Scala analysis: xen06-polansky-study-4
Octave IV 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: 7/6 266.870906 septimal minor third 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect 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: 27/16 905.865003 Pythagorean major 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: 28/27 62.9609 Archytas' 1/3-tone 4: 15/14 119.4428 major diatonic semitone 5: 16/15 111.7313 minor diatonic semitone 6: 33/32 53.2729 undecimal comma, al-Farabi's 1/4-tone 7: 12/11 150.6371 undecimal neutral second, 3/4-tone 8: 16/15 111.7313 minor diatonic semitone 9: 135/128 92.1787 major chroma, major limma 10: 28/27 62.9609 Archytas' 1/3-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 33/32 53.273 cents undecimal comma, al-Farabi's 1/4-tone 1: 2 28/27 62.961 cents Archytas' 1/3-tone 1: 1 21/20 84.467 cents minor semitone 1: 1 135/128 92.179 cents major chroma, major limma 1: 3 16/15 111.731 cents minor diatonic semitone 1: 3 15/14 119.443 cents major diatonic semitone 1: 1 12/11 150.637 cents undecimal neutral second, 3/4-tone 2: 1 35/32 155.140 cents septimal neutral second 2: 1 11/10 165.004 cents Ptolemy's second, 4/5-tone 2: 3 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: 2 8/7 231.174 cents septimal whole tone 2: 1 64/55 262.368 cents 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: 2 32/27 294.135 cents Pythagorean minor third 3: 1 25/21 301.847 cents BP second, quasi-equal minor third 3: 4 6/5 315.641 cents minor third 3: 1 27/22 354.547 cents neutral third, Zalzal wosta of al-Farabi 4: 1 11/9 347.408 cents undecimal neutral third 4: 2 56/45 378.602 cents septimal narrow major third 4: 3 5/4 386.314 cents major third 4: 1 81/64 407.820 cents Pythagorean major third 4: 1 80/63 413.578 cents wide major third 4: 1 14/11 417.508 cents undecimal diminished fourth or major third 4: 1 32/25 427.373 cents classic diminished fourth 4: 2 9/7 435.084 cents septimal major third, BP third 5: 1 55/42 466.851 cents 5: 2 21/16 470.781 cents narrow fourth 5: 6 4/3 498.045 cents perfect fourth 5: 1 27/20 519.551 cents acute fourth 5: 1 15/11 536.951 cents undecimal augmented fourth 5: 1 48/35 546.815 cents septimal semi-augmented fourth 6: 1 11/8 551.318 cents undecimal semi-augmented fourth 6: 1 112/81 561.006 cents 6: 2 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: 2 10/7 617.488 cents Euler's tritone 6: 1 81/56 638.994 cents 6: 1 16/11 648.682 cents undecimal semi-diminished fifth 7: 1 35/24 653.185 cents septimal semi-diminished fifth 7: 1 22/15 663.049 cents undecimal diminished fifth 7: 1 40/27 680.449 cents grave fifth 7: 6 3/2 701.955 cents perfect fifth 7: 2 32/21 729.219 cents wide fifth 7: 1 84/55 733.149 cents 8: 2 14/9 764.916 cents septimal minor sixth 8: 1 25/16 772.627 cents classic augmented fifth 8: 1 11/7 782.492 cents undecimal augmented fifth 8: 1 63/40 786.422 cents narrow minor sixth 8: 1 128/81 792.180 cents Pythagorean minor sixth 8: 3 8/5 813.686 cents minor sixth 8: 2 45/28 821.398 cents septimale wide minor sixth 8: 1 18/11 852.592 cents undecimal neutral sixth 9: 1 44/27 845.453 cents neutral sixth 9: 4 5/3 884.359 cents major sixth, BP sixth 9: 1 42/25 898.153 cents quasi-equal major sixth 