Scala analysis: xen18-ayers-table-38
Harmonic Square Means in Tetrachords between 1/1 and 4/3 and 3/2 and 2/1
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 1296/1201 131.795481 2: 2048/1649 375.149180 3: 32/25 427.372572 classic diminished fourth 4: 4/3 498.044999 perfect fourth 5: 36/25 631.282574 classic diminished fifth 6: 3/2 701.955001 perfect fifth 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 4/3 498.0450 perfect fourth 1: 1296/1201 131.7955 2: 153728/133569 243.3537 3: 1649/1600 52.2234 4: 25/24 70.6724 classic chromatic semitone, minor chroma 5: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 6: 25/24 70.6724 classic chromatic semitone, minor chroma 7: 4/3 498.0450 perfect fourth
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 1649/1600 52.223 cents 1: 2 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 1 1296/1201 131.795 cents 1: 1 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 1: 1 153728/133569 243.354 cents 1: 1 4/3 498.045 cents perfect fourth 2: 1 1649/1536 122.896 cents 2: 2 9/8 203.910 cents major whole tone 2: 1 2402/2025 295.577 cents 2: 1 2048/1649 375.149 cents 2: 1 25/18 568.717 cents classic augmented fourth 2: 1 1728/1201 629.840 cents 3: 1 14841/12800 256.133 cents 3: 1 75/64 274.582 cents classic augmented second 3: 1 1201/972 366.250 cents 3: 1 32/25 427.373 cents classic diminished fourth 3: 1 1800/1201 700.513 cents 3: 1 3/2 701.955 cents perfect fifth 3: 1 8192/4947 873.194 cents 4: 1 4947/4096 326.806 cents 4: 1 4/3 498.045 cents perfect fourth 4: 1 1201/900 499.487 cents 4: 1 25/16 772.627 cents classic augmented fifth 4: 1 1944/1201 833.750 cents 4: 1 128/75 925.418 cents diminished seventh 4: 1 25600/14841 943.867 cents 5: 1 1201/864 570.160 cents 5: 1 36/25 631.283 cents classic diminished fifth 5: 1 1649/1024 824.851 cents 5: 1 2025/1201 904.423 cents 5: 2 16/9 996.090 cents Pythagorean minor seventh 5: 1 3072/1649 1077.104 cents 6: 1 3/2 701.955 cents perfect fifth 6: 1 133569/76864 956.646 cents 6: 1 50/27 1066.762 cents grave major seventh 6: 1 1201/648 1068.205 cents 6: 2 48/25 1129.328 cents classic diminished octave 6: 1 3200/1649 1147.777 cents Highest number of different intervals for one interval class: 7 Average number of different intervals per interval class: 6.33333 = 19/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 131.8 375.1 427.4 498.0 631.3 702.0 1200.0 131.8 : 243.4 295.6 366.2 499.5 570.2 1068.2 1200.0 375.1 : 52.2 122.9 256.1 326.8 824.9 956.6 1200.0 427.4 : 70.7 203.9 274.6 772.6 904.4 1147.8 1200.0 498.0 : 133.2 203.9 702.0 833.8 1077.1 1129.3 1200.0 631.3 : 70.7 568.7 700.5 943.9 996.1 1066.8 1200.0 702.0 : 498.0 629.8 873.2 925.4 996.1 1129.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 52.2234 .. 498.0450 cents 6400/4947, 445.8216 cents 1649/1600, 52.2234 cents 2: 122.8958 .. 629.8405 cents 2654208/1980449, 506.9447 cents 1649/2048,-375.1492 cents 3: 256.1334 .. 873.1942 cents 104857600/73418427, 617.0608 cents 1980449/2457600,-373.7071 cents 4: 326.8058 .. 943.8666 cents 104857600/73418427, 617.0608 cents 24472809/33554432,-546.3884 cents 5: 570.1595 .. 