Scala analysis: xen06-polansky-study-3
Octave III tuning for 'Piano Study #5 (For JPR)'
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 33/32 53.272943 undecimal comma, al-Farabi's 1/4-tone 2: 9/8 203.910002 major whole tone 3: 6/5 315.641287 minor third 4: 5/4 386.313714 major third 5: 21/16 470.780907 narrow fourth 6: 11/8 551.317942 undecimal semi-augmented fourth 7: 3/2 701.955001 perfect fifth 8: 8/5 813.686286 minor sixth 9: 13/8 840.527662 tridecimal neutral sixth 10: 7/4 968.825906 harmonic seventh 11: 15/8 1088.268715 classic major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 33/32 53.2729 undecimal comma, al-Farabi's 1/4-tone 2: 12/11 150.6371 undecimal neutral second, 3/4-tone 3: 16/15 111.7313 minor diatonic semitone 4: 25/24 70.6724 classic chromatic semitone, minor chroma 5: 21/20 84.4672 minor semitone 6: 22/21 80.5370 undecimal minor semitone 7: 12/11 150.6371 undecimal neutral second, 3/4-tone 8: 16/15 111.7313 minor diatonic semitone 9: 65/64 26.8414 13th-partial chroma 10: 14/13 128.2982 2/3-tone 11: 15/14 119.4428 major diatonic semitone 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 65/64 26.841 cents 13th-partial chroma 1: 1 33/32 53.273 cents undecimal comma, al-Farabi's 1/4-tone 1: 1 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 1 22/21 80.537 cents undecimal minor semitone 1: 1 21/20 84.467 cents minor semitone 1: 3 16/15 111.731 cents minor diatonic semitone 1: 1 15/14 119.443 cents major diatonic semitone 1: 1 14/13 128.298 cents 2/3-tone 1: 2 12/11 150.637 cents undecimal neutral second, 3/4-tone 2: 1 13/12 138.573 cents tridecimal 2/3-tone 2: 2 35/32 155.140 cents septimal neutral second 2: 2 11/10 165.004 cents Ptolemy's second, 4/5-tone 2: 1 10/9 182.404 cents minor whole tone 2: 1 9/8 203.910 cents major whole tone 2: 2 8/7 231.174 cents septimal whole tone 2: 1 15/13 247.741 cents tridecimal 5/4-tone 2: 2 64/55 262.368 cents 3: 1 55/48 235.677 cents undecimal semi-augmented whole tone 3: 2 7/6 266.871 cents septimal minor third 3: 1 75/64 274.582 cents classic augmented second 3: 1 33/28 284.447 cents undecimal minor third 3: 1 13/11 289.210 cents tridecimal minor third 3: 3 6/5 315.641 cents minor third 3: 1 40/33 333.041 cents 3: 1 128/105 342.905 cents septimal neutral third 3: 1 16/13 359.472 cents tridecimal neutral third 4: 1 11/9 347.408 cents undecimal neutral third 4: 1 26/21 369.747 cents 4: 4 5/4 386.314 cents major third 4: 1 33/26 412.745 cents tridecimal major third 4: 2 14/11 417.508 cents undecimal diminished fourth or major third 4: 2 32/25 427.373 cents classic diminished fourth 4: 1 9/7 435.084 cents septimal major third, BP third 5: 1 165/128 439.587 cents 5: 1 13/10 454.214 cents tridecimal semi-diminished fourth 5: 1 21/16 470.781 cents narrow fourth 5: 6 4/3 498.045 cents perfect fourth 5: 1 15/11 536.951 cents undecimal augmented fourth 5: 1 48/35 546.815 cents septimal semi-augmented fourth 5: 1 18/13 563.382 cents tridecimal augmented fourth 6: 1 