Scala analysis: xen18-ayers-table-11
7 Iterated 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: 16/15 111.731285 minor diatonic semitone 2: 8/7 231.174094 septimal whole tone 3: 16/13 359.472338 tridecimal neutral third 4: 4/3 498.044999 perfect fourth 5: 16/11 648.682058 undecimal semi-diminished fifth 6: 8/5 813.686286 minor sixth 7: 16/9 996.089998 Pythagorean minor seventh 8: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 9/8 203.9100 major whole tone 1: 16/15 111.7313 minor diatonic semitone 2: 15/14 119.4428 major diatonic semitone 3: 14/13 128.2982 2/3-tone 4: 13/12 138.5727 tridecimal 2/3-tone 5: 12/11 150.6371 undecimal neutral second, 3/4-tone 6: 11/10 165.0042 Ptolemy's second, 4/5-tone 7: 10/9 182.4037 minor whole tone 8: 9/8 203.9100 major whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 16/15 111.731 cents minor diatonic semitone 1: 1 15/14 119.443 cents major diatonic semitone 1: 1 14/13 128.298 cents 2/3-tone 1: 1 13/12 138.573 cents tridecimal 2/3-tone 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 2: 1 8/7 231.174 cents septimal whole tone 2: 1 15/13 247.741 cents tridecimal 5/4-tone 2: 1 7/6 266.871 cents septimal minor third 2: 1 13/11 289.210 cents tridecimal minor third 2: 2 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 3: 1 16/13 359.472 cents tridecimal neutral third 3: 1 5/4 386.314 cents major third 3: 1 14/11 417.508 cents undecimal diminished fourth or major third 3: 1 9/7 435.084 cents septimal major third, BP third 3: 1 13/10 454.214 cents tridecimal semi-diminished fourth 3: 2 4/3 498.045 cents perfect fourth 3: 1 11/8 551.318 cents undecimal semi-augmented fourth 4: 1 4/3 498.045 cents perfect fourth 4: 1 15/11 536.951 cents undecimal augmented fourth 4: 1 18/13 563.382 cents tridecimal augmented fourth 4: 1 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 4: 1 10/7 617.488 cents Euler's tritone 4: 1 13/9 636.618 cents tridecimal diminished fifth 4: 1 22/15 663.049 cents undecimal diminished fifth 4: 1 3/2 701.955 cents perfect fifth 5: 1 16/11 648.682 cents undecimal semi-diminished fifth 5: 2 3/2 701.955 cents perfect fifth 5: 1 20/13 745.786 cents tridecimal semi-augmented fifth 5: 1 14/9 764.916 cents septimal minor sixth 5: 1 11/7 782.492 cents undecimal augmented fifth 5: 1 8/5 813.686 cents minor sixth 5: 1 13/8 840.528 cents tridecimal neutral sixth 6: 1 8/5 813.686 cents minor sixth 6: 1 18/11 852.592 cents undecimal neutral sixth 6: 2 5/3 884.359 cents major sixth, BP sixth 6: 1 22/13 910.790 cents tridecimal major sixth 6: 1 12/7 933.129 cents septimal major sixth 6: 1 26/15 952.259 cents tridecimal semi-augmented sixth 6: 1 7/4 968.826 cents harmonic seventh 7: 1 16/9 996.090 cents Pythagorean minor seventh 7: 1 9/5 1017.596 cents just minor seventh, BP seventh 7: 1 20/11 1034.996 cents large minor seventh 7: 1 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 7: 1 24/13 1061.427 cents tridecimal neutral seventh 7: 1 13/7 1071.702 cents 16/3-tone 7: 1 28/15 1080.557 cents grave major seventh 7: 1 15/8 1088.269 cents classic major seventh Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 7.42857 = 52/7