9: 2 27/16 905.865 cents Pythagorean 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 10: 1 55/32 937.632 cents 10: 2 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: 3 9/5 1017.596 cents just minor seventh, BP seventh 10: 1 20/11 1034.996 cents large minor seventh 10: 1 64/35 1044.860 cents septimal neutral seventh 11: 1 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 11: 3 28/15 1080.557 cents grave major seventh 11: 3 15/8 1088.269 cents classic major seventh 11: 1 256/135 1107.821 cents octave - major chroma 11: 1 40/21 1115.533 cents acute major seventh 11: 2 27/14 1137.039 cents septimal major seventh 11: 1 64/33 1146.727 cents 33rd subharmonic Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 7.09091 = 78/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 84.5 203.9 266.9 386.3 498.0 551.3 702.0 813.7 905.9 968.8 1088.3 1200.0 84.5 : 119.4 182.4 301.8 413.6 466.9 617.5 729.2 821.4 884.4 1003.8 1115.5 1200.0 203.9 : 63.0 182.4 294.1 347.4 498.0 609.8 702.0 764.9 884.4 996.1 1080.6 1200.0 266.9 : 119.4 231.2 284.4 435.1 546.8 639.0 702.0 821.4 933.1 1017.6 1137.0 1200.0 386.3 : 111.7 165.0 315.6 427.4 519.6 582.5 702.0 813.7 898.2 1017.6 1080.6 1200.0 498.0 : 53.3 203.9 315.6 407.8 470.8 590.2 702.0 786.4 905.9 968.8 1088.3 1200.0 551.3 : 150.6 262.4 354.5 417.5 537.0 648.7 733.1 852.6 915.6 1035.0 1146.7 1200.0 702.0 : 111.7 203.9 266.9 386.3 498.0 582.5 702.0 764.9 884.4 996.1 1049.4 1200.0 813.7 : 92.2 155.1 274.6 386.3 470.8 590.2 653.2 772.6 884.4 937.6 1088.3 1200.0 905.9 : 63.0 182.4 294.1 378.6 498.0 561.0 680.4 792.2 845.5 996.1 1107.8 1200.0 968.8 : 119.4 231.2 315.6 435.1 498.0 617.5 729.2 782.5 933.1 1044.9 1137.0 1200.0 1088.3: 111.7 196.2 315.6 378.6 498.0 609.8 663.0 813.7 925.4 1017.6 1080.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 53.2729 .. 150.6371 cents 128/121, 97.3641 cents 33/32, 53.2729 cents 2: 155.1396 .. 262.3683 cents 2048/1925, 107.2287 cents 385/384, 4.5026 cents 3: 266.8709 .. 354.5471 cents 81/77, 87.6762 cents 385/384, 4.5026 cents 4: 347.4079 .. 435.0841 cents 81/77, 87.6762 cents 242/243,-7.1391 cents 5: 466.8507 .. 546.8154 cents 288/275, 79.9646 cents 55/54, 31.7667 cents 6: 551.3179 .. 648.6821 cents 128/121, 97.3641 cents 385/384, 4.5026 cents 7: 653.1846 .. 733.1493 cents 288/275, 79.9646 cents 385/384, 4.5026 cents 8: 764.9159 .. 852.5921 cents 81/77, 87.6762 cents 55/54, 31.7667 cents 9: 845.4529 .. 933.1291 cents 81/77, 87.6762 cents 242/243,-7.1391 cents 10: 937.6317 .. 1044.8604 cents 2048/1925, 107.2287 cents 385/384, 4.5026 cents 11: 1049.3629 .. 1146.7271 cents 128/121, 97.3641 cents 385/384, 4.5026 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 33/32, 53.2729 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 12/11, 150.6371 cents Average / Smallest step : 1.877125 Largest / Average step : 1.506371 Largest / Smallest step : 2.827647 Median interval of one step : 16/15, 111.7313 cents, amount: 3 Least squares average step : 98.87917 cents, oct.: 1186.55008 cents Scale is not proper Step pattern alph. order: ABCBDEFDGCBD Step pattern size order : CFBFEAGEDBFE Scale is a Constant Structure, by a margin of 4.50256 cents Scale diversity : 1.597325 Lumma stability : 0.164245 Lumma impropriety