65/48 524.886 cents 6: 2 11/8 551.318 cents undecimal semi-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 64/45 609.776 cents 2nd tritone 6: 2 10/7 617.488 cents Euler's tritone 6: 2 16/11 648.682 cents undecimal semi-diminished fifth 6: 1 96/65 675.114 cents 7: 1 13/9 636.618 cents tridecimal diminished fifth 7: 1 35/24 653.185 cents septimal semi-diminished fifth 7: 1 22/15 663.049 cents undecimal diminished fifth 7: 6 3/2 701.955 cents perfect fifth 7: 1 32/21 729.219 cents wide fifth 7: 1 20/13 745.786 cents tridecimal semi-augmented fifth 7: 1 256/165 760.413 cents 8: 1 14/9 764.916 cents septimal minor sixth 8: 2 25/16 772.627 cents classic augmented fifth 8: 2 11/7 782.492 cents undecimal augmented fifth 8: 1 52/33 787.255 cents tridecimal minor sixth 8: 4 8/5 813.686 cents minor sixth 8: 1 21/13 830.253 cents 8: 1 18/11 852.592 cents undecimal neutral sixth 9: 1 13/8 840.528 cents tridecimal neutral sixth 9: 1 105/64 857.095 cents septimal neutral sixth 9: 1 33/20 866.959 cents 9: 3 5/3 884.359 cents major sixth, BP sixth 9: 1 22/13 910.790 cents tridecimal major sixth 9: 1 56/33 915.553 cents 9: 1 128/75 925.418 cents diminished seventh 9: 2 12/7 933.129 cents septimal major sixth 9: 1 96/55 964.323 cents 10: 2 55/32 937.632 cents 10: 1 26/15 952.259 cents tridecimal semi-augmented sixth 10: 2 7/4 968.826 cents harmonic seventh 10: 1 16/9 996.090 cents Pythagorean minor seventh 10: 1 9/5 1017.596 cents just minor seventh, BP seventh 10: 2 20/11 1034.996 cents large minor seventh 10: 2 64/35 1044.860 cents septimal neutral seventh 10: 1 24/13 1061.427 cents tridecimal neutral seventh 11: 2 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 11: 1 13/7 1071.702 cents 16/3-tone 11: 1 28/15 1080.557 cents grave major seventh 11: 3 15/8 1088.269 cents classic major seventh 11: 1 40/21 1115.533 cents acute major seventh 11: 1 21/11 1119.463 cents undecimal major seventh 11: 1 48/25 1129.328 cents classic diminished octave 11: 1 64/33 1146.727 cents 33rd subharmonic 11: 1 128/65 1173.159 cents Highest number of different intervals for one interval class: 9 Average number of different intervals per interval class: 8.00000 = 8
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 53.3 203.9 315.6 386.3 470.8 551.3 702.0 813.7 840.5 968.8 1088.3 1200.0 53.3 : 150.6 262.4 333.0 417.5 498.0 648.7 760.4 787.3 915.6 1035.0 1146.7 1200.0 203.9 : 111.7 182.4 266.9 347.4 498.0 609.8 636.6 764.9 884.4 996.1 1049.4 1200.0 315.6 : 70.7 155.1 235.7 386.3 498.0 524.9 653.2 772.6 884.4 937.6 1088.3 1200.0 386.3 : 84.5 165.0 315.6 427.4 454.2 582.5 702.0 813.7 867.0 1017.6 1129.3 1200.0 470.8 : 80.5 231.2 342.9 369.7 498.0 617.5 729.2 782.5 933.1 1044.9 1115.5 1200.0 551.3 : 150.6 262.4 289.2 417.5 537.0 648.7 702.0 852.6 964.3 1035.0 1119.5 1200.0 702.0 : 111.7 138.6 266.9 386.3 498.0 551.3 702.0 813.7 884.4 968.8 1049.4 1200.0 813.7 : 26.8 155.1 274.6 386.3 439.6 590.2 702.0 772.6 857.1 937.6 1088.3 1200.0 840.5 : 128.3 247.7 359.5 412.7 563.4 675.1 745.8 830.3 910.8 1061.4 1173.2 1200.0 968.8 : 119.4 231.2 284.4 435.1 546.8 617.5 702.0 782.5 933.1 1044.9 1071.7 1200.0 1088.3: 111.7 165.0 315.6 427.4 498.0 582.5 663.0 813.7 925.4 952.3 1080.