Scala analysis: AlexMalcom1721
Alexander Malcoms's ~[1721] tuning, compiled by A . Sparschuh
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 17/16 104.955410 17th harmonic 2: 9/8 203.910002 major whole tone 3: 19/16 297.513016 19th harmonic 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 17/12 603.000409 2nd septendecimal tritone 7: 3/2 701.955001 perfect fifth 8: 19/12 795.558015 undevicesimal minor sixth 9: 5/3 884.358713 major sixth, BP sixth 10: 85/48 989.314122 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: 17/16 104.9554 17th harmonic 2: 18/17 98.9546 Arabic lute index finger 3: 19/18 93.6030 undevicesimal semitone 4: 20/19 88.8007 small undevicesimal semitone 5: 16/15 111.7313 minor diatonic semitone 6: 17/16 104.9554 17th harmonic 7: 18/17 98.9546 Arabic lute index finger 8: 19/18 93.6030 undevicesimal semitone 9: 20/19 88.8007 small undevicesimal semitone 10: 17/16 104.9554 17th harmonic 11: 18/17 98.9546 Arabic lute index finger 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 20/19 88.801 cents small undevicesimal semitone 1: 2 19/18 93.603 cents undevicesimal semitone 1: 3 18/17 98.955 cents Arabic lute index finger 1: 3 17/16 104.955 cents 17th harmonic 1: 2 16/15 111.731 cents minor diatonic semitone 2: 2 10/9 182.404 cents minor whole tone 2: 2 19/17 192.558 cents quasi-meantone 2: 1 85/76 193.756 cents 2: 1 64/57 200.532 cents 2: 3 9/8 203.910 cents major whole tone 2: 1 96/85 210.686 cents 2: 2 17/15 216.687 cents septendecimal whole tone 3: 2 20/17 281.358 cents septendecimal augmented second 3: 1 85/72 287.359 cents septendecimal minor third 3: 1 45/38 292.711 cents 3: 1 32/27 294.135 cents Pythagorean minor third 3: 2 19/16 297.513 cents 19th harmonic 3: 1 68/57 305.487 cents 3: 4 6/5 315.641 cents minor third 4: 4 5/4 386.314 cents major third 4: 1 64/51 393.090 cents 4: 1 34/27 399.090 cents septendecimal major third 4: 2 24/19 404.442 cents smaller undevicesimal major third 4: 2 19/15 409.244 cents undevicesimal ditone 4: 1 108/85 414.596 cents 4: 1 51/40 420.597 cents septendecimal diminished fourth 5: 1 45/34 485.268 cents 5: 1 85/64 491.269 cents 5: 7 4/3 498.045 cents perfect fourth 5: 1 114/85 508.199 cents 5: 1 51/38 509.397 cents 5: 1 27/20 519.551 cents acute fourth 6: 1 80/57 586.846 cents 6: 1 45/32 590.224 cents diatonic tritone 6: 1 38/27 591.648 cents 6: 3 24/17 597.000 cents 1st septendecimal tritone 6: 3 17/12 603.000 cents 2nd septendecimal tritone 6: 1 27/19 608.352 cents 6: 1 64/45 609.776 cents 2nd tritone 6: 1 57/40 613.154 cents 7: 1 40/27 680.449 cents grave fifth 7: 1 76/51 690.603 cents 7: 1 85/57 691.801 cents 7: 7 3/2 701.955 cents perfect fifth 7: 1 128/85 708.731 cents 7: 1 68/45 714.732 cents 8: 1 80/51 779.403 cents 8: 1 85/54 785.404 cents 8: 2 30/19 790.756 cents smaller undevicesimal minor sixth 8: 2 19/12 795.558 cents undevicesimal minor sixth 8: 1 27/17 800.910 cents septendecimal minor sixth 8: 1 51/32 806.910 cents 8: 4 8/5 813.686 cents minor sixth 9: 4 5/3 884.359 cents major sixth, BP sixth 9: 1 57/34 894.513 cents 9: 2 32/19 902.487 cents 19th subharmonic 9: 1 27/16 905.865 cents Pythagorean major sixth 9: 1 76/45 907.289 cents 9: 1 144/85 912.641 cents 9: 2 17/10 918.642 cents septendecimal diminished seventh 10: 2 30/17 983.313 cents septendecimal minor seventh 10: 1 85/48 989.314 