Scala analysis: xen10-wilson-purvi-01-05
Purvi modulation 5 from Figure 1, Pythagoras (9/8 9/8 256/243) all 3 permutations
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: 32/27 294.134997 Pythagorean minor third 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 128/81 792.179997 Pythagorean minor 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: 9/8 203.9100 major whole tone 2: 256/243 90.2250 limma, Pythagorean minor second 3: 9/8 203.9100 major whole tone 4: 9/8 203.9100 major whole tone 5: 256/243 90.2250 limma, Pythagorean minor second 6: 9/8 203.9100 major whole tone 7: 9/8 203.9100 major whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 256/243 90.225 cents limma, Pythagorean minor second 1: 5 9/8 203.910 cents major whole tone 2: 4 32/27 294.135 cents Pythagorean minor third 2: 3 81/64 407.820 cents Pythagorean major third 3: 6 4/3 498.045 cents perfect fourth 3: 1 729/512 611.730 cents Pythagorean tritone 4: 1 1024/729 588.270 cents Pythagorean diminished fifth 4: 6 3/2 701.955 cents perfect fifth 5: 3 128/81 792.180 cents Pythagorean minor sixth 5: 4 27/16 905.865 cents Pythagorean major sixth 6: 5 16/9 996.090 cents Pythagorean minor seventh 6: 2 243/128 1109.775 cents Pythagorean major seventh Highest number of different intervals for one interval class: 2 Average number of different intervals per interval class: 2.00000 = 2
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 203.9 294.1 498.0 702.0 792.2 996.1 1200.0 203.9 : 90.2 294.1 498.0 588.3 792.2 996.1 1200.0 294.1 : 203.9 407.8 498.0 702.0 905.9 1109.8 1200.0 498.0 : 203.9 294.1 498.0 702.0 905.9 996.1 1200.0 702.0 : 90.2 294.1 498.0 702.0 792.2 996.1 1200.0 792.2 : 203.9 407.8 611.7 702.0 905.9 1109.8 1200.0 996.1 : 203.9 407.8 498.0 702.0 905.9 996.1 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 203.9100 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents 2: 294.1350 .. 407.8200 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents 3: 498.0450 .. 611.7300 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents 4: 588.2700 .. 701.9550 cents 2187/2048, 113.6850 cents 524288/531441,-23.4600 cents 5: 792.1800 .. 905.8650 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents 6: 996.0900 .. 1109.7750 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 256/243, 90.2250 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.900012 Largest / Average step : 1.189475 Largest / Smallest step : 2.260017 Median interval of one step : 9/8, 203.9100 cents, amount: 5 Most common interval of one step : 9/8, 203.9100 cents, amount: 5 Least squares average step : 167.36839 cents, oct.: 1171.57875 cents Scale is not proper Scale has Myhill's property chromas: 2187/2048, 113.6850 cents and 531441/524288, 23.4600 cents, ratio: 4.845906 generators: 3 of 498.04500 cents and 4 of 701.95500 cents ET pair of linear temperament : 12&41 Scale is distributional even Scale is a mode of a 665-tone equal temperament with octave 2/1 degrees: 113 163 276 389 439 552 665 generators: 276 and 389 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABAABAA Step pattern size order : LSLLSLL Scale is a Constant Structure, by a margin of 23.46001 cents Ratio Whole / Half tone : 2.260017 Scale diversity : 1.288369 Lumma stability : 0.451125 Lumma impropriety factor : 0.019550 Rothenberg efficiency : 0.761905 redundancy: 0.238095 Efficiency x scale size : 5.333333 Number of different interval sizes : 12 = 2.00000 / class Number of one step interval sizes : 2 Highest interval variety : 2 Mean interval variety : 2.00000 = 2 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 531441/524288, 23.4600 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 6 Scale is a chain of 3 triads 0.0 498.045 996.090 cents Most common triad is 0.0 498.045 701.955 cents, amount: 5 Number of recognisable fifths : 6, average 701.9550 cents Scale is Mixolydian octave species of abstract tetrachord Scale is Lydian octave species of abstract tetrachord Scale is Hypodorian octave species of abstract tetrachord Scale contains two identical conjunct tetrachords Best major thirds form a closed circle Best minor thirds form a closed circle Scale is a complete genus musicum : 2^m 3^6 Scale contains a complete diamond : 1 3 9 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 