Scala analysis: MersenneStar
Marin Mersenne's dodecatonic 5-limit Star compiled by A.Sparschuh
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 25/24 70.672427 classic chromatic semitone, minor chroma 2: 10/9 182.403712 minor whole tone 3: 32/27 294.134997 Pythagorean minor third 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 25/18 568.717426 classic augmented fourth 7: 3/2 701.955001 perfect fifth 8: 25/16 772.627428 classic augmented fifth 9: 5/3 884.358713 major sixth, BP sixth 10: 16/9 996.089998 Pythagorean minor 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: 25/24 70.6724 classic chromatic semitone, minor chroma 2: 16/15 111.7313 minor diatonic semitone 3: 16/15 111.7313 minor diatonic semitone 4: 135/128 92.1787 major chroma, major limma 5: 16/15 111.7313 minor diatonic semitone 6: 25/24 70.6724 classic chromatic semitone, minor chroma 7: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 8: 25/24 70.6724 classic chromatic semitone, minor chroma 9: 16/15 111.7313 minor diatonic semitone 10: 16/15 111.7313 minor diatonic semitone 11: 135/128 92.1787 major chroma, major limma 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 3 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 2 135/128 92.179 cents major chroma, major limma 1: 6 16/15 111.731 cents minor diatonic semitone 1: 1 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 2: 4 10/9 182.404 cents minor whole tone 2: 6 9/8 203.910 cents major whole tone 2: 2 256/225 223.463 cents Neapolitan diminished third 3: 3 75/64 274.582 cents classic augmented second 3: 3 32/27 294.135 cents Pythagorean minor third 3: 6 6/5 315.641 cents minor third 4: 7 5/4 386.314 cents major third 4: 1 512/405 405.866 cents narrow diminished fourth 4: 1 81/64 407.820 cents Pythagorean major third 4: 3 32/25 427.373 cents classic diminished fourth 5: 1 675/512 478.492 cents wide augmented third 5: 9 4/3 498.045 cents perfect fourth 5: 2 27/20 519.551 cents acute fourth 6: 2 25/18 568.717 cents classic augmented fourth 6: 4 45/32 590.224 cents diatonic tritone 6: 4 64/45 609.776 cents 2nd tritone 6: 2 36/25 631.283 cents classic diminished fifth 7: 2 40/27 680.449 cents grave fifth 7: 9 3/2 701.955 cents perfect fifth 7: 1 1024/675 721.508 cents narrow diminished sixth 8: 3 25/16 772.627 cents classic augmented fifth 8: 1 128/81 792.180 cents Pythagorean minor sixth 8: 1 405/256 794.134 cents wide augmented fifth 8: 7 8/5 813.686 cents minor sixth 9: 6 5/3 884.359 cents major sixth, BP sixth 9: 3 27/16 905.865 cents Pythagorean major sixth 9: 3 128/75 925.418 cents diminished seventh 10: 2 225/128 976.537 cents augmented sixth 10: 6 16/9 996.090 cents Pythagorean minor seventh 10: 4 9/5 1017.596 cents just minor seventh, BP seventh 11: 1 50/27 1066.762 cents grave major seventh 11: 6 15/8 1088.269 cents classic major seventh 11: 2 256/135 1107.821 cents octave - major chroma 11: 3 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: 3.45455 = 38/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 70.7 182.4 294.1 386.3 498.0 568.7 702.0 772.6 884.4 996.1 1088.3 1200.0 70.7 : 111.7 223.5 315.6 427.4 498.0 631.3 702.0 813.7 925.4 1017.6 1129.3 1200.0 182.4 : 111.7 203.9 315.6 386.3 519.6 590.2 702.0 813.7 905.9 1017.6 1088.3 1200.0 294.1 : 92.2 203.9 274.6 407.8 478.5 590.2 702.0 794.1 905.9 976.5 1088.3 1200.0 386.3 : 111.7 182.4 315.6 386.3 498.0 609.8 702.0 813.7 884.4 996.1 1107.8 1200.0 498.0 : 70.7 203.9 274.6 386.3 498.0 590.2 702.0 772.6 884.4 996.1 1088.3 1200.0 568.7 : 133.2 203.9 315.6 427.4 519.6 631.3 702.0 813.7 925.4 1017.6 1129.3 1200.0 702.0 : 70.7 182.4 294.1 386.3 498.0 568.7 680.4 792.2 884.4 996.1 1066.8 1200.0 772.6 : 111.7 223.5 315.6 427.4 498.0 609.8 721.5 813.7 925.4 996.1 1129.3 1200.0 884.4 : 111.7 203.9 315.6 386.3 498.0 609.8 702.0 813.7 884.4 1017.6 1088.3 1200.0 996.1 : 92.2 203.9 274.6 386.3 498.0 590.2 702.0 772.6 905.9 976.5 1088.3 1200.0 1088.3: 111.7 182.4 294.1 405.9 498.0 609.8 680.4 813.7 884.4 996.1 1107.