Scala analysis: magic
magic chord test
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 28/25 196.198479 middle second 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: 10/7 617.487807 Euler's tritone 8: 8/5 813.686286 minor sixth 9: 5/3 884.358713 major sixth, BP sixth 10: 7/4 968.825906 harmonic seventh 11: 25/14 1003.801521 middle minor seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 28/25 196.1985 middle second 1: 28/25 196.1985 middle second 2: 50/49 34.9756 Erlich's decatonic comma, tritonic diesis 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: 50/49 34.9756 Erlich's decatonic comma, tritonic diesis 8: 28/25 196.1985 middle second 9: 25/24 70.6724 classic chromatic semitone, minor chroma 10: 21/20 84.4672 minor semitone 11: 50/49 34.9756 Erlich's decatonic comma, tritonic diesis 12: 28/25 196.1985 middle second
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 3 50/49 34.976 cents Erlich's decatonic comma, tritonic diesis 1: 2 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 3 21/20 84.467 cents minor semitone 1: 1 16/15 111.731 cents minor diatonic semitone 1: 3 28/25 196.198 cents middle second 2: 3 15/14 119.443 cents major diatonic semitone 2: 2 35/32 155.140 cents septimal neutral second 2: 1 10/9 182.404 cents minor whole tone 2: 1 28/25 196.198 cents middle second 2: 3 8/7 231.174 cents septimal whole tone 2: 1 7/6 266.871 cents septimal minor third 2: 1 784/625 392.397 cents two (middle second) 3: 2 125/112 190.115 cents classic augmented semitone 3: 1 8/7 231.174 cents septimal whole tone 3: 2 7/6 266.871 cents septimal minor third 3: 1 25/21 301.847 cents BP second, quasi-equal minor third 3: 3 6/5 315.641 cents minor third 3: 1 49/40 351.338 cents larger approximation to neutral third 3: 2 32/25 427.373 cents classic diminished fourth 4: 2 25/21 301.847 cents BP second, quasi-equal minor third 4: 1 49/40 351.338 cents larger approximation to neutral third 4: 5 5/4 386.314 cents major third 4: 1 32/25 427.373 cents classic diminished fourth 4: 1 64/49 462.348 cents 2 septatones or septatonic major third 4: 2 168/125 511.840 cents 5: 2 5/4 386.314 cents major third 5: 1 125/98 421.289 cents 5: 1 21/16 470.781 cents narrow fourth 5: 3 4/3 498.045 cents perfect fourth 5: 2 48/35 546.815 cents septimal semi-augmented fourth 5: 3 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 6: 1 125/98 421.289 cents 6: 1 75/56 505.757 cents marvelous 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: 3 10/7 617.488 cents Euler's tritone 6: 1 36/25 631.283 cents classic diminished fifth 6: 1 112/75 694.243 cents marvelous fifth 6: 1 196/125 778.711 cents 7: 3 10/7 617.488 cents Euler's tritone 7: 2 35/24 653.185 cents septimal semi-diminished fifth 7: 3 3/2 701.955 cents perfect fifth 7: 1 32/21 729.219 cents wide fifth 7: 1 196/125 778.711 cents 7: 2 8/5 813.686 cents minor sixth 8: 2 125/84 688.160 cents 8: 1 49/32 737.652 cents 8: 1 25/16 772.627 cents classic augmented fifth 8: 5 8/5 813.686 cents minor sixth 8: 1 80/49 848.662 cents smaller approximation to neutral sixth 8: 2 42/25 898.153 cents quasi-equal major sixth 9: 2 25/16 772.627 cents classic augmented fifth 9: 1 80/49 848.662 cents smaller approximation to neutral sixth 9: 3 5/3 884.359 cents major sixth, BP sixth 9: 1 42/25 898.153 cents quasi-equal major sixth 9: 2 12/7 933.129 cents septimal major sixth 9: 1 7/4 968.826 cents harmonic seventh 9: 2 224/125 1009.885 cents 10: 1 625/392 807.603 cents 