Scala analysis: farabi9
Al-Farabi 9 note ud scale
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: 27/22 354.547060 neutral third, Zalzal wosta of al-Farabi 3: 81/64 407.820003 Pythagorean major third 4: 4/3 498.044999 perfect fourth 5: 3/2 701.955001 perfect fifth 6: 18/11 852.592059 undecimal neutral sixth 7: 27/16 905.865003 Pythagorean major sixth 8: 16/9 996.089998 Pythagorean minor seventh 9: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 9/8 203.9100 major whole tone 1: 9/8 203.9100 major whole tone 2: 12/11 150.6371 undecimal neutral second, 3/4-tone 3: 33/32 53.2729 undecimal comma, al-Farabi's 1/4-tone 4: 256/243 90.2250 limma, Pythagorean minor second 5: 9/8 203.9100 major whole tone 6: 12/11 150.6371 undecimal neutral second, 3/4-tone 7: 33/32 53.2729 undecimal comma, al-Farabi's 1/4-tone 8: 256/243 90.2250 limma, Pythagorean minor second 9: 9/8 203.9100 major whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 33/32 53.273 cents undecimal comma, al-Farabi's 1/4-tone 1: 2 256/243 90.225 cents limma, Pythagorean minor second 1: 2 12/11 150.637 cents undecimal neutral second, 3/4-tone 1: 3 9/8 203.910 cents major whole tone 2: 2 88/81 143.498 cents 2nd undecimal neutral second 2: 2 9/8 203.910 cents major whole tone 2: 2 32/27 294.135 cents Pythagorean minor third 2: 2 27/22 354.547 cents neutral third, Zalzal wosta of al-Farabi 2: 1 81/64 407.820 cents Pythagorean major third 3: 2 32/27 294.135 cents Pythagorean minor third 3: 2 11/9 347.408 cents undecimal neutral third 3: 2 81/64 407.820 cents Pythagorean major third 3: 1 128/99 444.772 cents 3: 1 4/3 498.045 cents perfect fourth 3: 1 243/176 558.457 cents 4: 6 4/3 498.045 cents perfect fourth 4: 1 11/8 551.318 cents undecimal semi-augmented fourth 4: 1 729/512 611.730 cents Pythagorean tritone 4: 1 16/11 648.682 cents undecimal semi-diminished fifth 5: 1 11/8 551.318 cents undecimal semi-augmented fourth 5: 1 1024/729 588.270 cents Pythagorean diminished fifth 5: 1 16/11 648.682 cents undecimal semi-diminished fifth 5: 6 3/2 701.955 cents perfect fifth 6: 1 352/243 641.543 cents 6: 1 3/2 701.955 cents perfect fifth 6: 1 99/64 755.228 cents 6: 2 128/81 792.180 cents Pythagorean minor sixth 6: 2 18/11 852.592 cents undecimal neutral sixth 6: 2 27/16 905.865 cents Pythagorean major sixth 7: 1 128/81 792.180 cents Pythagorean minor sixth 7: 2 44/27 845.453 cents neutral sixth 7: 2 27/16 905.865 cents Pythagorean major sixth 7: 2 16/9 996.090 cents Pythagorean minor seventh 7: 2 81/44 1056.502 cents 2nd undecimal neutral seventh 8: 3 16/9 996.090 cents Pythagorean minor seventh 8: 2 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 8: 2 243/128 1109.775 cents Pythagorean major seventh 8: 2 64/33 1146.727 cents 33rd subharmonic Highest number of different intervals for one interval class: 6 Average number of different intervals per interval class: 4.75000 = 19/4
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 0.0 : 203.9 354.5 407.8 498.0 702.0 852.6 905.9 996.1 1200.0 203.9 : 150.6 203.9 294.1 498.0 648.7 702.0 792.2 996.1 1200.0 354.5 : 53.3 143.5 347.4 498.0 551.3 641.5 845.5 1049.4 1200.0 407.8 : 90.2 294.1 444.8 498.0 588.3 792.2 996.1 1146.7 1200.0 498.0 : 203.9 354.5 407.8 498.0 702.0 905.9 1056.5 1109.8 1200.0 702.0 : 150.6 203.9 294.1 498.0 702.0 852.6 905.9 996.1 1200.0 852.6 : 53.3 143.5 347.4 551.3 702.0 755.2 845.5 1049.4 1200.0 905.9 : 90.2 294.1 498.0 648.7 702.0 792.2 996.1 1146.7 1200.0 996.1 : 203.9 407.8 558.5 611.7 702.0 905.9 1056.5 1109.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 53.2729 .. 203.9100 cents 12/11, 150.6371 cents 33/32, 53.2729 cents 2: 143.4979 .. 407.8200 cents 6561/5632, 264.3221 cents 704/729,-60.4121 cents 3: 294.1350 .. 558.4571 cents 6561/5632, 264.3221 cents 2048/2187,-113.6850 cents 4: 498.0450 .. 648.6821 cents 12/11, 150.6371 cents 704/729,-60.4121 cents 5: 551.3179 .. 701.9550 cents 12/11, 150.6371 cents 121/128,-97.3641 cents 6: 641.5429 .. 905.8650 cents 6561/5632, 264.3221 cents 704/729,-60.4121 cents 7: 792.1800 .. 1056.5021 cents 6561/5632, 264.3221 cents 2048/2187,-113.6850 cents 8: 996.0900 .. 1146.7271 cents 12/11, 150.6371 cents 704/729,-60.4121 cents
