Scala analysis: hahn12
Hahn-reduced 12 note scale
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 15/14 119.442808 major diatonic semitone 2: 8/7 231.174094 septimal whole tone 3: 6/5 315.641287 minor third 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 7/5 582.512193 septimal or Huygens' tritone, BP fourth 7: 3/2 701.955001 perfect fifth 8: 8/5 813.686286 minor sixth 9: 5/3 884.358713 major sixth, BP sixth 10: 7/4 968.825906 harmonic 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: 15/14 119.4428 major diatonic semitone 2: 16/15 111.7313 minor diatonic semitone 3: 21/20 84.4672 minor semitone 4: 25/24 70.6724 classic chromatic semitone, minor chroma 5: 16/15 111.7313 minor diatonic semitone 6: 21/20 84.4672 minor semitone 7: 15/14 119.4428 major diatonic semitone 8: 16/15 111.7313 minor diatonic semitone 9: 25/24 70.6724 classic chromatic semitone, minor chroma 10: 21/20 84.4672 minor semitone 11: 15/14 119.4428 major 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 21/20 84.467 cents minor semitone 1: 4 16/15 111.731 cents minor diatonic semitone 1: 3 15/14 119.443 cents major diatonic semitone 2: 2 35/32 155.140 cents septimal neutral second 2: 2 10/9 182.404 cents minor whole tone 2: 2 28/25 196.198 cents middle second 2: 2 9/8 203.910 cents major whole tone 2: 4 8/7 231.174 cents septimal whole tone 3: 4 7/6 266.871 cents septimal minor third 3: 1 75/64 274.582 cents classic augmented second 3: 1 25/21 301.847 cents BP second, quasi-equal minor third 3: 4 6/5 315.641 cents minor third 3: 1 128/105 342.905 cents septimal neutral third 3: 1 60/49 350.617 cents smaller approximation to neutral third 4: 1 49/40 351.338 cents larger approximation to neutral third 4: 1 56/45 378.602 cents septimal narrow major third 4: 6 5/4 386.314 cents major third 4: 2 32/25 427.373 cents classic diminished fourth 4: 1 9/7 435.084 cents septimal major third, BP third 4: 1 64/49 462.348 cents 2 septatones or septatonic major third 5: 1 98/75 463.069 cents 5: 2 21/16 470.781 cents narrow fourth 5: 5 4/3 498.045 cents perfect fourth 5: 2 75/56 505.757 cents marvelous fourth 5: 2 48/35 546.815 cents septimal semi-augmented fourth 6: 1 25/18 568.717 cents classic augmented fourth 6: 4 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 6: 1 45/32 590.224 cents diatonic tritone 6: 1 64/45 609.776 cents 2nd tritone 6: 4 10/7 617.488 cents Euler's tritone 6: 1 36/25 631.283 cents classic diminished fifth 7: 2 35/24 653.185 cents septimal semi-diminished fifth 7: 2 112/75 694.243 cents marvelous fifth 7: 5 3/2 701.955 cents perfect fifth 7: 2 32/21 729.219 cents wide fifth 7: 1 75/49 736.931 cents BP fifth 8: 1 49/32 737.652 cents 8: 1 14/9 764.916 cents septimal minor sixth 8: 2 25/16 772.627 cents classic augmented fifth 8: 6 8/5 813.686 cents minor sixth 8: 1 45/28 821.398 cents septimale wide minor sixth 8: 1 80/49 848.662 cents smaller approximation to neutral sixth 9: 1 49/30 849.383 cents larger approximation to neutral sixth 9: 1 105/64 857.095 cents septimal neutral sixth 9: 4 5/3 884.359 cents major sixth, BP sixth 9: 1 42/25 898.153 cents