10: 1 12/7 933.129 cents septimal major sixth 10: 3 7/4 968.826 cents harmonic seventh 10: 1 25/14 1003.802 cents middle minor seventh 10: 1 9/5 1017.596 cents just minor seventh, BP seventh 10: 2 64/35 1044.860 cents septimal neutral seventh 10: 3 28/15 1080.557 cents grave major seventh 11: 3 25/14 1003.802 cents middle minor seventh 11: 1 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 11: 3 49/25 1165.024 cents BP eighth Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 6.36364 = 70/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 196.2 231.2 315.6 386.3 498.0 582.5 617.5 813.7 884.4 968.8 1003.8 1200.0 196.2 : 35.0 119.4 190.1 301.8 386.3 421.3 617.5 688.2 772.6 807.6 1003.8 1200.0 231.2 : 84.5 155.1 266.9 351.3 386.3 582.5 653.2 737.7 772.6 968.8 1165.0 1200.0 315.6 : 70.7 182.4 266.9 301.8 498.0 568.7 653.2 688.2 884.4 1080.6 1115.5 1200.0 386.3 : 111.7 196.2 231.2 427.4 498.0 582.5 617.5 813.7 1009.9 1044.9 1129.3 1200.0 498.0 : 84.5 119.4 315.6 386.3 470.8 505.8 702.0 898.2 933.1 1017.6 1088.3 1200.0 582.5 : 35.0 231.2 301.8 386.3 421.3 617.5 813.7 848.7 933.1 1003.8 1115.5 1200.0 617.5 : 196.2 266.9 351.3 386.3 582.5 778.7 813.7 898.2 968.8 1080.6 1165.0 1200.0 813.7 : 70.7 155.1 190.1 386.3 582.5 617.5 702.0 772.6 884.4 968.8 1003.8 1200.0 884.4 : 84.5 119.4 315.6 511.8 546.8 631.3 702.0 813.7 898.2 933.1 1129.3 1200.0 968.8 : 35.0 231.2 427.4 462.3 546.8 617.5 729.2 813.7 848.7 1044.9 1115.5 1200.0 1003.8: 196.2 392.4 427.4 511.8 582.5 694.2 778.7 813.7 1009.9 1080.6 1165.0 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 34.9756 .. 196.1985 cents 686/625, 161.2229 cents 50/49, 34.9756 cents 2: 119.4428 .. 392.3970 cents 10976/9375, 272.9541 cents 375/392,-76.7557 cents 3: 190.1152 .. 427.3726 cents 3584/3125, 237.2573 cents 78125/87808,-202.2817 cents 4: 301.8465 .. 511.8398 cents 3528/3125, 209.9932 cents 625/672,-125.5261 cents 5: 386.3137 .. 582.5122 cents 28/25, 196.1985 cents 625/672,-125.5261 cents 6: 421.2893 .. 778.7107 cents 19208/15625, 357.4213 cents 625/686,-161.2229 cents 7: 617.4878 .. 813.6863 cents 28/25, 196.1985 cents 625/686,-161.2229 cents 8: 688.1602 .. 898.1535 cents 3528/3125, 209.9932 cents 625/672,-125.5261 cents 9: 772.6274 .. 1009.8848 cents 3584/3125, 237.2573 cents 625/672,-125.5261 cents 10: 807.6030 .. 1080.5572 cents 10976/9375, 272.9541 cents 78125/87808,-202.2817 cents 11: 1003.8015 .. 1165.0244 cents 686/625, 161.2229 cents 375/392,-76.7557 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 50/49, 34.9756 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 28/25, 196.1985 cents Average / Smallest step : 2.859135 Largest / Average step : 1.961985 Largest / Smallest step : 5.609579 Median interval of one step : 21/20, 84.4672 cents, amount: 3 Least squares average step : 97.01113 cents, oct.: 1164.13353 cents Scale is not proper Step pattern alph. order: ABCDECBADCBA Step pattern size order : EACBDCAEBCAE Scale is not a Constant Structure Scale diversity : 1.494443 Lumma stability : 0.058293 Lumma impropriety factor : 0.771782 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 : 5 Highest interval variety : 8 Mean interval variety : 6.36364 = 70/11 Median interval variety : 5 Lowest interval variety : 5 Smallest interval difference : 3136/3125, 6.0832 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 : 6, average 696.0715 cents Number of appreciable fifths : 4, average 700.0271 cents Scale contains a complete diamond : 1 5 7 Formal