Scala analysis: xen10-wilson-purvi-05-03
Purvi modulation 3 from Figure 5, Didymus/Ptolemy (10/9 9/8 16/15) all 6 permutations
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: 5/4 386.313714 major third 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 5/3 884.358713 major sixth, BP sixth 6: 16/9 996.089998 Pythagorean minor seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 9/8 203.9100 major whole tone 1: 9/8 203.9100 major whole tone 2: 10/9 182.4037 minor whole tone 3: 16/15 111.7313 minor diatonic semitone 4: 9/8 203.9100 major whole tone 5: 10/9 182.4037 minor whole tone 6: 16/15 111.7313 minor diatonic semitone 7: 9/8 203.9100 major whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 16/15 111.731 cents minor diatonic semitone 1: 2 10/9 182.404 cents minor whole tone 1: 3 9/8 203.910 cents major whole tone 2: 2 32/27 294.135 cents Pythagorean minor third 2: 2 6/5 315.641 cents minor third 2: 2 5/4 386.314 cents major third 2: 1 81/64 407.820 cents Pythagorean major third 3: 5 4/3 498.045 cents perfect fourth 3: 1 27/20 519.551 cents acute fourth 3: 1 45/32 590.224 cents diatonic tritone 4: 1 64/45 609.776 cents 2nd tritone 4: 1 40/27 680.449 cents grave fifth 4: 5 3/2 701.955 cents perfect fifth 5: 1 128/81 792.180 cents Pythagorean minor sixth 5: 2 8/5 813.686 cents minor sixth 5: 2 5/3 884.359 cents major sixth, BP sixth 5: 2 27/16 905.865 cents Pythagorean major sixth 6: 3 16/9 996.090 cents Pythagorean minor seventh 6: 2 9/5 1017.596 cents just minor seventh, BP seventh 6: 2 15/8 1088.269 cents classic major seventh Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.33333 = 10/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 203.9 386.3 498.0 702.0 884.4 996.1 1200.0 203.9 : 182.4 294.1 498.0 680.4 792.2 996.1 1200.0 386.3 : 111.7 315.6 498.0 609.8 813.7 1017.6 1200.0 498.0 : 203.9 386.3 498.0 702.0 905.9 1088.3 1200.0 702.0 : 182.4 294.1 498.0 702.0 884.4 996.1 1200.0 884.4 : 111.7 315.6 519.6 702.0 813.7 1017.6 1200.0 996.1 : 203.9 407.8 590.2 702.0 905.9 1088.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 111.7313 .. 203.9100 cents 135/128, 92.1787 cents 16/15, 111.7313 cents 2: 294.1350 .. 407.8200 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents 3: 498.0450 .. 590.2237 cents 135/128, 92.1787 cents 256/243, 90.2250 cents 4: 609.7763 .. 701.9550 cents 135/128, 92.1787 cents 2048/2025, 19.5526 cents 5: 792.1800 .. 905.8650 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents 6: 996.0900 .. 1088.2687 cents 135/128, 92.1787 cents 256/243, 90.2250 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 16/15, 111.7313 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.534293 Largest / Average step : 1.189475 Largest / Smallest step : 1.825004 Median interval of one step : 10/9, 182.4037 cents, amount: 2 Most common interval of one step : 9/8, 203.9100 cents, amount: 3 Least squares average step : 171.97733 cents, oct.: 1203.84130 cents Scale is strictly proper Step pattern alph. order: ABCABCA Step pattern size order : LMSLMSL Scale is a Constant Structure, by a margin of 19.55257 cents Scale diversity : 1.415885 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.503263 Rothenberg efficiency : 0.540136 redundancy: 0.459864 Efficiency x scale size : 3.780952 Number of different interval sizes : 20 = 3.33333 / class Number of one step interval sizes : 3 Highest interval variety : 4 Mean interval variety : 3.33333 = 10/3 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 2048/2025, 19.5526 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 5 Most common triad is 0.0 498.045 701.955 cents, amount: 3 Number of recognisable fifths : 5, average 701.9550 cents Scale is Mixolydian octave species of abstract tetrachord Scale is Hypodorian octave species of abstract tetrachord Scale contains two identical conjunct tetrachords Best fifths form a closed circle Best major thirds form a closed circle Best minor thirds form a closed circle Scale contains a