Scala analysis: xen06-polansky-study-1
Octave I and II 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: 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: 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: 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: 21/20 84.4672 minor semitone 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: 2 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 3 21/20 84.467 cents minor semitone 1: 4 16/15 111.731 cents minor diatonic semitone 1: 3 15/14 119.443 cents major diatonic semitone 2: 1 35/32 155.140 cents septimal neutral second 2: 3 10/9 182.404 cents minor whole tone 2: 2 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: 2 7/6 266.871 cents septimal minor third 3: 1 75/64 274.582 cents classic augmented second 3: 1 32/27 294.135 cents Pythagorean minor third 3: 2 25/21 301.847 cents BP second, quasi-equal minor third 3: 6 6/5 315.641 cents minor third 4: 1 56/45 378.602 cents septimal narrow major third 4: 6 5/4 386.314 cents major third 4: 1 63/50 400.108 cents quasi-equal major third 4: 1 80/63 413.578 cents wide 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: 2 21/16 470.781 cents narrow fourth 5: 7 4/3 498.045 cents perfect fourth 5: 1 75/56 505.757 cents marvelous fourth 5: 1 27/20 519.551 cents acute fourth 5: 1 48/35 546.815 cents septimal semi-augmented fourth 6: 1 25/18 568.717 cents classic augmented fourth 6: 3 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 6: 2 45/32 590.224 cents diatonic tritone 6: 2 64/45 609.776 cents 2nd tritone 6: 3 10/7 617.488 cents Euler's tritone 6: 1 36/25 631.283 cents classic diminished fifth 7: 1 35/24 653.185 cents septimal semi-diminished fifth 7: 1 40/27 680.449 cents grave fifth 7: 1 112/75 694.243 cents marvelous fifth 7: 7 3/2 701.955 cents perfect fifth 7: 2 32/21 729.219 cents wide fifth 8: 1 14/9 764.916 cents septimal minor sixth 8: 2 25/16 772.627 cents classic augmented fifth 8: 1 63/40 786.422 cents narrow minor sixth 8: 1 100/63 799.892 cents quasi-equal minor sixth 8: 6 8/5 813.686 cents minor sixth 8: 1 45/28 821.398 cents septimale wide minor sixth 9: 6 5/3 884.359 cents major sixth, BP sixth 9: 2 42/25 898.153 cents quasi-equal major sixth 9: 1 27/16 905.865 cents Pythagorean major sixth 9: 1 128/75 925.418 cents diminished seventh 9: 2 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: 2 25/14 1003.802 cents middle minor seventh 10: 3 9/5 1017.596 cents just minor seventh, BP seventh 10: 1 64/35 1044.860 cents septimal neutral seventh 11: 3 28/15 1080.557 cents grave major seventh 11: 4 15/8 1088.269 cents classic major seventh 11: 3 40/21 1115.533 cents acute major seventh 11: 2 48/25 1129.328 cents classic diminished octave Highest number of different intervals for one interval class: 6 Average number of different intervals per interval class: 5.09091 = 56/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 84.5 203.9 315.6 386.3 498.0 582.5 702.0 813.7 884.4 968.8 1088.3 1200.0 84.5 : 119.4 231.2 301.8 413.6 498.0 617.5 729.2 799.9 884.4 1003.8 1115.5 1200.0 203.9 : 111.7 182.4 294.1 378.6 498.0 609.8 680.4 764.9 884.4 996.1 1080.6 1200.0 315.6 : 70.7 182.4 266.9 386.3 498.0 568.7 653.2 772.6 884.4 968.8 1088.3 1200.0 386.3 : 111.7 196.2 315.6 427.4 498.0 582.5 702.0 813.7 898.2 1017.6 1129.3 1200.0 498.0 : 84.5 203.9 315.6 386.3 470.8 590.2 702.0 786.4 905.9 1017.6 1088.3 1200.0 582.5 : 119.4 231.2 301.8 386.3 505.8 617.5 702.0 821.4 933.1 1003.8 1115.5 1200.0 702.0 : 111.7 182.4 266.9 386.3 498.0 582.5 702.0 813.7 884.4 996.1 1080.6 1200.0 813.7 : 70.7 155.1 274.6 386.3 470.8 590.2 702.0 772.6 884.4 968.8 1088.3 1200.0 884.4 : 84.5 203.9 315.6 400.1 519.6 631.3 702.0 813.7 898.2 1017.6 1129.3 1200.0 968.8 : 119.4 231.2 315.6 435.1 546.8 617.5 729.2 813.7 933.1 1044.9 1115.5 1200.0 1088.3: 111.7 196.2 315.6 427.4 498.0 609.8 694.2 813.7 925.4 996.1 1080.