Scala analysis: Descartes_Hexa
Descartes 3 nested ~1650 hexachords:voices(B)-flat,naturall and in [B]
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: 9/8 203.910002 major whole tone 3: 5/4 386.313714 major third 4: 4/3 498.044999 perfect fourth 5: 40/27 680.448711 grave fifth 6: 3/2 701.955001 perfect fifth 7: 5/3 884.358713 major sixth, BP sixth 8: 27/16 905.865003 Pythagorean major sixth 9: 16/9 996.089998 Pythagorean minor seventh 10: 9/5 1017.596288 just minor seventh, BP seventh 11: 15/8 1088.268715 classic major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 10/9 182.4037 minor whole tone 2: 81/80 21.5063 syntonic comma, Didymus comma 3: 10/9 182.4037 minor whole tone 4: 16/15 111.7313 minor diatonic semitone 5: 10/9 182.4037 minor whole tone 6: 81/80 21.5063 syntonic comma, Didymus comma 7: 10/9 182.4037 minor whole tone 8: 81/80 21.5063 syntonic comma, Didymus comma 9: 256/243 90.2250 limma, Pythagorean minor second 10: 81/80 21.5063 syntonic comma, Didymus comma 11: 25/24 70.6724 classic chromatic semitone, minor chroma 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 4 81/80 21.506 cents syntonic comma, Didymus comma 1: 1 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 1 256/243 90.225 cents limma, Pythagorean minor second 1: 2 16/15 111.731 cents minor diatonic semitone 1: 4 10/9 182.404 cents minor whole tone 2: 1 135/128 92.179 cents major chroma, major limma 2: 2 16/15 111.731 cents minor diatonic semitone 2: 1 10/9 182.404 cents minor whole tone 2: 5 9/8 203.910 cents major whole tone 2: 3 32/27 294.135 cents Pythagorean minor third 3: 1 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 3: 1 10/9 182.404 cents minor whole tone 3: 1 9/8 203.910 cents major whole tone 3: 1 729/640 225.416 cents acute major second 3: 1 32/27 294.135 cents Pythagorean minor third 3: 3 6/5 315.641 cents minor third 3: 1 100/81 364.807 cents grave major third 3: 2 5/4 386.314 cents major third 3: 1 320/243 476.539 cents grave fourth 4: 1 9/8 203.910 cents major whole tone 4: 1 32/27 294.135 cents Pythagorean minor third 4: 2 6/5 315.641 cents minor third 4: 2 5/4 386.314 cents major third 4: 1 81/64 407.820 cents Pythagorean major third 4: 5 4/3 498.045 cents perfect fourth 5: 1 6/5 315.641 cents minor third 5: 1 243/200 337.148 cents acute minor third 5: 1 5/4 386.314 cents major third 5: 1 81/64 407.820 cents Pythagorean major third 5: 1 320/243 476.539 cents grave fourth 5: 1 4/3 498.045 cents perfect fourth 5: 2 27/20 519.551 cents acute fourth 5: 1 25/18 568.717 cents classic augmented fourth 5: 1 64/45 609.776 cents 2nd tritone 5: 2 40/27 680.449 cents grave fifth 6: 1 81/64 407.820 cents Pythagorean major third 6: 3 4/3 498.045 cents perfect fourth 6: 1 27/20 519.551 cents acute fourth 6: 1 45/32 590.224 cents diatonic tritone 6: 1 64/45 609.776 cents 2nd tritone 6: 1 