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 26.8414 .. 150.6371 cents 768/715, 123.7957 cents 65/64, 26.8414 cents 2: 138.5727 .. 262.3683 cents 768/715, 123.7957 cents 143/144,-12.0644 cents 3: 235.6767 .. 359.4723 cents 768/715, 123.7957 cents 3025/3072,-26.6917 cents 4: 347.4079 .. 435.0841 cents 81/77, 87.6762 cents 143/144,-12.0644 cents 5: 439.5867 .. 563.3823 cents 768/715, 123.7957 cents 385/384, 4.5026 cents 6: 524.8864 .. 675.1136 cents 4608/4225, 150.2273 cents 845/864,-38.4960 cents 7: 636.6177 .. 760.4133 cents 768/715, 123.7957 cents 845/864,-38.4960 cents 8: 764.9159 .. 852.5921 cents 81/77, 87.6762 cents 385/384, 4.5026 cents 9: 840.5277 .. 964.3233 cents 768/715, 123.7957 cents 143/144,-12.0644 cents 10: 937.6317 .. 1061.4273 cents 768/715, 123.7957 cents 3025/3072,-26.6917 cents 11: 1049.3629 .. 1173.1586 cents 768/715, 123.7957 cents 143/144,-12.0644 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 65/64, 26.8414 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 12/11, 150.6371 cents Average / Smallest step : 3.725591 Largest / Average step : 1.506371 Largest / Smallest step : 5.612121 Median interval of one step : 16/15, 111.7313 cents, amount: 3 Most common interval of one step : 16/15, 111.7313 cents, amount: 3 Least squares average step : 97.94188 cents, oct.: 1175.30258 cents Scale is not proper Step pattern alph. order: ABCDEFBCGHIC Step pattern size order : BIFCEDIFAHGF Scale is a Constant Structure, by a margin of 4.50256 cents Scale diversity : 1.694094 Lumma stability : 0.052240 Lumma impropriety factor : 0.148861 Rothenberg efficiency : 0.236376 redundancy: 0.763624 Efficiency x scale size : 2.836508 Number of different interval sizes : 88 = 8.00000 / class Number of one step interval sizes : 9 Highest interval variety : 9 Mean interval variety : 8.00000 = 8 Median interval variety : 7 Lowest interval variety : 7 Smallest interval difference : 441/440, 3.9302 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 6 Number of recognisable fifths : 6, average 701.9550 cents Scale contains a complete diamond : 1 5 All steps are superparticular (epimoric) Length of longest harmonic part : 3 at degree 9 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 3 = 25.0000% Scale is differentially coherent in interval class 1 Scale is n-1 differentially coherent in interval classes 2 and 3 combined Scale is differentially coherent in interval classes 3 and 4 combined Scale is 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 6 and 7 combined -- Rational properties -- Prime limit : 13 Odd number limit : 165 (O: 165 U: 165) Highest odd numerator or denominator: 33 Scale harmonicity : 0.013029 Average absolute harmonicity : 0.172200 Specific harmonicity : 0.110911 Fundamental : 1/160, -7.3219 