Scala analysis: qujus1
scale 1 420 2,2,2,2 1.897095 0.538729 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: 28/15 1080.557192 grave major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 15/14 119.4428 major diatonic 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: 16/15 111.7313 minor diatonic semitone 12: 15/14 119.4428 major diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 28/27 62.961 cents Archytas' 1/3-tone 1: 4 21/20 84.467 cents minor semitone 1: 1 200/189 97.937 cents 1: 2 16/15 111.731 cents minor diatonic semitone 1: 4 15/14 119.443 cents major diatonic semitone 2: 4 10/9 182.404 cents minor whole tone 2: 2 28/25 196.198 cents middle second 2: 4 9/8 203.910 cents major whole tone 2: 2 8/7 231.174 cents septimal whole tone 3: 3 7/6 266.871 cents septimal minor third 3: 3 25/21 301.847 cents BP second, quasi-equal minor third 3: 6 6/5 315.641 cents minor third 4: 2 56/45 378.602 cents septimal narrow major third 4: 4 5/4 386.314 cents major third 4: 2 63/50 400.108 cents quasi-equal major third 4: 2 80/63 413.578 cents wide major third 4: 2 9/7 435.084 cents septimal major third, BP third 5: 1 21/16 470.781 cents narrow fourth 5: 8 4/3 498.045 cents perfect fourth 5: 1 75/56 505.757 cents marvelous fourth 5: 2 27/20 519.551 cents acute fourth 6: 6 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 6: 6 10/7 617.488 cents Euler's tritone 7: 2 40/27 680.449 cents grave fifth 7: 1 112/75 694.243 cents marvelous fifth 7: 8 3/2 701.955 cents perfect fifth 7: 1 32/21 729.219 cents wide fifth 8: 2 14/9 764.916 cents septimal minor sixth 8: 2 63/40 786.422 cents narrow minor sixth 8: 2 100/63 799.892 cents quasi-equal minor sixth 8: 4 8/5 813.686 cents minor sixth 8: 2 45/28 821.398 cents septimale wide minor sixth 9: 6 5/3 884.359 cents major sixth, BP sixth 9: 3 42/25 898.153 cents quasi-equal major sixth 9: 3 12/7 933.129 cents septimal major sixth 10: 2 7/4 968.826 cents harmonic seventh 10: 4 16/9 996.090 cents Pythagorean minor seventh 10: 2 25/14 1003.802 cents middle minor seventh 10: 4 9/5 1017.596 cents just minor seventh, BP seventh 11: 4 28/15 1080.557 cents grave major seventh 11: 2 15/8 1088.269 cents classic major seventh 11: 1 189/100 1102.063 cents 11: 4 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: 5 Average number of different intervals per interval class: 4.00000 = 4
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 1080.6 1200.0 84.5 : 97.9 182.4 301.8 413.6 498.0 617.5 680.4 799.9 884.4 996.1 1115.5 1200.0 182.4 : 84.5 203.9 315.6 400.1 519.6 582.5 702.0 786.4 898.2 1017.6 1102.1 1200.0 266.9 : 119.4 231.2 315.6 435.1 498.0 617.5 702.0 813.7 933.1 1017.6 1115.5 1200.0 386.3 : 111.7 196.2 315.6 378.6 498.0 582.5 694.2 813.7 898.2 996.1 1080.6 1200.0 498.0 : 84.5 203.9 266.9 386.3 470.8 582.5 702.0 786.4 884.4 968.8 1088.3 1200.0 582.5 : 119.4 182.4 301.8 386.3 498.0 617.5 702.0 799.9 884.4 1003.8 1115.5 1200.0 702.0 : 63.0 182.4 266.9 378.6 498.0 582.5 680.4 764.9 884.4 996.1 1080.6 1200.0 764.9 : 119.4 203.9 315.6 435.1 519.6 617.5 702.0 821.4 933.1 1017.6 1137.0 1200.0 884.4 : 84.5 196.2 315.6 400.1 498.0 582.5 702.0 813.7 898.2 1017.6 1080.6 1200.0 968.8 : 111.7 231.2 315.6 413.6 498.0 617.5 729.2 813.7 933.1 996.1 1115.5 1200.0 1080.6: 119.4 203.9 301.8 386.3 505.8 617.5 702.0 821.4 884.4 1003.8 1088.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 62.9609 .. 119.4428 cents 405/392, 56.4819 cents 28/27, 62.9609 cents 2: 182.4037 .. 231.1741 cents 36/35, 48.7704 cents 28/27, 62.9609 cents 3: 266.8709 .. 315.6413 cents 36/35, 48.7704 cents 49/48, 35.6968 cents 4: 378.6022 .. 435.0841 cents 405/392, 56.4819 cents 28/27, 62.9609 cents 5: 470.7809 .. 