40/27 680.449 cents grave fifth 6: 3 3/2 701.955 cents perfect fifth 6: 1 128/81 792.180 cents Pythagorean minor sixth 7: 2 27/20 519.551 cents acute fourth 7: 1 45/32 590.224 cents diatonic tritone 7: 1 36/25 631.283 cents classic diminished fifth 7: 2 40/27 680.449 cents grave fifth 7: 1 3/2 701.955 cents perfect fifth 7: 1 243/160 723.461 cents acute fifth 7: 1 128/81 792.180 cents Pythagorean minor sixth 7: 1 8/5 813.686 cents minor sixth 7: 1 400/243 862.852 cents grave major sixth 7: 1 5/3 884.359 cents major sixth, BP sixth 8: 5 3/2 701.955 cents perfect fifth 8: 1 128/81 792.180 cents Pythagorean minor sixth 8: 2 8/5 813.686 cents minor sixth 8: 2 5/3 884.359 cents major sixth, BP sixth 8: 1 27/16 905.865 cents Pythagorean major sixth 8: 1 16/9 996.090 cents Pythagorean minor seventh 9: 1 243/160 723.461 cents acute fifth 9: 2 8/5 813.686 cents minor sixth 9: 1 81/50 835.193 cents acute minor sixth 9: 3 5/3 884.359 cents major sixth, BP sixth 9: 1 27/16 905.865 cents Pythagorean major sixth 9: 1 1280/729 974.584 cents grave minor seventh 9: 1 16/9 996.090 cents Pythagorean minor seventh 9: 1 9/5 1017.596 cents just minor seventh, BP seventh 9: 1 50/27 1066.762 cents grave major seventh 10: 3 27/16 905.865 cents Pythagorean major sixth 10: 5 16/9 996.090 cents Pythagorean minor seventh 10: 1 9/5 1017.596 cents just minor seventh, BP seventh 10: 2 15/8 1088.269 cents classic major seventh 10: 1 256/135 1107.821 cents octave - major chroma 11: 4 9/5 1017.596 cents just minor seventh, BP seventh 11: 2 15/8 1088.269 cents classic major seventh 11: 1 243/128 1109.775 cents Pythagorean major seventh 11: 1 48/25 1129.328 cents classic diminished octave 11: 4 160/81 1178.494 cents octave - syntonic comma Highest number of different intervals for one interval class: 10 Average number of different intervals per interval class: 7.09091 = 78/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 182.4 203.9 386.3 498.0 680.4 702.0 884.4 905.9 996.1 1017.6 1088.3 1200.0 182.4 : 21.5 203.9 315.6 498.0 519.6 702.0 723.5 813.7 835.2 905.9 1017.6 1200.0 203.9 : 182.4 294.1 476.5 498.0 680.4 702.0 792.2 813.7 884.4 996.1 1178.5 1200.0 386.3 : 111.7 294.1 315.6 498.0 519.6 609.8 631.3 702.0 813.7 996.1 1017.6 1200.0 498.0 : 182.4 203.9 386.3 407.8 498.0 519.6 590.2 702.0 884.4 905.9 1088.3 1200.0 680.4 : 21.5 203.9 225.4 315.6 337.1 407.8 519.6 702.0 723.5 905.9 1017.6 1200.0 702.0 : 182.4 203.9 294.1 315.6 386.3 498.0 680.4 702.0 884.4 996.1 1178.5 1200.0 884.4 : 21.5 111.7 133.2 203.9 315.6 498.0 519.6 702.0 813.7 996.1 1017.6 1200.0 905.9 : 90.2 111.7 182.4 294.1 476.5 498.0 680.4 792.2 974.6 996.1 1178.5 1200.0 996.1 : 21.5 92.2 203.9 386.3 407.8 590.2 702.0 884.4 905.9 1088.3 1109.8 1200.0 1017.6: 70.7 182.4 364.8 386.3 568.7 680.4 862.9 884.4 1066.8 1088.3 1178.5 1200.0 1088.3: 111.7 294.1 315.6 498.0 609.8 792.2 813.7 996.1 1017.6 1107.8 1129.