Scala analysis: xen10-wilson-purvi-08b-05
Purvi modulation 5 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: 11/9 347.407941 undecimal neutral third 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 44/27 845.452940 neutral sixth 6: 16/9 996.089998 Pythagorean minor seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 9/8 203.9100 major whole tone 1: 22/21 80.5370 undecimal minor semitone 2: 7/6 266.8709 septimal minor third 3: 12/11 150.6371 undecimal neutral second, 3/4-tone 4: 9/8 203.9100 major whole tone 5: 88/81 143.4979 2nd undecimal neutral second 6: 12/11 150.6371 undecimal neutral second, 3/4-tone 7: 9/8 203.9100 major whole tone
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 33/28 284.447 cents undecimal minor third 2: 1 32/27 294.135 cents Pythagorean minor third 2: 2 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: 2 18/11 852.592 cents undecimal neutral sixth 5: 1 27/16 905.865 cents Pythagorean major sixth 5: 1 56/33 915.553 cents 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: 4.66667 = 14/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 80.5 347.4 498.0 702.0 845.5 996.1 1200.0 80.5 : 266.9 417.5 621.4 764.9 915.6 1119.5 1200.0 347.4 : 150.6 354.5 498.0 648.7 852.6 933.1 1200.0 498.0 : 203.9 347.4 498.0 702.0 782.5 1049.4 1200.0 702.0 : 143.5 294.1 498.0 578.6 845.5 996.1 1200.0 845.5 : 150.6 354.5 435.1 702.0 852.6 1056.5 1200.0 996.1 : 203.9 284.4 551.3 702.0 905.9 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: 284.4470 .. 417.5080 cents 392/363, 133.0609 cents 99/98, 17.5761 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 .. 915.5530 cents 392/363, 133.0609 cents 99/98, 17.5761 cents 6: 933.1291 .. 1119.4630 cents 49/44, 186.3339 cents 99/98, 17.5761 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 : 169.15080 cents, oct.: 1184.05563 cents Scale is not proper Step pattern alph. order: ABCDECD Step pattern size order : AECDBCD Scale is a Constant Structure, by a margin of 17.57613 cents Scale diversity : 1.579089 Lumma stability : 0.192815 Lumma impropriety factor : 0.035697 Rothenberg efficiency : 0.408163 redundancy: 0.591837 Efficiency x scale size : 2.857143 Number of different interval sizes : 28 = 4.66667 / class Number of one step interval sizes : 5 Highest interval variety : 5 Mean interval variety : 4.66667 = 14/3 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 Best major 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 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 : 0.022658 Average absolute harmonicity : 0.243326 Specific harmonicity : 0.124741 Fundamental : 1/378, -8.5622 octaves, 0.6921 Hz. Guide tone : 528, 9.0444 octaves, 138138.298 Hz. Exponens Consonantiae : 1.995840E+05, 17.60664 octaves Euler's gradus suavitatis : 30 Sum of Mann's harmonic distance : 87.0, average 12.42857 Mersenne's string divisions : 6411300 Sum of van Prooijen's expressibility: 5.70346, average 0.81478 Sum of Tenney's harmonic distance : 12.05182, average 1.72169 Vogel's harmonic complexity : 18.57143 Wille's k value : 283 Wilson's harmonic complexity : 30 Rectangular lattice diameter : 6 Triangular lattice diameter : 4 Lattice compactness : 77.97655, average 2.78488 Lattice compactness (without 2's) : 51.85671, average 1.85203 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -1 .. 4 1.28571 11 1 3: -3 .. 1 -1.14286 10 3 7: -1 .. 0 -0.14286 1 11: 0 .. 1 0.42857 3 3 Average exponent except 2's :-6 / 7 =-0.85714 Average absolute exponent except 2's: 14 / 7 = 2.00000 Scale is weakly epimorphic with val: <7 10 21 22| Scale is JI-epimorphic with val: <7 11 19 24| Scale is weakly epimorphic with val: <7 12 21 25| Scale is weakly epimorphic with val: <7 12 22 23|
