Scala analysis: xen10-wilson-purvi-08a-07
Purvi modulation 7 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: 7/6 266.870906 septimal minor third 2: 14/11 417.507964 undecimal diminished fourth or major third 3: 4/3 498.044999 perfect fourth 4: 14/9 764.915905 septimal minor sixth 5: 7/4 968.825906 harmonic seventh 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: 7/6 266.8709 septimal minor third 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: 9/8 203.9100 major whole tone 6: 12/11 150.6371 undecimal neutral second, 3/4-tone 7: 22/21 80.5370 undecimal minor semitone
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 : 266.9 417.5 498.0 764.9 968.8 1119.5 1200.0 266.9 : 150.6 231.2 498.0 702.0 852.6 933.1 1200.0 417.5 : 80.5 347.4 551.3 702.0 782.5 1049.4 1200.0 498.0 : 266.9 470.8 621.4 702.0 968.8 1119.5 1200.0 764.9 : 203.9 354.5 435.1 702.0 852.6 933.1 1200.0 968.8 : 150.6 231.2 498.0 648.7 729.2 996.1 1200.0 1119.5: 80.5 347.4 498.0 578.6 845.5 1049.4 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 : 182.97566 cents, oct.: 1280.82964 cents Scale is not proper Step pattern alph. order: ABCADBC Step pattern size order : DBADCBA 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 Phrygian 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: 21 Scale harmonicity : 0.024711 Average absolute harmonicity : 0.213368 Specific harmonicity : 0.130134 Fundamental : 1/396, -8.6294 octaves, 0.6607 Hz. Guide tone : 84, 6.3923 octaves, 21976.547 Hz. Exponens Consonantiae : 3.326400E+04, 15.02167 octaves Euler's gradus suavitatis : 27 Sum of Mann's harmonic distance : 57.0, average 8.14286 Mersenne's string divisions : 55255200 Sum of van Prooijen's expressibility: 5.48517, average 0.78360 Sum of Tenney's harmonic distance : 11.10212, average 1.58602 Vogel's harmonic complexity : 19.85714 Wille's k value : 346 Wilson's harmonic complexity : 27 Rectangular lattice diameter : 5 Triangular lattice diameter : 3 Lattice compactness : 64.82677, average 2.31524 Lattice compactness (without 2's) : 42.06590, average 1.50235 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -2 .. 2 0.28571 8 1 3: -2 .. 1 -0.42857 5 1 7: 0 .. 1 0.71429 5 3 11: -1 .. 0 -0.28571 2 2 Average exponent except 2's : 0 / 7 = 0.00000 Average absolute exponent except 2's: 12 / 7 = 1.71429 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 1 1 N T3 SD: 96.9334 c. M: 155.1879 c. 8: 2 1 0 2 1 1 1 N D3 SD: 45.1128 c. M: 69.4630 c. 9: 2 1 1 2 1 1 1 P M ME S SD: 31.2362 c. M: 52.7963 c. 10: 2 1 1 2 2 1 1 P D3 S SD: 33.8220 c. M: 57.5080 c. Pseudo Rast 11: 2 2 1 2 2 1 1 P D3 S SD: 29.1782 c. M: 48.6891 c. 12: 3 1 1 3 2 1 1 N SD: 23.8729 c. M:-35.0841 c. Chromatic Phrygian 14: 3 2 1 3 2 2 1 P T3 SD: 13.2255 c. M: 25.9688 c. 17: 4 2 1 4 3 2 1 N SD: 11.3426 c. M:-19.4094 c. 26: 6 3 2 6 4 3 2 N SD: 10.1799 c. M:-19.6995 c. 27: 6 3 2 6 5 3 2 N SD: 9.4711 c. M: 