Scala analysis: west
"The common Western scale of just intonation"
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: 9/8 203.910002 major 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: 9/5 1017.596288 just minor seventh, BP seventh 11: 17/9 1101.045408 septendecimal major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 18/17 98.9546 Arabic lute index finger 1: 21/20 84.4672 minor semitone 2: 15/14 119.4428 major diatonic semitone 3: 16/15 111.7313 minor diatonic 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: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 11: 85/81 83.4491 12: 18/17 98.9546 Arabic lute index finger
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 1 85/81 83.449 cents 1: 2 21/20 84.467 cents minor semitone 1: 1 18/17 98.955 cents Arabic lute index finger 1: 3 16/15 111.731 cents minor diatonic semitone 1: 2 15/14 119.443 cents major diatonic semitone 1: 1 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 2: 4 10/9 182.404 cents minor whole tone 2: 1 189/170 183.422 cents 2: 1 28/25 196.198 cents middle second 2: 3 9/8 203.910 cents major whole tone 2: 1 17/15 216.687 cents septendecimal whole tone 2: 2 8/7 231.174 cents septimal whole tone 3: 2 7/6 266.871 cents septimal minor third 3: 1 85/72 287.359 cents septendecimal minor third 3: 1 32/27 294.135 cents Pythagorean minor third 3: 2 25/21 301.847 cents BP second, quasi-equal minor third 3: 1 81/68 302.865 cents Persian wosta 3: 5 6/5 315.641 cents minor third 4: 1 56/45 378.602 cents septimal narrow major third 4: 5 5/4 386.314 cents major third 4: 1 34/27 399.090 cents septendecimal major third 4: 1 63/50 400.108 cents quasi-equal major third 4: 1 80/63 413.578 cents wide major third 4: 1 108/85 414.596 cents 4: 1 32/25 427.373 cents classic diminished fourth 4: 1 9/7 435.084 cents septimal major third, BP third 5: 1 21/16 470.781 cents narrow fourth 5: 1 45/34 485.268 cents 5: 7 4/3 498.045 cents perfect fourth 5: 1 85/63 518.533 cents 5: 2 27/20 519.551 cents acute fourth 6: 2 25/18 568.717 cents classic augmented fourth 6: 2 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 6: 1 45/32 590.224 cents diatonic tritone 6: 1 24/17 597.000 cents 1st septendecimal tritone 6: 1 17/12 603.000 cents 2nd septendecimal tritone 6: 1 64/45 609.776 cents 2nd tritone 6: 2 10/7 617.488 cents Euler's tritone 6: 2 36/25 631.283 cents classic diminished fifth 7: 2 40/27 680.449 cents grave fifth 7: 1 126/85 681.467 cents 7: 7 3/2 701.955 cents perfect fifth 7: 1 68/45 714.732 cents 7: 1 32/21 729.219 cents wide fifth 8: 1 14/9 764.916 cents septimal minor sixth 8: 1 25/16 772.627 cents classic augmented fifth 8: 1 85/54 785.404 cents 8: 1 63/40 786.422 cents narrow minor sixth 8: 1 100/63 799.892 cents quasi-equal minor sixth 8: 1 27/17 800.910 cents septendecimal minor sixth 8: 5 8/5 813.686 cents minor sixth 8: 1 45/28 821.398 cents septimale wide minor sixth 9: 5 5/3 884.359 cents major sixth, BP sixth 9: 1 136/81 897.135 cents 9: 2 42/25 898.153 cents quasi-equal major sixth 9: 1 27/16 905.865 cents Pythagorean major sixth 9: 1 144/85 912.641 cents 9: 2 12/7 933.129 cents septimal major sixth 10: 2 7/4 968.826 cents harmonic seventh 10: 1 30/17 983.313 cents septendecimal minor seventh 10: 3 16/9 