Scala analysis: xen10-wilson-purvi-02b-01
Purvi modulation 1 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: 16/15 111.731285 minor diatonic semitone 2: 6/5 315.641287 minor third 3: 32/25 427.372572 classic diminished fourth 4: 3/2 701.955001 perfect fifth 5: 8/5 813.686286 minor sixth 6: 128/75 925.417571 diminished seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 75/64 274.5824 classic augmented second 1: 16/15 111.7313 minor diatonic semitone 2: 9/8 203.9100 major whole tone 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: 16/15 111.7313 minor diatonic semitone 7: 75/64 274.5824 classic augmented second
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 : 111.7 315.6 427.4 702.0 813.7 925.4 1200.0 111.7 : 203.9 315.6 590.2 702.0 813.7 1088.3 1200.0 315.6 : 111.7 386.3 498.0 609.8 884.4 996.1 1200.0 427.4 : 274.6 386.3 498.0 772.6 884.4 1088.3 1200.0 702.0 : 111.7 223.5 498.0 609.8 813.7 925.4 1200.0 813.7 : 111.7 386.3 498.0 702.0 813.7 1088.3 1200.0 925.4 : 274.6 386.3 590.2 702.0 976.5 1088.3 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 : 163.24206 cents, oct.: 1142.69442 cents Scale is not proper Scale has trivalence property Step pattern alph. order: ABACAAC Step pattern size order : SMSLSSL 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 Hypophrygian octave species of abstract tetrachord Best fifths form a closed circle Scale is fully connected 5-limit major-minor triadic Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 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 : 3 Inversional symmetry on degrees : 5 Inversional symmetry on intervals : 1-2 -- 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 : 0.236524 Specific harmonicity : 0.194647 Fundamental : 1/150, -7.2288 octaves, 1.7442 Hz. Guide tone : 384, 8.5850 octaves, 100464.217 Hz. Exponens Consonantiae : 5.760000E+04, 15.81378 octaves Euler's gradus suavitatis : 21 Sum of Mann's harmonic distance : 161.5, average 23.07143 Mersenne's string divisions : 256471215 Sum of van Prooijen's expressibility: 6.32415, average 0.90345 Sum of Tenney's harmonic distance : 13.42393, average 1.91770 Vogel's harmonic complexity : 11.85714 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: -1 .. 7 2.85714 22 1 3: -1 .. 1 0.00000 4 1 5: -2 .. 0 -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 0 2 N T3 SD: 51.6472 c. M:-86.9131 c. 8: 1 1 1 2 0 1 2 N D3 SD: 36.3150 c. M: 63.6863 c. 9: 1 1 1 2 1 1 2 P M ME S SD: 27.0176 c. M: 48.9746 c. 10: 1 2 1 2 1 1 2 P M ME S SD: 31.6672 c. M:-52.6274 c. 12: 1 2 1 3 1 1 3 N T3 S SD: 16.7704 c. M: 27.3726 c. 19: 2 3 2 4 2 2 4 P T3 S SD: 12.0593 c. M:-21.9508 c. 22: 2 4 2 5 2 2 5 N T3 S SD: 6.5211 c. M:-11.6314 c. 31: 3 5 3 7 3 3 7 N T3 S SD: 3.7394 c. M: 5.9639 c. 53: 5 9 5 12 5 5 12 N T3 S SD: 1.7813 c. M:-2.8843 c. 65: 6 11 6 15 6 6 15 N T3 S SD: 1.6606 c. M: 2.7572 c. 87: 8 15 8 20 8 8 20 N T3 S SD: 1.0959 c. M:-1.6001 c. 118: 11 20 11 27 11 11 27 N T3 S SD: 0.2124 c. M: 0.3870 c. 323: 30 55 30 74 30 30 74 N T3 S SD: 0.1998 c. M: 0.3402 c. 376: 35 64 35 86 35 35 86 N T3 S SD: 0.1877 c. M:-0.3162 c. 441: 41 75 41 101 41 41 101 N T3 S SD: 0.1358 c. M: 0.2475 c. 494: 46 84 46 113 46 46 113 N T3 S SD: 0.0970 c. M:-0.1578 c. 612: 57 104 57 140 57 57 140 N T3 S SD: 0.0480 c. M:-0.0784 c. 730: 68 124 68 167 68 68 167 N T3 S SD: 0.0359 c. M:-0.0619 c. 1053: 98 179 98 241 98 98 241 N T3 S SD: 0.0364 c. M: 0.0614 c. 1171: 109 199 109 268 109 109 268 N T3 S SD: 0.0313 c. M: 0.0546 c. 1665: 155 283 155 381 155 155 381 N T3 S SD: 0.0197 c. M:-0.0344 c. 1783: 166 303 166 408 166 166 408 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: 0.0000 cents 3 x x 4 x x 5:6 S SD: 27.2446 cents 4 x x 5:6 x 7:8 S SD: 14.9376 cents 5 x 6 x x 8:9:10 S SD: 23.0447 cents 6 x 7:8:9 x 10:12 S SD: 19.0357 cents 7 x 8:9 x 11:12:14 SD: 18.1403 cents 8:9 x 10:12:13:14:16 S SD: 18.8470 cents 9:10:11:12 x 14:15:18 S SD: 20.4550 cents 10:11:12:13:15:16:17:20 SD: 8.5766 cents 14:15:17:18:21:22:24:28 SD: 5.6627 cents 20:21:24:26:30:32:34:40 SD: 5.5507 cents 24:26:29:31:36:38:41:48 SD: 5.4252 cents 25:27:30:32:38:40:43:50 SD: 4.8861 cents 26:28:31:33:39:42:44:52 SD: 4.7397 cents 28:30:34:36:42:45:48:56 SD: 3.6635 cents 30:32:36:38:45:48:51:60 SD: 2.7647 cents 40:43:48:51:60:64:68:80 SD: 2.3619 cents 44:47:53:56:66:70:75:88 SD: 2.2503 cents 50:53:60:64:75:80:85:100 SD: 1.8279 cents 58:62:70:74:87:93:99:116 SD: 1.7939 cents 60:64:72:77:90:96:102:120 SD: 1.1622 cents 75:80:90:96:113:120:128:150 SD: 1.0968 cents 90:96:108:115:135:144:154:180 SD: 0.7736 cents 120:128:144:154:180:192:205:240 SD: 0.6870 cents 136:145:163:174:204:218:232:272 SD: 0.5806 cents 150:160:180:192:225:240:256:300 SD: 0.0000 cents