Scala analysis: 12highschool3
Third 12-note Highschool scale
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: 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: 16/15 111.7313 minor diatonic semitone 2: 15/14 119.4428 major 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: 1 28/25 196.198 cents middle second 2: 3 9/8 203.910 cents major whole tone 2: 1 256/225 223.463 cents Neapolitan diminished third 2: 3 8/7 231.174 cents septimal whole tone 3: 3 7/6 266.871 cents septimal minor third 3: 2 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: 2 128/105 342.905 cents septimal neutral third 4: 1 49/40 351.338 cents larger approximation to neutral third 4: 7 5/4 386.314 cents major third 4: 3 32/25 427.373 cents classic diminished fourth 4: 1 64/49 462.348 cents 2 septatones or septatonic major third 5: 3 21/16 470.781 cents narrow fourth 5: 6 4/3 498.045 cents perfect fourth 5: 1 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: 3 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 6: 2 45/32 590.224 cents diatonic tritone 6: 2 64/45 609.776 cents 2nd tritone 6: 3 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: 1 112/75 694.243 cents marvelous fifth 7: 6 3/2 701.955 cents perfect fifth 7: 3 32/21 729.219 cents wide fifth 8: 1 49/32 737.652 cents 8: 3 25/16 772.627 cents classic augmented fifth 8: 7 8/5 813.686 cents minor sixth 8: 1 80/49 848.662 cents smaller approximation to neutral sixth 9: 2 105/64 857.095 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: 2 128/75 925.418 cents diminished seventh 9: 3 12/7 933.129 cents septimal major sixth 10: 3 7/4 968.826 cents harmonic seventh 10: 1 225/128 976.537 cents augmented sixth 10: 3 16/9 996.090 cents Pythagorean minor seventh 10: 1 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: 4.72727 = 52/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 111.7 231.2 315.6 386.3 498.0 582.5 702.0 813.7 884.4 968.8 1088.3 1200.0 111.7 : 119.4 203.9 274.6 386.3 470.8 590.2 702.0 772.6 857.1 976.5 1088.3 1200.0 231.2 : 84.5 155.1 266.9 351.3 470.8 582.5 653.2 737.7 857.1 968.8 1080.6 1200.0 315.6 : 70.7 182.4 266.9 386.3 498.0 568.7 653.2 772.6 884.4 996.1 1115.5 1200.0 386.3 : 111.7 196.2 315.6 427.4 498.0 582.5 702.0 813.7 925.4 1044.9 1129.3 1200.0 498.0 : 84.5 203.9 315.6 386.3 470.8 590.2 702.0 813.7 933.1 1017.6 1088.3 1200.0 582.5 : 119.4 231.2 301.8 386.3 505.8 617.5 729.2 848.7 933.1 1003.8 1115.5 1200.0 702.0 : 111.7 182.4 266.9 386.3 498.0 609.8 729.2 813.7 884.4 996.1 1080.6 1200.0 813.7 : 70.7 155.1 274.6 386.3 498.0 617.5 702.0 772.6 884.4 968.8 1088.3 1200.0 884.4 : 84.5 203.9 315.6 427.4 546.8 631.3 702.0 813.7 898.2 1017.6 1129.3 1200.0 968.8 : 119.4 231.2 342.9 462.3 546.8 617.5 729.2 813.7 933.1 1044.9 1115.5 1200.0 1088.3: 111.7 223.5 342.9 427.4 498.0 609.8 694.2 813.7 925.4 996.1 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 .. 342.9054 cents 256/245, 76.0345 cents 49/48, 35.6968 cents 4: 351.3381 .. 462.3482 cents 2560/2401, 111.0101 cents 1029/1024, 8.4327 cents 5: 470.7809 .. 546.8154 cents 256/245, 76.0345 cents 1029/1024, 8.4327 cents 6: 568.7174 .. 631.2826 cents 648/625, 62.5651 cents 875/864, 21.9020 cents 7: 653.1846 .. 