996.090 cents Pythagorean minor seventh 10: 1 25/14 1003.802 cents middle minor seventh 10: 1 340/189 1016.578 cents 10: 4 9/5 1017.596 cents just minor seventh, BP seventh 11: 1 50/27 1066.762 cents grave major seventh 11: 2 28/15 1080.557 cents grave major seventh 11: 3 15/8 1088.269 cents classic major seventh 11: 1 17/9 1101.045 cents septendecimal major seventh 11: 2 40/21 1115.533 cents acute major seventh 11: 1 162/85 1116.551 cents 11: 2 48/25 1129.328 cents classic diminished octave Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 6.54545 = 72/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 84.5 203.9 315.6 386.3 498.0 582.5 702.0 813.7 884.4 1017.6 1101.0 1200.0 84.5 : 119.4 231.2 301.8 413.6 498.0 617.5 729.2 799.9 933.1 1016.6 1115.5 1200.0 203.9 : 111.7 182.4 294.1 378.6 498.0 609.8 680.4 813.7 897.1 996.1 1080.6 1200.0 315.6 : 70.7 182.4 266.9 386.3 498.0 568.7 702.0 785.4 884.4 968.8 1088.3 1200.0 386.3 : 111.7 196.2 315.6 427.4 498.0 631.3 714.7 813.7 898.2 1017.6 1129.3 1200.0 498.0 : 84.5 203.9 315.6 386.3 519.6 603.0 702.0 786.4 905.9 1017.6 1088.3 1200.0 582.5 : 119.4 231.2 301.8 435.1 518.5 617.5 702.0 821.4 933.1 1003.8 1115.5 1200.0 702.0 : 111.7 182.4 315.6 399.1 498.0 582.5 702.0 813.7 884.4 996.1 1080.6 1200.0 813.7 : 70.7 203.9 287.4 386.3 470.8 590.2 702.0 772.6 884.4 968.8 1088.3 1200.0 884.4 : 133.2 216.7 315.6 400.1 519.6 631.3 702.0 813.7 898.2 1017.6 1129.3 1200.0 1017.6: 83.4 182.4 266.9 386.3 498.0 568.7 680.4 764.9 884.4 996.1 1066.8 1200.0 1101.0: 99.0 183.4 302.9 414.6 485.3 597.0 681.5 800.9 912.6 983.3 1116.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 70.6724 .. 133.2376 cents 648/625, 62.5651 cents 25/24, 70.6724 cents 2: 182.4037 .. 231.1741 cents 36/35, 48.7704 cents 250/243, 49.1661 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: 568.7174 .. 631.2826 cents 648/625, 62.5651 cents 250/243, 49.1661 cents 7: 680.4487 .. 729.2191 cents 36/35, 48.7704 cents 250/243, 49.1661 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: 1066.7624 .. 1129.3276 cents 648/625, 62.5651 cents 250/243, 49.1661 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 : 27/25, 133.2376 cents Average / Smallest step : 1.414979 Largest / Average step : 1.332376 Largest / Smallest step : 1.885284 Median interval of one step : 105.34294 cents, amount: 0 Most common interval of one step : 16/15, 111.7313 cents, amount: 3 Least squares average step : 100.06097 cents, oct.: 1200.73163 cents Scale is strictly proper Step pattern alph. order: ABCDCABCDEFG Step pattern size order : CFEAECFEAGBD Scale is a Constant Structure, by a margin of 35.69681 cents Scale diversity : 1.439071 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.505599 Rothenberg efficiency : 0.275210 redundancy: 0.724790 Efficiency x scale size : 3.302525 Number of different interval sizes : 72 = 6.54545 / class Number of one step interval sizes : 7 Highest interval variety : 8 Mean interval variety : 6.54545 = 72/11 Median interval variety : 5 Lowest interval variety : 5 Smallest interval difference : 1701/1700, 1.0181 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 : 8, average 703.5521 cents Number of appreciable fifths : 7, average 701.9550 cents Best fifths form a closed circle Scale