complete diamond : 1 3 9 All steps are superparticular (epimoric) Length of longest harmonic part : 2 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 5 = 71.4286% Scale is differentially coherent in interval class 1 Scale is n-1 differentially coherent in interval classes 2 and 3 combined Scale is differentially coherent in interval classes 3 and 4 combined Scale is n-1 differentially coherent in interval classes 4 and 5 combined Limited inverse transpositions with margin 92.1787 cents: 1 -- Rational properties -- Prime limit : 5 Odd number limit : 81 (O: 81 U: 81) Highest odd numerator or denominator: 9 Scale harmonicity : 0.041551 Average absolute harmonicity : 0.277642 Specific harmonicity : 0.245085 Fundamental : 1/72, -6.1699 octaves, 3.6337 Hz. Guide tone : 720, 9.4919 octaves, 188370.407 Hz. Exponens Consonantiae : 5.184000E+04, 15.66178 octaves Euler's gradus suavitatis : 20 Sum of Mann's harmonic distance : 37.0, average 5.28571 Mersenne's string divisions : 214200 Sum of van Prooijen's expressibility: 4.26067, average 0.60867 Sum of Tenney's harmonic distance : 8.65118, average 1.23588 Vogel's harmonic complexity : 10.14286 Wille's k value : 40 Wilson's harmonic complexity : 17 Rectangular lattice diameter : 5 Triangular lattice diameter : 4 Lattice compactness : 91.15287, average 3.25546 Lattice compactness (without 2's) : 46.94155, average 1.67648 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -3 .. 4 0.14286 13 1 3: -2 .. 2 -0.14286 7 4 5: 0 .. 1 0.28571 2 2 Average exponent except 2's : 1 / 7 = 0.14286 Average absolute exponent except 2's: 9 / 7 = 1.28571 Scale is JI-epimorphic with val: <7 11 16| = patent
FIT/MODE
7: 1 1 1 1 1 1 1 SP B ME I SD: 27.4299 c. M: 43.4566 c. Lebeng 12: 2 2 1 2 2 1 2 P M ME S SD: 8.1957 c. M:-15.6413 c. G.Hypophrygian, G.Ionian (Iastian), M.Mixolydian, G.M.Hypoionian, Hypermixolydian, Mischung 3, Khamaj That, Mela Harikambhoji, Raga Balahamsa, Bhim, Devamanohari, Harini, Janjhuti, Kaamaai, Khambhavati, Surati, Gregorian nr.7, Enharmonic Byzantine Liturgical, Rast descending: Greece, Ching, Shang: China 19: 3 3 2 3 3 2 3 SP M ME S SD: 9.0657 c. M:-14.4363 c. Nineteen-tone Mixolydian 22: 4 3 2 4 3 2 4 P SD: 9.7447 c. M:-14.2718 c. Murchhana Suddhasadja, Raga Harikambhoji, Palaiyazh 29: 5 4 3 5 4 3 5 SP SD: 8.0397 c. M: 15.3932 c. 31: 5 5 3 5 5 3 5 SP M DE S SD: 6.5964 c. M:-10.3616 c. Intense Diatonic Hypophrygian, M.Mixolydian 34: 6 5 3 6 5 3 6 P SD: 4.8101 c. M:-7.8547 c. 41: 7 6 4 7 6 4 7 SP SD: 3.2972 c. M: 6.3099 c. 46: 8 7 4 8 7 4 8 P SD: 3.5639 c. M:-4.9906 c. 53: 9 8 5 9 8 5 9 SP SD: 0.7391 c. M: 1.4081 c. 118: 20 18 11 20 18 11 20 SP SD: 0.3469 c. M:-0.5202 c. 171: 29 26 16 29 26 16 29 SP SD: 0.2793 c. M:-0.4012 c. 289: 49 44 27 49 44 27 49 SP SD: 0.2764 c. M:-0.4498 c. 323: 55 49 30 55 49 30 55 SP SD: 0.2608 c. M:-0.4244 c. 376: 64 57 35 64 57 35 64 SP SD: 0.2446 c. M:-0.3453 c. 388: 66 59 36 66 59 36 66 SP SD: 0.1799 c. M:-0.2842 c. 441: 75 67 41 75 67 41 75 SP SD: 0.1070 c. M:-0.1716 c. 494: 84 75 46 84 75 46 84 SP SD: 0.1043 c. M: 0.1482 c. 559: 95 85 52 95 85 52 95 SP SD: 0.0476 c. M:-0.0906 c. 612: 104 93 57 104 93 57 104 SP SD: 0.0236 c. M: 0.0450 c. 1171: 199 178 109 199 178 109 199 SP SD: 0.0148 c. M:-0.0228 c. 1783: 303 271 166 303 271 166 303 SP SD: 0.0099 c. M:-0.0160 c.
FIT/HARMONIC
1 x x x x x x 2 S SD: 0.0000 cents 2 x x x 3 x x 4 S SD: 0.0000 cents 3 x x 4 x 5 x 6 S SD: 0.0000 cents 4 x 5 x 6 x 7:8 S SD: 6.8160 cents 5 x 6:7 x 8:9:10 S SD: 26.5228 cents 6:7 x 8:9:10:11:12 S SD: 13.7458 cents 7:8:9 x 11:12 x 14 SD: 21.8967 cents 8:9:10:11:12:13:14:16 S SD: 10.5970 cents 10:11 x 13:15:17:18:20 SD: 11.8704 cents 12:14:15:16:18:20:21:24 S SD: 9.8015 cents 15:17:19:20:23:25:27:30 SD: 7.2835 cents 16:18:20:21:24:27:28:32 S SD: 6.3071 cents 18:20:23:24:27:30:32:36 SD: 6.2440 cents 23:26:29:31:35:38:41:46 SD: 5.5619 cents 24:27:30:32:36:40:43:48 SD: 1.9247 cents 48:54:60:64:72:80:85:96 SD: 0.9680 cents 72:81:90:96:108:120:128:144 S SD: 0.0000 cents