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 70.6724 .. 133.2376 cents 648/625, 62.5651 cents 25/24, 70.6724 cents 2: 182.4037 .. 223.4626 cents 128/125, 41.0589 cents 250/243, 49.1661 cents 3: 274.5824 .. 315.6413 cents 128/125, 41.0589 cents 16875/16384, 51.1199 cents 4: 386.3137 .. 427.3726 cents 128/125, 41.0589 cents 25/24, 70.6724 cents 5: 478.4924 .. 519.5513 cents 128/125, 41.0589 cents 16875/16384, 51.1199 cents 6: 568.7174 .. 631.2826 cents 648/625, 62.5651 cents 250/243, 49.1661 cents 7: 680.4487 .. 721.5076 cents 128/125, 41.0589 cents 250/243, 49.1661 cents 8: 772.6274 .. 813.6863 cents 128/125, 41.0589 cents 16875/16384, 51.1199 cents 9: 884.3587 .. 925.4176 cents 128/125, 41.0589 cents 25/24, 70.6724 cents 10: 976.5374 .. 1017.5963 cents 128/125, 41.0589 cents 16875/16384, 51.1199 cents 11: 1066.7624 .. 1129.3276 cents 648/625, 62.5651 cents 250/243, 49.1661 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 : 27/25, 133.2376 cents Average / Smallest step : 1.414979 Largest / Average step : 1.332376 Largest / Smallest step : 1.885284 Median interval of one step : 16/15, 111.7313 cents, amount: 6 Most common interval of one step : 16/15, 111.7313 cents, amount: 6 Least squares average step : 98.69455 cents, oct.: 1184.33461 cents Scale is strictly proper Step pattern alph. order: ABBCBADABBCB Step pattern size order : ACCBCADACCBC Scale is a Constant Structure, by a margin of 49.16614 cents Scale diversity : 1.347288 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.569861 Rothenberg efficiency : 0.411409 redundancy: 0.588591 Efficiency x scale size : 4.936905 Number of different interval sizes : 38 = 3.45455 / class Number of one step interval sizes : 4 Highest interval variety : 4 Mean interval variety : 3.45455 = 38/11 Median interval variety : 3 Lowest interval variety : 3 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: 6 Number of recognisable fifths : 9, average 701.9550 cents Scale contains two identical disjunct hexachords Best fifths form a closed circle Scale contains a complete diamond : 1 3 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 3 = 25.0000% Scale is differentially coherent in interval classes 1 and 2 combined Scale is differentially coherent in interval classes 2 and 3 combined Scale is n-1 differentially coherent in interval classes 3 and 4 combined Scale is n-1 differentially coherent in interval classes 8 and 9 combined -- Rational properties -- Prime limit : 5 Odd number limit : 675 (O: 675 U: 675) Highest odd numerator or denominator: 27 Scale harmonicity : 0.030801 Average absolute harmonicity : 0.185647 Specific harmonicity : 0.168081 Fundamental : 1/432, -8.7549 octaves, 0.6056 Hz. Guide tone : 2400, 11.2288 octaves, 627901.357 Hz. Exponens Consonantiae : 1.036800E+06, 19.98371 octaves Euler's gradus suavitatis : 26 Sum of Mann's harmonic distance : 145.5, average 12.12500 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 11.06194, average 0.92183 Sum of Tenney's harmonic distance : 21.79721, average 1.81643 Vogel's harmonic complexity : 14.58333 Wille's k value : 337 Wilson's harmonic complexity : 22 Rectangular lattice diameter : 6 Triangular lattice diameter : 5 Lattice compactness : 300.18082, average 3.84847 Lattice compactness (without 2's) : 150.70153, average 1.93207 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -4 .. 5 -0.08333 27 1 3: -3 .. 1 -0.83333 14 4 5: 0 .. 2 0.83333 10 7 Average exponent except 2's : 0 / 12 = 0.00000 Average absolute exponent except 2's: 24 / 12 = 2.00000 Scale is JI-epimorphic with val: <12 19 28| = patent
