Scala analysis: xen10-wilson-purvi-05-02
Purvi modulation 2 from Figure 5, Didymus/Ptolemy (10/9 9/8 16/15) all 6 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: 5/4 386.313714 major third 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 5/3 884.358713 major sixth, BP sixth 6: 15/8 1088.268715 classic major seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 9/8 203.9100 major whole tone 2: 10/9 182.4037 minor whole tone 3: 16/15 111.7313 minor diatonic semitone 4: 9/8 203.9100 major whole tone 5: 10/9 182.4037 minor whole tone 6: 9/8 203.9100 major whole tone 7: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 16/15 111.731 cents minor diatonic semitone 1: 2 10/9 182.404 cents minor whole tone 1: 3 9/8 203.910 cents major whole tone 2: 1 32/27 294.135 cents Pythagorean minor third 2: 3 6/5 315.641 cents minor third 2: 3 5/4 386.314 cents major third 3: 5 4/3 498.045 cents perfect fourth 3: 1 27/20 519.551 cents acute fourth 3: 1 45/32 590.224 cents diatonic tritone 4: 1 64/45 609.776 cents 2nd tritone 4: 1 40/27 680.449 cents grave fifth 4: 5 3/2 701.955 cents perfect fifth 5: 3 8/5 813.686 cents minor sixth 5: 3 5/3 884.359 cents major sixth, BP sixth 5: 1 27/16 905.865 cents Pythagorean major sixth 6: 3 16/9 996.090 cents Pythagorean minor seventh 6: 2 9/5 1017.596 cents just minor seventh, BP seventh 6: 2 15/8 1088.269 cents classic major seventh Highest number of different intervals for one interval class: 3 Average number of different intervals per interval class: 3.00000 = 3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 203.9 386.3 498.0 702.0 884.4 1088.3 1200.0 203.9 : 182.4 294.1 498.0 680.4 884.4 996.1 1200.0 386.3 : 111.7 315.6 498.0 702.0 813.7 1017.6 1200.0 498.0 : 203.9 386.3 590.2 702.0 905.9 1088.3 1200.0 702.0 : 182.4 386.3 498.0 702.0 884.4 996.1 1200.0 884.4 : 203.9 315.6 519.6 702.0 813.7 1017.6 1200.0 1088.3: 111.7 315.6 498.0 609.8 813.7 996.1 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 111.7313 .. 203.9100 cents 135/128, 92.1787 cents 16/15, 111.7313 cents 2: 294.1350 .. 386.3137 cents 135/128, 92.1787 cents 256/243, 90.2250 cents 3: 498.0450 .. 590.2237 cents 135/128, 92.1787 cents 16/15, 111.7313 cents 4: 609.7763 .. 701.9550 cents 135/128, 92.1787 cents 2048/2025, 19.5526 cents 5: 813.6863 .. 905.8650 cents 135/128, 92.1787 cents 16/15, 111.7313 cents 6: 996.0900 .. 1088.2687 cents 135/128, 92.1787 cents 256/243, 90.2250 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 16/15, 111.7313 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.534293 Largest / Average step : 1.189475 Largest / Smallest step : 1.825004 Median interval of one step : 10/9, 182.4037 cents, amount: 2 Most common interval of one step : 9/8, 203.9100 cents, amount: 3 Least squares average step : 175.92784 cents, oct.: 1231.49491 cents Scale is strictly proper Scale has trivalence property Step pattern alph. order: ABCABAC Step pattern size order : LMSLMLS Scale is a Constant Structure, by a margin of 19.55257 cents Scale diversity : 1.374079 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.539106 Rothenberg efficiency : 0.544218 redundancy: 0.455782 Efficiency x scale size : 3.809524 Number of different interval sizes : 18 = 3.00000 / class Number of one step interval sizes : 3 Highest interval variety : 3 Mean interval variety : 3.00000 = 3 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 2048/2025, 19.5526 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 5 Scale is a chain of 3 triads 0.0 386.314 701.955 cents Most common triad is 0.0 498.045 701.955 cents, amount: 3 Number of recognisable fifths : 5, average 701.9550 cents Scale is Lydian octave species of abstract tetrachord Best fifths form a closed circle Best major thirds form a closed circle Best minor thirds form a closed circle Scale is fully connected 5-limit major-minor triadic Scale contains a complete diamond : 1 3 All steps are superparticular (epimoric) Length of longest harmonic part : 2 at degree 0 Length of longest subharmonic