SHOW DATA
Number of notes : 9 -- Interval properties -- Smallest interval : 33/32, 53.2729 cents, class 1 Average step (divided formal octave): 133.3333 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 2.502834 Largest / Average step : 1.529325 Largest / Smallest step : 3.827647 Median interval of one step : 12/11, 150.6371 cents, amount: 2 Most common interval of one step : 9/8, 203.9100 cents, amount: 3 Least squares average step : 132.85525 cents, oct.: 1195.69726 cents Scale is not proper Step pattern alph. order: ABCDABCDA Step pattern size order : DCABDCABD Scale is not a Constant Structure Scale diversity : 1.416249 Lumma stability : 0.088788 Lumma impropriety factor : 0.460087 Rothenberg efficiency : 0.521605 redundancy: 0.478395 Efficiency x scale size : 4.694444 Number of different interval sizes : 28 = 3.50000 / class Number of one step interval sizes : 4 Highest interval variety : 6 Mean interval variety : 4.75000 = 19/4 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 243/242, 7.1391 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 7 Most common triad is 0.0 498.045 701.955 cents, amount: 5 Number of recognisable fifths : 7, average 701.9550 cents Scale contains two identical conjunct pentachords Scale contains a complete diamond : 1 3 9 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 5 = 55.5556% Scale is n-1 differentially coherent in interval classes 3 and 4 combined Scale is differentially coherent in interval classes 4 and 5 combined -- Rational properties -- Prime limit : 11 Odd number limit : 729 (O: 729 U: 729) Highest odd numerator or denominator: 81 Scale harmonicity : 0.022634 Average absolute harmonicity : 0.215007 Specific harmonicity : 0.137505 Fundamental : 1/6336, -12.6294 octaves, 0.0413 Hz. Guide tone : 1296, 10.3399 octaves, 339066.733 Hz. Exponens Consonantiae : 8.211456E+06, 22.96921 octaves Euler's gradus suavitatis : 33 Sum of Mann's harmonic distance : 161.5, average 17.94444 Mersenne's string divisions : 77906325 Sum of van Prooijen's expressibility: 8.67533, average 0.96393 Sum of Tenney's harmonic distance : 17.59466, average 1.95496 Vogel's harmonic complexity : 23.77778 Wille's k value : 4009 Wilson's harmonic complexity : 29 Rectangular lattice diameter : 7 Triangular lattice diameter : 6 Lattice compactness : 199.59261, average 4.43539 Lattice compactness (without 2's) : 109.43319, average 2.43185 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -6 .. 4 -0.77778 23 1 3: -2 .. 4 1.33333 18 6 11: -1 .. 0 -0.22222 2 2 Average exponent except 2's : 10 / 9 = 1.11111 Average absolute exponent except 2's: 20 / 9 = 2.22222 Scale is weakly epimorphic with val: <9 13 33| Scale is weakly epimorphic with val: <9 14 30|
FIT/MODE
9: 1 1 1 0 2 1 0 1 2 N T3 SD: 67.9338 c. M: 105.8650 c. 10: 2 1 0 1 2 1 1 0 2 N SD: 31.0230 c. M:-54.1350 c. 11: 2 1 1 0 2 2 0 1 2 N SD: 32.5266 c. M: 61.6814 c. 12: 2 2 0 1 2 2 0 1 2 N SD: 22.2292 c. M:-47.4079 c. 13: 2 2 0 1 3 1 1 1 2 N SD: 25.5761 c. M: 38.5892 c. 14: 2 2 1 1 2 2 1 1 2 P D3 S SD: 22.5930 c. M:-36.9921 c. 16: 3 2 0 2 2 2 1 1 3 N SD: 22.6910 c. M: 32.8200 c. 17: 3 2 1 1 3 2 1 1 3 N SD: 7.9794 c. M:-15.7094 c. 24: 4 3 1 2 4 3 1 2 4 N SD: 4.2317 c. M: 7.8200 c. 41: 7 5 2 3 7 5 2 3 7 N SD: 1.9379 c. M: 3.8116 c. 65: 11 8 3 5 11 8 3 5 11 N SD: 1.8751 c. M: 3.7778 c. 94: 16 12 4 7 16 12 4 7 16 N SD: 1.3699 c. M:-2.8997 c. 118: 20 15 5 9 20 15 5 9 20 N SD: 0.8815 c. M:-1.6452 c. 135: 23 17 6 10 23 17 6 10 23 N SD: 0.6721 c. M:-1.0689 c. 159: 27 20 7 12 27 20 7 12 27 N SD: 0.1661 c. M: 0.2728 c. 359: 61 45 16 27 61 45 16 27 61 N SD: 0.1075 c. M: 0.2295 c. 518: 88 65 23 39 88 65 23 39 88 N SD: 0.0660 c. M: 0.1069 c. 971: 165 122 43 73 165 122 43 73 165 N SD: 0.0651 c. M:-0.1388 c. 1012: 172 127 45 76 172 127 45 76 172 N SD: 0.0427 c. M:-0.0851 c. 1171: 199 147 52 88 199 147 52 88 199 N SD: 0.0199 c. M:-0.0365 c. 1330: 226 167 59 100 226 167 59 100 226 N SD: 0.0186 c. M:-0.0395 c. 1689: 287 212 75 127 287 212 75 127 287 N SD: 0.0084 c. M: 0.0178 c.