cents 10: 3 16/9 996.090 cents Pythagorean minor seventh 10: 1 57/32 999.468 cents undevicesimal minor seventh 10: 1 152/85 1006.244 cents 10: 2 34/19 1007.442 cents quasi-mean seventh 10: 2 9/5 1017.596 cents just minor seventh, BP seventh 11: 2 15/8 1088.269 cents classic major seventh 11: 3 32/17 1095.045 cents 17th subharmonic 11: 3 17/9 1101.045 cents septendecimal major seventh 11: 2 36/19 1106.397 cents smaller undevicesimal major seventh 11: 2 19/10 1111.199 cents undevicesimal major seventh 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 : 105.0 203.9 297.5 386.3 498.0 603.0 702.0 795.6 884.4 989.3 1088.3 1200.0 105.0 : 99.0 192.6 281.4 393.1 498.0 597.0 690.6 779.4 884.4 983.3 1095.0 1200.0 203.9 : 93.6 182.4 294.1 399.1 498.0 591.6 680.4 785.4 884.4 996.1 1101.0 1200.0 297.5 : 88.8 200.5 305.5 404.4 498.0 586.8 691.8 790.8 902.5 1007.4 1106.4 1200.0 386.3 : 111.7 216.7 315.6 409.2 498.0 603.0 702.0 813.7 918.6 1017.6 1111.2 1200.0 498.0 : 105.0 203.9 297.5 386.3 491.3 590.2 702.0 806.9 905.9 999.5 1088.3 1200.0 603.0 : 99.0 192.6 281.4 386.3 485.3 597.0 702.0 800.9 894.5 983.3 1095.0 1200.0 702.0 : 93.6 182.4 287.4 386.3 498.0 603.0 702.0 795.6 884.4 996.1 1101.0 1200.0 795.6 : 88.8 193.8 292.7 404.4 509.4 608.4 702.0 790.8 902.5 1007.4 1106.4 1200.0 884.4 : 105.0 203.9 315.6 420.6 519.6 613.2 702.0 813.7 918.6 1017.6 1111.2 1200.0 989.3 : 99.0 210.7 315.6 414.6 508.2 597.0 708.7 813.7 912.6 1006.2 1095.0 1200.0 1088.3: 111.7 216.7 315.6 409.2 498.0 609.8 714.7 813.7 907.3 996.1 1101.0 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 88.8007 .. 111.7313 cents 76/75, 22.9306 cents 20/19, 88.8007 cents 2: 182.4037 .. 216.6867 cents 51/50, 34.2830 cents 25/24, 70.6724 cents 3: 281.3583 .. 315.6413 cents 51/50, 34.2830 cents 300/289, 64.6716 cents 4: 386.3137 .. 420.5967 cents 51/50, 34.2830 cents 25/24, 70.6724 cents 5: 485.2683 .. 519.5513 cents 51/50, 34.2830 cents 300/289, 64.6716 cents 6: 586.8457 .. 613.1543 cents 3249/3200, 26.3086 cents 1600/1539, 67.2944 cents 7: 680.4487 .. 714.7317 cents 51/50, 34.2830 cents 1600/1539, 67.2944 cents 8: 779.4033 .. 813.6863 cents 51/50, 34.2830 cents 300/289, 64.6716 cents 9: 884.3587 .. 918.6417 cents 51/50, 34.2830 cents 25/24, 70.6724 cents 10: 983.3133 .. 1017.5963 cents 51/50, 34.2830 cents 300/289, 64.6716 cents 11: 1088.2687 .. 1111.1993 cents 76/75, 22.9306 cents 25/24, 70.6724 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 20/19, 88.8007 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 16/15, 111.7313 cents Average / Smallest step : 1.126117 Largest / Average step : 1.117313 Largest / Smallest step : 1.258225 Median interval of one step : 18/17, 98.9546 cents, amount: 3 Least squares average step : 99.32349 cents, oct.: 1191.88193 cents Scale is strictly proper Step pattern alph. order: ABCDEABCDABE Step pattern size order : DCBAEDCBADCE Scale is a Constant Structure, by a margin of 64.67161 cents Scale diversity : 1.253748 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.711305 Rothenberg efficiency : 0.271886 redundancy: 0.728114 Efficiency x scale size : 3.262626 Number of different interval sizes : 72 = 6.54545 / class Number of one step interval sizes : 5 Highest interval variety : 8 Mean interval variety : 6.54545 = 72/11 Median interval variety : 5 Lowest interval variety : 5 Smallest interval difference : 1445/1444, 1.1985 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 : 11, average 701.7774 cents Number of appreciable fifths : 10, average 700.4820 cents Best fifths form a closed circle Scale contains a complete diamond : 1 3 All steps are superparticular (epimoric) Length of longest harmonic part : 5 at degree 4 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 Limited transpositions with margin 34.2830 cents: 1 6 Limited inverse transpositions with margin 34.2830 cents: 10 11 -- Rational properties -- Prime limit : 19 Odd number limit : 85 (O: 85 U: 85) Highest odd numerator or denominator: 85 Scale harmonicity : 0.011817 Average absolute harmonicity : 0.171039 Specific harmonicity : 0.074028 Fundamental : 1/48, -5.5850 octaves, 5.4505 Hz. Guide tone : 58140, 15.8272 octaves, 15210910.367 Hz. Exponens Consonantiae : 2.790720E+06, 21.41221 octaves Euler's gradus suavitatis : 51 Sum of Mann's harmonic distance : 166.5, average 13.87500 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 11.43034, average 0.95253 Sum of Tenney's harmonic distance : 21.76767, average 1.81397 Vogel's harmonic complexity : 14.00000 Wille's k value : 42 Wilson's harmonic complexity : 50 Rectangular lattice diameter : 6 Triangular lattice diameter : 3 Lattice compactness : 235.95610, average 3.02508 Lattice compactness (without 2's) : 125.03416, average 1.60300 Number of different primes : 5 Prime exponents' range, average, count, tones@limit: 2: -4 .. 2 -1.83333 28 1 3: -1 .. 2 -0.08333 9 3 5: 0 .. 1 0.33333 4 3 17: 0 .. 1 0.25000 3 3 19: 0 .. 1 0.16667 2 2 Average exponent except 2's : 8 / 12 = 0.66667 Average absolute exponent except 2's: 18 / 12 = 1.50000 Scale is JI-epimorphic with val: <12 19 28 49 51| = patent
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 7.9905 c. M:-15.6413 c. Twelve-tone Chromatic (1/11-comma) 33: 3 3 2 3 3 3 2 3 2 3 3 3 SP D3 SD: 9.7656 c. M:-15.1814 c. 34: 3 3 2 3 3 3 3 3 2 3 3 3 SP B ME SD: 7.2914 c. M:-16.2067 c. Jubilee-12 41: 4 3 3 3 4 4 3 3 3 4 3 4 SP D3 SD: 6.2662 c. M:-12.1178 c. 44: 4 3 4 3 4 4 4 3 3 4 4 4 SP D3 SD: 6.7295 c. M: 13.0009 c. 46: 4 4 3 4 4 4 4 3 4 4 4 4 SP D3 S SD: 5.8664 c. M: 12.9493 c. 53: 5 4 4 4 5 5 4 4 4 5 4 5 SP D3 SD: 4.1913 c. M:-8.3203 c. 57: 5 5 4 4 6 5 4 5 4 5 5 5 SP SD: 5.2305 c. M:-7.5259 c. 60: 5 5 5 4 6 5 5 5 4 5 5 6 SP T3 SD: 4.9698 c. M: 9.3141 c. 65: 6 5 5 5 6 6 5 5 5 6 5 6 SP D3 SD: 3.4769 c. M:-7.6090 c. 66: 6 5 5 5 6 6 6 5 5 5 6 6 SP B SD: 5.2667 c. M: 7.4959 c. 68: 6 6 5 5 6 6 6 5 5 6 6 6 SP D3 SD: 3.5795 c. M:-7.8547 c. 69: 6 6 5 5 7 6 5 6 5 6 6 6 SP SD: 4.4470 c. M:-7.3835 c. 72: 6 6 6 5 7 6 6 6 5 6 6 7 SP T3 SD: 3.5645 c. M: 5.9808 c. 77: 7 6 6 6 7 7 6 6 6 7 6 7 SP D3 SD: 3.4652 c. M:-8.0885 c. 80: 7 7 6 6 7 7 7 6 6 7 7 7 SP D3 SD: 3.3029 c. M:-6.7313 c. 84: 7 7 7 6 8 7 7 7 6 7 7 8 SP T3 SD: 2.9452 c. M: 4.9554 c. 87: 8 7 7 6 8 8 7 7 6 8 7 8 SP SD: 3.3093 c. M:-5.9353 c. 89: 8 7 7 7 8 8 7 7 7 7 8 8 SP SD: 3.1679 c. M:-5.5289 c. 92: 8 8 7 7 8 8 8 7 7 8 7 9 SP SD: 3.1301 c. M: 5.6600 c. 94: 8 8 7 7 9 8 8 7 7 8 8 9 SP SD: 3.1947 c. M: 6.3354 c. 96: 8 8 8 7 9 8 8 8 7 8 8 9 SP T3 SD: 2.8514 c. M: 4.9554 c. 99: 9 8 8 7 9 9 8 8 7 9 8 9 SP SD: 3.0907 c. M:-5.5173 c. 101: 9 8 8 8 9 9 8 8 7 9 9 9 SP SD: 3.0482 c. M:-5.7655 c. 104: 9 9 8 7 10 9 9 8 8 9 8 10 SP SD: 3.0016 