FIT/MODE
7: 0 2 1 1 1 1 1 N T3 SD: 34.2807 c. M: 80.5370 c. 10: 1 2 1 2 1 1 2 P M ME S SD: 22.9887 c. M:-39.4630 c. 13: 1 3 1 3 1 2 2 N D3 S SD: 23.5072 c. M:-36.5065 c. 14: 1 3 2 2 2 2 2 P T3 SD: 15.8868 c. M:-32.4814 c. 17: 1 4 2 3 2 2 3 N SD: 5.6659 c. M: 9.9488 c. 31: 2 7 4 5 4 4 5 N SD: 5.4729 c. M:-10.3616 c. 41: 3 9 5 7 5 5 7 N SD: 3.3770 c. M:-7.2678 c. 48: 3 11 6 8 6 6 8 N SD: 3.4014 c. M: 5.5370 c. 58: 4 13 7 10 7 7 10 N SD: 2.5330 c. M:-4.3162 c. 72: 5 16 9 12 9 9 12 N SD: 2.8821 c. M:-4.5471 c. 75: 5 17 9 13 9 9 13 N SD: 2.7505 c. M:-4.5921 c. 77: 5 17 10 13 9 10 13 N SD: 2.5439 c. M: 4.5508 c. 87: 6 19 11 15 10 11 15 N SD: 2.4368 c. M: 4.0736 c. 89: 6 20 11 15 11 11 15 N SD: 2.0739 c. M:-3.9853 c. 94: 6 21 12 16 11 12 16 N SD: 2.1233 c. M: 3.9413 c. 101: 7 22 13 17 12 13 17 N SD: 1.8601 c. M: 2.8535 c. 118: 8 26 15 20 14 15 20 N SD: 0.9025 c. M: 1.6452 c. 135: 9 30 17 23 16 17 23 N SD: 0.5711 c. M: 1.0085 c. 224: 15 50 28 38 27 28 38 N SD: 0.5080 c. M:-0.9756 c. 294: 20 65 37 50 35 37 50 N SD: 0.5032 c. M:-1.0956 c. 301: 20 67 38 51 36 38 51 N SD: 0.4712 c. M: 0.8028 c. 311: 21 69 39 53 37 39 53 N SD: 0.3737 c. M: 0.5916 c. 342: 23 76 43 58 41 43 58 N SD: 0.2057 c. M:-0.4012 c. 359: 24 80 45 61 43 45 61 N SD: 0.1698 c. M: 0.3142 c. 477: 32 106 60 81 57 60 81 N SD: 0.1280 c. M: 0.2381 c. 701: 47 156 88 119 84 88 119 N SD: 0.1280 c. M:-0.2010 c. 836: 56 186 105 142 100 105 142 N SD: 0.0712 c. M: 0.1543 c. 1012: 68 225 127 172 121 127 172 N SD: 0.0420 c. M:-0.0954 c. 1371: 92 305 172 233 164 172 233 N SD: 0.0394 c. M:-0.0756 c. 1489: 100 331 187 253 178 187 253 N SD: 0.0374 c. M: 0.0607 c. 1506: 101 335 189 256 180 189 256 N SD: 0.0429 c. M: 0.0741 c. 1713: 115 381 215 291 205 215 291 N SD: 0.0460 c. M:-0.0812 c. 1848: 124 411 232 314 221 232 314 N SD: 0.0095 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: 12.9686 cents 4 x 5 x 6 x 7:8 S SD: 11.8769 cents 5 x 6:7 x 8:9:10 S SD: 19.6117 cents 6 x 7:8:9:10:11:12 S SD: 17.3509 cents 7 x x 9 x 11:12:14 SD: 27.2629 cents 8:9:10:11:12:13:14:16 S SD: 20.3741 cents 9:10:11:12:14:15:16:18 S SD: 17.9878 cents 10:11:12:13:15:16:18:20 SD: 15.3441 cents 12:13:15:16:18:20:21:24 SD: 12.0702 cents 13:14:16:17:20:21:23:26 SD: 10.8596 cents 14:15:17:19:21:23:25:28 SD: 7.6066 cents 18:19:22:24:27:29:32:36 SD: 3.3872 cents 22:23:27:29:33:36:39:44 SD: 3.2903 cents 36:38:44:48:54:59:64:72 SD: 2.3340 cents 40:42:49:53:60:65:71:80 SD: 1.9182 cents 54:57:66:72:81:88:96:108 SD: 1.8666 cents 60:63:73:80:90:98:107:120 SD: 1.5797 cents 62:65:76:83:93:101:110:124 SD: 1.3436 cents 81:85:99:108:122:132:144:162 SD: 1.0976 cents 84:88:103:112:126:137:149:168 SD: 0.9942 cents 86:90:105:115:129:140:153:172 SD: 0.8677 cents 108:113:132:144:162:176:192:216 SD: 0.3125 cents 234:245:286:312:351:381:416:468 SD: 0.2599 cents 252:264:308:336:378:411:448:504 SD: 0.2007 cents 270:283:330:360:405:440:480:540 SD: 0.1249 cents 378:396:462:504:567:616:672:756 SD: 0.0000 cents