519.5513 cents 36/35, 48.7704 cents 49/48, 35.6968 cents 6: 582.5122 .. 617.4878 cents 50/49, 34.9756 cents 28/27, 62.9609 cents 7: 680.4487 .. 729.2191 cents 36/35, 48.7704 cents 28/27, 62.9609 cents 8: 764.9159 .. 821.3978 cents 405/392, 56.4819 cents 49/48, 35.6968 cents 9: 884.3587 .. 933.1291 cents 36/35, 48.7704 cents 28/27, 62.9609 cents 10: 968.8259 .. 1017.5963 cents 36/35, 48.7704 cents 49/48, 35.6968 cents 11: 1080.5572 .. 1137.0391 cents 405/392, 56.4819 cents 28/27, 62.9609 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 : 15/14, 119.4428 cents Average / Smallest step : 1.588287 Largest / Average step : 1.194428 Largest / Smallest step : 1.897095 Median interval of one step : 104.83390 cents, amount: 0 Least squares average step : 98.07241 cents, oct.: 1176.86897 cents Scale is strictly proper Step pattern alph. order: ABACDACECADC Step pattern size order : BCBEDBEAEBDE Scale is a Constant Structure, by a margin of 35.69681 cents Scale diversity : 1.365342 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.538729 Rothenberg efficiency : 0.356421 redundancy: 0.643579 Efficiency x scale size : 4.277056 Number of different interval sizes : 44 = 4.00000 / class Number of one step interval sizes : 5 Highest interval variety : 5 Mean interval variety : 4.00000 = 4 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 225/224, 7.7115 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 8 Most common triad is 0.0 498.045 701.955 cents, amount: 4 Number of recognisable fifths : 9, average 701.0982 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 : 2 at degree 10 Formal octave complements present : 3 = 25.0000% Scale is n-1 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 Scale is n-1 differentially coherent in interval classes 4 and 5 combined Scale is n-1 differentially coherent in interval classes 5 and 6 combined Scale is n-1 differentially coherent in interval classes 7 and 8 combined Limited inverse transpositions : 3 Inversional symmetry on intervals : 1-2 7-8 -- Rational properties -- Prime limit : 7 Odd number limit : 189 (O: 189 U: 189) Highest odd numerator or denominator: 21 Scale harmonicity : 0.028502 Average absolute harmonicity : 0.180143 Specific harmonicity : 0.143544 Fundamental : 1/180, -7.4919 octaves, 1.4535 Hz. Guide tone : 420, 8.7142 octaves, 109882.737 Hz. Exponens Consonantiae : 7.560000E+04, 16.20610 octaves Euler's gradus suavitatis : 25 Sum of Mann's harmonic distance : 97.0, average 8.08333 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 9.29427, average 0.77452 Sum of Tenney's harmonic distance : 18.55107, average 1.54592 Vogel's harmonic complexity : 14.91667 Wille's k value : 157 Wilson's harmonic complexity : 26 Rectangular lattice diameter : 6 Triangular lattice diameter : 4 Lattice compactness : 201.26958, average 2.58038 Lattice compactness (without 2's) : 134.20408, average 1.72057 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -2 .. 2 -0.08333 15 1 3: -2 .. 1 -0.50000 10 2 5: -1 .. 1 0.00000 6 3 7: 0 .. 1 0.50000 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 19 25 34| 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 25 35| Scale is weakly epimorphic with val: <12 20 26 31| Scale is weakly epimorphic with val: <12 20 26 33| Scale is weakly epimorphic with val: <12 20 26 35| Scale is weakly epimorphic with val: <12 20 26 37| Scale is weakly epimorphic with val: <12 20 27 33| Scale is weakly epimorphic with val: <12 20 27 37| Scale is weakly epimorphic with val: <12 20 29 31| Scale is weakly epimorphic with val: <12 20 29 35| Scale is weakly epimorphic with val: <12 20 30 31| Scale is weakly epimorphic with