Scala analysis: al-farabi_chrom
Al Farabi's Chromatic c700 AD
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 9/8 203.910002 major whole tone 2: 27/20 519.551289 acute fourth 3: 729/512 611.730005 Pythagorean tritone 4: 3/2 701.955001 perfect fifth 5: 9/5 1017.596288 just minor seventh, BP seventh 6: 19/10 1111.199302 undevicesimal major seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 20/19 88.8007 small undevicesimal semitone 1: 9/8 203.9100 major whole tone 2: 6/5 315.6413 minor third 3: 135/128 92.1787 major chroma, major limma 4: 256/243 90.2250 limma, Pythagorean minor second 5: 6/5 315.6413 minor third 6: 19/18 93.6030 undevicesimal semitone 7: 20/19 88.8007 small undevicesimal semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 20/19 88.801 cents small undevicesimal semitone 1: 1 256/243 90.225 cents limma, Pythagorean minor second 1: 1 135/128 92.179 cents major chroma, major limma 1: 1 19/18 93.603 cents undevicesimal semitone 1: 1 9/8 203.910 cents major whole tone 1: 2 6/5 315.641 cents minor third 2: 2 10/9 182.404 cents minor whole tone 2: 1 45/38 292.711 cents 2: 1 512/405 405.866 cents narrow diminished fourth 2: 1 81/64 407.820 cents Pythagorean major third 2: 1 19/15 409.244 cents undevicesimal ditone 2: 1 27/20 519.551 cents acute fourth 3: 1 5/4 386.314 cents major third 3: 3 4/3 498.045 cents perfect fourth 3: 1 4864/3645 499.469 cents 3: 1 27/19 608.352 cents 3: 1 729/512 611.730 cents Pythagorean tritone 4: 1 1024/729 588.270 cents Pythagorean diminished fifth 4: 1 38/27 591.648 cents 4: 1 3645/2432 700.531 cents 4: 3 3/2 701.955 cents perfect fifth 4: 1 8/5 813.686 cents minor sixth 5: 1 40/27 680.449 cents grave fifth 5: 1 30/19 790.756 cents smaller undevicesimal minor sixth 5: 1 128/81 792.180 cents Pythagorean minor sixth 5: 1 405/256 794.134 cents wide augmented fifth 5: 1 76/45 907.289 cents 5: 2 9/5 1017.596 cents just minor seventh, BP seventh 6: 2 5/3 884.359 cents major sixth, BP sixth 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 1 36/19 1106.397 cents smaller undevicesimal major seventh 6: 1 256/135 1107.821 cents octave - major chroma 6: 1 243/128 1109.775 cents Pythagorean major seventh 6: 1 19/10 1111.199 cents undevicesimal major seventh Highest number of different intervals for one interval class: 6 Average number of different intervals per interval class: 5.66667 = 17/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 203.9 519.6 611.7 702.0 1017.6 1111.2 1200.0 203.9 : 315.6 407.8 498.0 813.7 907.3 996.1 1200.0 519.6 : 92.2 182.4 498.0 591.6 680.4 884.4 1200.0 611.7 : 90.2 405.9 499.5 588.3 792.2 1107.8 1200.0 702.0 : 315.6 409.2 498.0 702.0 1017.6 1109.8 1200.0 1017.6: 93.6 182.4 386.3 702.0 794.1 884.4 1200.0 1111.2: 88.8 292.7 608.4 700.5 790.8 1106.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 88.8007 .. 315.6413 cents 57/50, 226.8406 cents 20/19, 88.8007 cents 2: 182.4037 .. 519.5513 cents 243/200, 337.1476 cents 25/27,-133.2376 cents 3: 386.3137 .. 611.7300 cents 729/640, 225.4163 cents 25/27,-133.2376 cents 4: 588.2700 .. 813.6863 cents 729/640, 225.4163 cents 524288/531441,-23.4600 cents 5: 680.4487 .. 1017.5963 cents 243/200, 337.1476 cents 25/27,-133.2376 cents 6: 884.3587 .. 