Scala analysis: trithagorean13--tritavewith5_3generator
Tritave scale with a 5/3 generator.
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 27/25 133.237575 large limma, BP small semitone, Zarlino semitone 2: 729/625 266.475150 two (large limma, BP small semitone, Zarlino semitone) 3: 19683/15625 399.712725 4: 3125/2187 617.883563 5: 125/81 751.121138 6: 5/3 884.358713 major sixth, BP sixth 7: 9/5 1017.596288 just minor seventh, BP seventh 8: 243/125 1150.833863 octave - maximal diesis 9: 6561/3125 1284.071438 ripple +1 octave 10: 15625/6561 1502.242276 11: 625/243 1635.479851 12: 25/9 1768.717426 classic augmented eleventh, BP twelfth 13: 3/1 1901.955001 perfect 12th
SHOW/INTERVAL
0: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 1: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 2: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 3: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 4: 48828125/43046721 218.1708 5: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 6: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 7: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 8: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 9: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 10: 48828125/43046721 218.1708 11: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 12: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 13: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 11 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 1: 2 48828125/43046721 218.171 cents 2: 9 729/625 266.475 cents two (large limma, BP small semitone, Zarlino semitone) 2: 4 1953125/1594323 351.408 cents 3: 7 19683/15625 399.713 cents 3: 6 78125/59049 484.646 cents 4: 5 531441/390625 532.950 cents 4: 8 3125/2187 617.884 cents 5: 3 14348907/9765625 666.188 cents 5: 10 125/81 751.121 cents 6: 1 387420489/244140625 799.425 cents 6: 12 5/3 884.359 cents major sixth, BP sixth 7: 12 9/5 1017.596 cents just minor seventh, BP seventh 7: 1 244140625/129140163 1102.530 cents 8: 10 243/125 1150.834 cents octave - maximal diesis 8: 3 9765625/4782969 1235.767 cents Fifives comma +1 octave 9: 8 6561/3125 1284.071 cents ripple +1 octave 9: 5 390625/177147 1369.005 cents 10: 6 177147/78125 1417.309 cents 10: 7 15625/6561 1502.242 cents 11: 4 4782969/1953125 1550.547 cents 11: 9 625/243 1635.480 cents 12: 2 129140163/48828125 1683.784 cents 12: 11 25/9 1768.717 cents classic augmented eleventh, BP twelfth 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 8 9 10 11 12 13 0.0 : 133.2 266.5 399.7 617.9 751.1 884.4 1017.6 1150.8 1284.1 1502.2 1635.5 1768.7 1902.0 133.2 : 133.2 266.5 484.6 617.9 751.1 884.4 1017.6 1150.8 1369.0 1502.2 1635.5 1768.7 1902.0 266.5 : 133.2 351.4 484.6 617.9 751.1 884.4 1017.6 1235.8 1369.0 1502.2 1635.5 1768.7 1902.0 399.7 : 218.2 351.4 484.6 617.9 751.1 884.4 1102.5 1235.8 1369.0 1502.2 1635.5 1768.7 1902.0 617.9 : 133.2 266.5 399.7 533.0 666.2 884.4 1017.6 1150.8 1284.1 1417.3 1550.5 1683.8 1902.0 751.1 : 133.2 