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 70.6724 .. 119.4428 cents 36/35, 48.7704 cents 25/24, 70.6724 cents 2: 155.1396 .. 231.1741 cents 256/245, 76.0345 cents 49/48, 35.6968 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 .. 546.8154 cents 256/245, 76.0345 cents 49/48, 35.6968 cents 6: 568.7174 .. 631.2826 cents 648/625, 62.5651 cents 875/864, 21.9020 cents 7: 653.1846 .. 729.2191 cents 256/245, 76.0345 cents 875/864, 21.9020 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 .. 1044.8604 cents 256/245, 76.0345 cents 49/48, 35.6968 cents 11: 1080.5572 .. 1129.3276 cents 36/35, 48.7704 cents 49/48, 35.6968 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 : 15/14, 119.4428 cents Average / Smallest step : 1.414979 Largest / Average step : 1.194428 Largest / Smallest step : 1.690091 Median interval of one step : 16/15, 111.7313 cents, amount: 4 Most common interval of one step : 16/15, 111.7313 cents, amount: 4 Least squares average step : 99.09443 cents, oct.: 1189.13322 cents Scale is strictly proper Step pattern alph. order: ABCDCABCDABC Step pattern size order : BDCACBDCABDC Scale is a Constant Structure, by a margin of 21.90205 cents Scale diversity : 1.451468 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.437710 Rothenberg efficiency : 0.303412 redundancy: 0.696588 Efficiency x scale size : 3.640945 Number of different interval sizes : 56 = 5.09091 / class Number of one step interval sizes : 4 Highest interval variety : 6 Mean interval variety : 5.09091 = 56/11 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 225/224, 7.7115 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 700.9911 cents Best fifths form a closed circle Scale contains a complete diamond : 1 3 5 All steps are superparticular (epimoric) 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 differentially coherent in interval class 1 Scale is n-1 differentially coherent in interval class 2 Scale is 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 -- Rational properties -- Prime limit : 7 Odd number limit : 75 (O: 75 U: 75) Highest odd numerator or denominator: 21 Scale harmonicity : 0.026965 Average absolute harmonicity : 0.192759 Specific harmonicity : 0.153315 Fundamental : 1/120, -6.9069 octaves, 2.1802 Hz. Guide tone : 2520, 11.2992 octaves, 659296.425 Hz. Exponens Consonantiae : 3.024000E+05, 18.20610 octaves Euler's gradus suavitatis : 27 Sum of Mann's harmonic distance : 80.0, average 6.66667 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 8.89287, average 0.74107 Sum of Tenney's harmonic distance : 17.26565, average 1.43880 Vogel's harmonic complexity : 12.83333 Wille's k value : 112 Wilson's harmonic complexity : 26 Rectangular lattice diameter : 6 Triangular lattice diameter : 3 Lattice compactness : 228.54328, average 2.93004 Lattice compactness (without 2's) : 126.80401, average 1.62569 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -3 .. 3 -0.50000 20 1 3: -1 .. 2 0.33333 8 3 5: -1 .. 1 -0.08333 7 5 7: 0 .. 1 0.25000 3 3 Average exponent except 2's : 6 / 12 = 0.50000 Average absolute exponent except 2's: 18 / 12 = 1.50000 Scale is JI-epimorphic with val: <12 19 28 34| = patent