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 21.5063 .. 182.4037 cents 800/729, 160.8974 cents 81/80, 21.5063 cents 2: 92.1787 .. 294.1350 cents 4096/3645, 201.9563 cents 243/256,-90.2250 cents 3: 133.2376 .. 476.5387 cents 8000/6561, 343.3011 cents 729/800,-160.8974 cents 4: 203.9100 .. 498.0450 cents 32/27, 294.1350 cents 2187/2560,-272.6287 cents 5: 315.6413 .. 680.4487 cents 100/81, 364.8074 cents 9/10,-182.4037 cents 6: 407.8200 .. 792.1800 cents 8192/6561, 384.3600 cents 2187/2560,-272.6287 cents 7: 519.5513 .. 884.3587 cents 100/81, 364.8074 cents 2187/2560,-272.6287 cents 8: 701.9550 .. 996.0900 cents 32/27, 294.1350 cents 9/10,-182.4037 cents 9: 723.4613 .. 1066.7624 cents 8000/6561, 343.3011 cents 2187/2560,-272.6287 cents 10: 905.8650 .. 1107.8213 cents 4096/3645, 201.9563 cents 729/800,-160.8974 cents 11: 1017.5963 .. 1178.4937 cents 800/729, 160.8974 cents 243/256,-90.2250 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 81/80, 21.5063 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 10/9, 182.4037 cents Average / Smallest step : 4.649803 Largest / Average step : 1.824037 Largest / Smallest step : 8.481412 Median interval of one step : 100.97814 cents, amount: 0 Least squares average step : 108.16074 cents, oct.: 1297.92883 cents Scale is not proper Step pattern alph. order: ABACABABDBEC Step pattern size order : EAEDEAEACABD Scale is not a Constant Structure Scale diversity : 1.377064 Lumma stability : 0.035844 Lumma impropriety factor : 0.846369 Rothenberg efficiency : 0.390344 redundancy: 0.609656 Efficiency x scale size : 4.684127 Number of different interval sizes : 40 = 3.63636 / class Number of one step interval sizes : 5 Highest interval variety : 10 Mean interval variety : 7.09091 = 78/11 Median interval variety : 5 Lowest interval variety : 5 Smallest interval difference : 32805/32768, 1.9537 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 9 Most common triad is 0.0 498.045 701.955 cents, amount: 7 Number of recognisable fifths : 9, average 701.9550 cents Scale contains a complete diamond : 1 3 9 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 7 = 58.3333% Scale is n-1 differentially coherent in interval class 1 Scale is differentially coherent in interval classes 2 and 3 combined Scale is differentially coherent in interval classes 3 and 4 combined Scale is n-1 differentially coherent in interval class 4 -- Rational properties -- Prime limit : 5 Odd number limit : 729 (O: 729 U: 729) Highest odd numerator or denominator: 27 Scale harmonicity : 0.027174 Average absolute harmonicity : 0.194244 Specific harmonicity : 0.170340 Fundamental : 1/2160, -11.0768 octaves, 0.1211 Hz. Guide tone : 2160, 11.0768 octaves, 565111.221 Hz. Exponens Consonantiae : 4.665600E+06, 22.15363 octaves Euler's gradus