major thirds form a closed circle Best minor thirds form a closed circle Scale is a complete genus musicum : 2^m 13^2 Steps are decreasing Formal octave complements present : 1 = 33.3333% Limited inverse transpositions : 1 Inversional symmetry on degrees : 2 Inversional symmetry on intervals : 0-1 -- Rational properties -- Prime limit : 13 Odd number limit : 169 (O: 169 U: 169) Highest odd numerator or denominator: 169 Scale harmonicity : 0.019118 Average absolute harmonicity : 0.353081 Specific harmonicity : 0.087874 Fundamental : 1/128, -7.0000 octaves, 2.0439 Hz. Guide tone : 338, 8.4009 octaves, 88429.441 Hz. Exponens Consonantiae : 4.326400E+04, 15.40088 octaves Euler's gradus suavitatis : 33 Sum of Mann's harmonic distance : 160.5, average 53.50000 Mersenne's string divisions : 2079 Sum of van Prooijen's expressibility: 3.34183, average 1.11394 Sum of Tenney's harmonic distance : 6.65316, average 2.21772 Vogel's harmonic complexity : 17.00000 Wille's k value : 84 Wilson's harmonic complexity : 26 Rectangular lattice diameter : 2 Triangular lattice diameter : 2 Lattice compactness : 27.93481, average 4.65580 Lattice compactness (without 2's) : 7.00000, average 1.16667 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -7 .. 1 -3.00000 11 1 13: 0 .. 2 1.00000 3 2 Average exponent except 2's : 3 / 3 = 1.00000 Average absolute exponent except 2's: 3 / 3 = 1.00000 Scale is weakly epimorphic with val: <3 10| Scale is JI-epimorphic with val: <3 11| = patent
FIT/MODE
3: 1 1 1 SP B ME I SD: 52.3210 c. M: 81.0553 c. 7: 3 2 2 SP M ME S SD: 21.4501 c. M:-33.2304 c. 10: 4 3 3 SP M ME S SD: 0.6812 c. M: 1.0553 c. 117: 47 35 35 SP M DE S SD: 0.6429 c. M:-0.9960 c. 127: 51 38 38 SP M DE S SD: 0.5386 c. M:-0.8344 c. 137: 55 41 41 SP M DE S SD: 0.4496 c. M:-0.6965 c. 147: 59 44 44 SP M DE S SD: 0.3727 c. M:-0.5773 c. 157: 63 47 47 SP M DE S SD: 0.3055 c. M:-0.4733 c. 167: 67 50 50 SP M DE S SD: 0.2465 c. M:-0.3818 c. 177: 71 53 53 SP M DE S SD: 0.1940 c. M:-0.3006 c. 187: 75 56 56 SP M DE S SD: 0.1472 c. M:-0.2281 c. 197: 79 59 59 SP M DE S SD: 0.1052 c. M:-0.1630 c. 207: 83 62 62 SP M DE S SD: 0.0672 c. M:-0.1041 c. 217: 87 65 65 SP M DE S SD: 0.0327 c. M:-0.0507 c. 227: 91 68 68 SP M DE S SD: 0.0013 c. M:-0.0019 c.
FIT/HARMONIC
1 x x 2 S SD: 0.0000 cents 2 x 3:4 S SD: 69.2863 cents 3:4:5:6 H SD: 15.6695 cents 10:13:16:20 SD: 12.6531 cents 13:17:21:26 SD: 6.5153 cents 16:21:26:32 SD: 3.4248 cents 19:25:31:38 SD: 3.0591 cents 34:45:55:68 SD: 2.9701 cents 35:46:57:70 SD: 2.9284 cents 37:49:60:74 SD: 2.1231 cents 40:53:65:80 SD: 2.0452 cents 50:66:81:100 SD: 1.7836 cents 53:70:86:106 SD: 0.8602 cents 56:74:91:112 SD: 0.4876 cents 72:95:117:144 SD: 0.3795 cents 128:169:208:256 SD: 0.0000 cents