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 0.0 : 111.7 231.2 359.5 498.0 648.7 813.7 996.1 1200.0 111.7 : 119.4 247.7 386.3 537.0 702.0 884.4 1088.3 1200.0 231.2 : 128.3 266.9 417.5 582.5 764.9 968.8 1080.6 1200.0 359.5 : 138.6 289.2 454.2 636.6 840.5 952.3 1071.7 1200.0 498.0 : 150.6 315.6 498.0 702.0 813.7 933.1 1061.4 1200.0 648.7 : 165.0 347.4 551.3 663.0 782.5 910.8 1049.4 1200.0 813.7 : 182.4 386.3 498.0 617.5 745.8 884.4 1035.0 1200.0 996.1 : 203.9 315.6 435.1 563.4 702.0 852.6 1017.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 111.7313 .. 203.9100 cents 135/128, 92.1787 cents 16/15, 111.7313 cents 2: 231.1741 .. 386.3137 cents 35/32, 155.1396 cents 64/63, 27.2641 cents 3: 359.4723 .. 551.3179 cents 143/128, 191.8456 cents 64/65,-26.8414 cents 4: 498.0450 .. 701.9550 cents 9/8, 203.9100 cents 32/33,-53.2729 cents 5: 648.6821 .. 840.5277 cents 143/128, 191.8456 cents 32/33,-53.2729 cents 6: 813.6863 .. 968.8259 cents 35/32, 155.1396 cents 64/65,-26.8414 cents 7: 996.0900 .. 1088.2687 cents 135/128, 92.1787 cents 64/63, 27.2641 cents
SHOW DATA
Number of notes : 8 -- Interval properties -- Smallest interval : 16/15, 111.7313 cents, class 1 Average step (divided formal octave): 150.0000 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.342507 Largest / Average step : 1.359400 Largest / Smallest step : 1.825004 Median interval of one step : 144.60486 cents, amount: 0 Least squares average step : 138.93546 cents, oct.: 1111.48370 cents Scale is not proper Step pattern alph. order: ABCDEFGH Step pattern size order : ABCDEFGH Scale is sum-free (all different intervals) Scale is not a Constant Structure Scale diversity : 1.621543 Lumma stability : 0.231659 Lumma impropriety factor : 0.133524 Rothenberg efficiency : 0.285714 redundancy: 0.714286 Efficiency x scale size : 2.285714 Number of different interval sizes : 48 = 6.85714 / class Number of one step interval sizes : 8 Highest interval variety : 8 Mean interval variety : 7.42857 = 52/7 Median interval variety : 7 Lowest interval variety : 7 Smallest interval difference : 225/224, 7.7115 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 3 Number of recognisable fifths : 3, average 701.9550 cents 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 : 1 = 12.5000% Scale is differentially coherent in interval class 1 -- Rational properties -- Prime limit : 13 Odd number limit : 15 (O: 15 U: 15) Highest odd numerator or denominator: 15 Scale harmonicity : 0.015071 Average absolute harmonicity : 0.207866 Specific