Scala analysis: qujus3
scale 3 840 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: 10/9 182.403712 minor whole tone 3: 7/6 266.870906 septimal 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: 14/9 764.915905 septimal minor sixth 9: 5/3 884.358713 major sixth, BP sixth 10: 7/4 968.825906 harmonic 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: 200/189 97.9365 3: 21/20 84.4672 minor semitone 4: 15/14 119.4428 major diatonic semitone 5: 16/15 111.7313 minor diatonic semitone 6: 21/20 84.4672 minor semitone 7: 15/14 119.4428 major diatonic semitone 8: 28/27 62.9609 Archytas' 1/3-tone 9: 15/14 119.4428 major diatonic semitone 10: 21/20 84.4672 minor semitone 11: 160/147 146.7069 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 : 84.5 182.4 266.9 386.3 498.0 582.5 702.0 764.9 884.4 968.8 1115.5 1200.0 84.5 : 97.9 182.4 301.8 413.6 498.0 617.5 680.4 799.9 884.4 1031.1 1115.5 1200.0 182.4 : 84.5 203.9 315.6 400.1 519.6 582.5 702.0 786.4 933.1 1017.6 1102.1 1200.0 266.9 : 119.4 231.2 315.6 435.1 498.0 617.5 702.0 848.7 933.1 1017.6 1115.5 1200.0 386.3 : 111.7 196.2 315.6 378.6 498.0 582.5 729.2 813.7 898.2 996.1 1080.6 1200.0 498.0 : 84.5 203.9 266.9 386.3 470.8 617.5 702.0 786.4 884.4 968.8 1088.3 1200.0 582.5 : 119.4 182.4 301.8 386.3 533.0 617.5 702.0 799.9 884.4 1003.8 1115.5 1200.0 702.0 : 63.0 182.4 266.9 413.6 498.0 582.5 680.4 764.9 884.4 996.1 1080.6 1200.0 764.9 : 119.4 203.9 350.6 435.1 519.6 617.5 702.0 821.4 933.1 1017.6 1137.0 1200.0 884.4 : 84.5 231.2 315.6 400.1 498.0 582.5 702.0 813.7 898.2 1017.6 1080.6 1200.0 968.8 : 146.7 231.2 315.6 413.6 498.0 617.5 729.2 813.7 933.1 996.1 1115.5 1200.0 1115.5: 84.5 168.9 266.9 351.3 470.8 582.5 667.0 786.4 849.4 968.8 1053.3 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 : 98.66431 cents, oct.: 1183.97171 cents Scale is strictly proper Step pattern alph. order: ABACDACECAFA Step pattern size order : BCBEDBEAEBFB 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 Most common triad is 0.0 498.045 701.955 cents, amount: 3 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 : 2 at degree 3 Length of longest subharmonic part : 1 at degree 0 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 -- 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 : 0.179969 Specific harmonicity : 0.132094 Fundamental : 1/1260, -10.2992 octaves, 0.2076 Hz. Guide tone : 840, 9.7142 octaves, 219765.475 Hz. Exponens Consonantiae : 1.058400E+06, 20.01345 octaves Euler's gradus suavitatis : 32 Sum of Mann's harmonic distance : 106.0, average 8.83333 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 : 18.85210, average 1.57101 Vogel's harmonic complexity : 21.66667 Wille's k value : 1102 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 0.00000 16 1 3: -2 .. 