Scala analysis: qujus2
scale 2 840 2,2,2,2 2.330127 0.344658 strictly proper
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 21/20 84.467193 minor semitone 2: 8/7 231.174094 septimal whole tone 3: 6/5 315.641287 minor third 4: 9/7 435.084095 septimal major third, BP third 5: 4/3 498.044999 perfect fourth 6: 10/7 617.487807 Euler's tritone 7: 3/2 701.955001 perfect fifth 8: 8/5 813.686286 minor sixth 9: 12/7 933.129094 septimal major sixth 10: 9/5 1017.596288 just minor seventh, BP seventh 11: 40/21 1115.532807 acute major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 21/20 84.4672 minor semitone 1: 21/20 84.4672 minor semitone 2: 160/147 146.7069 3: 21/20 84.4672 minor semitone 4: 15/14 119.4428 major diatonic semitone 5: 28/27 62.9609 Archytas' 1/3-tone 6: 15/14 119.4428 major diatonic semitone 7: 21/20 84.4672 minor semitone 8: 16/15 111.7313 minor diatonic semitone 9: 15/14 119.4428 major diatonic semitone 10: 21/20 84.4672 minor semitone 11: 200/189 97.9365 12: 21/20 84.4672 minor semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 28/27 62.961 cents Archytas' 1/3-tone 1: 5 21/20 84.467 cents minor semitone 1: 1 200/189 97.937 cents 1: 1 16/15 111.731 cents minor diatonic semitone 1: 3 15/14 119.443 cents major diatonic semitone 1: 1 160/147 146.707 cents 2: 1 441/400 168.934 cents two (minor semitone) 2: 4 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: 3 8/7 231.174 cents septimal whole tone 3: 4 7/6 266.871 cents septimal minor third 3: 2 25/21 301.847 cents BP second, quasi-equal minor third 3: 5 6/5 315.641 cents minor 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: 3 5/4 386.314 cents major third 4: 2 63/50 400.108 cents quasi-equal major third 4: 3 80/63 413.578 cents wide major third 4: 2 9/7 435.084 cents septimal major third, BP third 5: 2 21/16 470.781 cents narrow fourth 5: 7 4/3 498.045 cents perfect fourth 5: 2 27/20 519.551 cents acute fourth 5: 1 200/147 533.021 cents 6: 6 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 6: 6 10/7 617.488 cents Euler's tritone 7: 1 147/100 666.979 cents 7: 2 40/27 680.449 cents grave fifth 7: 7 3/2 701.955 cents perfect fifth 7: 2 32/21 729.219 cents wide fifth 8: 2 14/9 764.916 cents septimal minor sixth 8: 3 63/40 786.422 cents narrow minor sixth 8: 2 100/63 799.892 cents quasi-equal minor sixth 8: 3 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: 5 5/3 884.359 cents major sixth, BP sixth 9: 2 42/25 898.153 cents quasi-equal major sixth 9: 4 12/7 933.129 cents septimal major sixth 10: 3 7/4 968.826 cents harmonic seventh 10: 3 16/9 996.090 cents Pythagorean minor seventh 10: 1 25/14 1003.802 cents middle minor seventh 10: 4 9/5 1017.596 cents just minor seventh, BP seventh 10: 1 800/441 1031.066 cents 11: 1 147/80 1053.293 cents 11: 3 28/15 1080.557 cents grave major seventh 11: 1 15/8 1088.269 cents classic major seventh 11: 1 189/100 1102.063 cents 11: 5 40/21 1115.533 cents acute