1111.1993 cents 57/50, 226.8406 cents 25/27,-133.2376 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 20/19, 88.8007 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 6/5, 315.6413 cents Average / Smallest step : 1.930487 Largest / Average step : 1.841241 Largest / Smallest step : 3.554491 Median interval of one step : 19/18, 93.6030 cents, amount: 1 Most common interval of one step : 6/5, 315.6413 cents, amount: 2 Least squares average step : 186.00857 cents, oct.: 1302.05999 cents Scale is not proper Step pattern alph. order: ABCDBEF Step pattern size order : EFCBFDA Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 16.70397 cents Scale diversity : 1.445811 Lumma stability : 0.148001 Lumma impropriety factor : 0.463675 Rothenberg efficiency : 0.341497 redundancy: 0.658503 Efficiency x scale size : 2.390476 Number of different interval sizes : 34 = 5.66667 / class Number of one step interval sizes : 6 Highest interval variety : 6 Mean interval variety : 5.66667 = 17/3 Median interval variety : 5 Lowest interval variety : 5 Smallest interval difference : 1216/1215, 1.4243 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 3 Number of recognisable fifths : 4, average 701.5989 cents Length of longest harmonic part : 2 at degree 5 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 1 = 14.2857% Scale is n-1 differentially coherent in interval classes 1 and 2 combined -- Rational properties -- Prime limit : 19 Odd number limit : 3645 (O: 3645 U: 3645) Highest odd numerator or denominator: 729 Scale harmonicity : 0.015036 Average absolute harmonicity : 0.229003 Specific harmonicity : 0.107309 Fundamental : 1/2560, -11.3219 octaves, 0.1022 Hz. Guide tone : 27702, 14.7577 octaves, 7247551.410 Hz. Exponens Consonantiae : 7.091712E+07, 26.07963 octaves Euler's gradus suavitatis : 45 Sum of Mann's harmonic distance : 678.0, average 96.85714 Mersenne's string divisions : 355211430 Sum of van Prooijen's expressibility: 7.95845, average 1.13692 Sum of Tenney's harmonic distance : 15.17287, average 2.16755 Vogel's harmonic complexity : 18.42857 Wille's k value : 1822 Wilson's harmonic complexity : 42 Rectangular lattice diameter : 8 Triangular lattice diameter : 7 Lattice compactness : 120.70786, average 4.31099 Lattice compactness (without 2's) : 71.71930, average 2.56140 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -9 .. 1 -2.14286 17 1 3: 0 .. 6 2.00000 14 3 5: -1 .. 0 -0.42857 3 2 19: 0 .. 1 0.14286 1 1 Average exponent except 2's : 12 / 7 = 1.71429 Average absolute exponent except 2's: 18 / 7 = 2.57143 Scale is weakly epimorphic with val: <7 10 15 31| Scale is weakly epimorphic with val: <7 10 18 33| Scale is JI-epimorphic with val: <7 11 17 30| Scale is weakly epimorphic with val: <7 12 16 27| Scale is weakly epimorphic with val: <7 12 18 33|
FIT/MODE
7: 1 2 0 1 2 0 1 N T3 SD: 50.4122 c. M: 97.4443 c. 8: 1 2 1 1 2 0 1 N T3 SD: 46.2678 c. M: 69.5513 c. 10: 2 2 1 1 2 1 1 P D3 S SD: 32.9951 c. M: 57.5963 c. 11: 2 3 1 0 3 1 1 N G SD: 30.8157 c. M: 47.4095 c. 12: 2 3 1 1 3 1 1 N SD: 11.7959 c. M: 19.5513 c. Chromatic Mixolydian inverse 25: 4 7 2 2 6 2 2 N SD: 10.9108 c. M:-18.0450 c. 26: 4 7 2 2 7 2 2 N SD: 10.4251 c. M: 19.2946 c. 27: 5 7 2 2 7 2 2 N SD: 10.2860 c. M:-18.3122 c. 28: 5 7 2 2 8 2 2 N SD: 9.7594 c. M: 16.2407 c. 33: 6 8 3 2 9 3 2 N SD: 10.2500 c. M:-16.0734 c. 34: 6 9 2 3 9 3 2 N SD: 9.8477 c. M:-18.2125 c. 35: 6 9 3 2 10 2 3 N SD: 9.5773 c. M: 16.2407 c. 37: 6 10 3 3 9 3 3 N SD: 8.1215 c. M: 12.1909 c. 39: 7 10 3 3 10 3 3 N SD: 5.4453 c. M:-11.4746 c. 41: 7 11 3 3 11 3 3 N SD: 3.9591 c. M:-7.2780 c. 53: 9 14 4 4 14 4 4 N SD: 0.9541 c. M: 1.7653 c. 171: 29 45 13 13 45 13 13 N SD: 1.0426 c. M: 2.4274 c. 178: 30 