Scala analysis: xen10-wilson-purvi-08a-01
Purvi modulation 1 from Figure 8a, Ptolemy (12/11 22/21 7/6), (7/6 12/11 22/21)
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: 27/22 354.547060 neutral third, Zalzal wosta of al-Farabi 3: 9/7 435.084095 septimal major third, BP third 4: 3/2 701.955001 perfect fifth 5: 18/11 852.592059 undecimal neutral sixth 6: 12/7 933.129094 septimal major sixth 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 7/6 266.8709 septimal minor third 1: 9/8 203.9100 major whole tone 2: 12/11 150.6371 undecimal neutral second, 3/4-tone 3: 22/21 80.5370 undecimal minor semitone 4: 7/6 266.8709 septimal minor third 5: 12/11 150.6371 undecimal neutral second, 3/4-tone 6: 22/21 80.5370 undecimal minor semitone 7: 7/6 266.8709 septimal minor third
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 22/21 80.537 cents undecimal minor semitone 1: 2 12/11 150.637 cents undecimal neutral second, 3/4-tone 1: 1 9/8 203.910 cents major whole tone 1: 2 7/6 266.871 cents septimal minor third 2: 2 8/7 231.174 cents septimal whole tone 2: 2 11/9 347.408 cents undecimal neutral third 2: 1 27/22 354.547 cents neutral third, Zalzal wosta of al-Farabi 2: 1 14/11 417.508 cents undecimal diminished fourth or major third 2: 1 21/16 470.781 cents narrow fourth 3: 1 9/7 435.084 cents septimal major third, BP third 3: 4 4/3 498.045 cents perfect fourth 3: 1 11/8 551.318 cents undecimal semi-augmented fourth 3: 1 63/44 621.418 cents 4: 1 88/63 578.582 cents 4: 1 16/11 648.682 cents undecimal semi-diminished fifth 4: 4 3/2 701.955 cents perfect fifth 4: 1 14/9 764.916 cents septimal minor sixth 5: 1 32/21 729.219 cents wide fifth 5: 1 11/7 782.492 cents undecimal augmented fifth 5: 1 44/27 845.453 cents neutral sixth 5: 2 18/11 852.592 cents undecimal neutral sixth 5: 2 7/4 968.826 cents harmonic seventh 6: 2 12/7 933.129 cents septimal major sixth 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 2 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 6: 2 21/11 1119.463 cents undecimal major seventh Highest number of different intervals for one interval class: 5 Average number of different intervals per interval class: 4.33333 = 13/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 203.9 354.5 435.1 702.0 852.6 933.1 1200.0 203.9 : 150.6 231.2 498.0 648.7 729.2 996.1 1200.0 354.5 : 80.5 347.4 498.0 578.6 845.5 1049.4 1200.0 435.1 : 266.9 417.5 498.0 764.9 968.8 1119.5 1200.0 702.0 : 150.6 231.2 498.0 702.0 852.6 933.1 1200.0 852.6 : 80.5 347.4 551.3 702.0 782.5 1049.4 1200.0 933.1 : 266.9 470.8 621.4 702.0 968.8 1119.5 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 80.5370 .. 266.8709 cents 49/44, 186.3339 cents 22/21, 80.5370 cents 2: 231.1741 .. 470.7809 cents 147/128, 239.6068 cents 48/49,-35.6968 cents 3: 435.0841 .. 621.4180 cents 49/44, 186.3339 cents 48/49,-35.6968 cents 4: 578.5820 .. 764.9159 cents 49/44, 186.3339 cents 3872/3969,-42.8359 cents 5: 729.2191 .. 968.8259 cents 147/128, 239.6068 cents 48/49,-35.6968 cents 6: 933.1291 .. 1119.4630 cents 49/44, 186.3339 cents 48/49,-35.6968 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 22/21, 80.5370 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 7/6, 266.8709 cents Average / Smallest step : 2.128568 Largest / Average step : 1.556747 Largest / Smallest step : 3.313642 Median interval of one step : 12/11, 150.6371 cents, amount: 2 Least squares average step : 166.34151 cents, oct.: 1164.39056 cents Scale is not proper Step pattern alph. order: ABCDBCD Step pattern size order : CBADBAD Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 17.57613 cents Scale diversity : 1.575227 Lumma stability : 0.134228 Lumma impropriety factor : 0.154686 Rothenberg efficiency : 0.421769 redundancy: 0.578231 Efficiency x scale size : 2.952381 Number of different interval sizes : 26 = 4.33333 / class Number of one step interval sizes : 4 Highest interval variety : 5 Mean interval variety : 4.33333 = 13/3 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 243/242, 7.1391 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 4 Number of recognisable fifths : 4, average 701.9550 cents Scale is Hypodorian octave species of abstract tetrachord All steps are superparticular (epimoric) 