Scala analysis: xen12-chalmers-tritriadic-dm-1-5-27
Tritriadic D->M scale built from 1:5:27
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 729/640 225.416291 acute major second 2: 5/4 386.313714 major third 3: 27/20 519.551289 acute fourth 4: 40/27 680.448711 grave fifth 5: 27/16 905.865003 Pythagorean major sixth 6: 50/27 1066.762425 grave major seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 1: 729/640 225.4163 acute major second 2: 800/729 160.8974 grave whole tone 3: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 4: 800/729 160.8974 grave whole tone 5: 729/640 225.4163 acute major second 6: 800/729 160.8974 grave whole tone 7: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 1: 3 800/729 160.897 cents grave whole tone 1: 2 729/640 225.416 cents acute major second 2: 3 32/27 294.135 cents Pythagorean minor third 2: 1 19683/16000 358.654 cents 2: 3 5/4 386.314 cents major third 3: 1 25600/19683 455.032 cents 3: 5 27/20 519.551 cents acute fourth 3: 1 1000/729 547.211 cents 4: 1 729/500 652.789 cents 4: 5 40/27 680.449 cents grave fifth 4: 1 19683/12800 744.968 cents 5: 3 8/5 813.686 cents minor sixth 5: 1 32000/19683 841.346 cents 5: 3 27/16 905.865 cents Pythagorean major sixth 6: 2 1280/729 974.584 cents grave minor seventh 6: 3 729/400 1039.103 cents acute minor seventh 6: 2 50/27 1066.762 cents grave 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 : 225.4 386.3 519.6 680.4 905.9 1066.8 1200.0 225.4 : 160.9 294.1 455.0 680.4 841.3 974.6 1200.0 386.3 : 133.2 294.1 519.6 680.4 813.7 1039.1 1200.0 519.6 : 160.9 386.3 547.2 680.4 905.9 1066.8 1200.0 680.4 : 225.4 386.3 519.6 745.0 905.9 1039.1 1200.0 905.9 : 160.9 294.1 519.6 680.4 813.7 974.6 1200.0 1066.8: 133.2 358.7 519.6 652.8 813.7 1039.1 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 133.2376 .. 225.4163 cents 135/128, 92.1787 cents 27/25, 133.2376 cents 2: 294.1350 .. 386.3137 cents 135/128, 92.1787 cents 20480/19683, 68.7187 cents 3: 455.0324 .. 547.2111 cents 135/128, 92.1787 cents 20480/19683, 68.7187 cents 4: 652.7889 .. 744.9676 cents 135/128, 92.1787 cents 531441/500000, 105.5777 cents 5: 813.6863 .. 905.8650 cents 135/128, 92.1787 cents 20480/19683, 68.7187 cents 6: 974.5837 .. 1066.7624 cents 135/128, 92.1787 cents 20480/19683, 68.7187 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 27/25, 133.2376 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 729/640, 225.4163 cents Average / Smallest step : 1.286638 Largest / Average step : 1.314928 Largest / Smallest step : 1.691837 Median interval of one step : 800/729, 160.8974 cents, amount: 3 Most common interval of one step : 800/729, 160.8974 cents, amount: 3 Least squares average step : 175.77423 cents, oct.: 1230.41960 cents Scale is strictly proper Scale has trivalence property Scale is a mode of a 1432-tone equal temperament with octave 2/1 degrees: 269 461 620 812 1081 1273 1432 Least number of segments generator : 717 of 600.838 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 1229 of 1029.888 cents and inv. generated superset size: 46 = 39 more = 657.143% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABCBABC Step pattern size order : LMSMLMS Scale is a Constant Structure, by a margin of 68.71871 cents Scale diversity : 1.396467 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 : 20000/19683, 27.6598 cents Most common intervals : 27/20, 519.5513 cents & inv., amount: 5 Scale is a chain of 3 triads 0.0 294.135 680.449 cents Most common triad is 0.0 519.551 680.449 cents, amount: 3 Number of recognisable fifths : 0 Scale contains two identical disjunct tetrachords Best fifths form a closed circle Best minor thirds form a closed circle Scale contains a complete diamond : 5 27 Formal octave complements present : 3 = 42.8571% Scale is n-1 differentially coherent in interval classes 1 and 2 combined Scale is n-1 differentially coherent in interval class 2 Limited inverse transpositions