harmonicity : 0.129827 Fundamental : 1/45045, -15.4591 octaves, 0.0058 Hz. Guide tone : 16, 4.0000 octaves, 4186.009 Hz. Exponens Consonantiae : 7.207200E+05, 19.45908 octaves Euler's gradus suavitatis : 41 Sum of Mann's harmonic distance : 75.0, average 9.37500 Mersenne's string divisions : 55221075 Sum of van Prooijen's expressibility: 6.30686, average 0.78836 Sum of Tenney's harmonic distance : 13.83261, average 1.72908 Vogel's harmonic complexity : 38.50000 Wille's k value : 22522 Wilson's harmonic complexity : 42 Rectangular lattice diameter : 6 Triangular lattice diameter : 2 Lattice compactness : 85.91018, average 2.38639 Lattice compactness (without 2's) : 49.52520, average 1.37570 Number of different primes : 6 Prime exponents' range, average, count, tones@limit: 2: 1 .. 4 3.12500 25 1 3: -2 .. 0 -0.50000 4 2 5: -1 .. 0 -0.25000 2 2 7: -1 .. 0 -0.12500 1 1 11: -1 .. 0 -0.12500 1 1 13: -1 .. 0 -0.12500 1 1 Average exponent except 2's :-9 / 8 =-1.12500 Average absolute exponent except 2's: 9 / 8 = 1.12500 Scale is weakly epimorphic with val: <8 13 20 22 27 31| Scale is weakly epimorphic with val: <8 13 20 23 27 30|
FIT/MODE
8: 1 0 1 1 1 1 1 2 N T3 SD: 60.7846 c. M: 96.0900 c. 9: 1 1 1 1 1 1 2 1 P M ME SD: 35.5422 c. M:-70.5767 c. 10: 1 1 1 1 1 2 1 2 N D3 SD: 24.5892 c. M: 48.6821 c. 13: 1 2 1 1 2 2 2 2 N SD: 23.8974 c. M:-45.7490 c. 15: 1 2 1 2 2 2 2 3 N SD: 23.8056 c. M: 39.4723 c. 16: 2 1 2 2 2 2 2 3 P T3 SD: 21.6094 c. M:-38.2687 c. 17: 1 2 2 2 2 2 3 3 N SD: 21.6607 c. M: 41.1430 c. 18: 2 1 2 2 3 2 3 3 N SD: 21.3084 c. M: 31.3783 c. 19: 2 2 2 2 2 3 3 3 N SD: 14.4035 c. M:-21.4575 c. 22: 2 2 3 2 3 3 3 4 N SD: 11.0922 c. M:-22.3458 c. 24: 2 3 2 3 3 3 4 4 N SD: 9.9368 c. M:-18.8259 c. 29: 3 3 3 3 4 4 4 5 N SD: 11.1660 c. M:-17.1018 c. 30: 3 3 3 3 4 4 5 5 N SD: 9.6834 c. M: 18.0450 c. 31: 3 3 3 4 4 4 5 5 N SD: 6.7661 c. M: 11.0852 c. 41: 4 4 4 5 5 6 6 7 N SD: 4.5199 c. M: 8.2528 c. 46: 4 5 5 5 6 6 7 8 N SD: 4.5589 c. M: 7.3835 c. 53: 5 5 6 6 7 7 8 9 N SD: 3.4889 c. M:-7.9217 c. 57: 5 6 6 7 7 8 8 10 N SD: 5.1263 c. M:-7.3663 c. 58: 5 6 6 7 7 8 9 10 N SD: 5.6136 c. M: 8.2830 c. 61: 6 6 6 7 8 8 10 10 N SD: 5.4109 c. M:-7.1887 c. 63: 6 6 7 7 8 9 9 11 N SD: 3.3257 c. M: 5.6138 c. 72: 7 7 8 8 9 10 11 12 N SD: 3.7167 c. M:-7.1943 c. 74: 7 7 8 9 9 10 11 13 N SD: 3.6286 c. M: 6.9008 c. 77: 7 8 8 9 10 10 12 13 N SD: 2.7873 c. M:-5.8634 c. 87: 8 9 9 10 11 12 13 15 N SD: 1.7650 c. M:-3.3087 c. 128: 12 13 13 15 16 18 19 22 N SD: 2.0943 c. M: 3.2223 c. 130: 12 13 14 15 16 18 20 22 N SD: 1.1472 c. M: 2.5282 c. 181: 17 18 19 21 23 25 27 31 N SD: 1.1922 c. M:-1.7833 c. 183: 17 18 20 21 23 25 28 31 N SD: 0.8150 c. M: 1.6659 c. 217: 20 22 23 25 27 30 33 37 N SD: 0.9010 c. M: 1.6774 c. 224: 21 22 24 26 28 31 34 38 N SD: 0.5333 c. M: 0.8170 c. 270: 25 27 29 31 34 37 41 46 N SD: 0.3859 c. M: 0.6202 c. 441: 