octaves, 1.6352 Hz. Guide tone : 360360, 18.4591 octaves, 94279388.712 Hz. Exponens Consonantiae : 5.765760E+07, 25.78101 octaves Euler's gradus suavitatis : 49 Sum of Mann's harmonic distance : 117.0, average 9.75000 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 10.54553, average 0.87879 Sum of Tenney's harmonic distance : 20.35458, average 1.69622 Vogel's harmonic complexity : 14.08333 Wille's k value : 82 Wilson's harmonic complexity : 47 Rectangular lattice diameter : 7 Triangular lattice diameter : 3 Lattice compactness : 249.15304, average 3.19427 Lattice compactness (without 2's) : 112.74181, average 1.44541 Number of different primes : 6 Prime exponents' range, average, count, tones@limit: 2: -5 .. 3 -1.75000 31 1 3: 0 .. 2 0.58333 7 2 5: -1 .. 1 0.00000 4 4 7: 0 .. 1 0.16667 2 2 11: 0 .. 1 0.16667 2 2 13: 0 .. 1 0.08333 1 1 Average exponent except 2's : 12 / 12 = 1.00000 Average absolute exponent except 2's: 16 / 12 = 1.33333 Scale is JI-epimorphic with val: <12 19 28 34 42 45|
FIT/MODE
12: 1 1 1 1 1 1 1 1 0 2 1 1 N T3 SD: 27.0762 c. M:-48.6821 c. 13: 0 2 1 1 1 1 1 2 0 1 2 1 N D3 SD: 30.3646 c. M: 55.8012 c. 15: 1 2 1 1 1 1 2 1 1 1 2 1 P D3 SD: 21.4850 c. M:-39.4723 c. 16: 1 2 1 1 1 1 2 2 0 2 1 2 N T3 SD: 20.4375 c. M: 38.2687 c. 18: 1 2 2 1 1 1 3 1 1 2 1 2 N D3 SD: 18.9934 c. M:-31.3783 c. 19: 1 2 2 1 1 2 2 2 0 2 2 2 N SD: 14.8155 c. M: 28.6756 c. 21: 1 3 2 1 1 2 2 2 1 2 2 2 N SD: 15.4975 c. M:-27.2159 c. 22: 1 3 2 1 2 1 3 2 0 3 2 2 N SD: 11.3526 c. M: 22.3458 c. 24: 1 3 2 2 1 2 3 2 1 2 3 2 P T3 SD: 11.7773 c. M: 20.7809 c. 28: 1 4 2 2 2 2 3 3 1 3 2 3 P SD: 11.5632 c. M:-16.8884 c. 30: 1 4 3 2 2 2 4 2 1 3 3 3 N SD: 10.0732 c. M:-18.0450 c. 31: 1 4 3 2 2 2 4 3 1 3 3 3 N SD: 7.3811 c. M: 14.5633 c. 41: 2 5 4 2 3 3 5 4 1 4 4 4 N SD: 4.7530 c. M:-8.2528 c. 46: 2 6 4 3 3 3 6 4 1 5 5 4 N SD: 4.0887 c. M:-7.3835 c. 53: 2 7 5 3 4 3 7 5 1 6 5 5 N SD: 3.9476 c. M: 7.9899 c. 61: 3 7 6 4 4 4 8 5 2 6 6 6 N SD: 5.1691 c. M: 7.1887 c. 63: 3 8 6 3 5 4 8 6 1 7 6 6 N SD: 4.3607 c. M:-8.1682 c. 66: 3 8 6 4 5 4 9 6 1 7 7 6 N SD: 4.4774 c. M:-7.1359 c. 68: 3 9 6 4 5 4 9 6 2 7 7 6 N SD: 4.2732 c. M:-7.8547 c. 72: 3 9 7 4 5 5 9 7 1 8 7 7 N SD: 3.5129 c. M: 7.1943 c. 77: 3 10 7 5 5 5 10 7 2 8 8 7 N SD: 3.4364 c. M: 6.5197 c. 83: 4 10 8 5 6 5 11 7 2 9 8 8 N SD: 3.7281 c. M:-6.4787 c. 84: 4 10 8 5 6 6 10 8 2 9 8 8 N SD: 2.7343 c. M:-5.8249 c. 87: 4 11 8 5 6 6 11 8 2 9 9 8 N SD: 1.6930 c. M: 3.3087 c. 128: 6 16 12 7 9 9 16 12 3 13 13 12 N SD: 2.2638 c. M:-3.2223 c. 130: 6 16 12 8 9 9 16 12 3 14 13 12 N SD: 1.2945 c. M:-2.5282 c. 140: 6 18 13 8 10 9 18 13 3 15 14 13 N SD: 1.2531 c. M: 2.7465 c. 183: 8 23 17 11 13 12 23 17 4 20 18 17 N SD: 0.8619 c. M:-1.6659 c. 224: 10 28 21 13 16 15 28 21 5 24 22 21 N SD: 0.5267 c. M:-0.8170 c. 270: 12 34 25 16 19 18 34 25 6 29 27 25 N SD: 0.3459 c. M:-0.6202 c. 494: 22 62 46 29 35 33 62 46 11 53 49 46 N SD: 0.2015 c. M:-0.4742 c. 742: 33 93 69 44 52 50 93 69 17 79 74 69 N SD: 0.1988 c. M:-0.4427 c. 764: 34 