FIT/HARMONIC
1 x x x x x x x x 2 S SD: 0.0000 cents 2 x x x x 3 x x x 4 S SD: 0.0000 cents 3 x x x 4 x x 5 x 6 S SD: 7.1688 cents 4 x x 5 x 6 x x 7:8 S SD: 8.6813 cents 5 x 6 x 7 x 8 x 9:10 S SD: 20.6151 cents 6:7 x x 8:9 x 10:11:12 S SD: 14.2054 cents 7:8 x 9 x x 11:12 x 14 SD: 16.9044 cents 8:9 x 10:11:12:13 x 14:16 S SD: 9.2465 cents 9:10:11 x 12:14 x 15:16:18 S SD: 10.0408 cents 10:11:12 x 13:15:16:17:18:20 SD: 10.5237 cents 11:12 x 14:15:17:18:19:20:22 SD: 12.6595 cents 12:14 x 15:16:18 x 20:21:24 S SD: 10.7214 cents 13:15:16 x 17:20:21:22:23:26 SD: 9.3454 cents 14:16:17:18:19:21:23:24:25:28 SD: 6.6802 cents 15:17:18:19:20:23 x 25:27:30 SD: 7.9567 cents 16:18 x 20:21:24:26:27:28:32 S SD: 5.7210 cents 18:20:22:23:24:27:29:30:32:36 SD: 4.9375 cents 20:23 x 25:27:30:33:34:36:40 SD: 7.0767 cents 21:24:26:27:28:32:34:35:37:42 SD: 6.5770 cents 22:25:27:28:29:33:36:37:39:44 SD: 3.2326 cents 30:34:37:38:40:45:49:51:53:60 SD: 2.5564 cents 41:46:50:52:55:62:67:69:73:82 SD: 2.4790 cents 42:47:52:53:56:63:69:71:75:84 SD: 2.3756 cents 45:51:55:57:60:68:74:76:80:90 SD: 2.3681 cents 48:54:59:61:64:72:79:81:85:96 SD: 1.5846 cents 57:64:70:72:76:86:93:96:101:114 SD: 1.5541 cents 64:72:79:81:85:96:105:108:114:128 SD: 1.4799 cents 71:80:87:90:95:107:116:120:126:142 SD: 1.3553 cents 72:81:88:91:96:108:118:122:128:144 SD: 1.1879 cents 80:90:98:101:107:120:131:135:142:160 SD: 0.9065 cents 102:115:125:129:136:153:167:172:181:204 SD: 0.6548 cents 144:162:177:182:192:216:236:243:256:288 SD: 0.4957 cents 154:173:189:195:205:231:252:260:274:308 SD: 0.4651 cents 162:182:199:205:216:243:265:273:288:324 SD: 0.4190 cents 176:198:216:223:235:264:288:297:313:352 SD: 0.3546 cents 234:263:287:296:312:351:383:395:416:468 SD: 0.2537 cents 256:288:314:324:341:384:419:432:455:512 SD: 0.2273 cents 294:331:361:372:392:441:481:496:523:588 SD: 0.2271 cents 369:415:453:467:492:554:604:623:656:738 SD: 0.2227 cents 384:432:471:486:512:576:628:648:683:768 SD: 0.1833 cents 410:461:503:519:547:615:671:692:729:820 SD: 0.1828 cents 426:479:523:539:568:639:697:719:757:852 SD: 0.1630 cents 432:486:530:547:576:648:707:729:768:864 SD: 0.1127 cents 528:594:648:668:704:792:864:891:939:1056 SD: 0.0992 cents 576:648:707:729:768:864:943:972:1024:1152 SD: 0.0960 cents