factor : 0.011899 Rothenberg efficiency : 0.248623 redundancy: 0.751377 Efficiency x scale size : 2.983478 Number of different interval sizes : 78 = 7.09091 / class Number of one step interval sizes : 7 Highest interval variety : 8 Mean interval variety : 7.09091 = 78/11 Median interval variety : 6 Lowest interval variety : 6 Smallest interval difference : 441/440, 3.9302 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 6 Most common triad is 0.0 498.045 701.955 cents, amount: 3 Number of recognisable fifths : 6, 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 n-1 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 -- Rational properties -- Prime limit : 11 Odd number limit : 135 (O: 135 U: 135) Highest odd numerator or denominator: 27 Scale harmonicity : 0.016968 Average absolute harmonicity : 0.187146 Specific harmonicity : 0.134730 Fundamental : 1/240, -7.9069 octaves, 1.0901 Hz. Guide tone : 83160, 16.3436 octaves, 21756782.010 Hz. Exponens Consonantiae : 1.995840E+07, 24.25049 octaves Euler's gradus suavitatis : 40 Sum of Mann's harmonic distance : 102.0, average 8.50000 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 9.96769, average 0.83064 Sum of Tenney's harmonic distance : 19.27159, average 1.60597 Vogel's harmonic complexity : 14.91667 Wille's k value : 202 Wilson's harmonic complexity : 40 Rectangular lattice diameter : 8 Triangular lattice diameter : 4 Lattice compactness : 250.38294, average 3.21004 Lattice compactness (without 2's) : 138.64802, average 1.77754 Number of different primes : 5 Prime exponents' range, average, count, tones@limit: 2: -4 .. 3 -1.25000 27 1 3: -1 .. 3 0.50000 10 4 5: -1 .. 1 0.00000 4 3 7: 0 .. 1 0.25000 3 3 11: 0 .. 1 0.08333 1 1 Average exponent except 2's : 10 / 12 = 0.83333 Average absolute exponent except 2's: 18 / 12 = 1.50000 Scale is weakly epimorphic with val: <12 19 25 34 42| Scale is JI-epimorphic with val: <12 19 28 34 42| = patent
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 20.9168 c. M:-48.6821 c. Twelve-tone Chromatic (1/11-comma) 13: 1 1 1 1 1 1 2 1 1 1 1 1 P M ME SD: 23.5081 c. M:-46.5587 c. 15: 1 2 0 2 1 1 2 1 1 1 2 1 P D3 SD: 20.2062 c. M:-36.0900 c. 17: 1 2 1 2 1 1 2 2 1 1 2 1 P D3 SD: 21.2350 c. M:-41.1430 c. 18: 1 2 1 2 1 1 3 1 2 1 1 2 N D3 SD: 20.8015 c. M: 31.3783 c. 19: 1 2 1 2 2 1 2 2 1 1 2 2 P D3 SD: 14.4699 c. M: 21.6545 c. 22: 2 2 1 2 2 1 3 2 2 1 2 2 N T3 SD: 11.7438 c. M:-24.6237 c. 24: 2 2 1 3 2 1 3 2 2 1 3 2 P T3 SD: 10.9986 c. M: 18.8259 c. 28: 2 3 1 3 3 1 3 3 2 2 2 3 P T3 SD: 10.6818 c. M:-16.8884 c. 29: 2 3 1 3 3 1 4 3 2 1 3 3 N SD: 10.7787 c. M: 18.5950 c. 31: 2 3 2 3 3 1 4 3 2 2 3 3 P SD: 6.9453 c. M: 15.5424 c. 36: 3 3 2 4 3 2 4 3 3 2 4 3 SP T3 SD: 9.3592 c. M:-15.5328 c. 37: 3 3 2 4 3 2 5 3 3 2 4 3 P SD: 8.2695 c. M:-14.4340 c. 40: 3 4 2 4 4 1 5 4 3 2 4 4 N SD: 7.6337 c. M:-11.9550 c. 41: 3 4 2 4 4 2 5 4 3 2 4 4 P SD: 3.5905 c. M: 5.8259 c. 53: 4 5 3 5 5 2 7 5 4 3 5 5 N SD: 3.5604 c. M: 7.9217 c. 72: 5 7 4 7 7 3 9 7 5 4 7 7 P SD: 2.9886 c. M: 5.8650 c. 87: 6 9 4 9 8 4 11 8 7 4 9 8 N SD: 2.4577 c. M: 4.8019 c. 89: 6 9 5 9 8 4 11 8 7 5 9 8 P SD: 2.8291 c. M: 4.6975 c. 94: 7 9 5 9 9 4 12 9 7 5 9 9 N SD: 2.3402 c. M:-4.8945 c. 99: 7 10 5 10 9 5 12 9 8 5 10 9 P SD: 2.4035 