Scala analysis: xen10-wilson-purvi-08c-03
Purvi modulation 3 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: 21/11 1119.462965 undecimal major seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 22/21 80.5370 undecimal minor semitone 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: 7/6 266.8709 septimal minor third 7: 22/21 80.5370 undecimal minor semitone
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 1119.5 1200.0 150.6 : 203.9 347.4 551.3 702.0 968.8 1049.4 1200.0 354.5 : 143.5 347.4 498.0 764.9 845.5 996.1 1200.0 498.0 : 203.9 354.5 621.4 702.0 852.6 1056.5 1200.0 702.0 : 150.6 417.5 498.0 648.7 852.6 996.1 1200.0 852.6 : 266.9 347.4 498.0 702.0 845.5 1049.4 1200.0 1119.5: 80.5 231.2 435.1 578.6 782.5 933.1 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 : 175.29589 cents, oct.: 1227.07121 cents Scale is not proper Step pattern alph. order: ABCBADE Step pattern size order : CDBDCEA 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 minor thirds 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 2 and 3 combined Scale is n-1 differentially coherent in interval class 3 Scale is n-1 differentially coherent in interval classes 4 and 5 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 : 0.234354 Specific harmonicity : 0.124940 Fundamental : 1/66, -6.0444 octaves, 3.9640 Hz. Guide tone : 756, 9.5622 octaves, 197788.927 Hz. Exponens Consonantiae : 4.989600E+04, 15.60664 octaves Euler's gradus suavitatis : 28 Sum of Mann's harmonic distance : 74.0, average 10.57143 Mersenne's string divisions : 15687840 Sum of van Prooijen's expressibility: 5.79061, average 0.82723 Sum of Tenney's harmonic distance : 11.71300, average 1.67329 Vogel's harmonic complexity : 13.28571 Wille's k value : 49 Wilson's harmonic complexity : 30 Rectangular lattice diameter : 6 Triangular lattice diameter : 4 Lattice compactness : 68.63086, average 2.45110 Lattice compactness (without 2's) : 49.77494, average 1.77768 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -1 .. 2 0.57143 8 1 3: -1 .. 3 1.00000 9 2 7: 0 .. 1 0.14286 1 11: -1 .. 0 -0.57143 4 4 Average exponent except 2's : 4 / 7 = 0.57143 Average absolute exponent except 2's: 14 / 7 = 2.00000 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 2 0 N T3 SD: 32.9575 c. M:-80.5370 c. 8: 1 1 1 2 1 1 1 P M ME I SD: 45.7712 c. M: 69.4630 c. 10: 1 2 1 2 1 2 1 P M ME I SD: 21.8289 c. M: 39.4630 c. 13: 2 2 1 3 1 3 1 N D3 S SD: 25.7748 c. M:-36.5065 c. 14: 2 2 2 2 2 3 1 P T3 SD: 12.7838 c. M:-20.7915 c. 17: 2 3 2 3 2 4 1 N SD: 6.0062 c. M:-9.9488 c. 31: 4 5 4 5 4 7 2 N SD: 4.1397 c. M: 6.1600 c. 41: 5 7 5 7 5 9 3 N SD: 3.7291 c. M: 7.2678 c. 48: 6 8 6 8 6 11 3 N SD: 3.0730 c. M:-5.5370 c. 58: 7 10 7 10 7 13 4 N SD: 3.1565 c. M: 5.8095 c. 70: 9 12 8 12 9 15 5 N SD: 3.6290 c. M:-5.4529 c. 72: 9 12 9 12 9 16 5 N SD: 2.4861 c. M: 4.5471 c. 87: 11 15 10 15 11 19 6 N SD: 2.1984 c. M:-4.0736 c. 89: 11 15 11 15 11 20 6 N SD: 2.1625 c. M: 3.9853 c. 94: 12 16 11 16 12 21 6 N SD: 2.3288 c. M:-3.9413 c. 101: 13 17 12 17 13 22 7 N SD: 2.2385 c. M:-3.8184 c. 104: 13 18 12 18 13 23 7 N SD: 1.6509 c. M:-3.1452 c. 118: 15 20 14 20 15 26 8 N SD: 1.1378 c. M:-1.9053 c. 135: 17 23 16 23 17 30 9 N SD: 0.5635 c. M:-1.0085 c. 207: 26 35 25 35 26 46 14 N SD: 0.5258 c. M: 0.9239 c. 294: 37 50 35 50 37 65 20 N SD: 0.5196 c. M: 1.0956 c. 301: 38 51 36 51 38 67 20 N SD: 0.5271 c. M:-0.8580 c. 311: 39 53 37 53 39 69 21 N SD: 0.3051 c. M: 0.4919 c. 342: 43 58 41 58 43 76 23 N SD: 0.1660 c. M:-0.2401 c. 477: 60 81 57 81 60 106 32 N SD: 0.1647 c. M:-0.3063 c. 494: 62 84 59 84 62 110 33 N SD: 0.1533 c. M:-0.3751 c. 653: 82 111 78 111 82 145 44 N SD: 0.1368 c. M: 0.3205 c. 670: 84 114 80 114 84 149 45 N SD: 0.1102 c. M: 0.1893 c. 701: 88 119 84 119 88 156 47 N SD: 0.1030 c. M: 0.1961 c. 836: 105 142 100 142 105 186 56 N SD: 0.0710 c. M:-0.1543 c. 1012: 127 172 121 172 127 225 68 N SD: 0.0422 c. M: 0.0954 c. 1371: 172 233 164 233 172 305 92 N SD: 0.0509 c. M: 0.0900 c. 1489: 187 253 178 253 187 331 100 N SD: 0.0450 c. M:-0.0681 c. 1506: 189 256 180 256 189 335 101 N SD: 0.0358 c. M:-0.0589 c. 1713: 215 291 205 291 215 381 115 N SD: 0.0416 c. M: 0.0812 c. 1848: 232 314 221 314 232 411 124 N SD: 0.0092 c. M:-0.0176 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 6:7 x 8 S SD: 30.1241 cents 5 x 6:7 x 8 x 10 S SD: 25.2017 cents 6 x 7:8:9:10:11:12 S SD: 19.4438 cents 7:8 x 9 x 11:13:14 SD: 26.5672 cents 8:9:10:11:12:13:15:16 SD: 12.6198 cents 9:10:11:12:14:15:17:18 SD: 11.4040 cents 10:11:12:13:15:16:19:20 SD: 10.3245 cents 12:13:15:16:18:20:23:24 SD: 6.7170 cents 18:20:22:24:27:29:34:36 SD: 6.5842 cents 20:22:25:27:30:33:38:40 SD: 6.3129 cents 21:23:26:28:32:34:40:42 SD: 5.2993 cents 22:24:27:29:33:36:42:44 SD: 2.8265 cents 30:33:37:40:45:49:57:60 SD: 2.7020 cents 36:39:44:48:54:59:69:72 SD: 2.2617 cents 44:48:54:59:66:72:84:88 SD: 1.4012 cents 66:72:81:88:99:108:126:132 SD: 0.0000 cents