Scala analysis: xen10-wilson-purvi-02b-07
Purvi modulation 7 from Figure 2b, Helmholtz (16/15 75/64 16/15)
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 75/64 274.582429 classic augmented second 2: 5/4 386.313714 major third 3: 4/3 498.044999 perfect fourth 4: 25/16 772.627428 classic augmented fifth 5: 5/3 884.358713 major sixth, BP sixth 6: 15/8 1088.268715 classic major seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 75/64 274.5824 classic augmented second 2: 16/15 111.7313 minor diatonic semitone 3: 16/15 111.7313 minor diatonic semitone 4: 75/64 274.5824 classic augmented second 5: 16/15 111.7313 minor diatonic semitone 6: 9/8 203.9100 major whole tone 7: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 4 16/15 111.731 cents minor diatonic semitone 1: 1 9/8 203.910 cents major whole tone 1: 2 75/64 274.582 cents classic augmented second 2: 1 256/225 223.463 cents Neapolitan diminished third 2: 2 6/5 315.641 cents minor third 2: 4 5/4 386.314 cents major third 3: 1 32/25 427.373 cents classic diminished fourth 3: 4 4/3 498.045 cents perfect fourth 3: 2 45/32 590.224 cents diatonic tritone 4: 2 64/45 609.776 cents 2nd tritone 4: 4 3/2 701.955 cents perfect fifth 4: 1 25/16 772.627 cents classic augmented fifth 5: 4 8/5 813.686 cents minor sixth 5: 2 5/3 884.359 cents major sixth, BP sixth 5: 1 225/128 976.537 cents augmented sixth 6: 2 128/75 925.418 cents diminished seventh 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 4 15/8 1088.269 cents classic major seventh Highest number of different intervals for one interval class: 3 Average number of different intervals per interval class: 3.00000 = 3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 274.6 386.3 498.0 772.6 884.4 1088.3 1200.0 274.6 : 111.7 223.5 498.0 609.8 813.7 925.4 1200.0 386.3 : 111.7 386.3 498.0 702.0 813.7 1088.3 1200.0 498.0 : 274.6 386.3 590.2 702.0 976.5 1088.3 1200.0 772.6 : 111.7 315.6 427.4 702.0 813.7 925.4 1200.0 884.4 : 203.9 315.6 590.2 702.0 813.7 1088.3 1200.0 1088.3: 111.7 386.3 498.0 609.8 884.4 996.1 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 111.7313 .. 274.5824 cents 1125/1024, 162.8511 cents 16/15, 111.7313 cents 2: 223.4626 .. 386.3137 cents 1125/1024, 162.8511 cents 16384/16875,-51.1199 cents 3: 427.3726 .. 590.2237 cents 1125/1024, 162.8511 cents 128/125, 41.0589 cents 4: 609.7763 .. 772.6274 cents 1125/1024, 162.8511 cents 2048/2025, 19.5526 cents 5: 813.6863 .. 976.5374 cents 1125/1024, 162.8511 cents 128/125, 41.0589 cents 6: 925.4176 .. 1088.2687 cents 1125/1024, 162.8511 cents 16384/16875,-51.1199 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 : 75/64, 274.5824 cents Average / Smallest step : 1.534293 Largest / Average step : 1.601731 Largest / Smallest step : 2.457525 Median interval of one step : 16/15, 111.7313 cents, amount: 4 Most common interval of one step : 16/15, 111.7313 cents, amount: 4 Least squares average step : 178.45186 cents, oct.: 1249.16302 cents Scale is not proper Scale has trivalence property Step pattern alph. order: ABBABCB Step pattern size order : LSSLSMS Scale is a Constant Structure, by a margin of 19.55257 cents Scale diversity : 1.413252 Lumma stability : 0.270944 Lumma impropriety factor : 0.085200 Rothenberg efficiency : 0.506122 redundancy: 0.493878 Efficiency x scale size : 3.542857 Number of different interval sizes : 18 = 3.00000 / class Number of one step interval sizes : 3 Highest interval variety : 3 Mean interval variety : 3.00000 = 3 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 2048/2025, 19.5526 cents Number of recognisable fifths : 4, average 701.9550 cents Scale is Lydian octave species of abstract tetrachord Best fifths form a closed circle Scale is fully connected 5-limit major-minor triadic Formal octave complements present : 1 = 14.2857% Scale is differentially coherent in interval classes 1 and 2 combined Scale is n-1 differentially coherent in