Scala analysis: diadie1
First Diadie 2048/2025 128/125 scale = lumma5r.scl
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 16/15 111.731285 minor diatonic semitone 2: 9/8 203.910002 major whole tone 3: 75/64 274.582429 classic augmented second 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 45/32 590.223716 diatonic tritone 7: 3/2 701.955001 perfect fifth 8: 8/5 813.686286 minor sixth 9: 5/3 884.358713 major sixth, BP sixth 10: 225/128 976.537429 augmented sixth 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: 16/15 111.7313 minor diatonic semitone 2: 135/128 92.1787 major chroma, major limma 3: 25/24 70.6724 classic chromatic semitone, minor chroma 4: 16/15 111.7313 minor diatonic semitone 5: 16/15 111.7313 minor diatonic semitone 6: 135/128 92.1787 major chroma, major limma 7: 16/15 111.7313 minor diatonic semitone 8: 16/15 111.7313 minor diatonic semitone 9: 25/24 70.6724 classic chromatic semitone, minor chroma 10: 135/128 92.1787 major chroma, major limma 11: 16/15 111.7313 minor diatonic semitone 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 3 135/128 92.179 cents major chroma, major limma 1: 7 16/15 111.731 cents minor diatonic semitone 2: 2 1125/1024 162.851 cents double augmented prime 2: 2 10/9 182.404 cents minor whole tone 2: 4 9/8 203.910 cents major whole tone 2: 4 256/225 223.463 cents Neapolitan diminished third 3: 4 75/64 274.582 cents classic augmented second 3: 2 32/27 294.135 cents Pythagorean minor third 3: 5 6/5 315.641 cents minor third 3: 1 4096/3375 335.194 cents double diminished fourth 4: 8 5/4 386.314 cents major third 4: 4 32/25 427.373 cents classic diminished fourth 5: 2 675/512 478.492 cents wide augmented third 5: 8 4/3 498.045 cents perfect fourth 5: 1 27/20 519.551 cents acute fourth 5: 1 512/375 539.104 cents double diminished fifth 6: 6 45/32 590.224 cents diatonic tritone 6: 6 64/45 609.776 cents 2nd tritone 7: 1 375/256 660.896 cents double augmented fourth 7: 1 40/27 680.449 cents grave fifth 7: 8 3/2 701.955 cents perfect fifth 7: 2 1024/675 721.508 cents narrow diminished sixth 8: 4 25/16 772.627 cents classic augmented fifth 8: 8 8/5 813.686 cents minor sixth 9: 1 3375/2048 864.806 cents double augmented fifth 9: 5 5/3 884.359 cents major sixth, BP sixth 9: 2 27/16 905.865 cents Pythagorean major sixth 9: 4 128/75 925.418 cents diminished seventh 10: 4 225/128 976.537 cents augmented sixth 10: 4 16/9 996.090 cents Pythagorean minor seventh 10: 2 9/5 1017.596 cents just minor seventh, BP seventh 10: 2 2048/1125 1037.149 cents double diminished octave 11: 7 15/8 1088.269 cents classic major seventh 11: 3 256/135 1107.821 cents octave - major chroma 11: 2 48/25 1129.328 cents classic diminished octave Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.27273 = 36/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 111.7 203.9 274.6 386.3 498.0 590.2 702.0 813.7 884.4 976.5 1088.3 1200.0 111.7 : 92.2 162.9 274.6 386.3 478.5 590.2 702.0 772.6 864.8 976.5 1088.3 1200.0 203.9 : 70.7 182.4 294.1 386.3 498.0 609.8 680.4 772.6 884.4 996.1 1107.8 1200.0 274.6 : 111.7 223.5 315.6 427.4 539.1 609.8 702.0 813.7 925.4 1037.1 1129.3 1200.0 386.3 : 111.7 203.9 315.6 427.4 498.0 590.2 702.0 813.7 925.4 1017.6 1088.3 1200.0 498.0 : 92.2 203.9 315.6 386.3 478.5 590.2 702.0 813.7 905.9 976.5 1088.3 1200.0 590.2 : 111.7 223.5 294.1 386.3 498.0 609.8 721.5 813.7 884.4 996.1 1107.8 1200.0 702.0 : 111.7 182.4 274.6 386.3 498.0 609.8 702.0 772.6 884.4 996.1 1088.3 1200.0 813.7 : 70.7 162.9 274.6 386.3 498.0 590.2 660.9 772.6 884.4 976.5 1088.3 1200.0 884.4 : 92.2 203.9 315.6 427.4 519.6 590.2 702.0 813.7 905.9 1017.6 1129.3 1200.0 976.5 : 111.7 223.5 335.2 427.4 498.0 609.8 721.5 813.7 925.4 1037.1 1107.8 1200.0 1088.3: 111.7 223.5 315.6 386.3 498.0 609.8 702.0 813.7 925.4 996.1 1088.