729.2191 cents 256/245, 76.0345 cents 875/864, 21.9020 cents 8: 737.6518 .. 848.6619 cents 2560/2401, 111.0101 cents 1029/1024, 8.4327 cents 9: 857.0946 .. 933.1291 cents 256/245, 76.0345 cents 1029/1024, 8.4327 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 : 99.22027 cents, oct.: 1190.64323 cents Scale is strictly proper Step pattern alph. order: ABCDACBADCBA Step pattern size order : CDBACBDCABDC Scale is a Constant Structure, by a margin of 8.43272 cents Scale diversity : 1.454551 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.301389 Rothenberg efficiency : 0.304052 redundancy: 0.695948 Efficiency x scale size : 3.648629 Number of different interval sizes : 52 = 4.72727 / class Number of one step interval sizes : 4 Highest interval variety : 6 Mean interval variety : 4.72727 = 52/11 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 225/224, 7.7115 cents Most common intervals : 5/4, 386.3137 cents & inv., amount: 7 Most common triad is 0.0 386.314 813.686 cents, amount: 3 Number of recognisable fifths : 7, average 700.8534 cents Scale contains a complete diamond : 1 3 5 15 All steps are superparticular (epimoric) Length of longest harmonic part : 2 at degree 6 Length of longest subharmonic part : 2 at degree 0 Formal octave complements present : 11 = 91.6667% Scale is differentially coherent in interval class 1 Scale is n-1 differentially coherent in interval classes 2 and 3 combined -- Rational properties -- Prime limit : 7 Odd number limit : 225 (O: 225 U: 225) Highest odd numerator or denominator: 15 Scale harmonicity : 0.021880 Average absolute harmonicity : 0.191506 Specific harmonicity : 0.149542 Fundamental : 1/840, -9.7142 octaves, 0.3115 Hz. Guide tone : 1680, 10.7142 octaves, 439530.950 Hz. Exponens Consonantiae : 1.411200E+06, 20.42849 octaves Euler's gradus suavitatis : 32 Sum of Mann's harmonic distance : 74.0, average 6.16667 Mersenne's string divisions : 256936680 Sum of van Prooijen's expressibility: 8.63760, average 0.71980 Sum of Tenney's harmonic distance : 16.91347, average 1.40946 Vogel's harmonic complexity : 18.66667 Wille's k value : 787 Wilson's harmonic complexity : 30 Rectangular lattice diameter : 6 Triangular lattice diameter : 4 Lattice compactness : 242.83231, average 3.11323 Lattice compactness (without 2's) : 120.17815, average 1.54075 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -3 .. 4 0.50000 22 1 3: -1 .. 1 0.00000 6 2 5: -1 .. 1 -0.08333 7 6 7: -1 .. 1 0.08333 3 3 Average exponent except 2's : 0 / 12 = 0.00000 Average absolute exponent except 2's: 16 / 12 = 1.33333 Scale is weakly epimorphic with val: <12 17 28 34| Scale is JI-epimorphic with val: <12 19 28 34| = patent Scale is weakly epimorphic with val: <12 21 25 31| Scale is weakly epimorphic with val: <12 21 26 32| Scale is weakly epimorphic with val: <12 21 28 34| Scale is weakly epimorphic with val: <12 21 29 35| Scale is weakly epimorphic with val: <12 21 31 37|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 16.8233 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: 12.1023 c. M:-21.4575 c. 22: 2 2 2 1 2 2 2 2 1 2 2 2 P D3 I SD: 9.4437 c. M:-17.4878 c. 31: 3 3 2 2 3 2 3 3 2 2 3 3 P SD: 3.7701 c. M: 5.9639 c. 53: 5 5 4 3 5 4 5 5 3 4 5 5 SP SD: 2.8174 c. M:-6.1671 c. 56: 5 6 4 