major seventh 11: 1 27/14 1137.039 cents septimal major seventh 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 : 84.5 231.2 315.6 435.1 498.0 617.5 702.0 813.7 933.1 1017.6 1115.5 1200.0 84.5 : 146.7 231.2 350.6 413.6 533.0 617.5 729.2 848.7 933.1 1031.1 1115.5 1200.0 231.2 : 84.5 203.9 266.9 386.3 470.8 582.5 702.0 786.4 884.4 968.8 1053.3 1200.0 315.6 : 119.4 182.4 301.8 386.3 498.0 617.5 702.0 799.9 884.4 968.8 1115.5 1200.0 435.1 : 63.0 182.4 266.9 378.6 498.0 582.5 680.4 764.9 849.4 996.1 1080.6 1200.0 498.0 : 119.4 203.9 315.6 435.1 519.6 617.5 702.0 786.4 933.1 1017.6 1137.0 1200.0 617.5 : 84.5 196.2 315.6 400.1 498.0 582.5 667.0 813.7 898.2 1017.6 1080.6 1200.0 702.0 : 111.7 231.2 315.6 413.6 498.0 582.5 729.2 813.7 933.1 996.1 1115.5 1200.0 813.7 : 119.4 203.9 301.8 386.3 470.8 617.5 702.0 821.4 884.4 1003.8 1088.3 1200.0 933.1 : 84.5 182.4 266.9 351.3 498.0 582.5 702.0 764.9 884.4 968.8 1080.6 1200.0 1017.6: 97.9 182.4 266.9 413.6 498.0 617.5 680.4 799.9 884.4 996.1 1115.5 1200.0 1115.5: 84.5 168.9 315.6 400.1 519.6 582.5 702.0 786.4 898.2 1017.6 1102.1 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 62.9609 .. 146.7069 cents 360/343, 83.7460 cents 28/27, 62.9609 cents 2: 168.9344 .. 231.1741 cents 3200/3087, 62.2397 cents 64827/64000, 22.2275 cents 3: 266.8709 .. 350.6169 cents 360/343, 83.7460 cents 49/48, 35.6968 cents 4: 351.3381 .. 435.0841 cents 360/343, 83.7460 cents 2401/2400, 0.7212 cents 5: 470.7809 .. 533.0206 cents 3200/3087, 62.2397 cents 49/48, 35.6968 cents 6: 582.5122 .. 617.4878 cents 50/49, 34.9756 cents 1029/1000, 49.4916 cents 7: 666.9794 .. 729.2191 cents 3200/3087, 62.2397 cents 1029/1000, 49.4916 cents 8: 764.9159 .. 848.6619 cents 360/343, 83.7460 cents 49/48, 35.6968 cents 9: 849.3831 .. 933.1291 cents 360/343, 83.7460 cents 2401/2400, 0.7212 cents 10: 968.8259 .. 1031.0656 cents 3200/3087, 62.2397 cents 49/48, 35.6968 cents 11: 1053.2931 .. 1137.0391 cents 360/343, 83.7460 cents 64827/64000, 22.2275 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 28/27, 62.9609 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 160/147, 146.7069 cents Average / Smallest step : 1.588287 Largest / Average step : 1.467069 Largest / Smallest step : 2.330127 Median interval of one step : 91.20186 cents, amount: 0 Most common interval of one step : 21/20, 84.4672 cents, amount: 5 Least squares average step : 101.68829 cents, oct.: 1220.25948 cents Scale is strictly proper Step pattern alph. order: ABACDCAECAFA Step pattern size order : BFBEAEBDEBCB Scale is a Constant Structure, by a margin of 0.72120 cents Scale diversity : 1.472597 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.344658 Rothenberg efficiency : 0.308430 redundancy: 0.691570 Efficiency x scale size : 3.701154 Number of different interval sizes : 52 = 4.72727 / class Number of one step interval sizes : 6 Highest interval variety : 6 Mean interval variety : 4.72727 = 52/11 