Scala analysis: xen15-gilson-generalized-pythagorean-3-2-5
Generalized Pythagorean Scale, 3/2 stacked 5=3+2 times
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 9/8 203.910002 major whole tone 2: 81/64 407.820003 Pythagorean major third 3: 3/2 701.955001 perfect fifth 4: 27/16 905.865003 Pythagorean major sixth 5: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 32/27 294.1350 Pythagorean minor third 1: 9/8 203.9100 major whole tone 2: 9/8 203.9100 major whole tone 3: 32/27 294.1350 Pythagorean minor third 4: 9/8 203.9100 major whole tone 5: 32/27 294.1350 Pythagorean minor third
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 3 9/8 203.910 cents major whole tone 1: 2 32/27 294.135 cents Pythagorean minor third 2: 1 81/64 407.820 cents Pythagorean major third 2: 4 4/3 498.045 cents perfect fourth 3: 4 3/2 701.955 cents perfect fifth 3: 1 128/81 792.180 cents Pythagorean minor sixth 4: 2 27/16 905.865 cents Pythagorean major sixth 4: 3 16/9 996.090 cents Pythagorean minor seventh 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 4 5 0.0 : 203.9 407.8 702.0 905.9 1200.0 203.9 : 203.9 498.0 702.0 996.1 1200.0 407.8 : 294.1 498.0 792.2 996.1 1200.0 702.0 : 203.9 498.0 702.0 905.9 1200.0 905.9 : 294.1 498.0 702.0 996.1 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 203.9100 .. 294.1350 cents 256/243, 90.2250 cents 9/8, 203.9100 cents 2: 407.8200 .. 498.0450 cents 256/243, 90.2250 cents 2187/2048, 113.6850 cents 3: 701.9550 .. 792.1800 cents 256/243, 90.2250 cents 9/8, 203.9100 cents 4: 905.8650 .. 996.0900 cents 256/243, 90.2250 cents 2187/2048, 113.6850 cents
SHOW DATA
Number of notes : 5 -- Interval properties -- Smallest interval : 9/8, 203.9100 cents, class 1 Average step (divided formal octave): 240.0000 cents Largest one step interval : 32/27, 294.1350 cents Average / Smallest step : 1.176990 Largest / Average step : 1.225562 Largest / Smallest step : 1.442475 Median interval of one step : 9/8, 203.9100 cents, amount: 3 Most common interval of one step : 9/8, 203.9100 cents, amount: 3 Least squares average step : 231.79773 cents, oct.: 1158.98864 cents Scale is strictly proper Scale has Myhill's property chromas: 256/243, 90.2250 cents and 2187/2048, 113.6850 cents, ratio: 1.260017 generators: 498.04500 cents and 3 of 701.95500 cents ET pair of linear temperament : 12&41 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 665-tone equal temperament with octave 2/1 degrees: 113 226 389 502 665 generators: 276 and 389 Number of contiguous 1-step segments: 0 Step pattern alph. order: AABAB Step pattern size order : SSLSL Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 113.68501 cents Ratio Whole / Half tone : 2.260017 Scale diversity : 1.281255 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.699250 Rothenberg efficiency : 0.800000 redundancy: 0.200000 Efficiency x scale size : 4.000000 Number of different interval sizes : 8 = 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 : 256/243, 90.2250 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 4 Scale is a chain of 2 triads 0.0 498.045 996.090 cents Most common triad is 0.0 498.045 701.955 cents, amount: 3 Number of recognisable fifths : 4, average 701.9550 cents Best fifths form a closed circle Scale is a complete genus