Scala analysis: xen10-wilson-purvi-08c-02
Purvi modulation 2 from Figure 8c, Ptolemy (22/21 7/6 12/11), (12/11 7/6 22/21)
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 12/11 150.637059 undecimal neutral second, 3/4-tone 2: 27/22 354.547060 neutral third, Zalzal wosta of al-Farabi 3: 4/3 498.044999 perfect fourth 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: 12/11 150.6371 undecimal neutral second, 3/4-tone 2: 9/8 203.9100 major whole tone 3: 88/81 143.4979 2nd undecimal neutral second 4: 9/8 203.9100 major whole tone 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: 1 22/21 80.537 cents undecimal minor semitone 1: 1 88/81 143.498 cents 2nd undecimal neutral second 1: 2 12/11 150.637 cents undecimal neutral second, 3/4-tone 1: 2 9/8 203.910 cents major whole tone 1: 1 7/6 266.871 cents septimal minor third 2: 1 8/7 231.174 cents septimal whole tone 2: 3 11/9 347.408 cents undecimal neutral third 2: 2 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 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 11/7 782.492 cents undecimal augmented fifth 5: 2 44/27 845.453 cents neutral sixth 5: 3 18/11 852.592 cents undecimal neutral sixth 5: 1 7/4 968.826 cents harmonic seventh 6: 1 12/7 933.129 cents septimal major sixth 6: 2 16/9 996.090 cents Pythagorean minor seventh 6: 2 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 6: 1 81/44 1056.502 cents 2nd undecimal neutral seventh 6: 1 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 : 150.6 354.5 498.0 702.0 852.6 933.1 1200.0 150.6 : 203.9 347.4 551.3 702.0 782.5 1049.4 1200.0 354.5 : 143.5 347.4 498.0 578.6 845.5 996.1 1200.0 498.0 : 203.9 354.5 435.1 702.0 852.6 1056.5 1200.0 702.0 : 150.6 231.2 498.0 648.7 852.6 996.1 1200.0 852.6 : 80.5 347.4 498.0 702.0 845.5 1049.4 1200.0 933.1 : 266.9 417.5 621.4 764.9 968.8 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 .. 417.5080 cents 49/44, 186.3339 cents 48/49,-35.6968 cents 3: 435.0841 .. 621.4180 cents 49/44, 186.3339 cents 99/98, 17.5761 cents 4: 578.5820 .. 764.9159 cents 49/44, 186.3339 cents 3872/3969,-42.8359 cents 5: 782.4920 .. 968.8259 cents 49/44, 186.3339 cents 99/98, 17.5761 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 : 167.31015 cents, oct.: 1171.17105 cents Scale is not proper Step pattern alph. order: ABCBADE Step pattern size order : CDBDCAE Scale is a Constant Structure, by a margin of 17.57613 cents Scale diversity : 1.559442 Lumma stability : 0.163522 Lumma impropriety factor : 0.095191 Rothenberg efficiency : 0.423129 redundancy: 0.576871 Efficiency x scale size : 2.961905 Number of different interval sizes : 26 = 4.33333 / class Number of one step interval sizes : 5 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 Most common triad is 0.0 498.045 701.955 cents, amount: 2 Number of recognisable fifths : 4, average 701.9550 cents Best fifths form a closed circle 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 = 42.8571% Scale is n-1 differentially coherent in interval class 1 Scale is n-1 differentially coherent in interval classes 3 and 4 combined -- Rational properties -- Prime limit : 11 Odd number limit : 81 (O: 81 U: 81) Highest odd numerator or denominator: 27 Scale harmonicity : 0.023734 Average absolute harmonicity : 0.239319 Specific harmonicity : 0.129016 Fundamental : 1/462, -8.8517 octaves, 0.5663 Hz. Guide tone : 108, 6.7549 octaves, 28255.561 Hz. Exponens Consonantiae : 4.989600E+04, 15.60664 octaves Euler's gradus suavitatis : 28 Sum of Mann's harmonic distance : 67.5, average 9.64286 Mersenne's string divisions : 9314655 Sum of van Prooijen's expressibility: 5.31349, average 0.75907 Sum of Tenney's harmonic distance : 11.27367, average 1.61052 Vogel's harmonic complexity : 20.14286 Wille's k value : 346 Wilson's harmonic complexity : 30 Rectangular lattice diameter : 6 Triangular lattice diameter : 4 Lattice compactness : 70.45396, average 2.51621 Lattice compactness (without 2's) : 49.45710, average 1.76633 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -1 .. 