1077.1042 cents 2654208/1980449, 506.9447 cents 1980449/2457600,-373.7071 cents 6: 701.9550 .. 1147.7766 cents 6400/4947, 445.8216 cents 1649/2048,-375.1492 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 1649/1600, 52.2234 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 4/3, 498.0450 cents Average / Smallest step : 3.282601 Largest / Average step : 2.905262 Largest / Smallest step : 9.536818 Median interval of one step : 1296/1201, 131.7955 cents, amount: 1 Most common interval of one step : 25/24, 70.6724 cents, amount: 2 Least squares average step : 142.31810 cents, oct.: 996.22672 cents Scale is not proper Step pattern alph. order: ABCDEDF Step pattern size order : CEABDBF Scale is not a Constant Structure Scale diversity : 1.628404 Lumma stability : 0.087039 Lumma impropriety factor : 0.795174 Rothenberg efficiency : 0.326531 redundancy: 0.673469 Efficiency x scale size : 2.285714 Number of different interval sizes : 36 = 6.00000 / class Number of one step interval sizes : 6 Highest interval variety : 7 Mean interval variety : 6.33333 = 19/3 Median interval variety : 6 Lowest interval variety : 6 Smallest interval difference : 1201/1200, 1.4421 cents Number of recognisable fifths : 3, average 701.4743 cents Scale contains a complete diamond : 1 3 Formal octave complements present : 3 = 42.8571% -- Rational properties -- Prime limit : 1201 Odd number limit : 133569 (O: 133569 U: 133569) Highest odd numerator or denominator: 1649 Harmonicity values : limit above prime count Fundamental : 1/297067350, -28.1462 octaves, 0.0000 Hz. Guide tone : 165888, 17.3399 octaves, 43400541.777 Hz. Sum of Mann's harmonic distance : 3163.5, average 451.92857 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 10.04689, average 1.43527 Sum of Tenney's harmonic distance : 20.73639, average 2.96234 Vogel's harmonic complexity : 1144.28571 Wille's k value : 222800512 Lattice compactness : 145.54742, average 5.19812 Lattice compactness (without 2's) : 65.45877, average 2.33781 Number of different primes : 6 Prime exponents' range, average, count, tones@limit: 2: -1 .. 11 3.42857 26 1 3: -1 .. 4 0.85714 8 2 5: -2 .. 0 -0.57143 4 2 17: -1 .. 0 -0.14286 1 97: -1 .. 0 -0.14286 1 1 1201: -1 .. 0 -0.14286 1 1 Average exponent except 2's :-1 / 7 =-0.14286 Average absolute exponent except 2's: 15 / 7 = 2.14286
FIT/MODE
7: 1 1 0 1 1 0 3 N SD: 43.5016 c. M: 84.5154 c. 8: 1 2 0 0 1 1 3 N SD: 41.4751 c. M:-74.8508 c. 9: 1 2 0 1 1 0 4 N SD: 27.0231 c. M:-35.3841 c. 10: 1 2 0 1 1 1 4 N SD: 30.5603 c. M: 67.3726 c. 12: 1 3 0 1 1 1 5 N SD: 21.9220 c. M: 31.7955 c. 15: 2 3 0 1 2 1 6 N SD: 20.3173 c. M:-28.2045 c. 17: 2 3 1 1 2 1 7 N SD: 9.5835 c. M: 22.2080 c. 19: 2 4 1 1 2 1 8 N SD: 7.2289 c. M:-14.7327 c. 29: 3 6 1 2 3 2 12 N SD: 7.2427 c. M: 13.5795 c. 34: 4 7 1 2 4 2 14 N SD: 6.7714 c. M:-13.0861 c. 36: 4 7 2 2 4 2 15 N SD: 4.1696 c. M: 8.4825 c. 48: 5 10 2 3 5 3 20 N SD: 3.7596 c. M: 6.7955 c. 61: 7 12 3 3 7 4 25 N SD: 4.5857 c. M:-6.2417 c. 63: 7 13 2 4 7 4 26 N SD: 4.2838 c. M: 8.3250 c. 