musicum : 2^m 3^4 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 1 = 20.0000% Scale is differentially coherent in interval classes 1 and 2 combined Scale is n-1 differentially coherent in interval class 2 Scale is n-1 differentially coherent in interval class 3 Limited inverse transpositions : 2 Inversional symmetry on degrees : 1 Inversional symmetry on intervals : 3-4 -- Rational properties -- Prime limit : 3 Odd number limit : 81 (O: 81 U: 81) Highest odd numerator or denominator: 81 Scale harmonicity : 0.056604 Average absolute harmonicity : 0.307212 Specific harmonicity : 0.279069 Fundamental : 1/64, -6.0000 octaves, 4.0879 Hz. Guide tone : 162, 7.3399 octaves, 42383.342 Hz. Exponens Consonantiae : 1.036800E+04, 13.33985 octaves Euler's gradus suavitatis : 16 Sum of Mann's harmonic distance : 106.5, average 21.30000 Mersenne's string divisions : 317985 Sum of van Prooijen's expressibility: 4.77121, average 0.95424 Sum of Tenney's harmonic distance : 9.28666, average 1.85733 Vogel's harmonic complexity : 9.40000 Wille's k value : 40 Wilson's harmonic complexity : 12 Rectangular lattice diameter : 4 Triangular lattice diameter : 4 Lattice compactness : 57.76068, average 3.85071 Lattice compactness (without 2's) : 30.00000, average 2.00000 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -6 .. 1 -2.60000 15 1 3: 0 .. 4 2.00000 10 4 Average exponent except 2's : 10 / 5 = 2.00000 Average absolute exponent except 2's: 10 / 5 = 2.00000 Scale is weakly epimorphic with val: <5 7| Scale is JI-epimorphic with val: <5 8| = patent Scale is weakly epimorphic with val: <5 9|
FIT/MODE
5: 1 1 1 1 1 SP B ME I SD: 44.2010 c. M:-72.1800 c. 7: 1 1 2 1 2 P M ME S SD: 39.7815 c. M: 64.9629 c. Thai mode 1 8: 1 2 2 1 2 P M ME S SD: 37.4918 c. M: 53.9100 c. 9: 2 1 2 2 2 SP M ME S SD: 34.6391 c. M:-62.7567 c. Noll Pseudo-diatonic 11: 2 2 2 2 3 SP M ME S SD: 29.5439 c. M: 47.4095 c. 12: 2 2 3 2 3 SP M ME S SD: 4.7888 c. M: 7.8200 c. Major Pentatonic, Ryosen, Yona Nuki Major: Japan, Man Jue, Gong: China, Raga Bhopali (Bhup), Bilahari, Deskar, Kokila, Jait Kalyan, Mohanam, Peruvian Pentatonic 1, Ghana Pentatonic 2, Tizita Major: Ethiopia 29: 5 5 7 5 7 SP M DE S SD: 3.6578 c. M:-5.9731 c. 41: 7 7 10 7 10 SP M DE S SD: 1.1856 c. M:-1.9361 c. 53: 9 9 13 9 13 SP M DE S SD: 0.1671 c. M: 0.2728 c. 200: 34 34 49 34 49 SP M DE S SD: 0.1102 c. M:-0.1800 c. 253: 43 43 62 43 62 SP M DE S SD: 0.0521 c. M:-0.0851 c. 306: 52 52 75 52 75 SP M DE S SD: 0.0142 c. M:-0.0231 c. 359: 61 61 88 61 88 SP M DE S SD: 0.0126 c. M: 0.0206 c. 665: 113 113 163 113 163 SP M DE S SD: 0.0003 c. M: 0.0005 c.
FIT/HARMONIC
1 x x x x 2 S SD: 0.0000 cents 2 x x 3 x 4 S SD: 0.0000 cents 3 x 4 x 5:6 S SD: 30.9176 cents 4 x 5:6:7:8 S SD: 16.6332 cents 5 x 6 x 8:10 S SD: 43.4535 cents 6:7:8:9:10:12 S SD: 22.4207 cents 7:8:9:11:12:14 SD: 18.6721 cents 8:9:10:12:14:16 S SD: 13.3065 cents 10:11:13:15:17:20 SD: 12.3763 cents 14:16:18:21:24:28 S SD: 9.4446 cents 15:17:19:23:25:30 SD: 9.1118 cents 16:18:20:24:27:32 SD: 4.3013 cents 22:25:28:33:37:44 SD: 4.1506 cents 26:29:33:39:44:52 SD: 3.2824 cents 34:38:43:51:57:68 SD: 3.2208 cents 38:43:48:57:64:76 SD: 2.2336 cents 42:47:53:63:71:84 SD: 2.1875 cents 48:54:61:72:81:96 SD: 1.4220 cents 64:72:81:96:108:128 SD: 0.0000 cents