quasi-equal major sixth 9: 1 128/75 925.418 cents diminished seventh 9: 4 12/7 933.129 cents septimal major sixth 10: 4 7/4 968.826 cents harmonic seventh 10: 2 16/9 996.090 cents Pythagorean minor seventh 10: 2 25/14 1003.802 cents middle minor seventh 10: 2 9/5 1017.596 cents just minor seventh, BP seventh 10: 2 64/35 1044.860 cents septimal neutral seventh 11: 3 28/15 1080.557 cents grave major seventh 11: 4 15/8 1088.269 cents classic major seventh 11: 3 40/21 1115.533 cents acute major seventh 11: 2 48/25 1129.328 cents classic diminished octave Highest number of different intervals for one interval class: 6 Average number of different intervals per interval class: 5.27273 = 58/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 119.4 231.2 315.6 386.3 498.0 582.5 702.0 813.7 884.4 968.8 1088.3 1200.0 119.4 : 111.7 196.2 266.9 378.6 463.1 582.5 694.2 764.9 849.4 968.8 1080.6 1200.0 231.2 : 84.5 155.1 266.9 351.3 470.8 582.5 653.2 737.7 857.1 968.8 1088.3 1200.0 315.6 : 70.7 182.4 266.9 386.3 498.0 568.7 653.2 772.6 884.4 1003.8 1115.5 1200.0 386.3 : 111.7 196.2 315.6 427.4 498.0 582.5 702.0 813.7 933.1 1044.9 1129.3 1200.0 498.0 : 84.5 203.9 315.6 386.3 470.8 590.2 702.0 821.4 933.1 1017.6 1088.3 1200.0 582.5 : 119.4 231.2 301.8 386.3 505.8 617.5 736.9 848.7 933.1 1003.8 1115.5 1200.0 702.0 : 111.7 182.4 266.9 386.3 498.0 617.5 729.2 813.7 884.4 996.1 1080.6 1200.0 813.7 : 70.7 155.1 274.6 386.3 505.8 617.5 702.0 772.6 884.4 968.8 1088.3 1200.0 884.4 : 84.5 203.9 315.6 435.1 546.8 631.3 702.0 813.7 898.2 1017.6 1129.3 1200.0 968.8 : 119.4 231.2 350.6 462.3 546.8 617.5 729.2 813.7 933.1 1044.9 1115.5 1200.0 1088.3: 111.7 231.2 342.9 427.4 498.0 609.8 694.2 813.7 925.4 996.1 1080.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 70.6724 .. 119.4428 cents 36/35, 48.7704 cents 25/24, 70.6724 cents 2: 155.1396 .. 231.1741 cents 256/245, 76.0345 cents 49/48, 35.6968 cents 3: 266.8709 .. 350.6169 cents 360/343, 83.7460 cents 49/48, 35.6968 cents 4: 351.3381 .. 462.3482 cents 2560/2401, 111.0101 cents 2401/2400, 0.7212 cents 5: 463.0694 .. 546.8154 cents 360/343, 83.7460 cents 2401/2400, 0.7212 cents 6: 568.7174 .. 631.2826 cents 648/625, 62.5651 cents 875/864, 21.9020 cents 7: 653.1846 .. 736.9306 cents 360/343, 83.7460 cents 875/864, 21.9020 cents 8: 737.6518 .. 848.6619 cents 2560/2401, 111.0101 cents 2401/2400, 0.7212 cents 9: 849.3831 .. 933.1291 cents 360/343, 83.7460 cents 2401/2400, 0.7212 cents 10: 968.8259 .. 1044.8604 cents 256/245, 76.0345 cents 49/48, 35.6968 cents 11: 1080.5572 .. 1129.3276 cents 36/35, 48.7704 cents 49/48, 35.6968 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 : 15/14, 119.4428 cents Average / Smallest step : 1.414979 Largest / Average step : 1.194428 Largest / Smallest step : 1.690091 Median interval of one step : 16/15, 111.7313 cents, amount: 4 Most common interval of one step : 16/15, 111.7313 cents, amount: 4 Least squares average step : 99.23213 cents, oct.: 1190.78560 cents Scale is strictly proper Step pattern alph. order: ABCDBCABDCAB Step pattern size order : DCBACBDCABDC Scale is a Constant