contains a complete diamond : 1 3 5 Length of longest harmonic part : 2 at degree 1 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 7 = 58.3333% Scale is n-1 differentially coherent in interval classes 1 and 2 combined Scale is n-1 differentially coherent in interval classes 2 and 3 combined Scale is n-1 differentially coherent in interval classes 3 and 4 combined -- Rational properties -- Prime limit : 17 Odd number limit : 189 (O: 189 U: 189) Highest odd numerator or denominator: 21 Scale harmonicity : 0.014312 Average absolute harmonicity : 0.188522 Specific harmonicity : 0.119872 Fundamental : 1/360, -8.4919 octaves, 0.7267 Hz. Guide tone : 42840, 15.3867 octaves, 11208039.217 Hz. Exponens Consonantiae : 1.542240E+07, 23.87852 octaves Euler's gradus suavitatis : 45 Sum of Mann's harmonic distance : 83.0, average 6.91667 Mersenne's string divisions : 251170920 Sum of van Prooijen's expressibility: 9.05637, average 0.75470 Sum of Tenney's harmonic distance : 17.57722, average 1.46477 Vogel's harmonic complexity : 16.00000 Wille's k value : 202 Wilson's harmonic complexity : 46 Rectangular lattice diameter : 8 Triangular lattice diameter : 4 Lattice compactness : 226.21300, average 2.90017 Lattice compactness (without 2's) : 146.50515, average 1.87827 Number of different primes : 5 Prime exponents' range, average, count, tones@limit: 2: -3 .. 3 -0.08333 15 1 3: -2 .. 2 0.25000 11 3 5: -1 .. 1 -0.25000 7 5 7: 0 .. 1 0.16667 2 2 17: 0 .. 1 0.08333 1 1 Average exponent except 2's : 3 / 12 = 0.25000 Average absolute exponent except 2's: 21 / 12 = 1.75000 Scale is weakly epimorphic with val: <12 17 28 37 49| Scale is weakly epimorphic with val: <12 19 27 33 48| Scale is JI-epimorphic with val: <12 19 28 34 49| = patent Scale is weakly epimorphic with val: <12 21 29 31 52| Scale is weakly epimorphic with val: <12 21 31 35 50|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 12.0578 c. M: 17.5963 c. Twelve-tone Chromatic (1/11-comma) 19: 1 2 2 1 2 1 2 2 1 2 1 2 P M ME SD: 12.4915 c. M: 27.3612 c. 26: 2 2 3 1 3 2 2 3 1 3 2 2 P D3 I SD: 11.7934 c. M: 19.2946 c. 27: 2 3 2 2 2 2 3 2 2 3 2 2 SP D3 SD: 9.5016 c. M:-18.3122 c. 28: 2 3 2 2 3 2 2 3 2 3 2 2 SP D3 SD: 12.0012 c. M:-17.4878 c. 29: 2 3 3 1 3 2 3 3 1 4 2 2 P SD: 10.9329 c. M:-16.8865 c. 31: 2 3 3 2 3 2 3 3 2 3 2 3 SP M ME SD: 7.6688 c. M: 17.1744 c. 34: 2 4 3 2 3 3 3 3 2 4 2 3 P D3 SD: 7.5808 c. M:-17.4878 c. 35: 2 4 3 2 4 2 3 4 2 4 2 3 P SD: 9.9475 c. M:-16.2407 c. 37: 3 3 4 2 3 3 4 3 2 4 3 3 SP SD: 8.3611 c. M:-12.8301 c. 38: 3 3 4 2 4 2 4 4 2 4 3 3 P T3 SD: 8.1291 c. M: 14.4363 c. 41: 3 4 4 2 4 3 4 4 2 5 3 3 SP SD: 5.3125 c. M:-11.1497 c. 46: 3 5 4 3 4 3 5 4 3 5 3 4 SP SD: 4.4652 c. M: 8.5991 c. 53: 4 5 5 3 5 4 5 5 3 6 4 4 SP SD: 3.5916 c. M:-8.3886 c. 57: 4 6 5 3 6 4 5 6 3 6 4 5 SP SD: 5.7363 c. M: 7.3663 c. 58: 4 6 5 4 5 4 6 5 4 6 4 5 SP SD: 4.1785 c. M: 6.7897 c. 60: 4 6 6 3 6 4 6 6 3 7 4 5 SP SD: 3.8056 c. M: 6.3137 c. 72: 5 7 7 4 7 5 7 7 4 8 5 6 SP SD: 1.9731 c. M: 3.9100 c. 87: 6 9 8 5 8 6 9 8 5 10 6 7 SP SD: 1.9798 c. M: 3.2018 c. 99: 7 10 9 6 9 7 10 9 6 11 7 8 SP SD: 1.1981 c. M:-2.1506 c. 159: 11 16 15 9 15 11 16 15 9 18 11 13 SP SD: 1.0767 c. M: 1.4483 c. 171: 12 17 16 10 16 12 17 16 10 19 12 14 SP SD: 0.3032 c. M:-0.7090 c. 354: 25 