Scala analysis: xen10-wilson-purvi-03a-03
Purvi modulation 3 from Figure 3a, Al-Farabi (8/7 8/7 49/48), (49/48 8/7 8/7)
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: 9/7 435.084095 septimal major third, BP third 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 12/7 933.129094 septimal major sixth 6: 7/4 968.825906 harmonic seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 8/7 231.1741 septimal whole tone 1: 9/8 203.9100 major whole tone 2: 8/7 231.1741 septimal whole tone 3: 28/27 62.9609 Archytas' 1/3-tone 4: 9/8 203.9100 major whole tone 5: 8/7 231.1741 septimal whole tone 6: 49/48 35.6968 slendro diesis, septimal 1/6-tone 7: 8/7 231.1741 septimal whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 49/48 35.697 cents slendro diesis, septimal 1/6-tone 1: 1 28/27 62.961 cents Archytas' 1/3-tone 1: 2 9/8 203.910 cents major whole tone 1: 3 8/7 231.174 cents septimal whole tone 2: 3 7/6 266.871 cents septimal minor third 2: 1 32/27 294.135 cents Pythagorean minor third 2: 3 9/7 435.084 cents septimal major third, BP third 3: 2 21/16 470.781 cents narrow fourth 3: 4 4/3 498.045 cents perfect fourth 3: 1 72/49 666.258 cents Arabic lute grave fifth 4: 1 49/36 533.742 cents Arabic lute acute fourth 4: 4 3/2 701.955 cents perfect fifth 4: 2 32/21 729.219 cents wide fifth 5: 3 14/9 764.916 cents septimal minor sixth 5: 1 27/16 905.865 cents Pythagorean major sixth 5: 3 12/7 933.129 cents septimal major sixth 6: 3 7/4 968.826 cents harmonic seventh 6: 2 16/9 996.090 cents Pythagorean minor seventh 6: 1 27/14 1137.039 cents septimal major seventh 6: 1 96/49 1164.303 cents Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.33333 = 10/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 203.9 435.1 498.0 702.0 933.1 968.8 1200.0 203.9 : 231.2 294.1 498.0 729.2 764.9 996.1 1200.0 435.1 : 63.0 266.9 498.0 533.7 764.9 968.8 1200.0 498.0 : 203.9 435.1 470.8 702.0 905.9 1137.0 1200.0 702.0 : 231.2 266.9 498.0 702.0 933.1 996.1 1200.0 933.1 : 35.7 266.9 470.8 702.0 764.9 968.8 1200.0 968.8 : 231.2 435.1 666.3 729.2 933.1 1164.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 35.6968 .. 231.1741 cents 384/343, 195.4773 cents 49/48, 35.6968 cents 2: 266.8709 .. 435.0841 cents 54/49, 168.2132 cents 49/48, 35.6968 cents 3: 470.7809 .. 666.2582 cents 384/343, 195.4773 cents 49/48, 35.6968 cents 4: 533.7418 .. 729.2191 cents 384/343, 195.4773 cents 2401/2592,-132.5164 cents 5: 764.9159 .. 933.1291 cents 54/49, 168.2132 cents 49/48, 35.6968 cents 6: 968.8259 .. 1164.3032 cents 384/343, 195.4773 cents 49/48, 35.6968 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 49/48, 35.6968 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 8/7, 231.1741 cents Average / Smallest step : 4.802350 Largest / Average step : 1.348516 Largest / Smallest step : 6.476043 Median interval of one step : 9/8, 203.9100 cents, amount: 2 Most common interval of one step : 8/7, 231.1741 cents, amount: 3 Least squares average step : 173.24739 cents, oct.: 1212.73171 cents Scale is not proper Step pattern alph. order: ABCABDB Step pattern size order : CDBCDAD Scale is a Constant Structure, by a margin of 35.69681 cents Scale diversity : 1.307114 Lumma stability : 0.178484 Lumma impropriety factor : 0.110430 Rothenberg efficiency : 0.466667 redundancy: 0.533333 Efficiency x scale size : 3.266667 Number of different interval sizes : 20 = 3.33333 / class Number of one step interval sizes : 4 Highest interval variety : 4 Mean interval variety : 3.33333 = 10/3 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 64/63, 27.2641 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 All steps are superparticular (epimoric) Length of longest harmonic part : 2 at degree 6 Length of longest subharmonic part : 2 at degree 0 Formal octave complements present : 3 = 42.8571% 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 classes 3 and 4 combined Limited inverse transpositions with