Scala analysis: xen15-gilson-pythagorean-pentatonic
Pythagorean Intonation Pentatonic Scale (PIPS)
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: 4/3 498.044999 perfect fourth 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: 32/27 294.1350 Pythagorean minor third 3: 9/8 203.9100 major whole tone 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 : 203.9 498.0 702.0 905.9 1200.0 203.9 : 294.1 498.0 702.0 996.1 1200.0 498.0 : 203.9 407.8 702.0 905.9 1200.0 702.0 : 203.9 498.0 702.0 996.1 1200.0 905.9 : 294.1 498.0 792.2 996.1 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 : 235.07864 cents, oct.: 1175.39318 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: 2 of 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: 113 276 389 502 665 generators: 276 and 389 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABAAB Step pattern size order : SLSSL 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 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 Scale contains a complete diamond : 1 3 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 3 = 60.0000% 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 : 1 Inversional symmetry on degrees : 3 Inversional symmetry on intervals : 0-1 -- Rational properties -- Prime limit : 3 Odd number limit : 81 (O: 81 U: 81) Highest odd numerator or denominator: 27 Scale harmonicity : 0.060000 Average absolute harmonicity : 0.338069 Specific harmonicity : 0.313043 Fundamental : 1/48, -5.5850 octaves, 5.4505 Hz. Guide tone : 108, 6.7549 octaves, 28255.561 Hz. Exponens Consonantiae : 5.184000E+03, 12.33985 octaves Euler's gradus suavitatis : 15 Sum of Mann's harmonic distance : 37.5, average 7.50000 Mersenne's string divisions : 76755 Sum of van Prooijen's expressibility: 3.33985, average 0.66797 Sum of Tenney's harmonic distance : 6.65118, average 1.33024 Vogel's harmonic complexity : 9.00000 Wille's k value : 40 Wilson's harmonic complexity : 12 Rectangular lattice diameter : 4 Triangular lattice diameter : 4 Lattice compactness : 51.11264, average 3.40751 Lattice compactness (without 2's) : 27.00000, average 1.80000 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -4 .. 2 -1.00000 11 1 3: -1 .. 3 1.00000 7 4 Average exponent except 2's : 5 / 5 = 1.00000 Average absolute exponent except 2's: 7 / 5 = 1.40000 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: 31.2549 c. M:-54.1350 c. 7: 1 2 1 1 2 P M ME S SD: 28.1297 c. M: 48.7221 c. Thai mode 5 12: 2 3 2 2 3 SP M ME S SD: 3.3862 c. M: 5.8650 c. Scottish Pentatonic, Blues Major, Ritusen, Ritsu (Gagaku): Japan, Zhi, Zheng: China, Ujo, P'yongjo: Korea, Bac: Vietnam, Lai Soutsanaen, Lai Po Sai, Lai Soi: Laos, Raga Arabhi, Devakriya, Durga, Jaldhar Kedar, Suddha Saveri, Major complement 29: 5 7 5 5 7 SP M DE S SD: 2.5864 c. M:-4.4798 c. 41: 7 10 7 7 10 SP M DE S SD: 0.8384 c. M:-1.4521 c. 53: 9 13 9 9 13 SP M DE S SD: 0.1181 c. M: 0.2046 c. 200: 34 49 34 34 49 SP M DE S SD: 0.0779 c. M:-0.1350 c. 253: 43 62 43 43 62 SP M DE S SD: 0.0369 c. M:-0.0639 c. 306: 52 75 52 52 75 SP M DE S SD: 0.0100 c. M:-0.0174 c. 359: 61 88 61 61 88 SP M DE S SD: 0.0089 c. M: 0.0154 c. 665: 113 163 113 113 163 SP M DE S SD: 0.0002 c. M: 0.0003 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: 7.1688 cents 4 x 5:6:7:8 S SD: 32.0624 cents 5:6:7 x 8:10 S SD: 41.9192 cents 6:7:8:9:10:12 S SD: 13.3065 cents 10:11:13:15:17:20 SD: 11.9968 cents 14:16:19:21:24:28 SD: 9.8501 cents 15:17:20:23:25:30 SD: 9.1074 cents 16:18:21:24:27:32 SD: 5.4528 cents 22:25:29:33:37:44 SD: 5.3974 cents 24:27:32:36:41:48 SD: 4.2485 cents 30:34:40:45:51:60 SD: 3.6138 cents 32:36:43:48:54:64 SD: 2.6945 cents 42:47:56:63:71:84 SD: 1.9355 cents 48:54:64:72:81:96 SD: 0.0000 cents