c. M: 5.5445 c. 106: 9 9 8 8 10 9 9 8 8 9 9 10 SP SD: 2.3013 c. M: 4.4085 c. 118: 10 10 9 9 11 10 10 9 9 10 10 11 SP SD: 1.8415 c. M: 3.2605 c. 125: 11 10 10 9 12 11 10 10 9 11 10 12 SP SD: 1.6381 c. M: 3.4687 c. 137: 12 11 11 10 13 12 11 11 10 12 11 13 SP SD: 1.2600 c. M: 2.4501 c. 171: 15 14 13 13 16 15 14 13 13 15 14 16 SP SD: 1.1331 c. M: 2.7762 c. 183: 16 15 14 14 17 16 15 14 14 16 15 17 SP SD: 1.0393 c. M: 2.4310 c. 193: 17 16 15 14 18 17 16 15 14 17 16 18 SP SD: 0.7794 c. M: 1.4572 c. 205: 18 17 16 15 19 18 17 16 15 18 17 19 SP SD: 0.5316 c. M:-1.0236 c. 330: 29 27 26 24 31 29 27 26 24 29 27 31 SP SD: 0.5891 c. M: 0.9960 c. 342: 30 28 27 25 32 30 28 27 25 30 28 32 SP SD: 0.4540 c. M:-0.9332 c. 354: 31 29 28 26 33 31 29 28 26 31 29 33 SP SD: 0.4795 c. M:-1.0522 c. 376: 33 31 29 28 35 33 31 29 28 33 31 35 SP SD: 0.3805 c. M: 0.8772 c. 388: 34 32 30 29 36 34 32 30 29 34 32 36 SP SD: 0.3414 c. M: 0.7127 c. 460: 40 38 36 34 43 40 38 36 34 40 38 43 SP SD: 0.3483 c. M: 0.6185 c. 513: 45 42 40 38 48 45 42 40 38 45 42 48 SP SD: 0.3307 c. M: 0.5494 c. 525: 46 43 41 39 49 46 43 41 39 46 43 49 SP SD: 0.2884 c. M: 0.4814 c. 581: 51 48 45 43 54 51 48 45 43 51 48 54 SP SD: 0.2836 c. M:-0.5650 c. 593: 52 49 46 44 55 52 49 46 44 52 49 55 SP SD: 0.2539 c. M:-0.4745 c. 665: 58 55 52 49 62 58 55 52 49 58 55 62 SP SD: 0.2123 c. M: 0.4419 c. 677: 59 56 53 50 63 59 56 53 50 59 56 63 SP SD: 0.2088 c. M: 0.3764 c. 718: 63 59 56 53 67 63 59 56 53 63 59 67 SP SD: 0.1863 c. M:-0.3422 c. 730: 64 60 57 54 68 64 60 57 54 64 60 68 SP SD: 0.1402 c. M:-0.2873 c. 935: 82 77 73 69 87 82 77 73 69 82 77 87 SP SD: 0.1579 c. M:-0.2852 c. 947: 83 78 74 70 88 83 78 74 70 83 78 88 SP SD: 0.1866 c. M:-0.3374 c. 1041: 91 86 81 77 97 91 86 81 77 91 86 97 SP SD: 0.1363 c. M: 0.2651 c. 1053: 92 87 82 78 98 92 87 82 78 92 87 98 SP SD: 0.0853 c. M: 0.1514 c. 1395: 122 115 109 103 130 122 115 109 103 122 115 130 SP SD: 0.0708 c. M:-0.1409 c. 1771: 155 146 138 131 165 155 146 138 131 155 146 165 SP SD: 0.0621 c. M: 0.1125 c. 1783: 156 147 139 132 166 156 147 139 132 156 147 166 SP SD: 0.0239 c. M: 0.0448 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: 5.1221 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 8.6647 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 8.2540 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 11.2081 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 8.8494 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.2473 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 8.3743 cents 11:12 x 13:14:15:16 x 17:18:19:21:22 SD: 11.1032 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 5.7039 cents 13:14:15 x 16:17:18:20:21:22:23:24:26 S SD: 9.2855 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 7.0943 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.3484 cents 16:17:18:19:20:21:23:24:25:27:28:30:32 SD: 4.4019 cents 24:26:27:29:30:32:34:36:38:40:43:45:48 SD: 4.1220 cents 26:28:29:31:33:35:37:39:41:43:46:49:52 SD: 3.8710 cents 27:29:30:32:34:36:38:41:43:45:48:51:54 SD: 3.6134 cents 30:32:34:36:38:40:43:45:48:50:53:56:60 SD: 3.6117 cents 31:33:35:37:39:41:44:47:49:52:55:58:62 SD: 2.5791 cents 32:34:36:38:40:43:45:48:51:53:57:60:64 SD: 2.1962 cents 47:50:53:56:59:63:67:71:74:78:83:88:94 SD: 2.1043 cents 48:51:54:57:60:64:68:72:76:80:85:90:96 S SD: 0.0000 cents