Scala analysis: xen18-ayers-table-12
5 Generalized Harmonic Means between 1/1 and 2/1
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 12/11 150.637059 undecimal neutral second, 3/4-tone 2: 6/5 315.641287 minor third 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 12/7 933.129094 septimal major sixth 6: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 7/6 266.8709 septimal minor third 1: 12/11 150.6371 undecimal neutral second, 3/4-tone 2: 11/10 165.0042 Ptolemy's second, 4/5-tone 3: 10/9 182.4037 minor whole tone 4: 9/8 203.9100 major whole tone 5: 8/7 231.1741 septimal whole tone 6: 7/6 266.8709 septimal minor third
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 12/11 150.637 cents undecimal neutral second, 3/4-tone 1: 1 11/10 165.004 cents Ptolemy's second, 4/5-tone 1: 1 10/9 182.404 cents minor whole tone 1: 1 9/8 203.910 cents major whole tone 1: 1 8/7 231.174 cents septimal whole tone 1: 1 7/6 266.871 cents septimal minor third 2: 1 6/5 315.641 cents minor third 2: 1 11/9 347.408 cents undecimal neutral third 2: 1 5/4 386.314 cents major third 2: 1 14/11 417.508 cents undecimal diminished fourth or major third 2: 1 9/7 435.084 cents septimal major third, BP third 2: 1 4/3 498.045 cents perfect fourth 3: 1 4/3 498.045 cents perfect fourth 3: 1 11/8 551.318 cents undecimal semi-augmented fourth 3: 1 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 3: 1 10/7 617.488 cents Euler's tritone 3: 1 16/11 648.682 cents undecimal semi-diminished fifth 3: 1 3/2 701.955 cents perfect fifth 4: 1 3/2 701.955 cents perfect fifth 4: 1 14/9 764.916 cents septimal minor sixth 4: 1 11/7 782.492 cents undecimal augmented fifth 4: 1 8/5 813.686 cents minor sixth 4: 1 18/11 852.592 cents undecimal neutral sixth 4: 1 5/3 884.359 cents major sixth, BP sixth 5: 1 12/7 933.129 cents septimal major sixth 5: 1 7/4 968.826 cents harmonic seventh 5: 1 16/9 996.090 cents Pythagorean minor seventh 5: 1 9/5 1017.596 cents just minor seventh, BP seventh 5: 1 20/11 1034.996 cents large minor seventh 5: 1 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone Highest number of different intervals for one interval class: 6 Average number of different intervals per interval class: 6.00000 = 6
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 0.0 : 150.6 315.6 498.0 702.0 933.1 1200.0 150.6 : 165.0 347.4 551.3 782.5 1049.4 1200.0 315.6 : 182.4 386.3 617.5 884.4 1035.0 1200.0 498.0 : 203.9 435.1 702.0 852.6 1017.6 1200.0 702.0 : 231.2 498.0 648.7 813.7 996.1 1200.0 933.1 : 266.9 417.5 582.5 764.9 968.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 150.6371 .. 266.8709 cents 77/72, 116.2338 cents 12/11, 150.6371 cents 2: 315.6413 .. 498.0450 cents 10/9, 182.4037 cents 36/35, 48.7704 cents 3: 498.0450 .. 701.9550 cents 9/8, 203.9100 cents 1/1, 0.0000 cents 4: 701.9550 .. 884.3587 cents 10/9, 182.4037 cents 1/1, 0.0000 cents 5: 933.1291 .. 1049.3629 cents 77/72, 116.2338 cents 36/35, 48.7704 cents
SHOW DATA