suavitatis : 29 Sum of Mann's harmonic distance : 120.0, average 10.00000 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 10.20797, average 0.85066 Sum of Tenney's harmonic distance : 20.00672, average 1.66723 Vogel's harmonic complexity : 19.50000 Wille's k value : 1822 Wilson's harmonic complexity : 28 Rectangular lattice diameter : 8 Triangular lattice diameter : 6 Lattice compactness : 300.12437, average 3.84775 Lattice compactness (without 2's) : 182.76776, average 2.34318 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -4 .. 4 -0.16667 24 1 3: -3 .. 3 0.00000 18 5 5: -1 .. 1 0.33333 6 6 Average exponent except 2's : 4 / 12 = 0.33333 Average absolute exponent except 2's: 24 / 12 = 2.00000
FIT/MODE
12: 2 0 2 1 2 0 2 0 1 0 1 1 N SD: 11.7037 c. M:-19.5513 c. 19: 3 0 3 2 3 0 3 0 2 0 1 2 N SD: 11.4056 c. M: 21.6545 c. 26: 4 0 4 3 4 0 4 1 2 0 2 2 N SD: 13.2343 c. M:-19.4236 c. 27: 4 1 4 2 4 1 4 0 2 1 1 3 N SD: 13.0185 c. M: 21.6020 c. 28: 4 1 4 3 4 0 5 0 2 1 1 3 N SD: 11.4545 c. M: 16.8401 c. 29: 4 1 4 3 4 1 4 1 2 0 2 3 N SD: 12.4141 c. M: 24.4928 c. 31: 5 0 5 3 5 0 5 0 3 0 2 3 N SD: 9.4887 c. M:-16.3255 c. 33: 5 1 5 3 5 0 5 1 2 1 2 3 N SD: 9.5807 c. M:-14.2718 c. 34: 5 1 5 3 5 1 5 1 2 1 2 3 N SD: 6.4718 c. M:-11.7821 c. 41: 6 1 6 4 6 1 6 1 3 1 2 4 N SD: 4.5828 c. M: 7.2780 c. 46: 7 1 7 4 7 1 7 1 3 1 3 4 N SD: 4.0807 c. M:-7.3835 c. 53: 8 1 8 5 8 1 8 1 4 1 3 5 N SD: 0.9455 c. M: 1.4763 c. 118: 18 2 18 11 18 2 18 2 9 2 7 11 N SD: 0.5107 c. M:-0.9072 c. 171: 26 3 26 16 26 3 26 3 13 3 10 16 N SD: 0.3266 c. M: 0.6018 c. 335: 51 6 51 31 51 6 51 6 25 6 20 31 N SD: 0.3507 c. M:-0.6865 c. 388: 59 7 59 36 59 7 59 7 29 7 23 36 N SD: 0.2028 c. M:-0.3911 c. 441: 67 8 67 41 67 8 67 8 33 8 26 41 N SD: 0.1360 c. M:-0.2574 c. 559: 85 10 85 52 85 10 85 10 42 10 33 52 N SD: 0.0572 c. M:-0.1034 c. 612: 93 11 93 57 93 11 93 11 46 11 36 57 N SD: 0.0337 c. M: 0.0566 c. 1171: 178 21 178 109 178 21 178 21 88 21 69 109 N SD: 0.0167 c. M:-0.0319 c. 1783: 271 32 271 166 271 32 271 32 134 32 105 166 N SD: 0.0138 c. M:-0.0239 c.
FIT/HARMONIC
1 x x x x x x x x x x x 2 S SD: 0.0000 cents 2 x x x x x 3 x x x x x 4 S SD: 0.0000 cents 3 x x x 4 x x 5 x x x x 6 S SD: 0.0000 cents 4 x x 5 x x 6 x x 7 x x 8 S SD: 6.8160 cents 5 x x 6:7 x x 8 x x 9 x 10 S SD: 26.1717 cents 6 x 7 x 8 x 9:10 x x 11 x 12 S SD: 11.7535 cents 7 x 8:9 x 10:11 x 12 x x 13:14 S SD: 17.2556 cents 8 x 9:10:11 x 12:13 x 14 x 15:16 S SD: 9.2724 cents 9:10 x 11:12:13:14:15 x 16 x 17:18 S SD: 9.6623 cents 10:11 x x 13 x 15 x 17 x 18:19:20 SD: 7.7103 cents 11:12 x 14:15:16:17:18:19 x 20:21:22 S SD: 10.5297 cents 12:13:14:15:16 x 18:20 x 21:22:23:24 S SD: 9.5318 cents 13:14:15:16:17:19:20 x 22:23 x 24:26 SD: 9.9926 cents 14 x 16:18:19 x 21:23:24:25 x 26:28 SD: 8.4317 cents 15 x 17:19:20:22:23:25 x x 27:28:30 SD: 5.5546 cents 16 x 18:20:21 x 24 x 27:28:29:30:32 