with margin 92.1787 cents: 5 -- Rational properties -- Prime limit : 5 Odd number limit : 19683 (O: 19683 U: 19683) Highest odd numerator or denominator: 729 Scale harmonicity : 0.018797 Average absolute harmonicity : 0.200101 Specific harmonicity : 0.107454 Fundamental : 1/17280, -14.0768 octaves, 0.0151 Hz. Guide tone : 145800, 17.1536 octaves, 38145007.421 Hz. Exponens Consonantiae : 2.519424E+09, 31.23045 octaves Euler's gradus suavitatis : 41 Sum of Mann's harmonic distance : 807.5, average 115.35714 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 9.28715, average 1.32674 Sum of Tenney's harmonic distance : 18.80260, average 2.68609 Vogel's harmonic complexity : 23.57143 Wille's k value : 9841 Wilson's harmonic complexity : 42 Rectangular lattice diameter : 12 Triangular lattice diameter : 9 Lattice compactness : 159.12016, average 5.68286 Lattice compactness (without 2's) : 111.58913, average 3.98533 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -7 .. 3 -1.42857 20 1 3: -3 .. 6 0.85714 18 1 5: -1 .. 2 0.28571 6 5 Average exponent except 2's : 8 / 7 = 1.14286 Average absolute exponent except 2's: 24 / 7 = 3.42857 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: 35.2359 c. M: 53.9877 c. Lebeng 9: 2 1 1 1 2 1 1 P M ME S SD: 20.7823 c. M:-41.2504 c. 12: 2 2 1 2 2 2 1 P M ME S SD: 19.7735 c. M:-33.2376 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 16: 3 2 2 2 3 2 2 SP M ME S SD: 8.4762 c. M: 16.7624 c. Antiphrygian 25: 5 3 3 3 5 3 3 SP M DE S SD: 8.5713 c. M:-14.5837 c. Antiphrygian 28: 5 4 3 4 5 4 3 SP T3 SD: 5.8048 c. M: 11.1306 c. 37: 7 5 4 5 7 5 4 SP T3 SD: 2.0354 c. M:-3.5078 c. 53: 10 7 6 7 10 7 6 SP T3 SD: 1.3490 c. M: 2.6115 c. 90: 17 12 10 12 17 12 10 SP T3 SD: 0.6259 c. M:-1.2504 c. 143: 27 19 16 19 27 19 16 SP T3 SD: 0.7301 c. M:-1.1571 c. 171: 32 23 19 23 32 23 19 SP T3 SD: 0.4395 c. M: 0.8549 c. 208: 39 28 23 28 39 28 23 SP T3 SD: 0.3242 c. M:-0.5453 c. 261: 49 35 29 35 49 35 29 SP T3 SD: 0.0856 c. M: 0.1289 c. 559: 105 75 62 75 105 75 62 SP T3 SD: 0.0714 c. M:-0.1428 c. 612: 115 82 68 82 115 82 68 SP T3 SD: 0.0572 c. M: 0.0958 c. 820: 154 110 91 110 154 110 91 SP T3 SD: 0.0396 c. M:-0.0668 c. 1081: 203 145 120 145 203 145 120 SP T3 SD: 0.0360 c. M: 0.0694 c. 1171: 220 157 130 157 220 157 130 SP T3 SD: 0.0195 c. M:-0.0320 c. 1432: 269 192 159 192 269 192 159 SP T3 SD: 0.0018 c. M:-0.0027 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: 10.7531 cents 3 x x 4 x 5 x 6 S SD: 10.1382 cents 4 x 5 x 6:7 x 8 S SD: 16.6332 cents 5 x 6:7 x 8:9:10 S SD: 28.1942 cents 6:7 x 8:9:10:11:12 S SD: 9.7308 cents 7:8:9 x 10:12:13:14 S SD: 14.0865 cents 8:9:10:11:12:14:15:16 S SD: 11.3935 cents 10:11 x 14:15:17:19:20 SD: 16.8441 cents 11:13:14:15:16:19:20:22 SD: 13.5452 cents 12:14:15:16:18:20:22:24 S SD: 8.3407 cents 15:17:19:20:22:25:28:30 SD: 6.4208 cents 19:22:24:26:28:32:35:38 SD: 6.1668 cents 20:23:25:27:30:34:37:40 SD: 4.2916 cents 23:26:29:31:34:39:43:46 SD: 3.9413 cents 28:32:35:38:41:47:52:56 SD: 3.5968 cents 29:33:36:39:43:49:54:58 SD: 2.4265 cents 35:40:44:47:52:59:65:70 SD: 2.3342 cents 48:55:60:65:71:81:89:96 SD: 1.7246 cents 52:59:65:70:77:88:96:104 SD: 1.5887 cents 60:68:75:81:89:101:111:120 SD: 1.4442 cents 71:81:89:96:105:120:131:142 SD: 1.3949 cents 77:88:96:104:114:130:143:154 SD: 1.2815 cents 80:91:100:108:119:135:148:160 SD: 1.0871 cents 92:105:115:124:136:155:170:184 SD: 1.0647 cents 100:114:125:135:148:169:185:200 SD: 0.5458 cents 108:123:135:146:160:182:200:216 SD: 0.4813 cents 160:182:200:216:237:270:296:320 SD: 0.4219 cents 195:222:244:263:289:329:361:390 SD: 0.3919 cents 208:237:260:281:308:351:385:416 SD: 0.2558 cents 237:270:296:320:351:400:439:474 SD: 0.2411 cents 332:378:415:448:492:560:615:664 SD: 0.2183 cents 380:433:475:513:563:641:704:760 SD: 0.1684 cents 403:459:504:544:597:680:746:806 SD: 0.1628 cents 432:492:540:583:640:729:800:864 SD: 0.0928 cents 540:615:675:729:800:911:1000:1080 SD: 0.0776 cents