Scala analysis: xen12-chalmers-tritriadic-dm-3-5-15
Tritriadic D->M scale built from 3:5:15
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 10/9 182.403712 minor 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: 10/9 182.4037 minor whole tone 2: 9/8 203.9100 major 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 : 182.4 386.3 498.0 702.0 884.4 1088.3 1200.0 182.4 : 203.9 315.6 519.6 702.0 905.9 1017.6 1200.0 386.3 : 111.7 315.6 498.0 702.0 813.7 996.1 1200.0 498.0 : 203.9 386.3 590.2 702.0 884.4 1088.3 1200.0 702.0 : 182.4 386.3 498.0 680.4 884.4 996.1 1200.0 884.4 : 203.9 315.6 498.0 702.0 813.7 1017.6 1200.0 1088.3: 111.7 294.1 498.0 609.8 813.7 996.1 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 : 175.77423 cents, oct.: 1230.41960 cents Scale is strictly proper Scale has trivalence property Step pattern alph. order: ABCBABC Step pattern size order : MLSLMLS 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 0.0 315.641 701.955 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 Dorian octave species of abstract tetrachord Scale contains two identical disjunct tetrachords 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 3 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 class 2 Scale is differentially coherent in interval classes 3 and 4 combined Scale is n-1 differentially coherent in interval classes 4 and 5 combined 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 : 0.268251 Specific harmonicity : 0.256959 Fundamental : 1/72, -6.1699 octaves, 3.6337 Hz. Guide tone : 60, 5.9069 octaves, 15697.534 Hz. Exponens Consonantiae : 4.320000E+03, 12.07682 octaves Euler's gradus suavitatis : 16 Sum of Mann's harmonic distance : 37.0, average 5.28571 Mersenne's string divisions : 1101240 Sum of van Prooijen's expressibility: 4.48252, average 0.64036 Sum of Tenney's harmonic distance : 8.66891, average 1.23842 Vogel's harmonic complexity : 10.71429 Wille's k value : 67 Wilson's harmonic complexity : 14 Rectangular lattice diameter : 4 Triangular lattice diameter : 3 Lattice compactness : 73.81840, average 2.63637 Lattice compactness (without 2's) : 42.00604, average 1.50022 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -3 .. 2 -0.28571 10 1 3: -2 .. 1 -0.28571 6 2 5: 0 .. 1 0.57143 4 4 Average exponent except 2's : 2 / 7 = 0.28571 Average absolute exponent except 2's: 10 / 7 = 1.42857 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: 31.2614 c. M: 59.6973 c. Lebeng 9: 1 2 1 1 2 1 1 P M ME S SD: 33.7023 c. M: 49.0704 c. Pelog Saptanada: Sunda 10: 1 2 1 2 1 2 1 P M ME I SD: 32.2356 c. M: 62.4037 c. 12: 2 2 1 2 2 2 1 P M ME S SD: 11.2559 c. M:-17.5963 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: 7.7568 c. M: 14.5845 c. Nineteen-tone Major, Ionian 31: 5 5 3 5 5 5 3 SP M DE S SD: 5.7744 c. M:-11.1447 c. Thirty-one tone Major, Intense Diatonic Lydian, M.Ionian 34: 5 6 3 6 5 6 3 P T3 SD: 3.9275 c. M: 5.9331 c. 41: 6 7 4 7 6 7 4 SP T3 SD: 4.6124 c. M: 6.7940 c. 46: 7 8 4 8 7 8 4 P T3 SD: 3.7352 c. M:-7.3835 c. 53: 8 9 5 9 8 9 5 SP T3 SD: 1.0408 c. M: 1.4763 c. 118: 18 20 11 20 18 20 11 SP T3 SD: 0.3246 c. M:-0.6471 c. 171: 26 29 16 29 26 29 16 SP T3 SD: 0.2748 c. M: 0.5494 c. 289: 44 49 27 49 44 49 27 SP T3 SD: 0.2271 c. M: 0.3794 c. 323: 49 55 30 55 49 55 30 SP T3 SD: 0.2146 c. M: 0.3604 c. 376: 57 64 35 64 57 64 35 SP T3 SD: 0.2449 c. M: 0.4888 c. 388: 59 66 36 66 59 66 36 SP T3 SD: 0.2046 c. M:-0.3911 c. 441: 67 75 41 75 67 75 41 SP T3 SD: 0.0905 c. M:-0.1667 c. 559: 85 95 52 95 85 95 52 SP T3 SD: 0.0649 c. M:-0.1034 c. 612: 93 104 57 104 93 104 57 SP T3 SD: 0.0323 c. M: 0.0508 c. 1171: 178 199 109 199 178 199 109 SP T3 SD: 0.0165 c. M:-0.0319 c. 1783: 271 303 166 303 271 303 166 SP T3 SD: 0.0082 c. M: 0.0145 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: 15.4994 cents 7:8:9 x 11:12:13:14 SD: 19.6471 cents 8:9:10:11:12:13:15:16 SD: 10.3230 cents 10:11 x 13:15:17:19:20 SD: 10.4417 cents 12:13:15:16:18:20:23:24 SD: 8.2919 cents 15:17:19:20:23:25:28:30 SD: 8.0919 cents 16:18:20:21:24:27:30:32 S SD: 5.8351 cents 18:20:23:24:27:30:34:36 SD: 5.7341 cents 20:22:25:27:30:33:38:40 SD: 5.7032 cents 24:27:30:32:36:40:45:48 S SD: 3.0723 cents 36:40:45:48:54:60:68:72 SD: 1.8252 cents 48:53:60:64:72:80:90:96 SD: 1.5506 cents 63:70:79:84:95:105:118:126 SD: 1.5448 cents 64:71:80:85:96:107:120:128 SD: 1.2969 cents 72:80:90:96:108:120:135:144 S SD: 0.0000 cents