octave complements present : 11 = 91.6667% Scale is n-1 differentially coherent in interval classes 1 and 2 combined Scale is differentially coherent in interval classes 2 and 3 combined -- Rational properties -- Prime limit : 7 Odd number limit : 625 (O: 625 U: 625) Highest odd numerator or denominator: 25 Scale harmonicity : 0.017698 Average absolute harmonicity : 0.166989 Specific harmonicity : 0.114637 Fundamental : 1/2100, -11.0362 octaves, 0.1246 Hz. Guide tone : 4200, 12.0362 octaves, 1098827.374 Hz. Exponens Consonantiae : 8.820000E+06, 23.07235 octaves Euler's gradus suavitatis : 38 Sum of Mann's harmonic distance : 99.0, average 8.25000 Mersenne's string divisions : 324684360 Sum of van Prooijen's expressibility: 9.44927, average 0.78744 Sum of Tenney's harmonic distance : 18.91019, average 1.57585 Vogel's harmonic complexity : 22.41667 Wille's k value : 1837 Wilson's harmonic complexity : 40 Rectangular lattice diameter : 8 Triangular lattice diameter : 4 Lattice compactness : 221.83317, average 2.84401 Lattice compactness (without 2's) : 139.49014, average 1.78834 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -2 .. 3 0.66667 18 1 3: -1 .. 1 -0.08333 3 1 5: -2 .. 2 0.00000 10 4 7: -1 .. 1 0.00000 6 6 Average exponent except 2's :-1 / 12 =-0.08333 Average absolute exponent except 2's: 19 / 12 = 1.58333 Scale is weakly epimorphic with val: <12 18 25 34| Scale is weakly epimorphic with val: <12 18 26 35| Scale is weakly epimorphic with val: <12 18 28 31| Scale is weakly epimorphic with val: <12 18 28 33| Scale is weakly epimorphic with val: <12 18 28 35| Scale is weakly epimorphic with val: <12 18 28 37| Scale is weakly epimorphic with val: <12 18 29 32| Scale is weakly epimorphic with val: <12 18 31 34|
FIT/MODE
12: 2 0 1 1 1 1 0 2 1 1 0 2 N SD: 16.9606 c. M:-31.1741 c. 16: 3 0 1 1 2 1 0 3 1 1 0 3 N SD: 17.8391 c. M: 28.8015 c. 19: 3 1 1 1 2 1 1 3 1 1 1 3 N SD: 11.4349 c. M:-21.4575 c. 24: 4 1 1 2 2 2 0 4 2 1 1 4 N SD: 13.5927 c. M:-18.8259 c. 25: 4 1 2 1 2 2 1 4 1 2 1 4 N SD: 10.9588 c. M:-20.3587 c. 26: 4 1 2 1 3 2 0 5 1 2 1 4 N SD: 11.7885 c. M:-17.4878 c. 27: 4 1 2 2 2 2 1 4 2 2 1 4 N SD: 10.7364 c. M: 18.4207 c. 29: 5 1 2 1 3 2 1 5 1 2 1 5 N SD: 11.8915 c. M:-17.1018 c. 31: 5 1 2 2 3 2 1 5 2 2 1 5 N SD: 3.1960 c. M: 5.9639 c. 56: 9 2 4 3 5 4 2 9 3 4 2 9 N SD: 3.9711 c. M: 5.7873 c. 68: 11 2 5 4 6 5 2 11 4 5 2 11 N SD: 1.9531 c. M: 3.9274 c. 72: 12 2 5 4 7 5 2 12 4 5 2 12 N SD: 2.2961 c. M: 3.8015 c. 87: 14 3 6 5 8 6 3 14 5 6 3 14 N SD: 2.3967 c. M:-3.3087 c. 99: 16 3 7 6 9 7 3 16 6 7 3 16 N SD: 1.2657 c. M: 2.2591 c. 140: 23 4 10 8 13 10 4 23 8 10 4 23 N SD: 0.8263 c. M: 1.5016 c. 171: 28 5 12 10 16 12 5 28 10 12 5 28 N SD: 0.2635 c. M:-0.4049 c. 410: 67 12 29 24 38 29 12 67 24 29 12 67 N SD: 0.2393 c. M: 0.4840 c. 441: 72 13 31 26 41 31 13 72 26 31 13 72 N SD: 0.1541 c. M: 0.2801 c. 612: 100 18 43 36 57 43 18 100 36 43 18 100 N SD: 0.1174 c. M:-0.1985 c. 935: 153 27 66 55 87 66 27 153 55 66 27 153 N SD: 0.1210 c. M: 0.1652 c. 966: 158 28 68 57 90 68 28 158 57 68 28 158 N SD: 0.0879 c. M:-0.1182 c. 1137: 186 33 80 67 106 80 33 186 67 80 33 186 N SD: 0.0719 c. M:-0.1080 c. 1236: 202 36 87 73 115 87 36 202 73 87 36 202 N SD: 0.0802 c. M:-0.1073 c. 1308: 214 38 92 77 122 92 38 214 77 92 38 214 N SD: 0.0773 c. M: 0.1318 c. 1376: 225 40 97 81 128 97 40 225 81 97 40 225 N SD: 0.0489 c. M: 0.0799 c. 1407: 230 41 99 83 131 99 41 230 83 99 41 230 N SD: 0.0435 c. M: 0.0763 c. 1547: 253 45 109 