Scala analysis: xen10-wilson-purvi-08a-02
Purvi modulation 2 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: 12/7 933.129094 septimal major sixth 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 7/6 266.8709 septimal minor third 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: 22/21 80.5370 undecimal minor semitone 7: 7/6 266.8709 septimal minor third
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: 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 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: 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 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: 4.66667 = 14/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 203.9 354.5 498.0 702.0 852.6 933.1 1200.0 203.9 : 150.6 294.1 498.0 648.7 729.2 996.1 1200.0 354.5 : 143.5 347.4 498.0 578.6 845.5 1049.4 1200.0 498.0 : 203.9 354.5 435.1 702.0 905.9 1056.5 1200.0 702.0 : 150.6 231.2 498.0 702.0 852.6 996.1 1200.0 852.6 : 80.5 347.4 551.3 702.0 845.5 1049.4 1200.0 933.1 : 266.9 470.8 621.4 764.9 968.8 1119.5 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 : 167.69067 cents, oct.: 1173.83470 cents Scale is not proper Step pattern alph. order: ABCABDE Step pattern size order : DCBDCAE Scale is a Constant Structure, by a margin of 27.26409 cents Scale diversity : 1.600163 Lumma stability : 0.134228 Lumma impropriety factor : 0.154686 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 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 : 0.250210 Specific harmonicity : 0.137778 Fundamental : 1/1848, -10.8517 octaves, 0.1416 Hz. Guide tone : 108, 6.7549 octaves, 28255.561 Hz. Exponens Consonantiae : 1.995840E+05, 17.60664 octaves Euler's gradus suavitatis : 30 Sum of Mann's harmonic distance : 64.5, average 9.21429 Mersenne's string divisions : 6884745 Sum of van Prooijen's expressibility: 5.22634, average 0.74662 Sum of Tenney's harmonic distance : 11.01043, average 1.57292 Vogel's harmonic complexity : 23.42857 Wille's k value : 1039 Wilson's harmonic complexity : 30 Rectangular lattice diameter : 6 Triangular lattice diameter : 4 Lattice compactness : 82.35355, average 2.94120 Lattice compactness (without 2's) : 50.80682, average 1.81453 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -3 .. 2 0.14286 11 1 3: -1 .. 3 1.14286 10 3 7: -1 .. 0 -0.14286 1 1 11: -1 .. 0 -0.28571 2 2 Average exponent except 2's : 5 / 7 = 0.71429 Average absolute exponent except 2's: 13 / 7 = 1.85714 Scale is weakly epimorphic with val: <7 10 22 22| Scale is JI-epimorphic with val: <7 11 19 24| Scale is weakly epimorphic with val: <7 12 18 25| Scale is weakly epimorphic with val: <7 12 22 23|
FIT/MODE
7: 1 1 1 1 1 0 2 N T3 SD: 32.7629 c. M: 75.9862 c. 10: 2 1 1 2 1 1 2 P D3 I SD: 20.2275 c. M:-36.0900 c. 14: 2 2 2 2 2 1 3 P T3 SD: 16.1889 c. M: 32.4814 c. 17: 3 2 2 3 2 1 4 N SD: 7.2253 c. M: 15.4820 c. 24: 4 3 3 4 3 2 5 P SD: 6.9174 c. M:-16.8709 c. 31: 5 4 4 5 4 2 7 N SD: 5.5643 c. M: 10.3616 c. 41: 7 5 5 7 5 3 9 N SD: 2.3589 c. M: 3.8116 c. 72: 12 9 9 12 9 5 16 N SD: 2.6824 c. M: 4.5471 c. 77: 13 10 9 13 10 5 17 N SD: 2.4556 c. M:-4.5508 c. 87: 15 11 10 15 11 6 19 N SD: 2.9201 c. M:-4.8019 c. 89: 15 11 11 15 11 6 20 N SD: 2.3229 c. M: 3.9853 c. 94: 16 12 11 16 12 6 21 N SD: 1.5811 c. M:-2.8997 c. 118: 20 15 14 20 15 8 26 N SD: 1.2594 c. M:-2.4641 c. 125: 21 16 15 21 16 8 28 N SD: 1.4842 c. M: 2.3100 c. 135: 23 17 16 23 17 9 30 N SD: 0.5394 c. M:-1.0085 c. 176: 30 22 21 30 22 12 39 N SD: 0.4830 c. M:-0.9618 c. 301: 51 38 36 51 38 20 67 N SD: 0.3716 c. M: 0.5877 c. 311: 53 39 37 53 39 21 69 N SD: 0.4029 c. M:-0.6330 c. 342: 58 43 41 58 43 23 76 N SD: 0.2107 c. M: 0.4012 c. 359: 61 45 43 61 45 24 80 N SD: 0.2370 c. M: 0.5386 c. 477: 81 60 57 81 60 32 106 N SD: 0.1489 c. M:-0.2381 c. 494: 84 62 59 84 62 33 110 N SD: 0.1487 c. M: 0.3356 c. 535: 91 67 64 91 67 36 119 N SD: 0.1476 c. M: 0.2556 c. 670: 114 84 80 114 84 45 149 N SD: 0.1298 c. M:-0.2691 c. 701: 119 88 84 119 88 47 156 N SD: 0.1410 c. M: 0.2010 c. 836: 142 105 100 142 105 56 186 N SD: 0.0596 c. M: 0.1147 c. 1012: 172 127 121 172 127 68 225 N SD: 0.0348 c. M:-0.0725 c. 1848: 314 232 221 314 232 124 411 N SD: 0.0082 c. M: 0.0139 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 x 7:8 S SD: 11.9462 cents 5 x 6:7 x 8:9:10 S SD: 26.3034 cents 6:7 x 8:9:10 x 12 S SD: 14.1042 cents 7:8 x 9 x 11:12:14 SD: 19.6178 cents 8:9:10:11:12:13:14:16 S SD: 10.3677 cents 9:10:11:12:14:15 x 18 S SD: 12.3454 cents 10:11:12:13:15:16:17:20 SD: 11.6689 cents 13:15:16:17:20:21:22:26 SD: 11.0608 cents 14:16:17:19:21:23:24:28 SD: 6.4972 cents 18:20:22:24:27:29:31:36 SD: 5.1555 cents 22:25:27:29:33:36:38:44 SD: 4.2014 cents 30:34:37:40:45:49:51:60 SD: 3.0511 cents 38:43:47:51:57:62:65:76 SD: 3.0363 cents 39:44:48:52:59:64:67:78 SD: 2.4888 cents 48:54:59:64:72:79:82:96 SD: 1.7094 cents 66:74:81:88:99:108:113:132 SD: 0.8907 cents 88:99:108:117:132:144:151:176 SD: 0.7415 cents 102:115:125:136:153:167:175:204 SD: 0.6913 cents 144:162:177:192:216:236:247:288 SD: 0.5580 cents 154:173:189:205:231:252:264:308 SD: 0.5376 cents 162:182:199:216:243:265:278:324 SD: 0.4881 cents 168:189:206:224:252:275:288:336 SD: 0.2330 cents 294:331:361:392:441:481:504:588 SD: 0.2294 cents 330:371:405:440:495:540:566:660 SD: 0.2082 cents 360:405:442:480:540:589:617:720 SD: 0.1228 cents 462:520:567:616:693:756:792:924 SD: 0.1189 cents 528:594:648:704:792:864:905:1056 SD: 0.0390 cents