Number of notes : 6 -- Interval properties -- Smallest interval : 12/11, 150.6371 cents, class 1 Average step (divided formal octave): 200.0000 cents Largest one step interval : 7/6, 266.8709 cents Average / Smallest step : 1.327695 Largest / Average step : 1.334355 Largest / Smallest step : 1.771615 Median interval of one step : 193.15686 cents, amount: 0 Least squares average step : 186.25846 cents, oct.: 1117.55078 cents Scale is proper Step pattern alph. order: ABCDEF Step pattern size order : ABCDEF Scale is not a Constant Structure Scale diversity : 1.609657 Rothenberg stability : 0.866667 = 13/15 Lumma stability : 0.332346 Rothenberg efficiency : 0.355556 redundancy: 0.644444 Efficiency x scale size : 2.133333 Number of different interval sizes : 28 = 5.60000 / class Number of one step interval sizes : 6 Highest interval variety : 6 Mean interval variety : 6.00000 = 6 Median interval variety : 6 Lowest interval variety : 6 Smallest interval difference : 121/120, 14.3672 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 2 Most common triad is 0.0 498.045 701.955 cents, amount: 1 Number of recognisable fifths : 2, average 701.9550 cents Scale contains a complete diamond : 1 3 All steps are superparticular (epimoric) Steps are strictly increasing Length of longest harmonic part : 1 at degree 0 Scale is subharmonic, base degree is: 0 Formal octave complements present : 3 = 50.0000% Scale is differentially coherent in interval class 1 -- Rational properties -- Prime limit : 11 Odd number limit : 11 (O: 11 U: 11) Highest odd numerator or denominator: 11 Scale harmonicity : 0.023148 Average absolute harmonicity : 0.282832 Specific harmonicity : 0.171806 Fundamental : 1/2310, -11.1737 octaves, 0.1133 Hz. Guide tone : 12, 3.5850 octaves, 3139.507 Hz. Exponens Consonantiae : 2.772000E+04, 14.75864 octaves Euler's gradus suavitatis : 28 Sum of Mann's harmonic distance : 34.0, average 5.66667 Mersenne's string divisions : 504735 Sum of van Prooijen's expressibility: 3.53970, average 0.58995 Sum of Tenney's harmonic distance : 7.68034, average 1.28006 Vogel's harmonic complexity : 25.83333 Wille's k value : 1732 Wilson's harmonic complexity : 29 Rectangular lattice diameter : 5 Triangular lattice diameter : 2 Lattice compactness : 43.77602, average 2.08457 Lattice compactness (without 2's) : 28.43613, average 1.35410 Number of different primes : 5 Prime exponents' range, average, count, tones@limit: 2: -1 .. 2 1.16667 9 1 3: -1 .. 1 0.50000 5 2 5: -1 .. 0 -0.16667 1 1 7: -1 .. 0 -0.16667 1 1 11: -1 .. 0 -0.16667 1 1 Average exponent except 2's : 0 / 6 = 0.00000 Average absolute exponent except 2's: 8 / 6 = 1.33333 Scale is weakly epimorphic with val: <6 10 13 15 23| Scale is weakly epimorphic with val: <6 10 13 17 21| Scale is weakly epimorphic with val: <6 10 15 17 19| Scale is weakly epimorphic with val: <6 10 15 19 23| Scale is weakly epimorphic with val: <6 11 13 15 20| Scale is weakly epimorphic with val: <6 11 14 15 19| Scale is weakly epimorphic with val: <6 11 14 19 21| Scale is weakly epimorphic with val: <6 11 15 19 20|
FIT/MODE