17.5080 c. 28: 6 4 2 6 5 3 2 N SD: 10.9233 c. M:-16.8884 c. 31: 7 4 2 7 5 4 2 N SD: 5.4702 c. M:-9.2776 c. 41: 9 5 3 9 7 5 3 N SD: 4.6207 c. M: 7.7519 c. 46: 10 6 3 10 8 6 3 N SD: 4.3164 c. M: 8.3942 c. 57: 13 7 4 12 10 7 4 N SD: 4.9863 c. M:-7.2182 c. 58: 13 7 4 13 10 7 4 N SD: 2.3479 c. M: 3.7149 c. 72: 16 9 5 16 12 9 5 N SD: 1.6953 c. M: 2.7963 c. 89: 20 11 6 20 15 11 6 N SD: 1.9202 c. M:-3.6234 c. 104: 23 13 7 23 18 13 7 N SD: 1.7695 c. M: 3.3774 c. 118: 26 15 8 26 20 15 8 N SD: 1.6648 c. M: 2.7242 c. 121: 27 15 8 27 21 15 8 N SD: 1.6489 c. M:-3.0749 c. 130: 29 16 9 29 22 16 9 N SD: 1.3889 c. M: 2.5399 c. 135: 30 17 9 30 23 17 9 N SD: 0.3164 c. M:-0.5370 c. 342: 76 43 23 76 58 43 23 N SD: 0.1980 c. M: 0.4049 c. 477: 106 60 32 106 81 60 32 N SD: 0.1467 c. M: 0.2724 c. 612: 136 77 41 136 104 77 41 N SD: 0.1538 c. M: 0.2100 c. 670: 149 84 45 149 114 84 45 N SD: 0.1170 c. M: 0.1945 c. 805: 179 101 54 179 137 101 54 N SD: 0.1155 c. M: 0.1954 c. 836: 186 105 56 186 142 105 56 N SD: 0.1234 c. M:-0.1954 c. 877: 195 110 59 195 149 110 59 N SD: 0.1050 c. M: 0.1927 c. 1012: 225 127 68 225 172 127 68 N SD: 0.0754 c. M: 0.1167 c. 1147: 255 144 77 255 195 144 77 N SD: 0.0720 c. M: 0.1382 c. 1371: 305 172 92 305 233 172 92 N SD: 0.0582 c. M:-0.1019 c. 1506: 335 189 101 335 256 189 101 N SD: 0.0526 c. M:-0.0984 c. 1641: 365 206 110 365 279 206 110 N SD: 0.0604 c. M:-0.0983 c. 1713: 381 215 115 381 291 215 115 N SD: 0.0283 c. M:-0.0578 c. 1848: 411 232 124 411 314 232 124 N SD: 0.0146 c. M:-0.0245 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: 31.4805 cents 3 x x 4 x 5 x 6 S SD: 28.1557 cents 4 x 5 x 6:7 x 8 S SD: 17.5662 cents 5:6 x 7:8:9 x 10 S SD: 23.8917 cents 6:7 x 8:9 x 11:12 SD: 18.8447 cents 7:8:9 x 11:12:13:14 SD: 12.3012 cents 9 x 11:12:14:16:17:18 SD: 12.9062 cents 10:12:13 x 16:18:19:20 SD: 15.4122 cents 11:13:14:15:17:19:21:22 SD: 7.3569 cents 12:14:15:16:19:21:23:24 SD: 6.3230 cents 13:15 x 17:20:23:25:26 SD: 8.1304 cents 18:21:23:24:28:32:34:36 SD: 4.8012 cents 23:27:29:31:36:40:44:46 SD: 4.4536 cents 24:28:31:32:37:42:46:48 SD: 4.3847 cents 25:29:32:33:39:44:48:50 SD: 3.8266 cents 29:34:37:39:45:51:55:58 SD: 3.2953 cents 36:42:46:48:56:63:69:72 SD: 1.3852 cents 66:77:84:88:103:116:126:132 SD: 1.3357 cents 72:84:92:96:112:126:137:144 SD: 1.2769 cents 77:90:98:103:120:135:147:154 SD: 1.1283 cents 84:98:107:112:131:147:160:168 SD: 0.8697 cents 96:112:122:128:149:168:183:192 SD: 0.7594 cents 120:140:153:160:187:210:229:240 SD: 0.6317 cents 132:154:168:176:205:231:252:264 SD: 0.4018 cents 144:168:183:192:224:252:275:288 SD: 0.3773 cents 180:210:229:240:280:315:344:360 SD: 0.2794 cents 216:252:275:288:336:378:412:432 SD: 0.2330 cents 252:294:321:336:392:441:481:504 SD: 0.2153 cents 264:308:336:352:411:462:504:528 SD: 0.2007 cents 360:420:458:480:560:630:687:720 SD: 0.1388 cents 396:462:504:528:616:693:756:792 S SD: 0.0000 cents