Structure, by a margin of 0.72120 cents Scale diversity : 1.466379 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.275684 Rothenberg efficiency : 0.276359 redundancy: 0.723641 Efficiency x scale size : 3.316306 Number of different interval sizes : 58 = 5.27273 / class Number of one step interval sizes : 4 Highest interval variety : 6 Mean interval variety : 5.27273 = 58/11 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 2401/2400, 0.7212 cents Most common intervals : 5/4, 386.3137 cents & inv., amount: 6 Number of recognisable fifths : 7, average 699.7517 cents Scale contains a complete diamond : 1 3 5 All steps are superparticular (epimoric) Length of longest harmonic part : 2 at degree 0 Length of longest subharmonic part : 2 at degree 11 Formal octave complements present : 9 = 75.0000% Scale is differentially coherent in interval class 1 Scale is differentially coherent in interval classes 2 and 3 combined Scale is n-1 differentially coherent in interval classes 3 and 4 combined -- Rational properties -- Prime limit : 7 Odd number limit : 105 (O: 105 U: 105) Highest odd numerator or denominator: 15 Scale harmonicity : 0.022369 Average absolute harmonicity : 0.189223 Specific harmonicity : 0.140407 Fundamental : 1/840, -9.7142 octaves, 0.3115 Hz. Guide tone : 840, 9.7142 octaves, 219765.475 Hz. Exponens Consonantiae : 7.056000E+05, 19.42849 octaves Euler's gradus suavitatis : 31 Sum of Mann's harmonic distance : 73.0, average 6.08333 Mersenne's string divisions : 240360120 Sum of van Prooijen's expressibility: 8.63760, average 0.71980 Sum of Tenney's harmonic distance : 16.85548, average 1.40462 Vogel's harmonic complexity : 19.00000 Wille's k value : 787 Wilson's harmonic complexity : 30 Rectangular lattice diameter : 6 Triangular lattice diameter : 3 Lattice compactness : 225.71889, average 2.89383 Lattice compactness (without 2's) : 122.69819, average 1.57305 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -3 .. 3 0.08333 19 1 3: -1 .. 1 0.16667 6 2 5: -1 .. 1 0.08333 7 5 7: -1 .. 1 0.00000 4 4 Average exponent except 2's : 3 / 12 = 0.25000 Average absolute exponent except 2's: 17 / 12 = 1.41667 Scale is JI-epimorphic with val: <12 19 28 34| = patent
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 17.4085 c. M:-31.1741 c. Twelve-tone Chromatic (1/11-comma) 19: 2 2 1 1 2 1 2 2 1 1 2 2 N SD: 11.5186 c. M:-21.4575 c. 22: 2 2 2 1 2 2 2 2 1 2 2 2 P D3 I SD: 9.8759 c. M:-17.4878 c. 31: 3 3 2 2 3 2 3 3 2 2 3 3 P SD: 3.6765 c. M: 5.9639 c. 53: 5 5 4 3 5 4 5 5 3 4 5 5 SP SD: 3.3160 c. M: 6.2353 c. 68: 7 6 5 4 6 5 7 6 4 5 7 6 P SD: 2.9358 c. M:-5.8489 c. 72: 7 7 5 4 7 5 7 7 4 5 7 7 P SD: 2.4079 c. M: 4.9354 c. 84: 8 8 6 5 8 6 8 8 5 6 8 8 SP SD: 2.3972 c. M: 5.1571 c. 87: 9 8 6 5 8 6 9 8 5 6 9 8 N SD: 2.3402 c. M:-4.6951 c. 99: 10 9 7 6 9 7 10 9 6 7 10 9 P SD: 1.2843 c. M:-2.6404 c. 130: 13 12 9 8 12 9 13 12 8 9 13 12 P SD: 1.0450 c. M:-1.7951 c. 140: 14 13 10 8 13 10 14 13 8 10 14 13 P SD: 0.7909 c. M: 1.5016 c. 171: 17 16 12 10 16 12 17 16 10 12 17 16 P SD: 0.2918 c. M: 0.5494 c. 441: 44 41 31 26 41 31 44 41 26 31 44 41 P SD: 0.1306 