Scala analysis: xen15-gilson-aristoxenus-chromatic-tonikon
Aristoxenus' Chromatic Tonikon (or Eratosthenes' Chromatic)
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 20/19 88.800698 small undevicesimal semitone 2: 10/9 182.403712 minor whole tone 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 30/19 790.755699 smaller undevicesimal minor sixth 6: 5/3 884.358713 major sixth, BP sixth 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 6/5 315.6413 minor third 1: 20/19 88.8007 small undevicesimal semitone 2: 19/18 93.6030 undevicesimal semitone 3: 6/5 315.6413 minor third 4: 9/8 203.9100 major whole tone 5: 20/19 88.8007 small undevicesimal semitone 6: 19/18 93.6030 undevicesimal semitone 7: 6/5 315.6413 minor third
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: 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 24/19 404.442 cents smaller undevicesimal major third 2: 2 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: 4 4/3 498.045 cents perfect fourth 3: 1 27/19 608.352 cents 3: 1 57/40 613.154 cents 4: 1 80/57 586.846 cents 4: 1 38/27 591.648 cents 4: 4 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: 2 30/19 790.756 cents smaller undevicesimal minor sixth 5: 1 19/12 795.558 cents undevicesimal minor sixth 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: 2 36/19 1106.397 cents smaller undevicesimal major seventh 6: 2 19/10 1111.199 cents undevicesimal major seventh Highest number of different intervals for one interval class: 5 Average number of different intervals per interval class: 4.33333 = 13/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 88.8 182.4 498.0 702.0 790.8 884.4 1200.0 88.8 : 93.6 409.2 613.2 702.0 795.6 1111.2 1200.0 182.4 : 315.6 519.6 608.4 702.0 1017.6 1106.4 1200.0 498.0 : 203.9 292.7 386.3 702.0 790.8 884.4 1200.0 702.0 : 88.8 182.4 498.0 586.8 680.4 996.1 1200.0 790.8 : 93.6 409.2 498.0 591.6 907.3 1111.2 1200.0 884.4 : 315.6 404.4 498.0 813.7 1017.6 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 .. 613.1543 cents 57/50, 226.8406 cents 25/27,-133.2376 cents 4: 586.8457 .. 813.6863 cents 57/50, 226.8406 cents 3200/3249,-26.3086 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: 2 Least squares average step : 160.11067 cents, oct.: 1120.77469 cents Scale is not proper Step pattern alph. order: ABCDABC Step pattern size order : ABDCABD Scale is a Constant Structure, by a margin of 16.70397 cents Scale diversity : 1.447552 Lumma stability : 0.148001 Lumma impropriety factor : 0.466049 Rothenberg efficiency : 0.421769 redundancy: 0.578231 Efficiency x scale size : 2.952381 Number of different interval sizes : 26 = 4.33333 / class Number of one step interval sizes : 4 Highest interval variety : 5 Mean interval variety : 4.33333 = 13/3 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 361/360, 4.8023 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 4 Number of recognisable fifths : 4, average 701.9550 cents Scale is Dorian octave species of abstract tetrachord Scale contains two identical disjunct tetrachords Scale contains a complete diamond : 1 3 All steps are superparticular (epimoric) Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 2 at degree 0 Formal octave complements present : 3 = 42.8571% Scale is differentially coherent in interval class 1 Scale is differentially coherent in interval classes 3 and 4 combined -- Rational properties -- Prime limit : 19 Odd number limit : 57 (O: 57 U: 57) Highest odd numerator or denominator: 19 Scale harmonicity : 0.019415 Average absolute harmonicity : 0.246000 Specific harmonicity : 0.100576 Fundamental : 1/342, -8.4179 octaves, 0.7650 Hz. Guide tone : 60, 5.9069 octaves, 15697.534 Hz. Exponens Consonantiae : 2.052000E+04, 