val: <12 20 30 33| Scale is weakly epimorphic with val: <12 20 30 35| Scale is weakly epimorphic with val: <12 20 30 37| Scale is weakly epimorphic with val: <12 20 31 33| 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: 20.3605 c. M:-35.0841 c. Twelve-tone Chromatic (1/11-comma) 17: 1 2 1 1 2 1 2 1 2 1 1 2 P M ME SD: 20.0971 c. M: 33.3725 c. 19: 1 2 1 2 2 1 2 1 2 1 2 2 P M ME SD: 11.6219 c. M: 21.4575 c. Mandelbaum Twelve out of 19 26: 2 2 2 2 3 2 2 2 2 2 2 3 SP D3 S SD: 12.0983 c. M:-19.6995 c. 27: 2 2 2 3 2 2 3 1 3 2 2 3 SP D3 SD: 8.1630 c. M: 13.8905 c. 31: 2 3 2 3 3 2 3 2 3 2 3 3 SP M ME SD: 5.6439 c. M:-11.1447 c. 41: 3 3 3 4 4 3 4 2 4 3 4 4 SP D3 SD: 3.8900 c. M: 6.7940 c. 53: 4 4 4 5 5 4 5 3 5 4 5 5 SP D3 SD: 3.9731 c. M:-6.2353 c. 58: 4 5 4 6 5 4 6 3 6 4 5 6 SP SD: 3.4870 c. M:-6.7897 c. 68: 5 5 5 7 6 5 7 3 7 5 6 7 SP SD: 3.5308 c. M: 6.0924 c. 72: 5 6 5 7 7 5 7 4 7 5 7 7 SP SD: 1.7291 c. M: 2.9804 c. 99: 7 8 7 10 9 7 10 5 10 7 9 10 SP SD: 0.9806 c. M: 1.7693 c. 171: 12 14 12 17 16 12 17 9 17 12 16 17 SP SD: 0.2086 c. M: 0.4049 c. 441: 31 36 31 44 41 31 44 23 44 31 41 44 SP SD: 0.1590 c. M: 0.2901 c. 612: 43 50 43 61 57 43 61 32 61 43 57 61 SP SD: 0.1315 c. M: 0.2100 c. 783: 55 64 55 78 73 55 78 41 78 55 73 78 SP SD: 0.1314 c. M: 0.2435 c. 935: 66 76 66 93 87 66 93 49 93 66 87 93 SP SD: 0.1173 c. M:-0.2387 c. 1065: 75 87 75 106 99 75 106 56 106 75 99 106 SP SD: 0.1147 c. M:-0.1882 c. 1106: 78 90 78 110 103 78 110 58 110 78 103 110 SP SD: 0.0853 c. M:-0.1621 c. 1236: 87 101 87 123 115 87 123 65 123 87 115 123 SP SD: 0.0812 c. M:-0.1326 c. 1277: 90 104 90 127 119 90 127 67 127 90 119 127 SP SD: 0.0715 c. M:-0.1060 c. 1407: 99 115 99 140 131 99 140 74 140 99 131 140 SP SD: 0.0626 c. M:-0.1161 c. 1547: 109 126 109 154 144 109 154 81 154 109 144 154 SP SD: 0.0570 c. M: 0.1154 c. 1578: 111 129 111 157 147 111 157 83 157 111 147 157 SP SD: 0.0567 c. M:-0.1058 c. 1677: 118 137 118 167 156 118 167 88 167 118 156 167 SP SD: 0.0489 c. M:-0.0906 c. 1718: 121 140 121 171 160 121 171 90 171 121 160 171 SP SD: 0.0504 c. M: 0.0987 c. 1848: 130 151 130 184 172 130 184 97 184 130 172 184 SP SD: 0.0371 c. M:-0.0638 c. 2019: 142 165 142 201 188 142 201 106 201 142 188 201 SP SD: 0.0358 c. M: 0.0690 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: 5.1990 cents 7 x x 8:9 x 10 x 11 x 12:13:14 S SD: 11.5509 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 7.3061 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 8.3179 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 10.7116 cents 11 x 12:13:14:15 x x 17:18:19:21:22 SD: 9.3999 cents 12 x 13:14:15:16:17:18:19:20:21:22:24 S SD: 5.9286 cents 13:14 x 15:16:17:18 x 20:22:23:24:26 S SD: 7.9323 cents 14:15 x 16:18:19:20:21:22:23:25:26:28 SD: 8.7526 cents 15:16:17:18:19:20:21 x 23:25:26:28:30 SD: 6.8743 cents 16:17:18:19:20:21:22:24:25:27:28:30:32 SD: 5.3483 cents 17:18:19:20:21:23:24 x 26:28:30:32:34 SD: 5.3720 cents 18:19:20:21:23:24:25:27:28:30:32:34:36 SD: 4.4758 cents 20:21:22:23:25:27:28:30:31:33:35:37:40 SD: 3.6968 cents 24:25:27:28:30:32:34:36:37:40:42:45:48 SD: 3.0880 cents 35:37:39:41:44:47:49:53:54:58:61:65:70 SD: 2.8366 cents 36:38:40:42:45:48:50:54:56:60:63:67:72 SD: 1.4443 cents 60:63:67:70:75:80:84:90:93:100:105:112:120 SD: 0.8855 cents 84:88:93:98:105:112:118:126:131:140:147:157:168 SD: 0.8846 cents 96:101:107:112:120:128:134:144:149:160:168:179:192 SD: 0.7741 cents 108:113:120:126:135:144:151:162:168:180:189:202:216 SD: 0.6149 cents 120:126:133:140:150:160:168:180:187:200:210:224:240 SD: 0.4435 cents 144:151:160:168:180:192:202:216:224:240:252:269:288 SD: 0.3602 cents 180:189:200:210:225:240:252:270:280:300:315:336:360 SD: 0.0000 cents