c. M:-0.2851 c. 612: 61 57 43 36 57 43 61 57 36 43 61 57 P SD: 0.1079 c. M:-0.1985 c. 935: 93 87 66 55 87 66 93 87 55 66 93 87 SP SD: 0.0973 c. M:-0.1616 c. 1106: 110 103 78 65 103 78 110 103 65 78 110 103 SP SD: 0.0714 c. M:-0.1280 c. 1236: 123 115 87 73 115 87 123 115 73 87 123 115 SP SD: 0.0769 c. M:-0.1073 c. 1277: 127 119 90 75 119 90 127 119 75 90 127 119 SP SD: 0.0750 c. M:-0.1033 c. 1376: 137 128 97 81 128 97 137 128 81 97 137 128 SP SD: 0.0604 c. M:-0.1034 c. 1407: 140 131 99 83 131 99 140 131 83 99 140 131 SP SD: 0.0437 c. M: 0.0763 c. 1547: 154 144 109 91 144 109 154 144 91 109 154 144 SP SD: 0.0379 c. M: 0.0665 c. 1578: 157 147 111 93 147 111 157 147 93 111 157 147 SP SD: 0.0376 c. M: 0.0558 c. 1848: 184 172 130 109 172 130 184 172 109 130 184 172 SP SD: 0.0375 c. M: 0.0569 c. 2019: 201 188 142 119 188 142 201 188 119 142 201 188 SP SD: 0.0275 c. M: 0.0456 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: 0.0000 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 9.7541 cents 6 x 7 x x 8 x 9 x 10 x 11:12 S SD: 8.8001 cents 7 x 8 x 9 x 10 x 11 x 12:13:14 S SD: 11.1791 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 6.1706 cents 9 x 10:11 x 12:13 x 14:15:16:17:18 S SD: 10.8867 cents 10:11 x 12 x 13:14:15:16:17:18:19:20 S SD: 8.9870 cents 11:12 x 13:14:15 x 17 x 18:19:21:22 SD: 10.7289 cents 12:13:14 x 15:16:17:18:19:20:21:23:24 SD: 5.6308 cents 13:14:15 x 16:17:18:20:21:22:23:24:26 S SD: 7.3530 cents 15:16:17:18:19:20:21:23:24:25:26:28:30 SD: 4.2303 cents 24:26:27:29:30:32:34:36:38:40:42:45:48 SD: 3.7279 cents 27:29:31:32:34:36:38:41:43:45:47:51:54 SD: 3.2859 cents 28:30:32:34:35:37:39:42:45:47:49:53:56 SD: 2.9105 cents 30:32:34:36:38:40:42:45:48:50:53:56:60 SD: 2.7937 cents 36:39:41:43:45:48:50:54:58:60:63:68:72 SD: 2.5871 cents 40:43:46:48:50:53:56:60:64:67:70:75:80 SD: 1.5409 cents 55:59:63:66:69:73:77:83:88:92:96:103:110 SD: 1.4316 cents 56:60:64:67:70:75:78:84:90:93:98:105:112 SD: 1.3498 cents 60:64:69:72:75:80:84:90:96:100:105:113:120 SD: 1.2768 cents 70:75:80:84:88:93:98:105:112:117:123:131:140 SD: 1.2380 cents 72:77:82:86:90:96:101:108:115:120:126:135:144 SD: 0.9571 cents 84:90:96:101:105:112:118:126:134:140:147:158:168 SD: 0.8460 cents 96:103:110:115:120:128:134:144:154:160:168:180:192 SD: 0.7546 cents 100:107:114:120:125:133:140:150:160:167:175:188:200 SD: 0.7270 cents 112:120:128:134:140:149:157:168:179:187:196:210:224 SD: 0.6441 cents 120:129:137:144:150:160:168:180:192:200:210:225:240 SD: 0.5031 cents 140:150:160:168:175:187:196:210:224:233:245:263:280 SD: 0.4291 cents 168:180:192:202:210:224:235:252:269:280:294:315:336 SD: 0.3292 cents 180:193:206:216:225:240:252:270:288:300:315:338:360 SD: 0.3116 cents 225:241:257:270:281:300:315:338:360:375:394:422:450 SD: 0.2838 cents 240:257:274:288:300:320:336:360:384:400:420:450:480 SD: 0.1704 cents 280:300:320:336:350:373:392:420:448:467:490:525:560 SD: 0.1650 cents 420:450:480:504:525:560:588:630:672:700:735:788:840 SD: 0.0916 cents 560:600:640:672:700:747:784:840:896:933:980:1050:1120 SD: 0.0825 cents 600:643:686:720:750:800:840:900:960:1000:1050:1125:1200 SD: 0.0681 cents