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 2401/2400, 0.7212 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 7 Number of recognisable fifths : 7, average 701.9550 cents 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 : 2 at degree 7 Formal octave complements present : 5 = 41.6667% 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 differentially coherent in interval classes 5 and 6 combined -- Rational properties -- Prime limit : 7 Odd number limit : 441 (O: 441 U: 441) Highest odd numerator or denominator: 21 Scale harmonicity : 0.021565 Average absolute harmonicity : 0.177687 Specific harmonicity : 0.132094 Fundamental : 1/420, -8.7142 octaves, 0.6229 Hz. Guide tone : 2520, 11.2992 octaves, 659296.425 Hz. Exponens Consonantiae : 1.058400E+06, 20.01345 octaves Euler's gradus suavitatis : 32 Sum of Mann's harmonic distance : 111.0, average 9.25000 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 9.44040, average 0.78670 Sum of Tenney's harmonic distance : 19.75519, average 1.64627 Vogel's harmonic complexity : 16.41667 Wille's k value : 220 Wilson's harmonic complexity : 33 Rectangular lattice diameter : 7 Triangular lattice diameter : 4 Lattice compactness : 212.13356, average 2.71966 Lattice compactness (without 2's) : 137.66818, average 1.76498 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -2 .. 3 1.08333 19 1 3: -1 .. 2 0.50000 10 2 5: -1 .. 1 -0.16667 6 3 7: -1 .. 1 -0.33333 6 6 Average exponent except 2's : 0 / 12 = 0.00000 Average absolute exponent except 2's: 22 / 12 = 1.83333 Scale is weakly epimorphic with val: <12 17 26 32| Scale is weakly epimorphic with val: <12 17 28 37| Scale is JI-epimorphic with val: <12 19 28 34| = patent Scale is weakly epimorphic with val: <12 20 25 31| Scale is weakly epimorphic with val: <12 20 27 33| Scale is weakly epimorphic with val: <12 20 29 35| Scale is weakly epimorphic with val: <12 20 31 37|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 20.0787 c. M: 35.0841 c. Twelve-tone Chromatic (1/11-comma) 14: 1 2 1 1 1 1 1 2 1 1 1 1 P B ME SD: 19.1431 c. M:-43.4566 c. 15: 1 2 1 1 1 2 1 1 2 1 1 1 P D3 SD: 18.1131 c. M: 35.0841 c. 16: 1 2 1 2 1 1 1 2 1 2 1 1 P B D3 SD: 19.8306 c. M: 33.1291 c. 19: 1 3 1 2 1 2 1 2 2 1 2 1 N D3 SD: 12.9990 c. M:-21.4575 c. 25: 2 3 2 2 1 3 2 2 2 2 2 2 P D3 SD: 12.9275 c. M: 21.1291 c. 26: 2 3 2 2 2 2 2 3 2 2 2 2 SP B ME SD: 11.0159 c. M: 19.6995 c. 27: 2 3 2 3 1 3 2 2 3 2 2 2 P D3 SD: 7.2239 c. M: 13.6863 c. 31: 2 4 2 3 2 3 2 3 3 2 3 2 P D3 SD: 5.9226 c. M: 11.1447 c. 41: 3 5 3 4 2 4 3 4 4 3 3 3 P SD: 3.9488 c. M:-6.7940 c. 53: 4 6 4 5 3 5 4 5 5 4 4 4 SP SD: 3.9554 c. M: 6.1671 c. 58: 4 7 4 6 3 6 4 5 6 4 5 4 P SD: 3.2505 c. M: 6.7897 c. 68: 5 8 5 7 3 7 5 6 7 5 5 5 P SD: 3.5012 c. M:-6.0924 c. 72: 5 9 5 7 4 7 5 7 7 5 6 5 P SD: 1.5668 c. M:-2.9804 c. 99: 7 12 7 10 5 10 7 9 10 7 8 7 P SD: 0.8443 c. M: 1.5651 c. 171: 12 21 12 17 9 17 12 16 17 12 14 12 P SD: 0.2174 c. M:-0.4049 