266.5 399.7 533.0 751.1 884.4 1017.6 1150.8 1284.1 1417.3 1550.5 1768.7 1902.0 884.4 : 133.2 266.5 399.7 617.9 751.1 884.4 1017.6 1150.8 1284.1 1417.3 1635.5 1768.7 1902.0 1017.6: 133.2 266.5 484.6 617.9 751.1 884.4 1017.6 1150.8 1284.1 1502.2 1635.5 1768.7 1902.0 1150.8: 133.2 351.4 484.6 617.9 751.1 884.4 1017.6 1150.8 1369.0 1502.2 1635.5 1768.7 1902.0 1284.1: 218.2 351.4 484.6 617.9 751.1 884.4 1017.6 1235.8 1369.0 1502.2 1635.5 1768.7 1902.0 1502.2: 133.2 266.5 399.7 533.0 666.2 799.4 1017.6 1150.8 1284.1 1417.3 1550.5 1683.8 1902.0 1635.5: 133.2 266.5 399.7 533.0 666.2 884.4 1017.6 1150.8 1284.1 1417.3 1550.5 1768.7 1902.0 1768.7: 133.2 266.5 399.7 533.0 751.1 884.4 1017.6 1150.8 1284.1 1417.3 1635.5 1768.7 1902.0 1902.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 133.2376 .. 218.1708 cents 1220703125/1162261467, 84.9333 cen 27/25, 133.2376 cents 2: 266.4751 .. 351.4084 cents 1220703125/1162261467, 84.9333 cen 48.30431 cents 3: 399.7127 .. 484.6460 cents 1220703125/1162261467, 84.9333 cen 48.30431 cents 4: 532.9503 .. 617.8836 cents 1220703125/1162261467, 84.9333 cen 48.30431 cents 5: 666.1879 .. 751.1211 cents 1220703125/1162261467, 84.9333 cen 48.30431 cents 6: 799.4254 .. 884.3587 cents 1220703125/1162261467, 84.9333 cen 48.30431 cents 7: 1017.5963 .. 1102.5296 cents 1220703125/1162261467, 84.9333 cen 27/25, 133.2376 cents 8: 1150.8339 .. 1235.7671 cents 1220703125/1162261467, 84.9333 cen 48.30431 cents 9: 1284.0714 .. 1369.0047 cents 1220703125/1162261467, 84.9333 cen 48.30431 cents 10: 1417.3090 .. 1502.2423 cents 1220703125/1162261467, 84.9333 cen 48.30431 cents 11: 1550.5466 .. 1635.4799 cents 1220703125/1162261467, 84.9333 cen 48.30431 cents 12: 1683.7842 .. 1768.7174 cents 1220703125/1162261467, 84.9333 cen 48.30431 cents
SHOW DATA
Number of notes : 13 -- Interval properties -- Smallest interval : 27/25, 133.2376 cents, class 1 Average step (divided formal octave): 146.3042 cents Largest one step interval : 48828125/43046721, 218.1708 cents Average / Smallest step : 1.098070 Largest / Average step : 1.491213 Largest / Smallest step : 1.637457 Median interval of one step : 27/25, 133.2376 cents, amount: 11 Most common interval of one step : 27/25, 133.2376 cents, amount: 11 Least squares average step : 146.82275 cents, oct.: 1908.69574 cents Scale is strictly proper Scale has Myhill's property chromas: 1220703125/1162261467, 84.9333 cents and 48.30431 cents, ratio: 1.758296 generators: 6 of 884.35871 cents and 7 of 1017.59629 cents ET pair of linear temperament : 13&15 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 1142-tone equal temperament with octave 3/1 degrees: 80 160 240 371 451 531 611 691 771 902 982 1062 1142 generators: 531 and 611 Number of contiguous 1-step segments: 0 Step pattern alph. order: AAABAAAAABAAA Step pattern size order : SSSLSSSSSLSSS Scale is a Constant Structure, by a margin of 48.30431 cents Scale diversity : 0.294993 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.464131 