96 71 45 54 51 96 71 17 82 76 71 N SD: 0.2000 c. M:-0.4233 c. 805: 36 101 75 47 57 54 101 75 18 86 80 75 N SD: 0.2455 c. M:-0.3917 c. 836: 37 105 78 49 59 56 105 78 19 89 83 78 N SD: 0.2214 c. M:-0.6207 c. 882: 39 111 82 52 62 59 111 82 20 94 88 82 N SD: 0.1606 c. M: 0.2975 c. 1012: 45 127 94 60 71 68 127 94 23 108 101 94 N SD: 0.1574 c. M:-0.2688 c. 1084: 48 136 101 64 76 73 136 101 24 116 108 101 N SD: 0.1657 c. M: 0.3063 c. 1106: 49 139 103 65 78 74 139 103 25 118 110 103 N SD: 0.1242 c. M:-0.3403 c. 1236: 55 155 115 73 87 83 155 115 28 132 123 115 N SD: 0.1130 c. M:-0.2490 c. 1308: 58 164 122 77 92 88 164 122 29 140 130 122 N SD: 0.1212 c. M: 0.2403 c. 1376: 61 173 128 81 97 92 173 128 31 147 137 128 N SD: 0.1051 c. M:-0.1700 c. 1506: 67 189 140 89 106 101 189 140 34 161 150 140 N SD: 0.1108 c. M:-0.1775 c. 1578: 70 198 147 93 111 106 198 147 35 169 157 147 N SD: 0.0793 c. M: 0.2235 c. 1802: 80 226 168 106 127 121 226 168 40 193 179 168 N SD: 0.0853 c. M: 0.1444 c. 1848: 82 232 172 109 130 124 232 172 41 198 184 172 N SD: 0.0833 c. M: 0.2679 c. 1889: 84 237 176 111 133 127 237 176 42 202 188 176 N SD: 0.0663 c. M:-0.0886 c. 2159: 96 271 201 127 152 145 271 201 48 231 215 201 N SD: 0.0595 c. M: 0.1386 c. 2190: 97 275 204 129 154 147 275 204 49 234 218 204 N SD: 0.0616 c. M: 0.1223 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: 17.2062 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: 11.5787 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 13.4956 cents 7 x 8 x x 9:10 x 11 x 12:13:14 S SD: 13.4906 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 0.0000 cents 9 x 10:11 x 12 x x 14:15:16:17:18 S SD: 10.7683 cents 10 x 11:12 x 13:14:15:16 x 18:19:20 S SD: 8.3633 cents 11 x 12:13:14 x 15:17 x 18:19:21:22 SD: 10.5515 cents 12 x x 14:15:16:17:18:19:20:21:23:24 SD: 9.7420 cents 13:14:15 x 16:17:18:20 x 21:23:24:26 SD: 10.7524 cents 14:15:16:17 x 18:19:21:22:23:25:26:28 SD: 9.1619 cents 15:16:17:18:19:20:21:23:24 x 26:28:30 SD: 7.9277 cents 16:17:18:19:20:21:22:24 x 26:28:30:32 S SD: 4.9791 cents 18:19:20:22:23:24:25:27:29 x 32:34:36 SD: 7.4091 cents 19:20:21:23:24:25:26:29:30:31:33:36:38 SD: 5.7896 cents 21:22:24:25:26:28:29:32 x 34:37:39:42 SD: 5.7111 cents 22:23:25:26:28:29:30:33:35:36:39:41:44 SD: 5.0348 cents 23:24:26:28:29:30:32:35:37 x 40:43:46 SD: 4.7362 cents 24:25:27:29:30:32:33:36:38:39:42:45:48 SD: 3.2471 cents 26:27:29:31:33:34:36:39 x 42:46:49:52 SD: 3.9802 cents 29:30:33:35:36:38:40:44:46:47:51:54:58 SD: 3.2845 cents 30:31:34:36:38:39:41:45:48:49:53:56:60 SD: 3.2207 cents 31:32:35:37:39:41:43:47 x 50:54:58:62 SD: 3.2086 cents 32:33:36:38:40:42:44:48:51:52:56:60:64 S SD: 1.6128 cents 56:58:63:67:70:74:77:84:90:91:98:105:112 SD: 1.3939 cents 64:66:72:77:80:84:88:96:102:104:112:120:128 SD: 0.6780 cents 96:99:108:115:120:126:132:144:154:156:168:180:192 SD: 0.4513 cents 128:132:144:154:160:168:176:192:205:208:224:240:256 SD: 0.4008 cents 160:165:180:192:200:210:220:240:256:260:280:300:320 S SD: 0.0000 cents