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 17.1379 c. M:-31.2826 c. Twelve-tone Chromatic (1/11-comma) 19: 1 2 2 1 2 1 2 1 2 2 1 2 P M ME SD: 10.6931 c. M:-21.6545 c. Meantone Chromatic (1/3-comma) 29: 2 2 3 2 3 2 3 2 2 3 2 3 SP M ME SD: 10.5884 c. M: 16.8865 c. 31: 2 3 3 2 3 2 3 2 3 3 2 3 SP M ME SD: 8.0238 c. M:-15.5424 c. Meantone Chromatic (53/220-comma) 33: 2 3 3 3 3 2 3 2 3 3 3 3 SP D3 SD: 9.3306 c. M: 14.2718 c. 34: 2 3 3 3 3 2 4 2 3 3 3 3 SP D3 SD: 5.3179 c. M: 11.7821 c. 46: 3 4 4 4 4 3 5 3 4 4 4 4 SP D3 SD: 5.4490 c. M:-9.9813 c. 53: 3 5 5 4 5 3 6 3 5 5 4 5 SP SD: 1.5892 c. M: 2.8161 c. 118: 7 11 11 9 11 7 13 7 11 11 9 11 SP SD: 0.4605 c. M:-0.7803 c. 171: 10 16 16 13 16 10 19 10 16 16 13 16 SP SD: 0.3951 c. M: 0.6976 c. 289: 17 27 27 22 27 17 32 17 27 27 22 27 SP SD: 0.3084 c. M:-0.6747 c. 323: 19 30 30 25 30 19 36 19 30 30 25 30 SP SD: 0.2914 c. M: 0.6365 c. 376: 22 35 35 29 35 22 42 22 35 35 29 35 SP SD: 0.3537 c. M: 0.6323 c. 388: 23 36 36 30 36 23 43 23 36 36 30 36 SP SD: 0.3032 c. M:-0.5684 c. 441: 26 41 41 34 41 26 49 26 41 41 34 41 SP SD: 0.1242 c. M: 0.2574 c. 559: 33 52 52 43 52 33 62 33 52 52 43 52 SP SD: 0.0987 c. M:-0.1812 c. 612: 36 57 57 47 57 36 68 36 57 57 47 57 SP SD: 0.0492 c. M: 0.0900 c. 1171: 69 109 109 90 109 69 130 69 109 109 90 109 SP SD: 0.0244 c. M:-0.0455 c. 1783: 105 166 166 137 166 105 198 105 166 166 137 166 SP SD: 0.0112 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 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: 6.8160 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 10.5852 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 7.9180 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 14.2040 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 7.3205 cents 9 x 10 x 11:12 x x 14:15:16:17:18 S SD: 5.2087 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 8.2013 cents 11 x 12:13:14 x 15 x 17:18:20:21:22 SD: 9.2109 cents 12 x 13:14:15:16:17:18:19:20:21:23:24 SD: 7.3613 cents 13 x 14:15:16:17:18 x 20:22:23:24:26 S SD: 9.5750 cents 14:15:16:17:18:19 x 21:22:23:25:26:28 SD: 9.4868 cents 15:16:17:18:19:20:21 x 23:25:27:28:30 SD: 6.8188 cents 16:17:18:19:20:21:22:24:25:27:28:30:32 SD: 5.2050 cents 18:19:20:21:23:24:25:27:28:30:32:34:36 SD: 4.5183 cents 21:22:23:25:26:28:29:32:33:35:37:39:42 SD: 4.1720 cents 24:25:27:28:30:32:33:36:38:40:43:45:48 SD: 3.9228 cents 26:27:29:31:33:35:36:39:41:43:46:49:52 SD: 3.4976 cents 27:28:30:32:34:36:38:41:42:45:48:51:54 SD: 3.1430 cents 28:29:31:33:35:37:39:42:44:47:50:53:56 SD: 2.7310 cents 36:38:40:43:45:48:50:54:56:60:64:68:72 SD: 2.5414 cents 39:41:43:46:49:52:54:59:61:65:69:73:78 SD: 2.5095 cents 42:44:47:50:53:56:58:63:66:70:75:79:84 SD: 2.4484 cents 43:45:48:51:54:57:60:65:67:72:76:81:86 SD: 2.3493 cents 44:46:49:52:55:59:61:66:69:73:78:83:88 SD: 1.7093 cents 48:50:53:57:60:64:67:72:75:80:85:90:96 SD: 1.3168 cents 64:67:71:76:80:85:89:96:100:107:114:120:128 SD: 1.1318 cents 72:75:80:85:90:96:100:108:113:120:128:135:144 SD: 0.8533 cents 96:100:107:114:120:128:133:144:150:160:171:180:192 SD: 0.7011 cents 120:125:133:142:150:160:167:180:188:200:213:225:240 SD: 0.6804 cents 126:131:140:149:158:168:175:189:197:210:224:236:252 SD: 0.6484 cents 136:142:151:161:170:181:189:204:213:227:242:255:272 SD: 0.6394 cents 144:150:160:171:180:192:200:216:225:240:256:270:288 SD: 0.2815 cents 216:225:240:256:270:288:300:324:338:360:384:405:432 SD: 0.2136 cents 288:300:320:341:360:384:400:432:450:480:512:540:576 SD: 0.1410 cents 432:450:480:512:540:576:600:648:675:720:768:810:864 SD: 0.0000 cents