91 144 109 45 253 91 109 45 253 N SD: 0.0412 c. M: 0.0665 c. 1578: 258 46 111 93 147 111 46 258 93 111 46 258 N SD: 0.0264 c. M:-0.0539 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: 42.2336 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: 21.1168 cents 5 x x 6 x x 7 x 8 x x 9:10 S SD: 2.7590 cents 6 x 7 x x 8 x 9 x 10 x 11:12 S SD: 17.0659 cents 7 x 8 x 9 x x 10:11 x 12:13:14 S SD: 13.7294 cents 8:9 x x 10 x 11 x 13 x 14 x 16 SD: 6.9782 cents 9:10 x 11 x 12 x 13:14:15 x 16:18 S SD: 7.9090 cents 10:11 x 12 x 13:14 x 16:17 x 18:20 S SD: 8.1585 cents 11:12 x 13:14:15 x 16 x 18:19:20:22 S SD: 10.3808 cents 12:13:14 x 15:16 x 17:19:20:21 x 24 SD: 7.9610 cents 13 x 15 x 16:17:18:19:21:22 x 23:26 SD: 8.0752 cents 14 x 16:17:18:19 x 20:22:23 x 25:28 SD: 8.1125 cents 15 x 17:18:19:20:21 x 24:25:26:27:30 SD: 3.4648 cents 16:18 x 19:20:21:22:23:26:27:28:29:32 SD: 5.7984 cents 17:19 x 20:21:23:24 x 27:28:30 x 34 SD: 6.3342 cents 18:20:21:22:23:24:25:26:29:30 x 32:36 SD: 6.2197 cents 19:21:22:23:24:25 x 27:30:32:33:34:38 SD: 5.1770 cents 20:22:23:24:25:27:28:29:32:33:35:36:40 SD: 4.3376 cents 21 x 24:25:26:28:29:30:34:35:37:38:42 SD: 4.1764 cents 22 x 25:26:28:29:31 x 35:37 x 39:44 SD: 5.8674 cents 23:26 x 28:29:31:32:33:37:38:40:41:46 SD: 4.1342 cents 24:27 x 29:30:32 x 34:38:40:42:43:48 SD: 2.7834 cents 26:29:30:31:33:35:36:37:42:43 x 46:52 SD: 4.5853 cents 27:30:31:32:34:36:38:39:43:45:47:48:54 SD: 3.2679 cents 28:31:32:34:35:37:39:40:45:47:49:50:56 SD: 3.0620 cents 29:32:33:35:36:39:41 x 46:48:51:52:58 SD: 4.0277 cents 30 x 34:36:38:40:42:43:48:50:53:54:60 SD: 3.1863 cents 31:35 x 37:39:41:43:44:50:52:54:55:62 SD: 3.5095 cents 32:36:37:38:40:43:45:46:51:53:56:57:64 SD: 3.0445 cents 33:37:38:40:41:44:46:47:53:55:58:59:66 SD: 2.3198 cents 35:39:40:42:44:47:49:50:56:58:61:63:70 SD: 2.1513 cents 38 x 43:46:48:51:53:54:61:63:67:68:76 SD: 3.4065 cents 39:44:45:47:49:52:55:56:62:65:68:70:78 SD: 2.6976 cents 40:45:46:48:50:53:56:57:64:67:70:71:80 SD: 1.8550 cents 48:54:55:58:60:64:67:69:77:80:84:86:96 SD: 1.7073 cents 51:57:58:61:64:68:71:73:82:85:89:91:102 SD: 1.5961 cents 56:63:64:67:70:75:78:80:90:93:98:100:112 SD: 1.4949 cents 60:67:69:72:75:80:84:86:96:100:105:107:120 SD: 1.1227 cents 75:84:86:90:94:100:105:107:120:125:131:134:150 SD: 0.7048 cents 100:112:114:120:125:133:140:143:160:167:175:179:200 SD: 0.6959 cents 117:131:134:140:146:156:164:167:187:195:205:209:234 SD: 0.6561 cents 120:134:137:144:150:160:168:171:192:200:210:214:240 SD: 0.6124 cents 132:148:151:158:165:176:185:189:211:220:231:236:264 SD: 0.5974 cents 135:151:154:162:169:180:189:193:216:225:236:241:270 SD: 0.4361 cents 140:157:160:168:175:187:196:200:224:233:245:250:280 SD: 0.3776 cents 168:188:192:202:210:224:235:240:269:280:294:300:336 SD: 0.3513 cents 215:241:246:258:269:287:301:307:344:358:376:384:430 SD: 0.3479 cents 224:251:256:269:280:299:314:320:358:373:392:400:448 SD: 0.3443 cents 225:252:257:270:281:300:315:321:360:375:394:402:450 SD: 0.2725 cents 240:269:274:288:300:320:336:343:384:400:420:429:480 SD: 0.2419 cents 285:319:326:342:356:380:399:407:456:475:499:509:570 SD: 0.2065 cents 300:336:343:360:375:400:420:429:480:500:525:536:600 SD: 0.1741 cents 420:470:480:504:525:560:588:600:672:700:735:750:840 SD: 0.1227 cents 525:588:600:630:656:700:735:750:840:875:919:938:1050 SD: 0.1024 cents 560:627:640:672:700:747:784:800:896:933:980:1000:1120 SD: 0.0944 cents 600:672:686:720:750:800:840:857:960:1000:1050:1071:1200 SD: 0.0867 cents