Scala analysis: xen18-ayers-table-23
Inverted Geometric Means Between 1/1 and 2/1 Produce a Symmetrical Scale
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 256/243 90.224996 limma, Pythagorean minor second 2: 9/8 203.910002 major whole tone 3: 32/27 294.134997 Pythagorean minor third 4: 4/3 498.044999 perfect fourth 5: 3/2 701.955001 perfect fifth 6: 27/16 905.865003 Pythagorean major sixth 7: 16/9 996.089998 Pythagorean minor seventh 8: 243/128 1109.775004 Pythagorean major seventh 9: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 256/243 90.2250 limma, Pythagorean minor second 1: 256/243 90.2250 limma, Pythagorean minor second 2: 2187/2048 113.6850 apotome 3: 256/243 90.2250 limma, Pythagorean minor second 4: 9/8 203.9100 major whole tone 5: 9/8 203.9100 major whole tone 6: 9/8 203.9100 major whole tone 7: 256/243 90.2250 limma, Pythagorean minor second 8: 2187/2048 113.6850 apotome 9: 256/243 90.2250 limma, Pythagorean minor second
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 4 256/243 90.225 cents limma, Pythagorean minor second 1: 2 2187/2048 113.685 cents apotome 1: 3 9/8 203.910 cents major whole tone 2: 1 65536/59049 180.450 cents Pythagorean diminished third 2: 4 9/8 203.910 cents major whole tone 2: 2 32/27 294.135 cents Pythagorean minor third 2: 2 81/64 407.820 cents Pythagorean major third 3: 4 32/27 294.135 cents Pythagorean minor third 3: 2 81/64 407.820 cents Pythagorean major third 3: 2 4/3 498.045 cents perfect fourth 3: 1 729/512 611.730 cents Pythagorean tritone 4: 2 8192/6561 384.360 cents Pythagorean diminished fourth 4: 1 81/64 407.820 cents Pythagorean major third 4: 2 4/3 498.045 cents perfect fourth 4: 2 729/512 611.730 cents Pythagorean tritone 4: 2 3/2 701.955 cents perfect fifth 5: 2 4/3 498.045 cents perfect fourth 5: 2 1024/729 588.270 cents Pythagorean diminished fifth 5: 2 3/2 701.955 cents perfect fifth 5: 1 128/81 792.180 cents Pythagorean minor sixth 5: 2 6561/4096 815.640 cents Pythagorean augmented fifth 6: 1 1024/729 588.270 cents Pythagorean diminished fifth 6: 2 3/2 701.955 cents perfect fifth 6: 2 128/81 792.180 cents Pythagorean minor sixth 6: 4 27/16 905.865 cents Pythagorean major sixth 7: 2 128/81 792.180 cents Pythagorean minor sixth 7: 2 27/16 905.865 cents Pythagorean major sixth 7: 4 16/9 996.090 cents Pythagorean minor seventh 7: 1 59049/32768 1019.550 cents Pythagorean augmented sixth 8: 3 16/9 996.090 cents Pythagorean minor seventh 8: 2 4096/2187 1086.315 cents Pythagorean diminished octave 8: 4 243/128 1109.775 cents Pythagorean major seventh Highest number of different intervals for one interval class: 5 Average number of different intervals per interval class: 4.00000 = 4
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 0.0 : 90.2 203.9 294.1 498.0 702.0 905.9 996.1 1109.8 1200.0 90.2 : 113.7 203.9 407.8 611.7 815.6 905.9 1019.6 1109.8 1200.0 203.9 : 90.2 294.1 498.0 702.0 792.2 905.9 996.1 1086.3 1200.0 294.1 : 203.9 407.8 611.7 702.0 815.6 905.9 996.1 1109.8 1200.0 498.0 : 203.9 407.8 498.0 611.7 702.0 792.2 905.9 996.1 1200.0 702.0 : 203.9 294.1 407.8 498.0 588.3 702.0 792.2 996.1 1200.0 905.9 : 90.2 203.9 294.1 384.4 498.0 588.3 792.2 996.1 1200.0 996.1 : 113.7 203.9 294.1 407.8 498.0 702.0 905.9 1109.8 1200.0 1109.8: 90.2 180.4 294.1 384.4 588.3 792.2 996.1 1086.