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 14.5585 c. M:-31.1741 c. Twelve-tone Chromatic (1/11-comma) 19: 1 2 2 1 2 1 2 2 1 1 2 2 P D3 SD: 12.0656 c. M: 21.4575 c. 22: 2 2 2 1 2 2 2 2 1 2 2 2 P D3 I SD: 11.9179 c. M:-24.6237 c. 24: 2 2 2 2 2 2 2 2 2 1 3 2 P T3 SD: 12.6688 c. M: 18.8259 c. 27: 2 3 2 2 2 2 3 2 2 2 2 3 P D3 SD: 11.2080 c. M: 21.6020 c. 28: 2 3 2 2 3 2 2 3 2 2 2 3 SP D3 SD: 12.9144 c. M:-17.4878 c. 30: 2 3 3 2 2 3 3 2 2 2 3 3 P D3 SD: 11.3696 c. M:-18.0450 c. 31: 2 3 3 2 3 2 3 3 2 2 3 3 SP D3 SD: 5.0587 c. M: 10.3616 c. 41: 3 4 4 2 4 3 4 4 2 3 4 4 P SD: 4.1390 c. M:-6.3099 c. 53: 4 5 5 3 5 4 5 5 3 4 5 5 SP SD: 2.9950 c. M:-6.1671 c. 72: 5 7 7 4 7 5 7 7 4 5 7 7 SP SD: 2.4796 c. M: 4.9354 c. 84: 6 8 8 5 8 6 8 8 5 6 8 8 SP SD: 2.0903 c. M: 3.9100 c. 87: 6 9 8 5 8 6 9 8 5 6 9 8 SP SD: 1.9271 c. M: 3.3087 c. 99: 7 10 9 6 9 7 10 9 6 7 10 9 SP SD: 1.3126 c. M:-2.6404 c. 130: 9 13 12 8 12 9 13 12 8 9 13 12 SP SD: 1.1276 c. M:-1.7951 c. 140: 10 14 13 8 13 10 14 13 8 10 14 13 SP SD: 0.9976 c. M:-1.8043 c. 171: 12 17 16 10 16 12 17 16 10 12 17 16 SP SD: 0.2977 c. M: 0.5494 c. 441: 31 44 41 26 41 31 44 41 26 31 44 41 SP SD: 0.1124 c. M: 0.1993 c. 612: 43 61 57 36 57 43 61 57 36 43 61 57 SP SD: 0.0898 c. M: 0.1985 c. 966: 68 96 90 57 90 68 96 90 57 68 96 90 SP SD: 0.0936 c. M: 0.1833 c. 1053: 74 105 98 62 98 74 105 98 62 74 105 98 SP SD: 0.0869 c. M: 0.1760 c. 1106: 78 110 103 65 103 78 110 103 65 78 110 103 SP SD: 0.0808 c. M:-0.1621 c. 1236: 87 123 115 73 115 87 123 115 73 87 123 115 SP SD: 0.0705 c. M:-0.1073 c. 1277: 90 127 119 75 119 90 127 119 75 90 127 119 SP SD: 0.0756 c. M:-0.1060 c. 1407: 99 140 131 83 131 99 140 131 83 99 140 131 SP SD: 0.0462 c. M: 0.0763 c. 1578: 111 157 147 93 147 111 157 147 93 111 157 147 SP SD: 0.0493 c. M: 0.1077 c. 1718: 121 171 160 101 160 121 171 160 101 121 171 160 SP SD: 0.0449 c. M:-0.0747 c. 1848: 130 184 172 109 172 130 184 172 109 130 184 172 SP SD: 0.0390 c. M: 0.0569 c. 2019: 142 201 188 119 188 142 201 188 119 142 201 188 SP SD: 0.0350 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: 9.7541 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 10.3979 cents 7 x 8 x 9 x 10 x 11 x 12:13:14 S SD: 11.8382 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 5.1441 cents 9 x 10:11 x 12:13 x 14:15:16:17:18 S SD: 9.7399 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 8.6686 cents 11 x 12:13:14:15 x 17 x 18:19:21:22 SD: 11.7530 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 7.8741 cents 13:14:15 x 16:17:18:20:21:22:23:24:26 S SD: 9.1052 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 8.3411 cents 15:16:17:18:19:20:21:23:24:25:26:28:30 SD: 4.7245 cents 20:21:23:24:25:27:28:30:32:33:35:38:40 SD: 4.3612 cents 23:24:26:28:29:31:32:35:37:38:40:43:46 SD: 4.2300 cents 24:25:27:29:30:32:34:36:38:40:42:45:48 SD: 2.7414 cents 39:41:44:47:49:52:55:59:62:65:68:73:78 SD: 2.2228 cents 40:42:45:48:50:53:56:60:64:67:70:75:80 SD: 1.1558 cents 72:76:81:86:90:96:101:108:115:120:126:135:144 SD: 1.0828 cents 80:84:90:96:100:107:112:120:128:133:140:150:160 SD: 0.5771 cents 120:126:135:144:150:160:168:180:192:200:210:225:240 S SD: 0.0000 cents