Scala analysis: xen10-wilson-purvi-08b-01
Purvi modulation 1 from Figure 8b, Ptolemy (7/6 22/21 12/11), (22/21 12/11 7/6)
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 22/21 80.537035 undecimal minor semitone 2: 8/7 231.174094 septimal whole tone 3: 9/7 435.084095 septimal major third, BP third 4: 3/2 701.955001 perfect fifth 5: 11/7 782.492036 undecimal augmented fifth 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: 22/21 80.5370 undecimal minor semitone 2: 12/11 150.6371 undecimal neutral second, 3/4-tone 3: 9/8 203.9100 major whole tone 4: 7/6 266.8709 septimal minor third 5: 22/21 80.5370 undecimal minor semitone 6: 12/11 150.6371 undecimal neutral second, 3/4-tone 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 : 80.5 231.2 435.1 702.0 782.5 933.1 1200.0 80.5 : 150.6 354.5 621.4 702.0 852.6 1119.5 1200.0 231.2 : 203.9 470.8 551.3 702.0 968.8 1049.4 1200.0 435.1 : 266.9 347.4 498.0 764.9 845.5 996.1 1200.0 702.0 : 80.5 231.2 498.0 578.6 729.2 933.1 1200.0 782.5 : 150.6 417.5 498.0 648.7 852.6 1119.5 1200.0 933.1 : 266.9 347.4 498.0 702.0 968.8 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 : 161.19423 cents, oct.: 1128.35961 cents Scale is not proper Step pattern alph. order: ABCDABD Step pattern size order : ABCDABD 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 Hypolydian 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 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 Scale is n-1 differentially coherent in interval classes 5 and 6 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 : 0.220735 Specific harmonicity : 0.130134 Fundamental : 1/42, -5.3923 octaves, 6.2292 Hz. Guide tone : 792, 9.6294 octaves, 207207.448 Hz. Exponens Consonantiae : 3.326400E+04, 15.02167 octaves Euler's gradus suavitatis : 27 Sum of Mann's harmonic distance : 59.5, average 8.50000 Mersenne's string divisions : 588240 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 : 11.28571 Wille's k value : 31 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: -1 .. 3 0.85714 8 1 3: -1 .. 2 0.42857 5 1 7: -1 .. 0 -0.71429 5 3 11: 0 .. 1 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: 0 1 1 2 0 1 2 N T3 SD: 69.6575 c. M: 96.7778 c. 9: 1 1 1 2 1 1 2 P M ME S SD: 31.2362 c. M:-52.7963 c. 10: 1 1 2 2 1 1 2 P D3 S SD: 33.8220 c. M:-57.5080 c. 11: 1 1 2 2 1 2 2 P D3 S SD: 29.1782 c. M:-48.6891 c. 12: 1 1 2 3 1 1 3 N SD: 23.8729 c. M: 35.0841 c. Chromatic Phrygian inverse 14: 1 2 2 3 1 2 3 P T3 SD: 13.2255 c. M:-25.9688 c. 17: 1 2 3 4 1 2 4 N SD: 11.3426 c. M: 19.4094 c. 26: 2 3 4 6 2 3 6 N SD: 10.1799 c. M: 19.6995 c. 27: 2 3 5 6 2 3 6 N SD: 9.4711 c. M:-17.5080 c. 28: 2 3 5 6 2 4 6 N SD: 10.9233 c. M: 16.8884 c. 31: 2 4 5 7 2 4 7 N SD: 5.4702 c. M: 9.2776 c. 41: 3 5 7 9 3 5 9 N SD: 4.6207 c. M:-7.7519 c. 46: 3 6 8 10 3 6 10 N SD: 4.3164 c. M:-8.3942 c. 57: 4 7 10 12 4 7 13 N SD: 4.9863 c. M: 7.2182 c. 58: 4 7 10 13 4 7 13 N SD: 2.3479 c. M:-3.7149 c. 72: 5 9 12 16 5 9 16 N SD: 1.6953 c. M:-2.7963 c. 89: 6 11 15 20 6 11 20 N SD: 1.9202 c. M: 3.6234 c. 104: 7 13 18 23 7 13 23 N SD: 1.7695 c. M:-3.3774 c. 118: 8 15 20 26 8 15 26 N SD: 1.6648 c. M:-2.7242 c. 121: 8 15 21 27 8 15 27 N SD: 1.6489 c. M: 3.0749 c. 130: 9 16 22 29 9 16 29 N SD: 1.3889 c. M:-2.5399 c. 135: 9 17 23 30 9 17 30 N SD: 0.3164 c. M: 0.5370 c. 342: 23 43 58 76 23 43 76 N SD: 0.1980 c. M:-0.4049 c. 477: 32 60 81 106 32 60 106 N SD: 0.1467 c. M:-0.2724 c. 612: 41 77 104 136 41 77 136 N SD: 0.1538 c. M:-0.2100 c. 670: 45 84 114 149 45 84 149 N SD: 0.1170 c. M:-0.1945 c. 805: 54 101 137 179 54 101 179 N SD: 0.1155 c. M:-0.1954 c. 836: 56 105 142 186 56 105 186 N SD: 0.1234 c. M: 0.1954 c. 877: 59 110 149 195 59 110 195 N SD: 0.1050 c. M:-0.1927 c. 1012: 68 127 172 225 68 127 225 N SD: 0.0754 c. M:-0.1167 c. 1147: 77 144 195 255 77 144 255 N SD: 0.0720 c. M:-0.1382 c. 1371: 92 172 233 305 92 172 305 N SD: 0.0582 c. M: 0.1019 c. 1506: 101 189 256 335 101 189 335 N SD: 0.0526 c. M: 0.0984 c. 1641: 110 206 279 365 110 206 365 N SD: 0.0604 c. M: 0.0983 c. 1713: 115 215 291 381 115 215 381 N SD: 0.0283 c. M: 0.0578 c. 1848: 124 232 314 411 124 232 411 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: 0.0000 cents 3 x x 4 x x 5:6 S SD: 26.5469 cents 4 x x 5:6 x 7:8 S SD: 15.1096 cents 5 x 6 x x 8:9:10 S SD: 30.8651 cents 6 x 7:8:9 x 10:12 S SD: 17.4550 cents 7 x 8:9 x 11:12:14 SD: 0.0000 cents 8 x 9:10:12:13:14:16 S SD: 14.6859 cents 9 x 10:12 x 14:15:18 S SD: 19.0054 cents 10 x 11:13:15:16:17:20 SD: 12.8316 cents 11:12:13:14 x 17:19:22 SD: 16.3251 cents 12:13:14:15:18:19:21:24 SD: 13.1444 cents 13:14:15:17 x 20:22:26 SD: 12.0911 cents 14:15:16:18:21:22:24:28 S SD: 5.5580 cents 20:21:23:26:30:31:34:40 SD: 5.0963 cents 21:22:24:27:32:33:36:42 SD: 3.8949 cents 28:29:32:36:42:44:48:56 SD: 2.8265 cents 42:44:48:54:63:66:72:84 S SD: 0.0000 cents