35 33 21 33 25 35 33 21 39 25 29 SP SD: 0.4027 c. M:-0.6495 c. 441: 31 44 41 26 41 31 44 41 26 49 31 36 SP SD: 0.3040 c. M:-0.9954 c. 472: 33 47 44 28 44 33 47 44 28 52 33 39 SP SD: 0.3668 c. M: 0.6471 c. 494: 35 49 46 29 46 35 49 46 29 55 34 41 SP SD: 0.3004 c. M: 0.6405 c. 571: 40 57 53 34 53 40 57 53 34 63 40 47 SP SD: 0.3083 c. M: 0.4334 c. 593: 42 59 55 35 55 42 59 55 35 66 41 49 SP SD: 0.2729 c. M:-0.5244 c. 665: 47 66 62 39 62 47 66 62 39 74 46 55 SP SD: 0.1902 c. M:-0.3450 c. 764: 54 76 71 45 71 54 76 71 45 85 53 63 SP SD: 0.1693 c. M:-0.3496 c. 836: 59 83 78 49 78 59 83 78 49 93 58 69 SP SD: 0.1481 c. M:-0.2629 c. 882: 62 88 82 52 82 62 88 82 52 98 61 73 SP SD: 0.1448 c. M: 0.3651 c. 935: 66 93 87 55 87 66 93 87 55 104 65 77 SP SD: 0.1204 c. M:-0.2387 c. 1007: 71 100 94 59 94 71 100 94 59 112 70 83 SP SD: 0.1407 c. M: 0.2164 c. 1053: 74 105 98 62 98 74 105 98 62 117 73 87 SP SD: 0.0921 c. M: 0.1907 c. 1224: 86 122 114 72 114 86 122 114 72 136 85 101 SP SD: 0.0725 c. M: 0.1593 c. 1236: 87 123 115 73 115 87 123 115 73 137 86 102 SP SD: 0.0719 c. M: 0.1206 c. 1407: 99 140 131 83 131 99 140 131 83 156 98 116 SP SD: 0.0553 c. M: 0.1123 c. 1578: 111 157 147 93 147 111 157 147 93 175 110 130 SP SD: 0.0621 c. M: 0.1077 c. 1988: 140 198 185 117 185 140 198 185 117 221 138 164 SP SD: 0.0582 c. M:-0.1141 c. 2060: 145 205 192 121 192 145 205 192 121 229 143 170 SP SD: 0.0628 c. M: 0.1001 c. 2072: 146 206 193 122 193 146 206 193 122 230 144 171 SP SD: 0.0520 c. M:-0.1133 c. 2159: 152 215 201 127 201 152 215 201 127 240 150 178 SP SD: 0.0474 c. M:-0.0971 c. 2171: 153 216 202 128 202 153 216 202 128 241 151 179 SP SD: 0.0476 c. M:-0.1021 c. 2330: 164 232 217 137 217 164 232 217 137 259 162 192 SP SD: 0.0488 c. M:-0.0861 c. 2342: 165 233 218 138 218 165 233 218 138 260 163 193 SP SD: 0.0369 c. M:-0.0759 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: 12.1926 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 0.0000 cents 6 x x 7 x 8 x 9 x 10:11 x 12 S SD: 9.7006 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 13.2155 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.1349 cents 9 x 10:11 x 12:13 x 14:15:16:17:18 S SD: 9.4543 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 6.8653 cents 11 x 12:13:14:15 x 17 x 18:20:21:22 SD: 11.3014 cents 12:13 x 14:15:16:17:18:19:20:22:23:24 S SD: 7.9784 cents 13:14:15 x 16:17:18:20:21:22:23:25:26 SD: 9.4513 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 8.1531 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 4.7871 cents 20:21:23:24:25:27:28:30:32:33:36:38:40 SD: 4.0106 cents 24:25:27:29:30:32:34:36:38:40:43:45:48 SD: 3.0162 cents 35:37:39:42:44:47:49:53:56:58:63:66:70 SD: 2.6843 cents 36:38:41:43:45:48:50:54:58:60:65:68:72 SD: 2.5841 cents 39:41:44:47:49:52:55:59:62:65:70:74:78 SD: 2.2789 cents 40:42:45:48:50:53:56:60:64:67:72:76:80 SD: 1.4324 cents 60:63:68:72:75:80:84:90:96:100:108:113:120 SD: 1.1464 cents 75:79:84:90:94:100:105:113:120:125:135:142:150 SD: 1.1375 cents 80:84:90:96:100:107:112:120:128:133:144:151:160 SD: 0.5868 cents 120:126:135:144:150:160:168:180:192:200:216:227:240 SD: 0.2120 cents 240:252:270:288:300:320:336:360:384:400:432:453:480 SD: 0.1061 cents 360:378:405:432:450:480:504:540:576:600:648:680:720 SD: 0.0000 cents