interval class 2 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 : 4 Inversional symmetry on degrees : 2 Inversional symmetry on intervals : 5-6 -- Rational properties -- Prime limit : 5 Odd number limit : 225 (O: 225 U: 225) Highest odd numerator or denominator: 75 Scale harmonicity : 0.038265 Average absolute harmonicity : 0.233229 Specific harmonicity : 0.194647 Fundamental : 1/192, -7.5850 octaves, 1.3626 Hz. Guide tone : 300, 8.2288 octaves, 78487.670 Hz. Exponens Consonantiae : 5.760000E+04, 15.81378 octaves Euler's gradus suavitatis : 21 Sum of Mann's harmonic distance : 115.0, average 16.42857 Mersenne's string divisions : 66062808 Sum of van Prooijen's expressibility: 6.32415, average 0.90345 Sum of Tenney's harmonic distance : 12.21981, average 1.74569 Vogel's harmonic complexity : 12.28571 Wille's k value : 112 Wilson's harmonic complexity : 16 Rectangular lattice diameter : 4 Triangular lattice diameter : 4 Lattice compactness : 102.71809, average 3.66850 Lattice compactness (without 2's) : 41.32248, average 1.47580 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -6 .. 2 -1.71429 18 1 3: -1 .. 1 0.00000 4 1 5: 0 .. 2 1.00000 7 5 Average exponent except 2's : 7 / 7 = 1.00000 Average absolute exponent except 2's: 11 / 7 = 1.57143 Scale is weakly epimorphic with val: <7 11 15| Scale is JI-epimorphic with val: <7 11 16| = patent Scale is weakly epimorphic with val: <7 12 17|
FIT/MODE
7: 1 1 1 1 1 1 1 SP B ME I SD: 59.3432 c. M: 103.1539 c. Lebeng 8: 2 1 0 2 1 1 1 N D3 SD: 36.3150 c. M:-63.6863 c. 9: 2 1 1 2 1 1 1 P M ME S SD: 27.0176 c. M:-48.9746 c. 10: 2 1 1 2 1 2 1 P M ME S SD: 31.6672 c. M: 52.6274 c. Sethares Neutral, Beatles-7, Dichotic-7 12: 3 1 1 3 1 2 1 N T3 S SD: 16.7704 c. M:-27.3726 c. 19: 4 2 2 4 2 3 2 P T3 S SD: 12.0593 c. M: 21.9508 c. 22: 5 2 2 5 2 4 2 N T3 S SD: 6.5211 c. M: 11.6314 c. 31: 7 3 3 7 3 5 3 N T3 S SD: 3.7394 c. M:-5.9639 c. 53: 12 5 5 12 5 9 5 N T3 S SD: 1.7813 c. M: 2.8843 c. 65: 15 6 6 15 6 11 6 N T3 S SD: 1.6606 c. M:-2.7572 c. 87: 20 8 8 20 8 15 8 N T3 S SD: 1.0959 c. M: 1.6001 c. 118: 27 11 11 27 11 20 11 N T3 S SD: 0.2124 c. M:-0.3870 c. 323: 74 30 30 74 30 55 30 N T3 S SD: 0.1998 c. M:-0.3402 c. 376: 86 35 35 86 35 64 35 N T3 S SD: 0.1877 c. M: 0.3162 c. 441: 101 41 41 101 41 75 41 N T3 S SD: 0.1358 c. M:-0.2475 c. 494: 113 46 46 113 46 84 46 N T3 S SD: 0.0970 c. M: 0.1578 c. 612: 140 57 57 140 57 104 57 N T3 S SD: 0.0480 c. M: 0.0784 c. 730: 167 68 68 167 68 124 68 N T3 S SD: 0.0359 c. M: 0.0619 c. 1053: 241 98 98 241 98 179 98 N T3 S SD: 0.0364 c. M:-0.0614 c. 1171: 268 109 109 268 109 199 109 N T3 S SD: 0.0313 c. M:-0.0546 c. 1665: 381 155 155 381 155 283 155 N T3 S SD: 0.0197 c. M: 0.0344 c. 1783: 408 166 166 408 166 303 166 N T3 S SD: 0.0068 c. M:-0.0110 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: 35.3362 cents 3 x x 4 x 5 x 6 S SD: 0.0000 cents 4 x 5 x 6:7 x 8 S SD: 27.5333 cents 5:6 x 7:8 x 9:10 S SD: 24.9006 cents 6:7 x 8:9:10:11:12 S SD: 13.5069 cents 7:8:9 x 11:12:13:14 SD: 13.9574 cents 8:9:10:11 x 13:15:16 SD: 16.4602 cents 9 x 11:12:14:15:17:18 SD: 6.9450 cents 10:12 x 13:16:17:19:20 SD: 13.9383 cents 11:13:14:15:17:18:21:22 SD: 10.1454 cents 12:14:15:16:19:20:23:24 SD: 6.4415 cents 16:19:20:21:25:27:30:32 SD: 5.9448 cents 18:21:23:24:28:30:34:36 SD: 5.9419 cents 21:25:26:28:33:35:39:42 SD: 5.3253 cents 23:27:29:31:36:38:43:46 SD: 4.1575 cents 24:28:30:32:38:40:45:48 SD: 3.4561 cents 33:39:41:44:52:55:62:66 SD: 3.3530 cents 35:41:44:47:55:58:66:70 SD: 3.3278 cents 36:42:45:48:56:60:68:72 SD: 2.3997 cents 39:46:49:52:61:65:73:78 SD: 2.0995 cents 48:56:60:64:75:80:90:96 SD: 1.1016 cents 75:88:94:100:117:125:141:150 SD: 1.0579 cents 87:102:109:116:136:145:163:174 SD: 0.6199 cents 105:123:131:140:164:175:197:210 SD: 0.5145 cents 135:158:169:180:211:225:253:270 SD: 0.5053 cents 144:169:180:192:225:240:270:288 SD: 0.3661 cents 192:225:240:256:300:320:360:384 SD: 0.0000 cents