1 -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 1 1 1 2 1 1 1 1 1 2 1 P B ME SD: 19.1431 c. M: 43.4566 c. 16: 1 1 2 1 2 1 1 1 2 1 2 1 P B D3 SD: 19.8306 c. M:-33.1291 c. 19: 1 2 1 2 2 1 2 1 2 1 3 1 N D3 SD: 12.9990 c. M: 21.4575 c. 25: 2 2 2 2 2 2 3 1 2 2 3 2 P D3 SD: 12.9275 c. M:-21.1291 c. 26: 2 2 2 2 3 2 2 2 2 2 3 2 SP B ME SD: 11.0159 c. M:-19.6995 c. 27: 2 2 2 3 2 2 3 1 3 2 3 2 P D3 SD: 7.2239 c. M:-13.6863 c. 31: 2 3 2 3 3 2 3 2 3 2 4 2 P D3 SD: 5.9226 c. M:-11.1447 c. 41: 3 3 3 4 4 3 4 2 4 3 5 3 P SD: 3.9488 c. M: 6.7940 c. 53: 4 4 4 5 5 4 5 3 5 4 6 4 SP SD: 3.9554 c. M:-6.1671 c. 58: 4 5 4 6 5 4 6 3 6 4 7 4 P SD: 3.2505 c. M:-6.7897 c. 68: 5 5 5 7 6 5 7 3 7 5 8 5 P SD: 3.5012 c. M: 6.0924 c. 72: 5 6 5 7 7 5 7 4 7 5 9 5 P SD: 1.5668 c. M: 2.9804 c. 99: 7 8 7 10 9 7 10 5 10 7 12 7 P SD: 0.8443 c. M:-1.5651 c. 171: 12 14 12 17 16 12 17 9 17 12 21 12 P SD: 0.2174 c. M: 0.4049 c. 441: 31 36 31 44 41 31 44 23 44 31 54 31 P SD: 0.1399 c. M: 0.2901 c. 612: 43 50 43 61 57 43 61 32 61 43 75 43 P SD: 0.1303 c. M: 0.2100 c. 1007: 71 82 71 100 94 71 100 53 100 71 123 71 SP SD: 0.1251 c. M: 0.2164 c. 1065: 75 87 75 106 99 75 106 56 106 75 130 75 SP SD: 0.1153 c. M:-0.1882 c. 1106: 78 90 78 110 103 78 110 58 110 78 135 78 SP SD: 0.0935 c. M:-0.1621 c. 1236: 87 101 87 123 115 87 123 65 123 87 151 87 SP SD: 0.0809 c. M:-0.1326 c. 1277: 90 104 90 127 119 90 127 67 127 90 156 90 SP SD: 0.0722 c. M:-0.1060 c. 1407: 99 115 99 140 131 99 140 74 140 99 172 99 SP SD: 0.0623 c. M:-0.1161 c. 1547: 109 126 109 154 144 109 154 81 154 109 189 109 SP SD: 0.0618 c. M: 0.1154 c. 1578: 111 129 111 157 147 111 157 83 157 111 193 111 SP SD: 0.0571 c. M:-0.1058 c. 1677: 118 137 118 167 156 118 167 88 167 118 205 118 SP SD: 0.0469 c. M:-0.0906 c. 1848: 130 151 130 184 172 130 184 97 184 130 226 130 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: 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: 16.8946 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 11.0283 cents 7 x x 8:9 x 10 x 11 x 12:13:14 S SD: 13.0778 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 8.0040 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 8.1606 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 10.2731 cents 11 x 12:13:14:15 x x 17:18:19:21:22 SD: 8.3583 cents 12 x 13:14:15:16:17:18:19:20:21:23:24 SD: 5.2979 cents 13:14 x 15:16:17:18 x 20:22:23:25:26 SD: 7.8744 cents 14:15 x 16:18:19:20:21:22:23:25:27:28 SD: 8.9321 cents 15:16:17:18:19:20:21 x 23:25:26:29:30 SD: 7.2627 cents 16:17:18:19:20:21:22:24:25:27:28:30:32 SD: 5.7753 cents 17:18:19:20:21:23:24 x 26:28:30:32:34 SD: 5.5311 cents 18:19:20:21:23:24:25:27:28:30:32:34:36 SD: 4.3099 cents 20:21:22:23:25:27:28:30:31:33:35:38:40 SD: 3.4818 cents 24:25:27:28:30:32:34:36:37:40:42:46:48 SD: 3.1513 cents 35:37:39:41:44:47:49:53:54:58:61:67:70 SD: 2.8319 cents 36:38:40:42:45:48:50:54:56:60:63:69:72 SD: 1.6459 cents 56:59:62:65:70:75:78:84:87:93:98:107:112 SD: 1.5848 cents 60:63:67:70:75:80:84:90:93:100:105:114:120 SD: 0.9563 cents 72:76:80:84:90:96:101:108:112:120:126:137:144 SD: 0.8270 cents 96:101:107:112:120:128:134:144:149:160:168:183:192 SD: 0.7655 cents 108:113:120:126:135:144:151:162:168:180:189:206:216 SD: 0.5800 cents 120:126:133:140:150:160:168:180:187:200:210:229:240 SD: 0.5193 cents 144:151:160:168:180:192:202:216:224:240:252:274:288 SD: 0.3753 cents 180:189:200:210:225:240:252:270:280:300:315:343:360 SD: 0.0601 cents