S SD: 4.4862 cents 17:19 x 21:23:25:26:28:29:30:31:32:34 SD: 5.7072 cents 18:20 x 23:24 x 27:30 x 32 x 34:36 SD: 5.0173 cents 19:21 x 24:25:28:29 x 32 x 34:36:38 SD: 5.4104 cents 20:22:23:25:27 x 30:33:34 x 36:38:40 SD: 5.6612 cents 21:23:24:26:28:31:32:35 x 37:38:39:42 SD: 4.9951 cents 22:24:25:28:29 x 33 x 37:39:40:41:44 SD: 5.6109 cents 23 x 26:29:31:34:35:38:39:41 x 43:46 SD: 4.0309 cents 24 x 27:30:32 x 36:40:41 x 43:45:48 SD: 2.5234 cents 25 x 28:31:33:37:38 x 42:44:45:47:50 SD: 3.8313 cents 26:29 x 33:35 x 39:43:44:46:47:49:52 SD: 3.7720 cents 27:30 x 34:36:40:41:45:46:48:49:51:54 SD: 3.2177 cents 28:31:32:35:37:41:42 x 47:50 x 53:56 SD: 4.2947 cents 29:32:33:36:39:43:44:48:49 x 52:54:58 SD: 3.6694 cents 30:33:34:38:40:44:45:50:51:53:54:56:60 SD: 3.3697 cents 31:34:35:39:41:46:47 x 52:55:56:58:62 SD: 3.3857 cents 32 x 36:40:43:47:48:53:54:57:58:60:64 SD: 2.3663 cents 33 x 37:41:44:49:50:55:56:59 x 62:66 SD: 2.5794 cents 34:38 x 43:45:50:51:57 x 60:61:64:68 SD: 3.4409 cents 35:39 x 44:47:52:53:58:59:62:63:66:70 SD: 2.5788 cents 36:40:41:45:48:53:54:60:61:64:65:68:72 SD: 2.3736 cents 37:41:42:46:49:55:56:62 x 66:67:69:74 SD: 2.9886 cents 38:42:43:48:51:56:57:63:64 x 68:71:76 SD: 2.8187 cents 39:43:44:49:52:58:59:65:66:69:70:73:78 SD: 2.1565 cents 41 x 46:51:55:61:62:68:69:73:74:77:82 SD: 2.2006 cents 42 x 47:53:56:62:63:70:71:75:76:79:84 SD: 2.1787 cents 43:48 x 54:57:64:65:72:73:76:77:81:86 SD: 2.7181 cents 44:49:50:55:59:65:66:73:74:78:79:83:88 SD: 2.1855 cents 45:50:51:56:60:67:68:75:76:80:81:84:90 SD: 1.9040 cents 48:53:54:60:64:71:72:80:81:85:86:90:96 SD: 1.2791 cents 50 x 56:63:67:74:75:83:84:89:90:94:100 SD: 1.9516 cents 51:57 x 64:68:76:77:85:86:91:92:96:102 SD: 1.9946 cents 54:60:61:68:72:80:81:90:91:96:97:101:108 SD: 1.3189 cents 59 x 66:74:79:87:89:98:100:105:106:111:118 SD: 2.0024 cents 60:67:68:75:80:89:90:100:101:107:108:113:120 SD: 1.5566 cents 63:70:71:79:84:93:95:105:106:112:113:118:126 SD: 1.2583 cents 64:71:72:80:85:95:96:107:108:114:115:120:128 SD: 0.8909 cents 72:80:81:90:96:107:108:120:122:128:130:135:144 SD: 0.8668 cents 81:90:91:101:108:120:122:135:137:144:146:152:162 SD: 0.8241 cents 96:107:108:120:128:142:144:160:162:171:173:180:192 SD: 0.6005 cents 120:133:135:150:160:178:180:200:203:213:216:225:240 SD: 0.5834 cents 135:150:152:169:180:200:203:225:228:240:243:253:270 SD: 0.4533 cents 144:160:162:180:192:213:216:240:243:256:259:270:288 SD: 0.2516 cents 216:240:243:270:288:320:324:360:365:384:389:405:432 SD: 0.2112 cents 288:320:324:360:384:427:432:480:486:512:518:540:576 SD: 0.1584 cents 360:400:405:450:480:533:540:600:608:640:648:675:720 SD: 0.1491 cents 384:427:432:480:512:569:576:640:648:683:691:720:768 SD: 0.1421 cents 432:480:486:540:576:640:648:720:729:768:778:810:864 SD: 0.0742 cents 576:640:648:720:768:853:864:960:972:1024:1037:1080:1152 SD: 0.0629 cents