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 203.9100 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents 2: 180.4500 .. 407.8200 cents 4782969/4194304, 227.3700 cents 524288/531441,-23.4600 cents 3: 294.1350 .. 611.7300 cents 19683/16384, 317.5950 cents 2048/2187,-113.6850 cents 4: 384.3600 .. 701.9550 cents 19683/16384, 317.5950 cents 4194304/4782969,-227.3700 cents 5: 498.0450 .. 815.6400 cents 19683/16384, 317.5950 cents 8/9,-203.9100 cents 6: 588.2700 .. 905.8650 cents 19683/16384, 317.5950 cents 4194304/4782969,-227.3700 cents 7: 792.1800 .. 1019.5500 cents 4782969/4194304, 227.3700 cents 2048/2187,-113.6850 cents 8: 996.0900 .. 1109.7750 cents 2187/2048, 113.6850 cents 524288/531441,-23.4600 cents
SHOW DATA
Number of notes : 9 -- Interval properties -- Smallest interval : 256/243, 90.2250 cents, class 1 Average step (divided formal octave): 133.3333 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.477787 Largest / Average step : 1.529325 Largest / Smallest step : 2.260017 Median interval of one step : 2187/2048, 113.6850 cents, amount: 2 Most common interval of one step : 256/243, 90.2250 cents, amount: 4 Least squares average step : 136.73132 cents, oct.: 1230.58184 cents Scale is not proper Scale is a mode of a 665-tone equal temperament with octave 2/1 degrees: 50 113 163 276 389 502 552 615 665 Least number of segments generator : 333 of 600.902 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 389 of 701.955 cents and inv. generated superset size: 11 = 2 more = 122.222% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABACCCABA Step pattern size order : SMSLLLSMS Scale is not a Constant Structure Scale diversity : 1.241231 Lumma stability : 0.150375 Lumma impropriety factor : 0.548875 Rothenberg efficiency : 0.582011 redundancy: 0.417989 Efficiency x scale size : 5.238095 Number of different interval sizes : 18 = 2.25000 / class Number of one step interval sizes : 3 Highest interval variety : 5 Mean interval variety : 4.00000 = 4 Median interval variety : 3.5 Lowest interval variety : 3 Smallest interval difference : 531441/524288, 23.4600 cents Most common intervals : 9/8, 203.9100 cents & inv., amount: 7 Most common triad is 0.0 203.910 996.090 cents, amount: 5 Number of recognisable fifths : 6, average 701.9550 cents Scale contains a complete diamond : 1 3 9 27 Formal octave complements present : 9 = 100.0000% Inversional symmetry on degrees : 0 Inversional symmetry on intervals : 4-5 -- Rational properties -- Prime limit : 3 Odd number limit : 59049 (O: 59049 U: 59049) Highest odd numerator or denominator: 243 Scale harmonicity : 0.024000 Average absolute harmonicity : 0.218940 Specific harmonicity : 0.150526 Fundamental : 1/31104, -14.9248 octaves, 0.0084 Hz. Guide tone : 62208, 15.9248 octaves, 16275203.166 Hz. Exponens Consonantiae : 1.934918E+09, 30.84963 octaves Euler's gradus suavitatis : 36 Sum of Mann's harmonic distance : 514.5, average 57.16667 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 10.49667, average 1.16630 Sum of Tenney's harmonic distance : 21.03272, average 2.33697 Vogel's harmonic complexity : 22.55556 Wille's k value : 29524 Wilson's harmonic complexity : 30 Rectangular lattice diameter : 10 Triangular lattice diameter : 10 Lattice compactness : 290.35425, average 6.45232 Lattice compactness (without 2's) : 158.00000, average 3.51111 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -7 .. 8 0.55556 35 1 3: -5 .. 5 0.00000 22 8 Average exponent except 2's : 0 / 9 = 0.00000 Average absolute exponent except 2's: 22 / 9 = 2.44444 Scale is weakly epimorphic with val: <9 13| Scale is weakly epimorphic with val: <9 14| = patent