41 44 47 51 55 61 67 75 N SD: 0.4029 c. M: 1.0630 c. 494: 46 49 53 57 62 68 75 84 N SD: 0.1604 c. M: 0.4049 c. 764: 71 76 82 88 96 105 116 130 N SD: 0.1851 c. M: 0.2840 c. 882: 82 88 94 102 111 121 134 150 N SD: 0.1792 c. M:-0.2975 c. 925: 86 92 99 107 116 127 141 157 N SD: 0.1786 c. M: 0.2809 c. 1012: 94 101 108 117 127 139 154 172 N SD: 0.1485 c. M: 0.2688 c. 1084: 101 108 116 125 136 149 165 184 N SD: 0.1575 c. M:-0.3063 c. 1106: 103 110 118 128 139 152 168 188 N SD: 0.1373 c. M: 0.3403 c. 1236: 115 123 132 143 155 170 188 210 N SD: 0.1166 c. M: 0.2490 c. 1299: 121 129 139 150 163 179 197 221 N SD: 0.1608 c. M: 0.2470 c. 1308: 122 130 140 151 164 180 199 222 N SD: 0.1349 c. M:-0.2403 c. 1376: 128 137 147 159 173 189 209 234 N SD: 0.1123 c. M: 0.1700 c. 1395: 130 139 149 161 175 192 212 237 N SD: 0.1020 c. M:-0.2238 c. 1506: 140 150 161 174 189 207 229 256 N SD: 0.1035 c. M: 0.1775 c. 1578: 147 157 169 182 198 217 240 268 N SD: 0.0920 c. M:-0.2235 c. 1665: 155 166 178 192 209 229 253 283 N SD: 0.0899 c. M:-0.1773 c. 1802: 168 179 193 208 226 248 274 306 N SD: 0.1004 c. M:-0.1444 c. 1826: 170 182 195 211 229 251 278 310 N SD: 0.0999 c. M:-0.1860 c. 1848: 172 184 198 213 232 254 281 314 N SD: 0.0980 c. M:-0.2679 c. 1889: 176 188 202 218 237 260 287 321 N SD: 0.0604 c. M: 0.0849 c. 2190: 204 218 234 253 275 301 333 372 N SD: 0.0508 c. M:-0.0851 c.
FIT/HARMONIC
1 x x x x x x x 2 S SD: 0.0000 cents 2 x x x x 3 x x 4 S SD: 26.6365 cents 3 x x x 4 x 5 x 6 S SD: 23.5575 cents 4 x x 5 x 6 x 7:8 S SD: 16.3970 cents 5 x x 6 x 7:8:9:10 S SD: 16.4464 cents 6 x 7 x 8:9:10:11:12 S SD: 18.2154 cents 7 x 8 x 9:10:11:12:14 S SD: 16.5616 cents 8 x 9:10:11:12:13:14:16 S SD: 13.2508 cents 9 x 10:11:12:13:14:16:18 S SD: 10.1501 cents 10:11 x 12:13:15:16:18:20 SD: 14.2720 cents 11:12:13:14:15:16:18:20:22 S SD: 14.1371 cents 12:13:14:15:16:17:19:21:24 SD: 9.5802 cents 13:14:15:16:17:19:21:23:26 SD: 5.7174 cents 18:19:21:22:24:26:29:32:36 SD: 5.6424 cents 22:23:25:27:29:32:35:39:44 SD: 5.3676 cents 26:28:30:32:35:38:42:46:52 SD: 4.3953 cents 27:29:31:33:36:39:43:48:54 SD: 2.9663 cents 30:32:34:37:40:44:48:53:60 SD: 2.9238 cents 35:37:40:43:47:51:56:62:70 SD: 2.6528 cents 42:45:48:52:56:61:67:75:84 SD: 2.0069 cents 48:51:55:59:64:70:77:85:96 SD: 1.5699 cents 57:61:65:70:76:83:91:101:114 SD: 1.3217 cents 75:80:86:92:100:109:120:133:150 SD: 1.1690 cents 77:82:88:95:103:112:123:137:154 SD: 1.0233 cents 90:96:103:111:120:131:144:160:180 SD: 0.5618 cents 147:157:168:181:196:214:235:261:294 SD: 0.4781 cents 183:195:209:225:244:266:293:325:366 SD: 0.4617 cents 195:208:223:240:260:284:312:347:390 SD: 0.3733 cents 225:240:257:277:300:327:360:400:450 SD: 0.2250 cents 315:336:360:388:420:458:504:560:630 SD: 0.1920 cents 330:352:377:406:440:480:528:587:660 SD: 0.1690 cents 420:448:480:517:560:611:672:747:840 SD: 0.1068 cents