14.32474 octaves Euler's gradus suavitatis : 32 Sum of Mann's harmonic distance : 65.0, average 9.28571 Mersenne's string divisions : 1452360 Sum of van Prooijen's expressibility: 5.16496, average 0.73785 Sum of Tenney's harmonic distance : 10.62435, average 1.51776 Vogel's harmonic complexity : 23.42857 Wille's k value : 256 Wilson's harmonic complexity : 33 Rectangular lattice diameter : 5 Triangular lattice diameter : 3 Lattice compactness : 61.44136, average 2.19433 Lattice compactness (without 2's) : 45.78208, average 1.63507 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -1 .. 2 0.85714 8 1 3: -2 .. 1 -0.28571 6 2 5: 0 .. 1 0.57143 4 2 19: -1 .. 0 -0.28571 2 2 Average exponent except 2's : 0 / 7 = 0.00000 Average absolute exponent except 2's: 12 / 7 = 1.71429 Scale is weakly epimorphic with val: <7 10 18 33| Scale is weakly epimorphic with val: <7 11 16 28| Scale is JI-epimorphic with val: <7 11 17 30| Scale is weakly epimorphic with val: <7 12 16 27|
FIT/MODE
7: 1 0 2 1 1 0 2 N T3 SD: 42.4656 c. M:-82.6279 c. 8: 1 0 2 2 0 1 2 N S SD: 40.2096 c. M:-61.1993 c. 9: 1 0 3 1 1 1 2 N SD: 36.5778 c. M: 49.0704 c. 12: 1 1 3 2 1 1 3 N SD: 10.5072 c. M:-17.5963 c. Chromatic Dorian, Mela Kanakangi, Raga Kanakambari 26: 2 2 7 4 2 2 7 N SD: 6.5063 c. M:-9.6473 c. 27: 2 2 7 5 2 2 7 N SD: 6.4924 c. M:-9.2443 c. 38: 3 3 10 6 3 3 10 N SD: 5.2248 c. M:-7.2182 c. 39: 3 3 10 7 3 3 10 N SD: 5.7529 c. M:-9.2443 c. 41: 3 3 11 7 3 3 11 N SD: 3.5395 c. M: 6.7940 c. 53: 4 4 14 9 4 4 14 N SD: 1.1599 c. M:-1.7653 c. 152: 11 12 40 26 11 12 40 N SD: 1.0069 c. M: 1.9586 c. 164: 12 13 43 28 12 13 43 N SD: 0.6558 c. M:-1.0071 c. 205: 15 16 54 35 15 16 54 N SD: 0.6341 c. M: 0.9958 c. 217: 16 17 57 37 16 17 57 N SD: 0.2786 c. M:-0.4339 c. 270: 20 21 71 46 20 21 71 N SD: 0.2129 c. M:-0.3554 c. 460: 34 36 121 78 34 36 121 N SD: 0.1887 c. M: 0.3209 c. 487: 36 38 128 83 36 38 128 N SD: 0.2067 c. M:-0.3037 c. 513: 38 40 135 87 38 40 135 N SD: 0.1340 c. M:-0.2006 c. 677: 50 53 178 115 50 53 178 N SD: 0.1318 c. M: 0.2092 c. 730: 54 57 192 124 54 57 192 N SD: 0.0437 c. M: 0.0708 c. 1000: 74 78 263 170 74 78 263 N SD: 0.0332 c. M:-0.0450 c. 1513: 112 118 398 257 112 118 398 N SD: 0.0257 c. M:-0.0383 c. 1730: 128 135 455 294 128 135 455 N SD: 0.0177 c. M:-0.0344 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 x 4 x x 5:6 S SD: 0.0000 cents 4 x x 5:6 x 7:8 S SD: 35.0166 cents 5 x 6:7 x 8 x 10 S SD: 39.8535 cents 6 x 7:8:9 x 10:12 S SD: 16.8934 cents 7 x 8:9 x 11:12:14 SD: 18.7504 cents 8 x 9:11:12 x 13:16 SD: 14.4522 cents 9 x 10:12 x 14:15:18 S SD: 5.1680 cents 10 x 11:13:15:16:17:20 SD: 10.4417 cents 11 x 12:15 x 17:18:22 SD: 14.0138 cents 12 x 13:16:18:19:20:24 SD: 7.3489 cents 13:14 x 17:20:21:22:26 SD: 13.8143 cents 14:15:16:19:21:22:23:28 SD: 10.0461 cents 15:16:17:20:23:24:25:30 SD: 8.6600 cents 16:17:18:21:24:25:27:32 SD: 6.7883 cents 18:19:20:24:27:28:30:36 S SD: 3.7546 cents 34:36:38:45:51:54:57:68 SD: 3.4275 cents 36:38:40:48:54:57:60:72 S SD: 0.9702 cents 90:95:100:120:135:142:150:180 SD: 0.7101 cents 114:120:127:152:171:180:190:228 SD: 0.6500 cents 150:158:167:200:225:237:250:300 SD: 0.5464 cents 152:160:169:203:228:240:253:304 SD: 0.5456 cents 171:180:190:228:257:270:285:342 SD: 0.4816 cents 188:198:209:251:282:297:313:376 SD: 0.4788 cents 190:200:211:253:285:300:317:380 SD: 0.4367 cents 192:202:213:256:288:303:320:384 SD: 0.4275 cents 216:227:240:288:324:341:360:432 SD: 0.4029 cents 226:238:251:301:339:357:377:452 SD: 0.3984 cents 228:240:253:304:342:360:380:456 SD: 0.3256 cents 252:265:280:336:378:398:420:504 SD: 0.2540 cents 288:303:320:384:432:455:480:576 SD: 0.1925 cents 306:322:340:408:459:483:510:612 SD: 0.1143 cents 342:360:380:456:513:540:570:684 S SD: 0.0000 cents