Scala analysis: xen10-wilson-purvi-08a-03
Purvi modulation 3 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: 9/8 203.910002 major whole 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: 9/8 203.9100 major whole tone 2: 12/11 150.6371 undecimal neutral second, 3/4-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 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 : 203.9 354.5 498.0 702.0 852.6 1119.5 1200.0 203.9 : 150.6 294.1 498.0 648.7 915.6 996.1 1200.0 354.5 : 143.5 347.4 498.0 764.9 845.5 1049.4 1200.0 498.0 : 203.9 354.5 621.4 702.0 905.9 1056.5 1200.0 702.0 : 150.6 417.5 498.0 702.0 852.6 996.1 1200.0 852.6 : 266.9 347.4 551.3 702.0 845.5 1049.4 1200.0 1119.5: 80.5 284.4 435.1 578.6 782.5 933.1 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: 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 : 175.67641 cents, oct.: 1229.73486 cents Scale is not proper Step pattern alph. order: ABCABDE Step pattern size order : DCBDCEA 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 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 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 : 0.245244 Specific harmonicity : 0.124741 Fundamental : 1/264, -8.0444 octaves, 0.9910 Hz. Guide tone : 756, 9.5622 octaves, 197788.927 Hz. Exponens Consonantiae : 1.995840E+05, 17.60664 octaves Euler's gradus suavitatis : 30 Sum of Mann's harmonic distance : 71.0, average 10.14286 Mersenne's string divisions : 11595360 Sum of van Prooijen's expressibility: 5.70346, average 0.81478 Sum of Tenney's harmonic distance : 11.44976, average 1.63568 Vogel's harmonic complexity : 16.57143 Wille's k value : 148 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: -3 .. 2 -0.14286 9 1 3: -1 .. 3 1.14286 10 3 7: 0 .. 1 0.14286 1 11: -1 .. 0 -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: 1 1 1 1 1 2 0 N T3 SD: 34.2807 c. M:-80.5370 c. 8: 1 1 1 2 1 1 1 P M ME I SD: 50.1012 c. M: 69.4630 c. 10: 2 1 1 2 1 2 1 P M ME S SD: 22.9887 c. M: 39.4630 c. Sethares Neutral, Beatles-7, Dichotic-7 14: 2 2 2 2 2 3 1 P T3 SD: 15.8868 c. M: 32.4814 c. 16: 3 2 2 2 2 4 1 P SD: 21.0721 c. M: 27.5921 c. 17: 3 2 2 3 2 4 1 N SD: 5.6659 c. M:-9.9488 c. 31: 5 4 4 5 4 7 2 N SD: 5.4729 c. M: 10.3616 c. 41: 7 5 5 7 5 9 3 N SD: 3.3770 c. M: 7.2678 c. 48: 8 6 6 8 6 11 3 N SD: 3.4014 c. M:-5.5370 c. 58: 10 7 7 10 7 13 4 N SD: 2.5330 c. M: 4.3162 c. 72: 12 9 9 12 9 16 5 N SD: 2.8821 c. M: 4.5471 c. 75: 13 9 9 13 9 17 5 N SD: 2.7505 c. M: 4.5921 c. 77: 13 10 9 13 10 17 5 N SD: 2.5439 c. M:-4.5508 c. 87: 15 11 10 15 11 19 6 N SD: 2.4368 c. M:-4.0736 c. 89: 15 11 11 15 11 20 6 N SD: 2.0739 c. M: 3.9853 c. 94: 16 12 11 16 12 21 6 N SD: 2.1233 c. M:-3.9413 c. 101: 17 13 12 17 13 22 7 N SD: 1.8601 c. M:-2.8535 c. 118: 20 15 14 20 15 26 8 N SD: 0.9025 c. M:-1.6452 c. 135: 23 17 16 23 17 30 9 N SD: 0.5711 c. M:-1.0085 c. 224: 38 28 27 38 28 50 15 N SD: 0.5080 c. M: 0.9756 c. 294: 50 37 35 50 37 65 20 N SD: 0.5032 c. M: 1.0956 c. 301: 51 38 36 51 38 67 20 N SD: 0.4712 c. M:-0.8028 c. 311: 53 39 37 53 39 69 21 N SD: 0.3737 c. M:-0.5916 c. 342: 58 43 41 58 43 76 23 N SD: 0.2057 c. M: 0.4012 c. 359: 61 45 43 61 45 80 24 N SD: 0.1698 c. M:-0.3142 c. 477: 81 60 57 81 60 106 32 N SD: 0.1280 c. M:-0.2381 c. 701: 119 88 84 119 88 156 47 N SD: 0.1280 c. M: 0.2010 c. 836: 142 105 100 142 105 186 56 N SD: 0.0712 c. M:-0.1543 c. 1012: 172 127 121 172 127 225 68 N SD: 0.0420 c. M: 0.0954 c. 1371: 233 172 164 233 172 305 92 N SD: 0.0394 c. M: 0.0756 c. 1489: 253 187 178 253 187 331 100 N SD: 0.0374 c. M:-0.0607 c. 1506: 256 189 180 256 189 335 101 N SD: 0.0429 c. M:-0.0741 c. 1713: 291 215 205 291 215 381 115 N SD: 0.0460 c. M: 0.0812 c. 1848: 314 232 221 314 232 411 124 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: 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 6:7 x 8:9:10:11:12 S SD: 16.5724 cents 7:8 x 9 x 11:13:14 SD: 21.8198 cents 8:9:10:11:12:13:15:16 SD: 10.0669 cents 14:16:17:19:21:23:27:28 SD: 6.9655 cents 18:20:22:24:27:29:34:36 SD: 5.6742 cents 22:25:27:29:33:36:42:44 SD: 3.7640 cents 30:34:37:40:45:49:57:60 SD: 2.5337 cents 48:54:59:64:72:79:92:96 SD: 1.7724 cents 57:64:70:76:86:93:109:114 SD: 1.7416 cents 58:65:71:77:87:95:111:116 SD: 1.6938 cents 66:74:81:88:99:108:126:132 SD: 0.8341 cents 88:99:108:117:132:144:168:176 SD: 0.7036 cents 96:108:118:128:144:157:183:192 SD: 0.5492 cents 144:162:177:192:216:236:275:288 SD: 0.5455 cents 154:173:189:205:231:252:294:308 SD: 0.5376 cents 162:182:199:216:243:265:309:324 SD: 0.4703 cents 168:189:206:224:252:275:321:336 SD: 0.3138 cents 198:223:243:264:297:324:378:396 SD: 0.2774 cents 264:297:324:352:396:432:504:528 SD: 0.0000 cents