47 14 13 47 14 13 N SD: 1.0839 c. M:-1.7531 c. 217: 37 57 17 16 57 17 16 N SD: 0.8601 c. M:-2.0949 c. 229: 39 60 18 17 60 18 17 N SD: 0.7390 c. M:-1.3704 c. 231: 39 61 18 17 61 18 17 N SD: 0.7856 c. M: 1.3126 c. 243: 41 64 19 18 64 19 18 N SD: 0.7696 c. M: 1.4409 c. 270: 46 71 21 20 71 21 20 N SD: 0.6729 c. M:-1.6033 c. 282: 48 74 22 21 74 22 21 N SD: 0.5385 c. M:-1.0360 c. 296: 50 78 23 22 78 23 22 N SD: 0.6048 c. M: 1.2073 c. 323: 55 85 25 24 85 25 24 N SD: 0.5899 c. M:-1.2731 c. 328: 56 86 25 25 86 26 24 N SD: 0.6555 c. M:-0.9958 c. 335: 57 88 26 25 88 26 25 N SD: 0.4487 c. M:-0.8073 c. 349: 59 92 27 26 92 27 26 N SD: 0.5302 c. M: 1.0447 c. 381: 65 100 29 29 100 30 28 N SD: 0.5064 c. M:-0.8144 c. 388: 66 102 30 29 102 30 29 N SD: 0.4254 c. M: 0.8900 c. 434: 74 114 33 33 114 34 32 N SD: 0.4207 c. M:-0.6983 c. 487: 83 128 37 37 128 38 36 N SD: 0.3818 c. M: 0.6417 c. 540: 92 142 41 41 142 42 40 N SD: 0.3746 c. M: 0.6189 c. 559: 95 147 43 42 147 44 41 N SD: 0.3005 c. M:-0.7864 c. 571: 97 150 44 43 150 45 42 N SD: 0.3208 c. M:-0.5345 c. 612: 104 161 47 46 161 48 45 N SD: 0.2161 c. M:-0.5654 c. 624: 106 164 48 47 164 49 46 N SD: 0.2213 c. M:-0.3392 c. 665: 113 175 51 50 175 52 49 N SD: 0.1639 c. M:-0.3796 c. 677: 115 178 52 51 178 53 50 N SD: 0.1455 c. M: 0.2086 c. 718: 122 189 55 54 189 56 53 N SD: 0.1485 c. M:-0.2310 c. 730: 124 192 56 55 192 57 54 N SD: 0.1011 c. M: 0.2232 c. 1012: 172 266 78 76 266 79 75 N SD: 0.1264 c. M: 0.2050 c. 1053: 179 277 81 79 277 82 78 N SD: 0.1115 c. M:-0.2358 c. 1106: 188 291 85 83 291 86 82 N SD: 0.1295 c. M:-0.2049 c. 1118: 190 294 86 84 294 87 83 N SD: 0.1171 c. M: 0.2870 c. 1342: 228 353 103 101 353 105 99 N SD: 0.1133 c. M:-0.2761 c. 1395: 237 367 107 105 367 109 103 N SD: 0.0901 c. M:-0.1985 c. 1448: 246 381 111 109 381 113 107 N SD: 0.0806 c. M: 0.1278 c. 1513: 257 398 116 114 398 118 112 N SD: 0.0957 c. M: 0.2297 c. 1624: 276 427 125 122 427 127 120 N SD: 0.0822 c. M:-0.1307 c. 1665: 283 438 128 125 438 130 123 N SD: 0.0960 c. M:-0.1619 c. 1677: 285 441 129 126 441 131 124 N SD: 0.0517 c. M:-0.0768 c. 1730: 294 455 133 130 455 135 128 N SD: 0.0276 c. M:-0.0619 c. 1783: 303 469 137 134 469 139 132 N SD: 0.0262 c. M:-0.0479 c.
FIT/HARMONIC
1 x x x x x x 2 S SD: 0.0000 cents 2 x x x 3 x x 4 S SD: 0.0000 cents 3 x 4 x x 5 x 6 S SD: 44.9874 cents 4 x 5 x 6:7 x 8 S SD: 35.4708 cents 5:6 x 7:8:9 x 10 S SD: 32.1381 cents 6:7:8 x 9:11 x 12 SD: 14.7455 cents 7:8:9:10:11 x 13:14 SD: 21.0538 cents 8:9:11 x 12:14:15:16 SD: 10.4263 cents 10:11 x 14:15:18:19:20 SD: 8.1092 cents 13:15:18 x 20:23:25:26 SD: 14.0350 cents 14:16:19:20:21:25:27:28 SD: 5.9211 cents 22:25:30:31:33:40:42:44 SD: 5.1527 cents 23:26:31:33:35:41:44:46 SD: 5.1444 cents 24:27:32:34:36:43:46:48 SD: 4.1203 cents 26:29:35:37:39:47:49:52 SD: 3.1902 cents 31:35:42:44:47:56:59:62 SD: 3.1824 cents 38:43:51:54:57:68:72:76 SD: 2.6441 cents 40:45:54:57:60:72:76:80 S SD: 0.2035 cents 368:414:497:524:552:662:699:736 SD: 0.1935 cents 392:441:529:558:588:706:745:784 SD: 0.1915 cents 408:459:551:581:612:734:775:816 SD: 0.1772 cents 432:486:583:615:648:778:821:864 SD: 0.1686 cents 448:504:605:638:672:806:851:896 SD: 0.1657 cents 472:531:637:672:708:850:897:944 SD: 0.1514 cents 512:576:691:729:768:922:973:1024 SD: 0.1387 cents 520:585:702:740:780:936:988:1040 SD: 0.1305 cents 552:621:745:786:828:994:1049:1104 SD: 0.1295 cents 560:630:756:797:840:1008:1064:1120 SD: 0.1066 cents 600:675:810:854:900:1080:1140:1200 SD: 0.0860 cents