3 5 4 6 5 3 4 6 5 P SD: 4.2856 c. M: 5.7873 c. 68: 6 7 5 4 6 5 7 6 4 5 7 6 SP SD: 3.1746 c. M:-5.8489 c. 72: 7 7 5 4 7 5 7 7 4 5 7 7 P SD: 2.6806 c. M: 4.9354 c. 84: 8 8 6 5 8 6 8 8 5 6 8 8 SP SD: 2.0184 c. M:-3.2021 c. 87: 8 9 6 5 8 6 9 8 5 6 9 8 P SD: 1.9492 c. M:-3.3087 c. 99: 9 10 7 6 9 7 10 9 6 7 10 9 SP SD: 1.4034 c. M:-2.6404 c. 118: 11 12 8 7 11 8 12 11 7 8 12 11 P SD: 1.3986 c. M: 2.8512 c. 130: 12 13 9 8 12 9 13 12 8 9 13 12 SP SD: 1.0693 c. M:-1.7951 c. 140: 13 14 10 8 13 10 14 13 8 10 14 13 SP SD: 0.7793 c. M: 1.5016 c. 171: 16 17 12 10 16 12 17 16 10 12 17 16 SP SD: 0.3294 c. M:-0.5494 c. 270: 25 27 19 16 25 19 27 25 16 19 27 25 SP SD: 0.3251 c. M:-0.6202 c. 410: 38 41 29 24 38 29 41 38 24 29 41 38 SP SD: 0.3440 c. M:-0.5118 c. 441: 41 44 31 26 41 31 44 41 26 31 44 41 SP SD: 0.1122 c. M: 0.1993 c. 612: 57 61 43 36 57 43 61 57 36 43 61 57 SP SD: 0.0973 c. M:-0.1985 c. 935: 87 93 66 55 87 66 93 87 55 66 93 87 SP SD: 0.0965 c. M:-0.1616 c. 966: 90 96 68 57 90 68 96 90 57 68 96 90 SP SD: 0.0868 c. M:-0.1182 c. 1106: 103 110 78 65 103 78 110 103 65 78 110 103 SP SD: 0.0664 c. M:-0.1280 c. 1236: 115 123 87 73 115 87 123 115 73 87 123 115 SP SD: 0.0800 c. M:-0.1073 c. 1277: 119 127 90 75 119 90 127 119 75 90 127 119 SP SD: 0.0742 c. M:-0.1033 c. 1376: 128 137 97 81 128 97 137 128 81 97 137 128 SP SD: 0.0667 c. M:-0.1034 c. 1407: 131 140 99 83 131 99 140 131 83 99 140 131 SP SD: 0.0422 c. M: 0.0763 c. 1547: 144 154 109 91 144 109 154 144 91 109 154 144 SP SD: 0.0387 c. M: 0.0665 c. 1578: 147 157 111 93 147 111 157 147 93 111 157 147 SP SD: 0.0382 c. M: 0.0558 c. 1848: 172 184 130 109 172 130 184 172 109 130 184 172 SP SD: 0.0380 c. M: 0.0569 c. 2019: 188 201 142 119 188 142 201 188 119 142 201 188 SP SD: 0.0269 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: 9.4015 cents 11:12 x 13:14:15 x 17 x 18:19:21:22 SD: 11.0356 cents 12:13:14 x 15:16:17:18:19:20:21:23:24 SD: 5.8852 cents 13:14:15 x 16:17:18:20:21:22:23:24:26 S SD: 7.4624 cents 15:16:17:18:19:20:21:23:24:25:26:28:30 SD: 4.1812 cents 20:21:23:24:25:27:28:30:32:33:35:38:40 SD: 3.8647 cents 27:29:31:32:34:36:38:41:43:45:47:51:54 SD: 3.4157 cents 28:30:32:34:35:37:39:42:45:47:49:53:56 SD: 2.9806 cents 30:32:34:36:38:40:42:45:48:50:53:56:60 SD: 2.7188 cents 35:37:40:42:44:47:49:53:56:58:61:66:70 SD: 2.6423 cents 36:38:41:43:45:48:50:54:58:60:63:68:72 SD: 2.5366 cents 40:43:46:48:50:53:56:60:64:67:70:75:80 SD: 1.8451 cents 48:51:55:58:60:64:67:72:77:80:84:90:96 SD: 1.3350 cents 60:64:69:72:75:80:84:90:96:100:105:113:120 SD: 1.1033 cents 72:77:82:86:90:96:101:108:115:120:126:135:144 SD: 0.9927 cents 90:96:103:108:113:120:126:135:144:150:158:169:180 SD: 0.8391 cents 100:107:114:120:125:133:140:150:160:167:175:188:200 SD: 0.8332 cents 104:111:119:125:130:139:146:156:166:173:182:195:208 SD: 0.7518 cents 105:112:120:126:131:140:147:158:168:175:184:197:210 SD: 0.5759 cents 120:128:137:144:150:160:168:180:192:200:210:225:240 SD: 0.1504 cents 300:320:343:360:375:400:420:450:480:500:525:563:600 SD: 0.1416 cents 420:448:480:504:525:560:588:630:672:700:735:788:840 SD: 0.0916 cents 540:576:617:648:675:720:756:810:864:900:945:1013:1080 SD: 0.0787 cents 600:640:686:720:750:800:840:900:960:1000:1050:1125:1200 SD: 0.0601 cents