c. M:-6.2578 c. 111: 8 11 6 11 10 5 14 10 9 6 11 10 N SD: 2.4304 c. M:-4.1471 c. 118: 8 12 6 12 11 5 15 11 9 6 12 11 N SD: 1.5526 c. M: 3.1113 c. 130: 9 13 7 13 12 6 16 12 10 7 13 12 P SD: 1.1723 c. M:-2.5282 c. 152: 11 15 8 15 14 7 19 14 12 8 15 14 N SD: 1.4025 c. M:-2.3749 c. 171: 12 17 9 17 16 8 21 16 13 9 17 16 P SD: 0.9501 c. M:-3.0680 c. 183: 13 18 10 18 17 8 23 17 14 10 18 17 N SD: 0.9023 c. M:-1.9816 c. 217: 15 22 11 22 20 10 27 20 17 11 22 20 N SD: 1.0281 c. M:-1.6774 c. 224: 16 22 12 22 21 10 28 21 17 12 22 21 N SD: 0.6553 c. M:-1.2471 c. 270: 19 27 14 27 25 12 34 25 21 14 27 25 N SD: 0.3866 c. M:-0.8017 c. 342: 24 34 18 34 32 15 43 32 26 18 34 32 N SD: 0.3668 c. M: 0.6018 c. 441: 31 44 23 44 41 20 55 41 34 23 44 41 N SD: 0.3354 c. M:-1.0630 c. 494: 35 49 26 49 46 22 62 46 38 26 49 46 N SD: 0.2375 c. M:-0.5530 c. 612: 43 61 32 61 57 27 77 57 47 32 61 57 N SD: 0.1364 c. M: 0.3376 c. 836: 59 83 44 83 78 37 105 78 64 44 83 78 N SD: 0.1393 c. M: 0.2304 c. 1053: 74 105 55 105 98 47 132 98 81 55 105 98 N SD: 0.1210 c. M:-0.2490 c. 1106: 78 110 58 110 103 49 139 103 85 58 110 103 N SD: 0.0804 c. M:-0.1621 c. 1236: 87 123 65 123 115 55 155 115 95 65 123 115 N SD: 0.0772 c. M:-0.1384 c. 1578: 111 157 83 157 147 70 198 147 121 83 157 147 N SD: 0.0661 c. M: 0.1616 c. 1848: 130 184 97 184 172 82 232 172 142 97 184 172 N SD: 0.0300 c. M: 0.0516 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: 7.1688 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: 15.1396 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 7.4090 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 13.9175 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 3.3552 cents 9 x 10 x 11:12 x x 14:15:16:17:18 S SD: 9.4570 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 9.9268 cents 11 x 12:13:14 x 15:17:18 x 19:21:22 SD: 11.2142 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 8.0481 cents 13:14 x 15:16:17:18:20:21:22:23:24:26 S SD: 7.9157 cents 14:15:16 x 18 x 19:21:22:24:25:26:28 SD: 8.9842 cents 15:16:17:18:19:20:21:23:24:25:26:28:30 SD: 6.9863 cents 16:17:18:19:20:21:22:24:26:27:28:30:32 S SD: 4.4273 cents 17:18:19:20:21:23 x 26:27:29:30:32:34 SD: 5.3845 cents 18:19:20:21:23:24:25:27:29:30:32:34:36 SD: 5.1432 cents 20:21 x 23:25:27:28:30:32:34:35:38:40 SD: 4.7631 cents 22:23:25:26:28:29:30:33:35:37:39:41:44 SD: 4.6921 cents 23:24:26:27:29:31:32:35:37:39:40:43:46 SD: 3.9194 cents 24:25:27:28:30:32:33:36:38:41:42:45:48 SD: 2.5956 cents 32:34:36:37:40:43:44:48:51:54:56:60:64 SD: 2.4837 cents 40:42:45:47:50:53:55:60:64:68:70:75:80 SD: 1.7338 cents 48:50:54:56:60:64:66:72:77:81:84:90:96 SD: 1.2092 cents 64:67:72:75:80:85:88:96:102:108:112:120:128 SD: 1.1116 cents 72:76:81:84:90:96:99:108:115:122:126:135:144 SD: 0.9967 cents 80:84:90:93:100:107:110:120:128:135:140:150:160 SD: 0.6849 cents 96:101:108:112:120:128:132:144:154:162:168:180:192 SD: 0.4718 cents 120:126:135:140:150:160:165:180:192:203:210:225:240 SD: 0.3558 cents 144:151:162:168:180:192:198:216:230:243:252:270:288 SD: 0.3151 cents 192:202:216:224:240:256:264:288:307:324:336:360:384 SD: 0.3010 cents 216:227:243:252:270:288:297:324:346:365:378:405:432 SD: 0.2883 cents 240:252:270:280:300:320:330:360:384:405:420:450:480 SD: 0.0000 cents