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 70.6724 .. 111.7313 cents 128/125, 41.0589 cents 25/24, 70.6724 cents 2: 162.8511 .. 223.4626 cents 262144/253125, 60.6114 cents 16875/16384, 51.1199 cents 3: 274.5824 .. 335.1939 cents 262144/253125, 60.6114 cents 16875/16384, 51.1199 cents 4: 386.3137 .. 427.3726 cents 128/125, 41.0589 cents 16875/16384, 51.1199 cents 5: 478.4924 .. 539.1039 cents 262144/253125, 60.6114 cents 16875/16384, 51.1199 cents 6: 590.2237 .. 609.7763 cents 2048/2025, 19.5526 cents 16875/16384, 51.1199 cents 7: 660.8961 .. 721.5076 cents 262144/253125, 60.6114 cents 16875/16384, 51.1199 cents 8: 772.6274 .. 813.6863 cents 128/125, 41.0589 cents 16875/16384, 51.1199 cents 9: 864.8061 .. 925.4176 cents 262144/253125, 60.6114 cents 16875/16384, 51.1199 cents 10: 976.5374 .. 1037.1489 cents 262144/253125, 60.6114 cents 16875/16384, 51.1199 cents 11: 1088.2687 .. 1129.3276 cents 128/125, 41.0589 cents 16875/16384, 51.1199 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 25/24, 70.6724 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 16/15, 111.7313 cents Average / Smallest step : 1.414979 Largest / Average step : 1.117313 Largest / Smallest step : 1.580974 Median interval of one step : 16/15, 111.7313 cents, amount: 7 Most common interval of one step : 16/15, 111.7313 cents, amount: 7 Least squares average step : 99.13670 cents, oct.: 1189.64039 cents Scale is strictly proper Step pattern alph. order: ABCAABAACBAA Step pattern size order : LMSLLMLLSMLL Scale is a Constant Structure, by a margin of 51.11986 cents Scale diversity : 1.361037 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.543786 Rothenberg efficiency : 0.395893 redundancy: 0.604107 Efficiency x scale size : 4.750722 Number of different interval sizes : 36 = 3.27273 / class Number of one step interval sizes : 3 Highest interval variety : 4 Mean interval variety : 3.27273 = 36/11 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 2048/2025, 19.5526 cents Number of recognisable fifths : 8, average 701.9550 cents Best fifths form a closed circle Scale is fully connected 5-limit major-minor triadic Scale contains a complete diamond : 1 3 15 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 classes 1 and 2 combined Scale is n-1 differentially coherent in interval classes 3 and 4 combined Limited inverse transpositions : 11 Inversional symmetry on intervals : 5-6 11-12 -- Rational properties -- Prime limit : 5 Odd number limit : 3375 (O: 3375 U: 3375) Highest odd numerator or denominator: 225 Scale harmonicity : 0.026178 Average absolute harmonicity : 0.187356 Specific harmonicity : 0.148193 Fundamental : 1/1920, -10.9069 octaves, 0.1363 Hz. Guide tone : 3600, 11.8138 octaves, 941852.035 Hz. Exponens Consonantiae : 6.912000E+06, 22.72067 octaves Euler's gradus suavitatis : 30 Sum of Mann's harmonic distance : 342.5, average 28.54167 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 12.23803, average 1.01984 Sum of Tenney's harmonic distance : 23.85326, average 1.98777 Vogel's harmonic complexity : 17.58333 Wille's k value : 1687 Wilson's harmonic complexity : 24 Rectangular lattice diameter : 6 Triangular lattice diameter : 6 Lattice compactness : 357.14954, average 4.57884 Lattice compactness (without 2's) : 149.60866, average 1.91806 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -7 .. 4 -1.41667 37 1 3: -1 .. 2 0.50000 12 3 5: -1 .. 