5 = 71.4286% 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 class 4 Limited inverse transpositions : 6 Inversional symmetry on degrees : 3 Inversional symmetry on intervals : 6-7 -- Rational properties -- Prime limit : 3 Odd number limit : 729 (O: 729 U: 729) Highest odd numerator or denominator: 81 Scale harmonicity : 0.038462 Average absolute harmonicity : 0.263954 Specific harmonicity : 0.215488 Fundamental : 1/648, -9.3399 octaves, 0.4037 Hz. Guide tone : 1152, 10.1699 octaves, 301392.651 Hz. Exponens Consonantiae : 7.464960E+05, 19.50978 octaves Euler's gradus suavitatis : 23 Sum of Mann's harmonic distance : 162.5, average 23.21429 Mersenne's string divisions : 110054175 Sum of van Prooijen's expressibility: 6.20258, average 0.88608 Sum of Tenney's harmonic distance : 13.12627, average 1.87518 Vogel's harmonic complexity : 14.14286 Wille's k value : 364 Wilson's harmonic complexity : 18 Rectangular lattice diameter : 6 Triangular lattice diameter : 6 Lattice compactness : 132.79540, average 4.74269 Lattice compactness (without 2's) : 69.00000, average 2.46429 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -3 .. 7 2.14286 23 1 3: -4 .. 2 -1.00000 13 6 Average exponent except 2's :-7 / 7 =-1.00000 Average absolute exponent except 2's: 13 / 7 = 1.85714 Scale is weakly epimorphic with val: <7 10| Scale is JI-epimorphic with val: <7 11| = patent Scale is weakly epimorphic with val: <7 12|
FIT/MODE
7: 1 1 1 1 1 1 1 SP B ME I SD: 36.3153 c. M:-64.9629 c. Lebeng 10: 2 0 2 2 1 1 2 P D3 S SD: 34.7924 c. M: 54.1350 c. 11: 2 1 2 1 1 2 2 N D3 SD: 31.2034 c. M:-47.4095 c. 12: 2 1 2 2 1 2 2 P M ME S SD: 4.3715 c. M:-7.8200 c. G.M.Hypodorian, G.M.Aeolian, G.Hyperphrygian, Natural Minor, Melodic Minor descending, Asavari That, Mela Natabhairavi, Raga Jaunpuri, Adana, Darbari, Dhanyasi, Jingla, Gregorian nr.2, Makam Buselik, Nihavend, Peruvian Minor, Se, Chiao: China, Geez, Ezel: Ethiopia, Kiourdi descending: Greece, Cushak: Armenia 29: 5 2 5 5 2 5 5 N M DE S SD: 3.3391 c. M: 5.9731 c. Twenty-nine tone Minor 41: 7 3 7 7 3 7 7 N M DE S SD: 1.0823 c. M: 1.9361 c. Forty-one tone Minor 53: 9 4 9 9 4 9 9 N M DE S SD: 0.1525 c. M:-0.2728 c. Makam Nihavend, Muberka, Fifty-three tone Minor 200: 34 15 34 34 15 34 34 N M DE S SD: 0.1006 c. M: 0.1800 c. 253: 43 19 43 43 19 43 43 N M DE S SD: 0.0476 c. M: 0.0851 c. 306: 52 23 52 52 23 52 52 N M DE S SD: 0.0129 c. M: 0.0231 c. 359: 61 27 61 61 27 61 61 N M DE S SD: 0.0115 c. M:-0.0206 c. 665: 113 50 113 113 50 113 113 N M DE S SD: 0.0003 c. M:-0.0005 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: 30.7262 cents 4 x 5 x 6 x 7:8 S SD: 24.0316 cents 5 x 6:7 x 8:9:10 S SD: 18.4632 cents 6 x 7:8:9 x 11:12 SD: 11.9689 cents 7:8 x 9 x 11:12:14 SD: 18.7247 cents 8:9 x 11:12:13:14:16 SD: 12.8223 cents 9:10:11:12 x 14:16:18 S SD: 10.5986 cents 10:11:12:13:15:16:18:20 SD: 9.9205 cents 11:12:13:15 x 17:20:22 SD: 14.3131 cents 12 x 14:16:18:19:21:24 SD: 6.4508 cents 13:15 x 17:20:21:23:26 SD: 13.4284 cents 14:16:17:19:21:22:25:28 SD: 8.5704 cents 15:17:18:20:23:24:27:30 SD: 7.8229 cents 16:18:19:21:24:25:28:32 SD: 6.1948 cents 22:25:26:29:33:35:39:44 SD: 4.2336 cents 26:29:31:35:39:41:46:52 SD: 3.7411 cents 32:36:38:43:48:51:57:64 SD: 2.9323 cents 38:43:45:51:57:60:68:76 SD: 2.7245 cents 42:47:50:56:63:66:75:84 SD: 2.4625 cents 48:54:57:64:72:76:85:96 SD: 1.1844 cents 72:81:85:96:108:114:128:144 SD: 1.0816 cents 81:91:96:108:122:128:144:162 SD: 1.0709 cents 86:97:102:115:129:136:153:172 SD: 1.0097 cents 96:108:114:128:144:152:171:192 SD: 0.8358 cents 102:115:121:136:153:161:181:204 SD: 0.7932 cents 112:126:133:149:168:177:199:224 SD: 0.7468 cents 114:128:135:152:171:180:203:228 SD: 0.6934 cents 136:153:161:181:204:215:242:272 SD: 0.5911 cents 150:169:178:200:225:237:267:300 SD: 0.5714 cents 152:171:180:203:228:240:270:304 SD: 0.5380 cents 162:182:192:216:243:256:288:324 SD: 0.3395 cents 216:243:256:288:324:341:384:432 SD: 0.2416 cents 264:297:313:352:396:417:469:528 SD: 0.2250 cents 312:351:370:416:468:493:555:624 SD: 0.2110 cents 336:378:398:448:504:531:597:672 SD: 0.1960 cents 384:432:455:512:576:607:683:768 SD: 0.1547 cents 432:486:512:576:648:683:768:864 SD: 0.1207 cents 486:547:576:648:729:768:864:972 SD: 0.1131 cents 600:675:711:800:900:948:1067:1200 SD: 0.0946 cents