part : 2 at degree 4 Formal octave complements present : 3 = 42.8571% Scale is differentially coherent in interval class 1 Scale is differentially coherent in interval class 2 Scale is differentially coherent in interval classes 3 and 4 combined Scale is differentially coherent in interval classes 4 and 5 combined Scale is n-1 differentially coherent in interval class 5 Limited inverse transpositions with margin 92.1787 cents: 2 -- Rational properties -- Prime limit : 5 Odd number limit : 45 (O: 45 U: 45) Highest odd numerator or denominator: 15 Scale harmonicity : 0.051546 Average absolute harmonicity : 0.274175 Specific harmonicity : 0.257372 Fundamental : 1/24, -4.5850 octaves, 10.9011 Hz. Guide tone : 180, 7.4919 octaves, 47092.602 Hz. Exponens Consonantiae : 4.320000E+03, 12.07682 octaves Euler's gradus suavitatis : 16 Sum of Mann's harmonic distance : 36.0, average 5.14286 Mersenne's string divisions : 985320 Sum of van Prooijen's expressibility: 4.48252, average 0.64036 Sum of Tenney's harmonic distance : 8.57200, average 1.22457 Vogel's harmonic complexity : 8.14286 Wille's k value : 22 Wilson's harmonic complexity : 14 Rectangular lattice diameter : 4 Triangular lattice diameter : 3 Lattice compactness : 78.00667, average 2.78595 Lattice compactness (without 2's) : 41.76998, average 1.49178 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -3 .. 2 -0.85714 12 1 3: -1 .. 2 0.28571 6 3 5: 0 .. 1 0.42857 3 3 Average exponent except 2's : 5 / 7 = 0.71429 Average absolute exponent except 2's: 9 / 7 = 1.28571 Scale is JI-epimorphic with val: <7 11 16| = patent
FIT/MODE
7: 1 1 1 1 1 1 1 SP B ME I SD: 33.3285 c. M: 59.6973 c. Lebeng 10: 2 1 1 2 1 2 1 P M ME S SD: 25.8629 c. M: 44.3587 c. Sethares Neutral, Beatles-7, Dichotic-7 12: 2 2 1 2 2 2 1 P M ME S SD: 9.2003 c. M:-15.6413 c. G.Lydian, M.Ionian, M.Hypolydian, Major, Bilaval That, Mela Shankarabharanam, Raga Atana, Ghana Heptatonic, Peruvian Major, Matzore, Rast ascending: Greece, 4th plagal Byzantine, Ararai: Ethiopia, Makam Cargah, Ajam Ashiran, Dastgah-e Mahur, Dastgah-e Rast Panjgah, Xin: China, DS2, Heptatonia prima 19: 3 3 2 3 3 3 2 SP M ME S SD: 9.0995 c. M: 14.5845 c. Nineteen-tone Major, Ionian 22: 4 3 2 4 3 4 2 P T3 SD: 8.1766 c. M:-14.2718 c. Twenty-two tone "Just" Major 29: 5 4 3 5 4 5 3 SP T3 SD: 9.2386 c. M: 15.3932 c. Twenty-nine tone "Just" Major 31: 5 5 3 5 5 5 3 SP M DE S SD: 5.5622 c. M: 10.3616 c. Thirty-one tone Major, Intense Diatonic Lydian, M.Ionian 34: 6 5 3 6 5 6 3 P T3 SD: 4.3829 c. M:-7.8547 c. Thirty-four tone "Just" Major 41: 7 6 4 7 6 7 4 SP T3 SD: 3.8489 c. M: 6.3099 c. Forty-one tone "Just" Major 46: 8 7 4 8 7 8 4 P T3 SD: 4.1494 c. M:-7.3835 c. Forty-six tone "Just" Major 53: 9 8 5 9 8 9 5 SP T3 SD: 0.9247 c. M: 1.4763 c. Fifty-three tone "Just" Major 118: 20 18 11 20 18 20 11 SP T3 SD: 0.2902 c. M: 0.5202 c. 171: 29 26 16 29 26 29 16 SP T3 SD: 0.3133 c. M: 0.5494 c. 289: 49 44 27 49 44 49 27 SP T3 SD: 0.2608 c. M: 0.4498 c. 323: 55 49 30 55 49 55 30 SP T3 SD: 0.2307 c. M:-0.4244 c. 376: 64 57 35 64 57 64 35 SP T3 SD: 0.2071 c. M:-0.3453 c. 388: 66 59 36 66 59 66 36 SP T3 SD: 0.2183 c. M:-0.3911 c. 441: 75 67 41 75 67 75 41 SP T3 SD: 0.1059 c. M:-0.1716 c. 494: 84 75 46 84 75 84 46 SP T3 SD: 0.0903 c. M: 0.1482 c. 559: 95 85 52 95 85 95 52 SP T3 SD: 0.0609 c. M:-0.1034 c. 612: 104 93 57 104 93 104 57 SP T3 SD: 0.0264 c. M: 0.0450 c. 1171: 199 178 109 199 178 199 109 SP T3 SD: 0.0178 c. M:-0.0319 c. 1783: 303 271 166 303 271 303 166 SP T3 SD: 0.0086 c. M:-0.0160 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: 0.0000 cents 4 x 5 x 6:7 x 8 S SD: 21.1168 cents 5 x 6:7 x 8:9:10 S SD: 29.7446 cents 6:7 x 8:9:10:11:12 S SD: 12.3353 cents 7:8:9 x 11:12:13:14 SD: 18.4550 cents 8:9:10:11:12:13:15:16 SD: 9.8552 cents 10:11 x 13:15:17:19:20 SD: 11.9443 cents 12:14:15:16:18:20:23:24 SD: 10.5094 cents 14:16:18:19:21:23:26:28 SD: 10.0568 cents 15:17:19:20:23:25:28:30 SD: 6.6951 cents 16:18:20:21:24:27:30:32 S SD: 4.9608 cents 24:27:30:32:36:40:45:48 S SD: 0.0000 cents