2 0.50000 10 8 Average exponent except 2's : 12 / 12 = 1.00000 Average absolute exponent except 2's: 22 / 12 = 1.83333 Scale is weakly epimorphic with val: <12 17 28| Scale is JI-epimorphic with val: <12 19 28| = patent Scale is weakly epimorphic with val: <12 21 28|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 13.5695 c. M:-25.4176 c. Twelve-tone Chromatic (1/11-comma) 22: 2 2 1 2 2 2 2 2 1 2 2 2 SP B ME SD: 7.2000 c. M:-14.2718 c. Jubilee-12 31: 3 2 2 3 3 2 3 3 2 2 3 3 SP D3 SD: 5.9069 c. M: 10.3616 c. Fokker 12-tone 41: 4 3 2 4 4 3 4 4 2 3 4 4 SP SD: 5.9766 c. M: 11.1678 c. 43: 4 3 3 4 4 3 4 4 3 3 4 4 SP D3 SD: 4.6646 c. M:-8.6645 c. 53: 5 4 3 5 5 4 5 5 3 4 5 5 SP SD: 1.5697 c. M: 2.9525 c. 65: 6 5 4 6 6 5 6 6 4 5 6 6 SP SD: 1.2713 c. M:-2.3406 c. 118: 11 9 7 11 11 9 11 11 7 9 11 11 SP SD: 0.2660 c. M: 0.5202 c. 376: 35 29 22 35 35 29 35 35 22 29 35 35 SP SD: 0.1777 c. M:-0.3453 c. 441: 41 34 26 41 41 34 41 41 26 34 41 41 SP SD: 0.1705 c. M:-0.3333 c. 494: 46 38 29 46 46 38 46 46 29 38 46 46 SP SD: 0.0791 c. M: 0.1482 c. 612: 57 47 36 57 57 47 57 57 36 47 57 57 SP SD: 0.0388 c. M: 0.0726 c. 1171: 109 90 69 109 109 90 109 109 69 90 109 109 SP SD: 0.0323 c. M:-0.0638 c. 1289: 120 99 76 120 120 99 120 120 76 99 120 120 SP SD: 0.0287 c. M:-0.0491 c. 1783: 166 137 105 166 166 137 166 166 105 137 166 166 SP SD: 0.0108 c. M:-0.0189 c. 1901: 177 146 112 177 177 146 177 177 112 146 177 177 SP SD: 0.0096 c. M:-0.0179 c. 2395: 223 184 141 223 223 184 223 223 141 184 223 223 SP SD: 0.0087 c. M: 0.0163 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 x 3 x x x x 4 S SD: 0.0000 cents 3 x x x x 4 x x x 5 x x 6 S SD: 0.0000 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 1.9279 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 11.7152 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 6.6104 cents 7 x 8 x 9 x 10 x 11 x 12:13:14 S SD: 11.9498 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 5.9864 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 9.3311 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 9.2559 cents 11:12 x 13:14:15 x 17 x 18:19:21:22 SD: 10.9929 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 4.7925 cents 13:14 x 15:16:17:18:20:21:22:23:24:26 S SD: 7.7706 cents 14:15:16 x 18:19:20:21:22:23:25:26:28 SD: 7.8598 cents 15:16:17:18:19:20:21:23:24:25:26:28:30 SD: 5.6095 cents 16:17:18:19:20:21:23:24:26:27:28:30:32 SD: 5.2740 cents 17:18:19:20:21:23:24:26:27:28:30:32:34 SD: 4.7489 cents 20:21 x 23:25:27:28:30:32:33:35:38:40 SD: 5.1567 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 5.1169 cents 23:25:26:27:29:31:32:35:37:38:40:43:46 SD: 4.8338 cents 24:26:27:28:30:32:34:36:38:40:42:45:48 S SD: 3.0406 cents 30:32:34:35:38:40:42:45:48:50:53:56:60 SD: 2.5592 cents 40:43:45:47:50:53:56:60:64:67:70:75:80 SD: 1.8894 cents 47:50:53:55:59:63:66:71:75:78:83:88:94 SD: 1.8333 cents 48:51:54:56:60:64:68:72:77:80:84:90:96 SD: 1.5554 cents 57:61:64:67:71:76:80:86:91:95:100:107:114 SD: 1.3220 cents 64:68:72:75:80:85:90:96:102:107:113:120:128 SD: 1.2524 cents 72:77:81:84:90:96:101:108:115:120:127:135:144 SD: 0.9958 cents 87:93:98:102:109:116:122:131:139:145:153:163:174 SD: 0.8794 cents 88:94:99:103:110:117:124:132:141:147:155:165:176 SD: 0.7482 cents 104:111:117:122:130:139:146:156:166:173:183:195:208 SD: 0.6556 cents 120:128:135:141:150:160:169:180:192:200:211:225:240 SD: 0.4417 cents 152:162:171:178:190:203:214:228:243:253:267:285:304 SD: 0.4116 cents 168:179:189:197:210:224:236:252:269:280:295:315:336 SD: 0.3042 cents 192:205:216:225:240:256:270:288:307:320:338:360:384 SD: 0.2725 cents 240:256:270:281:300:320:338:360:384:400:422:450:480 SD: 0.2528 cents 256:273:288:300:320:341:360:384:410:427:450:480:512 SD: 0.2316 cents 312:333:351:366:390:416:439:468:499:520:548:585:624 SD: 0.2296 cents 344:367:387:403:430:459:484:516:550:573:605:645:688 SD: 0.2069 cents 360:384:405:422:450:480:506:540:576:600:633:675:720 SD: 0.0934 cents 552:589:621:647:690:736:776:828:883:920:970:1035:1104 SD: 0.0925 cents 600:640:675:703:750:800:844:900:960:1000:1055:1125:1200 SD: 0.0657 cents