FIT/MODE
9: 0 1 1 1 2 1 1 1 1 N T3 SD: 68.0999 c. M: 105.8650 c. 10: 0 1 1 2 1 2 1 1 1 N SD: 62.6806 c. M: 101.9550 c. 11: 1 1 1 2 1 2 1 1 1 N D3 I SD: 29.4595 c. M:-47.4095 c. 12: 1 1 1 2 2 2 1 1 1 N I SD: 5.7554 c. M: 9.7750 c. 29: 2 3 2 5 5 5 2 3 2 N I SD: 4.3961 c. M: 7.4664 c. 41: 3 4 3 7 7 7 3 4 3 N I SD: 1.4249 c. M: 2.4201 c. 53: 4 5 4 9 9 9 4 5 4 N I SD: 0.2008 c. M: 0.3410 c. 200: 15 19 15 34 34 34 15 19 15 N I SD: 0.1325 c. M: 0.2250 c. 253: 19 24 19 43 43 43 19 24 19 N I SD: 0.0627 c. M: 0.1064 c. 306: 23 29 23 52 52 52 23 29 23 N I SD: 0.0170 c. M: 0.0289 c. 359: 27 34 27 61 61 61 27 34 27 N I SD: 0.0151 c. M: 0.0257 c. 665: 50 63 50 113 113 113 50 63 50 N I SD: 0.0003 c. M: 0.0006 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 5 x x 6 S SD: 7.1688 cents 4 x x 5 x 6 x 7 x 8 S SD: 24.0316 cents 5 x x 6:7 x 8:9 x 10 S SD: 25.7345 cents 6 x x 7:8:9:10:11 x 12 S SD: 10.5986 cents 7 x 8 x 9:11:12 x 13:14 SD: 19.2834 cents 8 x 9 x 11:12 x 14:15:16 SD: 10.5986 cents 9 x 10:11:12:14:15:16:17:18 S SD: 11.0421 cents 10 x 11:12:13:15:17:18:19:20 SD: 8.4086 cents 11 x 12:13:15:17:19:20:21:22 SD: 12.6744 cents 12:13 x 14:16:18:20:21:23:24 SD: 8.4413 cents 13:14:15 x 17:20:22:23:25:26 SD: 10.4689 cents 14:15:16:17:19:21:24:25:27:28 SD: 8.4952 cents 15:16:17:18:20:23:25:27:28:30 SD: 7.2986 cents 16:17:18:19:21:24:27:28:30:32 SD: 5.1849 cents 18:19:20:21:24:27:30:32:34:36 S SD: 4.6561 cents 22:23:25:26:29:33:37:39:42:44 SD: 3.5948 cents 32:34:36:38:43:48:54:57:61:64 SD: 2.4136 cents 38:40:43:45:51:57:64:68:72:76 SD: 2.1845 cents 42:44:47:50:56:63:71:75:80:84 SD: 2.0476 cents 48:51:54:57:64:72:81:85:91:96 SD: 1.8591 cents 54:57:61:64:72:81:91:96:103:108 SD: 1.2872 cents 72:76:81:85:96:108:122:128:137:144 SD: 1.2348 cents 80:84:90:95:107:120:135:142:152:160 SD: 1.0129 cents 90:95:101:107:120:135:152:160:171:180 SD: 0.8817 cents 96:101:108:114:128:144:162:171:182:192 SD: 0.6467 cents 130:137:146:154:173:195:219:231:247:260 SD: 0.6324 cents 144:152:162:171:192:216:243:256:273:288 SD: 0.5929 cents 150:158:169:178:200:225:253:267:285:300 SD: 0.4812 cents 168:177:189:199:224:252:284:299:319:336 SD: 0.4172 cents 206:217:232:244:275:309:348:366:391:412 SD: 0.4117 cents 222:234:250:263:296:333:375:395:421:444 SD: 0.4002 cents 226:238:254:268:301:339:381:402:429:452 SD: 0.3813 cents 248:261:279:294:331:372:419:441:471:496 SD: 0.3738 cents 258:272:290:306:344:387:435:459:490:516 SD: 0.3463 cents 264:278:297:313:352:396:446:469:501:528 SD: 0.2870 cents 282:297:317:334:376:423:476:501:535:564 SD: 0.2795 cents 297:313:334:352:396:446:501:528:564:594 SD: 0.2544 cents 318:335:358:377:424:477:537:565:604:636 SD: 0.2474 cents 336:354:378:398:448:504:567:597:638:672 SD: 0.1570 cents 432:455:486:512:576:648:729:768:820:864 SD: 0.0554 cents