6: 0 1 1 1 1 2 N T3 SD: 110.9019 c. M: 150.6371 c. 7: 1 1 1 1 1 2 P M ME SD: 35.2950 c. M: 75.9862 c. 8: 1 1 1 2 1 2 P D3 SD: 31.5152 c. M:-48.0450 c. 9: 1 1 2 1 2 2 P D3 SD: 29.4067 c. M: 48.9746 c. 10: 1 2 1 2 2 2 P D3 SD: 26.7071 c. M:-44.3587 c. 12: 1 2 2 2 2 3 P T3 SD: 25.5406 c. M: 50.6371 c. 13: 2 1 2 3 2 3 P G SD: 30.0559 c. M: 38.7182 c. 14: 2 2 2 2 3 3 P D3 S SD: 17.2971 c. M:-27.2159 c. 15: 2 2 2 3 3 3 P SD: 15.7054 c. M:-26.8709 c. 17: 2 2 3 3 3 4 P SD: 15.6427 c. M: 33.2883 c. 19: 2 3 3 3 4 4 P SD: 12.2373 c. M: 24.3213 c. 22: 3 3 3 4 4 5 P SD: 8.5675 c. M:-12.9993 c. 26: 3 4 4 4 5 6 P SD: 9.0436 c. M: 12.1755 c. 27: 3 4 4 5 5 6 P SD: 9.0153 c. M: 17.3037 c. 31: 4 4 5 5 6 7 P SD: 4.5403 c. M: 5.9639 c. 41: 5 6 6 7 8 9 P SD: 3.4322 c. M:-6.3099 c. 58: 7 8 9 10 11 13 P SD: 3.4315 c. M: 5.8095 c. 72: 9 10 11 12 14 16 P SD: 1.2344 c. M: 1.9550 c. 118: 15 16 18 20 23 26 P SD: 1.2902 c. M:-2.4641 c. 152: 19 21 23 26 29 34 P SD: 0.7902 c. M: 1.5501 c. 193: 24 27 29 33 37 43 P SD: 0.9281 c. M:-1.4572 c. 198: 25 27 30 34 38 44 P SD: 0.7489 c. M:-1.0753 c. 224: 28 31 34 38 43 50 P SD: 0.5198 c. M: 0.9862 c. 239: 30 33 36 41 46 53 P SD: 0.6995 c. M: 0.9739 c. 248: 31 34 38 42 48 55 P SD: 0.6398 c. M: 1.1252 c. 270: 34 37 41 46 52 60 P SD: 0.2635 c. M:-0.4741 c. 342: 43 47 52 58 66 76 P SD: 0.1834 c. M:-0.2401 c. 494: 62 68 75 84 95 110 P SD: 0.1555 c. M: 0.3356 c. 612: 77 84 93 104 118 136 P SD: 0.1642 c. M:-0.3433 c. 742: 93 102 113 126 143 165 P SD: 0.1533 c. M: 0.2774 c. 764: 96 105 116 130 147 170 P SD: 0.1197 c. M:-0.1483 c. 836: 105 115 127 142 161 186 P SD: 0.0866 c. M:-0.1482 c. 1084: 136 149 165 184 209 241 P SD: 0.0985 c. M: 0.1431 c. 1106: 139 152 168 188 213 246 P SD: 0.0849 c. M:-0.1767 c. 1236: 155 170 188 210 238 275 P SD: 0.0905 c. M: 0.1516 c. 1354: 170 186 206 230 261 301 P SD: 0.0727 c. M: 0.1317 c. 1506: 189 207 229 256 290 335 P SD: 0.0559 c. M: 0.1034 c. 1578: 198 217 240 268 304 351 P SD: 0.0506 c. M: 0.0667 c. 1848: 232 254 281 314 356 411 P SD: 0.0246 c. M: 0.0569 c. 2190: 275 301 333 372 422 487 P SD: 0.0320 c. M:-0.0479 c.
FIT/HARMONIC
1 x x x x x 2 S SD: 0.0000 cents 2 x x x 3 x 4 S SD: 0.0000 cents 3 x x 4 x 5:6 S SD: 16.2568 cents 4 x 5 x 6:7:8 S SD: 19.7940 cents 5 x 6:7:8:9:10 S SD: 32.7129 cents 6 x 7:8:9:10:12 S SD: 13.7943 cents 7:8 x 9:11:12:14 SD: 26.0281 cents 8:9:10:11:12:14:16 S SD: 18.2154 cents 9:10:11:12:14:15:18 S SD: 15.2396 cents 10:11:12:13:15:17:20 SD: 8.0579 cents 14:15:17:19:21:24:28 SD: 8.0481 cents 18:20:22:24:27:31:36 SD: 7.6052 cents 20:22:24:27:30:34:40 SD: 4.9408 cents 24:26:29:32:36:41:48 SD: 3.0063 cents 34:37:41:45:51:58:68 SD: 3.0062 cents 36:39:43:48:54:62:72 SD: 2.7590 cents 42:46:50:56:63:72:84 SD: 2.5675 cents 44:48:53:59:66:75:88 SD: 2.5623 cents 45:49:54:60:68:77:90 SD: 2.2598 cents 46:50:55:61:69:79:92 SD: 2.2224 cents 54:59:65:72:81:93:108 SD: 1.6628 cents 56:61:67:75:84:96:112 SD: 1.6050 cents 66:72:79:88:99:113:132 SD: 0.8156 cents 90:98:108:120:135:154:180 SD: 0.7564 cents 120:131:144:160:180:206:240 SD: 0.4478 cents 144:157:173:192:216:247:288 SD: 0.4089 cents 186:203:223:248:279:319:372 SD: 0.3167 cents 210:229:252:280:315:360:420 SD: 0.1145 cents 540:589:648:720:810:926:1080 SD: 0.0996 cents 570:622:684:760:855:977:1140 SD: 0.0943 cents