2 0.85714 10 1 3: -1 .. 3 1.00000 9 2 7: -1 .. 0 -0.14286 1 1 11: -1 .. 0 -0.42857 3 3 Average exponent except 2's : 3 / 7 = 0.42857 Average absolute exponent except 2's: 13 / 7 = 1.85714 Scale is weakly epimorphic with val: <7 10 18 22| Scale is JI-epimorphic with val: <7 11 19 24| Scale is weakly epimorphic with val: <7 12 22 23|
FIT/MODE
7: 1 1 1 1 1 0 2 N T3 SD: 31.3758 c. M: 75.9862 c. 9: 1 2 1 1 1 1 2 N D3 S SD: 32.9976 c. M: 52.5921 c. 10: 1 2 1 2 1 1 2 P M ME S SD: 18.8990 c. M: 30.6371 c. 14: 2 2 2 2 2 1 3 P T3 SD: 13.1573 c. M:-20.7915 c. 17: 2 3 2 3 2 1 4 N SD: 7.4951 c. M: 15.4820 c. 24: 3 4 3 4 3 2 5 P SD: 6.7620 c. M:-16.8709 c. 31: 4 5 4 5 4 2 7 N SD: 4.2598 c. M: 6.1600 c. 41: 5 7 5 7 5 3 9 N SD: 2.8402 c. M: 4.2956 c. 58: 7 10 7 10 7 4 13 N SD: 3.1441 c. M: 5.8095 c. 72: 9 12 9 12 9 5 16 N SD: 2.2515 c. M: 4.5471 c. 89: 11 15 11 15 11 6 20 N SD: 2.4024 c. M: 3.9853 c. 94: 12 16 11 16 12 6 21 N SD: 1.8480 c. M:-2.8997 c. 104: 13 18 12 18 13 7 23 N SD: 1.7416 c. M:-3.1452 c. 111: 14 19 13 19 14 7 25 N SD: 1.6989 c. M: 3.3994 c. 118: 15 20 14 20 15 8 26 N SD: 1.4374 c. M:-2.4641 c. 135: 17 23 16 23 17 9 30 N SD: 0.5313 c. M:-1.0085 c. 176: 22 30 21 30 22 12 39 N SD: 0.4833 c. M:-0.9618 c. 207: 26 35 25 35 26 14 46 N SD: 0.4765 c. M: 0.9239 c. 287: 36 49 34 49 36 19 64 N SD: 0.5172 c. M:-0.8536 c. 301: 38 51 36 51 38 20 67 N SD: 0.4403 c. M:-0.8580 c. 311: 39 53 37 53 39 21 69 N SD: 0.3402 c. M:-0.6330 c. 342: 43 58 41 58 43 23 76 N SD: 0.1721 c. M:-0.2401 c. 477: 60 81 57 81 60 32 106 N SD: 0.1814 c. M:-0.3063 c. 494: 62 84 59 84 62 33 110 N SD: 0.1396 c. M: 0.3356 c. 670: 84 114 80 114 84 45 149 N SD: 0.1079 c. M: 0.1893 c. 701: 88 119 84 119 88 47 156 N SD: 0.1188 c. M: 0.1961 c. 836: 105 142 100 142 105 56 186 N SD: 0.0593 c. M: 0.1147 c. 1012: 127 172 121 172 127 68 225 N SD: 0.0351 c. M:-0.0725 c. 1506: 189 256 180 256 189 101 335 N SD: 0.0364 c. M: 0.0614 c. 1713: 215 291 205 291 215 115 381 N SD: 0.0421 c. M: 0.0812 c. 1848: 232 314 221 314 232 124 411 N SD: 0.0078 c. M:-0.0123 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: 10.5889 cents 4 x 5 x 6 x 7:8 S SD: 11.9462 cents 5 x 6:7 x 8:9:10 S SD: 26.3034 cents 6 x 7:8:9:10 x 12 S SD: 18.6507 cents 7:8 x 9 x 11:12:14 SD: 24.7906 cents 8:9:10:11:12:13:14:16 S SD: 12.8611 cents 9:10:11:12:14:15 x 18 S SD: 12.9457 cents 10:11:12:13:15:16:17:20 SD: 10.4635 cents 12:13:15:16:18:20:21:24 SD: 8.3764 cents 14:15:17:19:21:23:24:28 SD: 6.8486 cents 18:20:22:24:27:29:31:36 SD: 6.1428 cents 21:23:26:28:32:34:36:42 SD: 5.2694 cents 22:24:27:29:33:36:38:44 SD: 3.3872 cents 30:33:37:40:45:49:51:60 SD: 3.1922 cents 36:39:44:48:54:59:62:72 SD: 2.3369 cents 44:48:54:59:66:72:75:88 SD: 1.9873 cents 52:57:64:69:78:85:89:104 SD: 1.8841 cents 54:59:66:72:81:88:93:108 SD: 1.8790 cents 56:61:69:75:84:92:96:112 SD: 1.8078 cents 66:72:81:88:99:108:113:132 SD: 0.3125 cents 220:240:270:293:330:360:377:440 SD: 0.2964 cents 242:264:297:323:363:396:415:484 SD: 0.2692 cents 264:288:324:352:396:432:453:528 SD: 0.2341 cents 308:336:378:411:462:504:528:616 SD: 0.2007 cents 330:360:405:440:495:540:566:660 SD: 0.1249 cents 396:432:486:528:594:648:679:792 SD: 0.0520 cents 462:504:567:616:693:756:792:924 SD: 0.0000 cents