Rothenberg efficiency : 0.717949 redundancy: 0.282051 Efficiency x scale size : 9.333333 Number of different interval sizes : 24 = 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 : 48.30431 cents Most common intervals : 5/3, 884.3587 cents & inv., amount: 12 Scale is a chain of 6 triads 0.0 884.359 1768.717 cents Most common triad is 0.0 884.359 1017.596 cents, amount: 11 Number of recognisable fifths : 0 Number of recognisable fourths : 6, average 484.6460 cents Number of appreciable fourths : 0 Scale contains two identical disjunct tetrachords Best fifths form a closed circle Best minor thirds form a closed circle Scale is a complete genus musicum : 3^m 5^12 Formal octave complements present : 13 = 100.0000% 2/1 octave complements present : 0 = 0.0000% Inversional symmetry on degrees : 0 Inversional symmetry on intervals : 6-7 -- Rational properties -- Prime limit : 5 Odd number limit : 387420489 (O: 387420489 U: 244140625) Highest odd numerator or denominator: 19683 Scale harmonicity : 0.008188 Average absolute harmonicity : 0.066072 Specific harmonicity : 0.047194 Fundamental : 1/102515625, -26.6113 octaves, 0.0000 Hz. Guide tone : 307546875, 28.1962 octaves, 80462125028.431 Hz. Exponens Consonantiae : 3.152836E+16, 54.80750 octaves Euler's gradus suavitatis : 83 Sum of Mann's harmonic distance : 37700.0, average 2900.00000 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 30.90049, average 2.37696 Sum of Tenney's harmonic distance : 58.46114, average 4.49701 Vogel's harmonic complexity : 55.61538 Wille's k value : 153773437 Wilson's harmonic complexity : 111 Rectangular lattice diameter : 29 Triangular lattice diameter : 17 Lattice compactness : 711.25226, average 7.81596 Lattice compactness (without 2's) : 711.25226, average 7.81596 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 3: -8 .. 9 0.53846 61 1 5: -6 .. 6 0.00000 42 12 Average exponent except 2's : 7 / 13 = 0.53846 Average absolute exponent except 2's: 103 / 13 = 7.92308 Scale is weakly epimorphic with val: <13 17| Scale is weakly epimorphic with val: <13 18| Scale is JI-epimorphic with val: <13 19| = patent Scale is weakly epimorphic with val: <13 20| Scale is weakly epimorphic with val: <13 21|
FIT/MODE
13: 1 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 24.4455 c. M: 39.2000 c. 15: 1 1 1 2 1 1 1 1 1 2 1 1 1 P M ME I SD: 12.0492 c. M: 19.3217 c. 28: 2 2 2 3 2 2 2 2 2 3 2 2 2 SP M ME I SD: 4.8947 c. M: 7.8491 c. 43: 3 3 3 5 3 3 3 3 3 5 3 3 3 SP M DE I SD: 1.0159 c. M: 1.6291 c. 114: 8 8 8 13 8 8 8 8 8 13 8 8 8 SP M DE I SD: 0.4358 c. M:-0.6989 c. 157: 11 11 11 18 11 11 11 11 11 18 11 11 11 SP M DE I SD: 0.0382 c. M: 0.0613 c. 671: 47 47 47 77 47 47 47 47 47 77 47 47 47 SP M DE I SD: 0.0294 c. M: 0.0471 c. 828: 58 58 58 95 58 58 58 58 58 95 58 58 58 SP M DE I SD: 0.0165 c. M: 0.0265 c. 985: 69 69 69 113 69 69 69 69 69 113 69 69 69 SP M DE I SD: 0.0078 c. M: 0.0125 c. 1142: 80 80 80 131 80 80 80 80 80 131 80 80 80 SP M DE I SD: 0.0015 c. M: 0.0024 c.