c. 441: 31 54 31 44 23 44 31 41 44 31 36 31 P SD: 0.1399 c. M:-0.2901 c. 612: 43 75 43 61 32 61 43 57 61 43 50 43 P SD: 0.1303 c. M:-0.2100 c. 1007: 71 123 71 100 53 100 71 94 100 71 82 71 SP SD: 0.1251 c. M:-0.2164 c. 1065: 75 130 75 106 56 106 75 99 106 75 87 75 SP SD: 0.1153 c. M: 0.1882 c. 1106: 78 135 78 110 58 110 78 103 110 78 90 78 SP SD: 0.0935 c. M:-0.1621 c. 1236: 87 151 87 123 65 123 87 115 123 87 101 87 SP SD: 0.0809 c. M: 0.1326 c. 1277: 90 156 90 127 67 127 90 119 127 90 104 90 SP SD: 0.0722 c. M:-0.1060 c. 1407: 99 172 99 140 74 140 99 131 140 99 115 99 SP SD: 0.0623 c. M: 0.1161 c. 1547: 109 189 109 154 81 154 109 144 154 109 126 109 SP SD: 0.0618 c. M:-0.1154 c. 1578: 111 193 111 157 83 157 111 147 157 111 129 111 SP SD: 0.0571 c. M: 0.1058 c. 1677: 118 205 118 167 88 167 118 156 167 118 137 118 SP SD: 0.0469 c. M: 0.0906 c. 1848: 130 226 130 184 97 184 130 172 184 130 151 130 SP SD: 0.0385 c. M: 0.0638 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: 16.2568 cents 4 x x x 5 x x 6 x 7 x x 8 S SD: 15.1096 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 6.9951 cents 6 x 7 x x 8 x 9 x 10:11 x 12 S SD: 11.3797 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 7.6855 cents 8 x 9 x 10:11 x 12:13:14 x 15:16 S SD: 11.6578 cents 9 x 10:11 x 12:13 x 14:15:16:17:18 S SD: 10.6476 cents 10 x 11:12:13 x 14:15:16:17:18:19:20 S SD: 7.8717 cents 11:12 x 13:14:15:16:17:18:19:20:21:22 S SD: 10.1828 cents 12:13:14 x 15:16:17:18:19:21:22:23:24 SD: 8.8693 cents 13:14:15:16:17 x 19:20:21:22:23:25:26 SD: 9.2701 cents 14:15:16:17:18:19:20:21:22:24:25:27:28 SD: 5.4057 cents 16:17:18:19 x 21:23:24:26:27:29:30:32 SD: 6.2338 cents 18:19:21:22:23:24:26:27:29:31:32:34:36 SD: 5.0878 cents 20:21:23:24:26:27:29:30:32:34:36:38:40 SD: 3.5725 cents 21:22:24:25:27:28:30:32:34:36:38:40:42 SD: 3.1758 cents 34:36:39:41:44:45:49:51:54:58:61:65:68 SD: 2.8281 cents 35:37:40:42:45:47:50:53:56:60:63:67:70 SD: 2.0973 cents 40:42:46:48:51:53:57:60:64:69:72:76:80 SD: 2.0378 cents 42:44:48:50:54:56:60:63:67:72:76:80:84 SD: 1.4810 cents 56:59:64:67:72:75:80:84:90:96:101:107:112 SD: 1.2390 cents 60:63:69:72:77:80:86:90:96:103:108:114:120 SD: 1.1316 cents 80:84:91:96:103:107:114:120:128:137:144:152:160 SD: 0.9928 cents 84:88:96:101:108:112:120:126:134:144:151:160:168 SD: 0.6406 cents 105:110:120:126:135:140:150:158:168:180:189:200:210 SD: 0.5625 cents 126:132:144:151:162:168:180:189:202:216:227:240:252 SD: 0.4916 cents 140:147:160:168:180:187:200:210:224:240:252:267:280 SD: 0.3142 cents 224:235:256:269:288:299:320:336:358:384:403:427:448 SD: 0.3102 cents 240:252:274:288:309:320:343:360:384:411:432:457:480 SD: 0.3016 cents 245:257:280:294:315:327:350:368:392:420:441:467:490 SD: 0.3007 cents 260:273:297:312:334:347:371:390:416:446:468:495:520 SD: 0.2835 cents 280:294:320:336:360:373:400:420:448:480:504:533:560 SD: 0.1573 cents 420:441:480:504:540:560:600:630:672:720:756:800:840 SD: 0.0000 cents