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 class 1 Scale is differentially coherent in interval classes 3 and 4 combined -- Rational properties -- Prime limit : 11 Odd number limit : 63 (O: 63 U: 63) Highest odd numerator or denominator: 27 Scale harmonicity : 0.024115 Average absolute harmonicity : 0.228744 Specific harmonicity : 0.131702 Fundamental : 1/616, -9.2668 octaves, 0.4247 Hz. Guide tone : 108, 6.7549 octaves, 28255.561 Hz. Exponens Consonantiae : 6.652800E+04, 16.02167 octaves Euler's gradus suavitatis : 28 Sum of Mann's harmonic distance : 69.0, average 9.85714 Mersenne's string divisions : 110155920 Sum of van Prooijen's expressibility: 5.70346, average 0.81478 Sum of Tenney's harmonic distance : 11.73059, average 1.67580 Vogel's harmonic complexity : 20.42857 Wille's k value : 346 Wilson's harmonic complexity : 27 Rectangular lattice diameter : 5 Triangular lattice diameter : 3 Lattice compactness : 74.16306, average 2.64868 Lattice compactness (without 2's) : 45.31798, average 1.61850 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -3 .. 2 -0.14286 9 1 3: 0 .. 3 1.57143 11 2 7: -1 .. 0 -0.28571 2 2 11: -1 .. 0 -0.28571 2 2 Average exponent except 2's : 7 / 7 = 1.00000 Average absolute exponent except 2's: 15 / 7 = 2.14286 Scale is weakly epimorphic with val: <7 10 22 26| Scale is JI-epimorphic with val: <7 11 19 24| Scale is weakly epimorphic with val: <7 11 20 23| Scale is weakly epimorphic with val: <7 12 22 23|
FIT/MODE
7: 1 1 0 2 1 0 2 N T3 SD: 47.4430 c. M: 92.2270 c. 8: 1 1 1 2 1 0 2 N D3 SD: 40.9835 c. M: 54.5471 c. 10: 2 1 1 2 1 1 2 P D3 I SD: 25.5116 c. M:-44.9159 c. 13: 2 2 1 3 1 1 3 N SD: 21.3733 c. M:-36.5065 c. 14: 2 2 1 3 2 1 3 P T3 SD: 15.1809 c. M: 32.4814 c. 17: 3 2 1 4 2 1 4 N SD: 8.3111 c. M: 15.4820 c. 31: 5 4 2 7 4 2 7 N SD: 6.2788 c. M: 10.3616 c. 41: 7 5 3 9 5 3 9 N SD: 2.7837 c. M:-3.9403 c. 58: 10 7 4 13 7 4 13 N SD: 2.4639 c. M: 4.3162 c. 72: 12 9 5 16 9 5 16 N SD: 2.6622 c. M: 4.5471 c. 77: 13 10 5 17 10 5 17 N SD: 2.4904 c. M:-4.5508 c. 89: 15 11 6 20 11 6 20 N SD: 2.6782 c. M: 3.9853 c. 94: 16 12 6 21 12 6 21 N SD: 1.6280 c. M:-2.8997 c. 118: 20 15 8 26 15 8 26 N SD: 1.5068 c. M:-2.4641 c. 135: 23 17 9 30 17 9 30 N SD: 0.5590 c. M:-1.0085 c. 287: 49 36 19 64 36 19 64 N SD: 0.6118 c. M:-0.9680 c. 301: 51 38 20 67 38 20 67 N SD: 0.4077 c. M: 0.5877 c. 342: 58 43 23 76 43 23 76 N SD: 0.1966 c. M: 0.4012 c. 477: 81 60 32 106 60 32 106 N SD: 0.1554 c. M:-0.2381 c. 535: 91 67 36 119 67 36 119 N SD: 0.1442 c. M: 0.2556 c. 670: 114 84 45 149 84 45 149 N SD: 0.1306 c. M:-0.2691 c. 836: 142 105 56 186 105 56 186 N SD: 0.0824 c. M: 0.1559 c. 971: 165 122 65 216 122 65 216 N SD: 0.0826 c. M:-0.1388 c. 1012: 172 127 68 225 127 68 225 N SD: 0.0490 c. M:-0.0938 c. 1371: 233 172 92 305 172 92 305 N SD: 0.0580 c. M: 0.0875 c. 1506: 256 189 101 335 189 101 335 N SD: 0.0421 c. M:-0.0741 c. 1713: 291 215 115 381 215 115 381 N SD: 0.0502 c. M: 0.0812 c. 1848: 314 232 124 411 232 124 411 N SD: 0.0106 c. M: 0.0192 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: 23.5070 cents 4 x 5 x 6 x 7:8 S SD: 11.9462 cents 5 x 6 x x 8:9:10 S SD: 25.2017 cents 6:7 x 8:9:10 x 12 S SD: 18.9074 cents 7:8 x 9 x 11:12:14 SD: 15.0431 cents 8:9:10 x 12:13:14:16 S SD: 8.2140 cents 9:10:11:12:14:15 x 18 S SD: 16.2025 cents 10:11:12:13:15:16:17:20 SD: 10.2188 cents 11:12 x 14:17:18:19:22 SD: 12.8978 cents 12:14:15 x 18:20:21:24 S SD: 14.1976 cents 13:15:16:17:20:21:22:26 SD: 10.8097 cents 14:16:17:18:21:23:24:28 SD: 4.8012 cents 22:25:27:28:33:36:38:44 SD: 3.9959 cents 31:35:38:40:47:51:53:62 SD: 3.2942 cents 32:36:39:41:48:52:55:64 SD: 2.6636 cents 39:44:48:50:59:64:67:78 SD: 2.5869 cents 48:54:59:62:72:79:82:96 SD: 2.0560 cents 49:55:60:63:74:80:84:98 SD: 1.9383 cents 56:63:69:72:84:92:96:112 SD: 1.3852 cents 66:74:81:85:99:108:113:132 SD: 0.9831 cents 80:90:98:103:120:131:137:160 SD: 0.6511 cents 88:99:108:113:132:144:151:176 SD: 0.3904 cents 154:173:189:198:231:252:264:308 SD: 0.3571 cents 168:189:206:216:252:275:288:336 SD: 0.2330 cents 256:288:314:329:384:419:439:512 SD: 0.2034 cents 360:405:442:463:540:589:617:720 SD: 0.1446 cents 448:504:550:576:672:733:768:896 SD: 0.0873 cents 528:594:648:679:792:864:905:1056 SD: 0.0651 cents