FIT/HARMONIC
1 x x x x x x x 2 x x x x 3 S SD: 24.5831 cents 2 x x x x 3 x x 4 x x 5 x 6 S SD: 21.2896 cents 3 x x 4 x x 5 x 6 x 7:8 x 9 S SD: 21.8912 cents 4 x x 5 x 6 x 7:8 x x 10:11:12 S SD: 14.3660 cents 5 x 6 x 7:8 x 9:10:11:12:13:14:15 S SD: 13.1419 cents 6 x 7:8 x 9:10:11:12:13:14:15:17:18 SD: 13.8848 cents 7 x 8:9:10:11:12:13:14:15:17:18:19:21 SD: 10.2764 cents 8 x 9:10:11:12:13:14:16:17:19:21:22:24 SD: 11.6865 cents 9:10 x 11:13:14:15:16:18:19:21:23:25:27 SD: 8.3499 cents 11:12:13:14:16:17:18:20:21:23:26:28:31:33 SD: 5.7335 cents 12:13:14:15:17:19:20:22:23:25:29:31:33:36 SD: 5.5462 cents 16:17:19:20:23:25:27:29:31:34:38:41:44:48 SD: 4.8378 cents 18:19:21:23:26:28:30:32:35:38:43:46:50:54 SD: 4.5024 cents 19:21:22:24:27:29:32:34:37:40:45:49:53:57 SD: 4.1194 cents 22:24:26:28:31:34:37:40:43:46:52:57:61:66 SD: 4.0364 cents 23:25:27:29:33:35:38:41:45:48:55:59:64:69 SD: 3.1827 cents 24:26:28:30:34:37:40:43:47:50:57:62:67:72 SD: 2.4242 cents 30:32:35:38:43:46:50:54:58:63:71:77:83:90 SD: 2.4017 cents 35:38:41:44:50:54:58:63:68:73:83:90:97:105 SD: 1.6341 cents 42:45:49:53:60:65:70:76:82:88:100:108:117:126 SD: 1.5389 cents 49:53:57:62:70:76:82:88:95:103:117:126:136:147 SD: 1.2824 cents 54:58:63:68:77:83:90:97:105:113:129:139:150:162 SD: 1.1634 cents 61:66:71:77:87:94:102:110:119:128:145:157:169:183 SD: 0.9878 cents 72:78:84:91:103:111:120:130:140:151:171:185:200:216 SD: 0.8637 cents 77:83:90:97:110:119:128:139:150:162:183:198:214:231 SD: 0.8260 cents 84:91:98:106:120:130:140:151:163:176:200:216:233:252 SD: 0.7548 cents 89:96:104:112:127:137:148:160:173:187:212:229:247:267 SD: 0.6298 cents 112:121:131:141:160:173:187:202:218:235:267:288:311:336 SD: 0.5865 cents 126:136:147:159:180:194:210:227:245:265:300:324:350:378 SD: 0.4740 cents 138:149:161:174:197:213:230:248:268:290:329:355:383:414 SD: 0.3841 cents 162:175:189:204:231:250:270:292:315:340:386:417:450:486 SD: 0.3655 cents 180:194:210:227:257:278:300:324:350:378:429:463:500:540 SD: 0.3644 cents 186:201:217:234:266:287:310:335:362:391:443:478:517:558 SD: 0.3575 cents 199:215:232:251:284:307:332:358:387:418:474:512:553:597 SD: 0.3147 cents 210:227:245:265:300:324:350:378:408:441:500:540:583:630 SD: 0.2914 cents 212:229:247:267:303:327:353:382:412:445:505:545:589:636 SD: 0.2697 cents 234:253:273:295:334:361:390:421:455:491:557:602:650:702 SD: 0.2683 cents 247:267:288:311:353:381:412:445:480:519:588:635:686:741 SD: 0.2670 cents 252:272:294:317:360:389:420:454:490:529:600:648:700:756 SD: 0.2483 cents 265:286:309:334:379:409:442:477:515:556:631:682:736:795 SD: 0.2392 cents 289:312:337:364:413:446:482:520:562:607:688:743:803:867 SD: 0.1650 cents 324:350:378:408:463:500:540:583:630:680:772:833:900:972 SD: 0.1306 cents 450:486:525:567:643:694:750:810:875:945:1072:1157:1250:1350 SD: 0.1217 cents 462:499:539:582:660:713:770:832:898:970:1100:1188:1283:1386 SD: 0.0977 cents 600:648:700:756:857:926:1000:1080:1166:1260:1429:1543:1667:1800 SD: 0.0942 cents