margin 195.4773 cents: 1 -- Rational properties -- Prime limit : 7 Odd number limit : 49 (O: 49 U: 49) Highest odd numerator or denominator: 9 Scale harmonicity : 0.029787 Average absolute harmonicity : 0.259901 Specific harmonicity : 0.182981 Fundamental : 1/168, -7.3923 octaves, 1.5573 Hz. Guide tone : 252, 7.9773 octaves, 65929.642 Hz. Exponens Consonantiae : 4.233600E+04, 15.36960 octaves Euler's gradus suavitatis : 24 Sum of Mann's harmonic distance : 39.0, average 5.57143 Mersenne's string divisions : 5969040 Sum of van Prooijen's expressibility: 4.55292, average 0.65042 Sum of Tenney's harmonic distance : 9.18647, average 1.31235 Vogel's harmonic complexity : 14.00000 Wille's k value : 94 Wilson's harmonic complexity : 23 Rectangular lattice diameter : 5 Triangular lattice diameter : 3 Lattice compactness : 80.25368, average 2.86620 Lattice compactness (without 2's) : 44.65633, average 1.59487 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -3 .. 2 -0.14286 11 1 3: -1 .. 2 0.71429 7 3 7: -1 .. 1 -0.14286 3 3 Average exponent except 2's : 4 / 7 = 0.57143 Average absolute exponent except 2's: 10 / 7 = 1.42857 Scale is weakly epimorphic with val: <7 10 22| Scale is JI-epimorphic with val: <7 11 20| = patent Scale is weakly epimorphic with val: <7 12 18|
FIT/MODE
7: 1 2 0 1 1 1 1 N T3 SD: 49.5680 c. M:-79.2016 c. 8: 1 2 0 1 2 0 2 N T3 SD: 55.6346 c. M: 101.9550 c. 9: 1 2 1 1 2 0 2 N D3 SD: 37.7251 c. M: 70.5767 c. 10: 2 2 0 2 2 0 2 N M DE S SD: 26.1072 c. M:-44.9159 c. 12: 2 2 1 2 2 1 2 P M ME S SD: 21.7886 c. M: 35.0841 c. G.Hypophrygian, G.Ionian (Iastian), M.Mixolydian, G.M.Hypoionian, Hypermixolydian, Mischung 3, Khamaj That, Mela Harikambhoji, Raga Balahamsa, Bhim, Devamanohari, Harini, Janjhuti, Kaamaai, Khambhavati, Surati, Gregorian nr.7, Enharmonic Byzantine Liturgical, Rast descending: Greece, Ching, Shang: China 14: 2 3 1 2 3 0 3 N SD: 18.4932 c. M: 32.4814 c. 17: 3 3 1 3 3 1 3 N M DE S SD: 10.9706 c. M:-19.4094 c. Safi al-Din's maqam `Ussaq 22: 4 4 1 4 4 1 4 N M DE S SD: 8.5378 c. M:-14.2718 c. 31: 5 6 2 5 6 1 6 N SD: 6.1537 c. M: 10.3616 c. 36: 6 7 2 6 7 1 7 N SD: 2.0943 c. M: 3.9100 c. 58: 10 11 3 10 11 2 11 N SD: 2.1045 c. M:-3.5879 c. 77: 13 15 4 13 15 2 15 N SD: 1.4486 c. M: 2.5921 c. 94: 16 18 5 16 18 3 18 N SD: 0.8160 c. M:-1.3869 c. 130: 22 25 7 22 25 4 25 N SD: 0.6982 c. M: 1.2379 c. 135: 23 26 7 23 26 4 26 N SD: 0.3154 c. M:-0.5344 c. 171: 29 33 9 29 33 5 33 N SD: 0.2527 c. M: 0.4049 c. 265: 45 51 14 45 51 8 51 N SD: 0.2088 c. M: 0.3671 c. 306: 52 59 16 52 59 9 59 N SD: 0.1338 c. M:-0.2100 c. 400: 68 77 21 68 77 12 77 N SD: 0.0972 c. M:-0.1741 c. 571: 97 110 30 97 110 17 110 N SD: 0.0360 c. M: 0.0578 c. 706: 120 136 37 120 136 21 136 N SD: 0.0316 c. M:-0.0560 c. 1277: 217 246 67 217 246 38 246 N SD: 0.0042 c. M:-0.0073 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: 16.2568 cents 4 x 5 x 6 x 7:8 S SD: 12.1926 cents 5:6 x 7:8 x 9:10 S SD: 37.1381 cents 6:7 x 8:9:10:11:12 S SD: 18.8774 cents 7:8:9 x 11:12 x 14 SD: 17.0053 cents 8:9:10:11:12 x 14:16 S SD: 12.0376 cents 9:10 x 12:14:15:16:18 S SD: 14.4803 cents 10:11:13 x 15:17:18:20 SD: 11.1406 cents 11:12:14:15:17:19 x 22 SD: 14.4360 cents 12:14:15:16:18 x 21:24 S SD: 13.2734 cents 13:15:17 x 20:22:23:26 SD: 12.4280 cents 14:16:18:19:21:24:25:28 SD: 7.7005 cents 15:17:19:20:23 x 26:30 SD: 8.4216 cents 16:18 x 21:24:27:28:32 S SD: 6.4262 cents 18:20:23:24:27:31:32:36 SD: 5.3160 cents 19:21:24:25:29 x 33:38 SD: 9.8333 cents 20:23:26:27:30:34:35:40 SD: 7.1231 cents 21:24:27:28:32:36:37:42 SD: 5.7577 cents 24:27:31:32:36:41:42:48 SD: 1.4300 cents 56:63:72:75:84:96:98:112 SD: 1.1016 cents 80:90:103:107:120:137:140:160 SD: 0.8830 cents 88:99:113:117:132:151:154:176 SD: 0.8047 cents 112:126:144:149:168:192:196:224 SD: 0.5527 cents 136:153:175:181:204:233:238:272 SD: 0.5204 cents 144:162:185:192:216:247:252:288 SD: 0.2386 cents 168:189:216:224:252:288:294:336 S SD: 0.0000 cents