65: 7 13 3 4 7 4 27 N SD: 2.9867 c. M: 5.9184 c. 70: 8 14 3 4 8 4 29 N SD: 2.5257 c. M:-5.3474 c. 82: 9 17 3 5 9 5 34 N SD: 2.4474 c. M:-5.3386 c. 84: 9 17 4 5 9 5 35 N SD: 2.4104 c. M: 3.7206 c. 89: 10 18 4 5 10 5 37 N SD: 2.3559 c. M:-4.0881 c. 99: 11 20 4 6 11 6 41 N SD: 1.5026 c. M: 3.1301 c. 118: 13 24 5 7 13 7 49 N SD: 0.5637 c. M:-1.1220 c. 253: 28 51 11 15 28 15 105 N SD: 0.4886 c. M:-1.0108 c. 272: 30 55 12 16 30 16 113 N SD: 0.4286 c. M:-0.5686 c. 371: 41 75 16 22 41 22 154 N SD: 0.4081 c. M:-0.8191 c. 390: 43 79 17 23 43 23 162 N SD: 0.3837 c. M: 0.5133 c. 458: 50 93 20 27 51 27 190 N SD: 0.3913 c. M: 0.7911 c. 511: 56 104 22 30 57 30 212 N SD: 0.3117 c. M:-0.5847 c. 528: 58 107 23 31 59 31 219 N SD: 0.2730 c. M:-0.5356 c. 547: 60 111 24 32 61 32 227 N SD: 0.2633 c. M:-0.5273 c. 576: 63 117 25 34 64 34 239 N SD: 0.2500 c. M: 0.5455 c. 646: 71 131 28 38 72 38 268 N SD: 0.1731 c. M:-0.2964 c. 665: 73 135 29 39 74 39 276 N SD: 0.1756 c. M:-0.2966 c. 730: 80 148 32 43 81 43 303 N SD: 0.1752 c. M: 0.3547 c. 764: 84 155 33 45 85 45 317 N SD: 0.1498 c. M:-0.2435 c. 783: 86 159 34 46 87 46 325 N SD: 0.1584 c. M:-0.3297 c. 800: 88 162 35 47 89 47 332 N SD: 0.1371 c. M:-0.2174 c. 848: 93 172 37 50 94 50 352 N SD: 0.1141 c. M: 0.1917 c. 918: 101 186 40 54 102 54 381 N SD: 0.0984 c. M:-0.2307 c. 1376: 151 279 60 81 153 81 571 N SD: 0.0941 c. M: 0.1492 c. 1494: 164 303 65 88 166 88 620 N SD: 0.0515 c. M: 0.0686 c. 1612: 177 327 70 95 179 95 669 N SD: 0.0389 c. M: 0.0773 c. 1766: 194 358 77 104 196 104 733 N SD: 0.0348 c. M: 0.0642 c.
FIT/HARMONIC
1 x x x x x x 2 S SD: 0.0000 cents 2 x x x x x 3:4 S SD: 0.0000 cents 3 x x x 4 x 5:6 S SD: 60.8012 cents 4 x 5 x x x 6:8 S SD: 3.7215 cents 5 x 6 x x 7:8:10 S SD: 33.9150 cents 6 x 7 x 8 x 9:12 S SD: 27.0696 cents 7:8 x 9 x 10:11:14 SD: 25.7776 cents 8:9:10 x 11 x 12:16 S SD: 18.0700 cents 9:10:11 x 12:13:14:18 SD: 14.2627 cents 10:11:12:13 x 14:15:20 S SD: 14.6657 cents 11:12 x 14:15:16:17:22 SD: 11.7139 cents 12:13:15 x 16:17:18:24 SD: 5.1920 cents 13:14:16 x 17:19:20:26 SD: 10.5012 cents 14:15:17:18:19:20:21:28 SD: 7.6450 cents 15:16 x 19:20:22:23:30 SD: 9.4107 cents 16:17:20 x 21:23:24:32 SD: 6.6613 cents 19:21 x 24:25:27:29:38 SD: 10.8063 cents 20:22:25:26:27:29:30:40 SD: 7.2196 cents 21:23:26:27:28:30:32:42 SD: 5.8600 cents 22:24:27:28:29:32:33:44 SD: 5.6622 cents 23:25 x 29:31:33:35:46 SD: 7.1640 cents 24:26:30:31:32:35:36:48 SD: 4.2786 cents 25:27:31:32:33:36:38:50 SD: 4.1358 cents 28:30:35:36:37:40:42:56 SD: 3.9592 cents 36:39:45:46:48:52:54:72 SD: 2.0608 cents 50:54:62:64:67:72:75:100 SD: 1.3105 cents 75:81:93:96:100:108:113:150 SD: 1.1826 cents 78:84:97:100:104:112:117:156 SD: 1.0044 cents 100:108:124:128:133:144:150:200 SD: 0.7609 cents 128:138:159:164:171:184:192:256 SD: 0.7265 cents 136:147:169:174:181:196:204:272 SD: 0.6676 cents 150:162:186:192:200:216:225:300 SD: 0.4425 cents 186:201:231:238:248:268:279:372 SD: 0.3923 cents 214:231:266:274:285:308:321:428 SD: 0.3912 cents 228:246:283:292:304:328:342:456 SD: 0.3151 cents 264:285:328:338:352:380:396:528 SD: 0.1817 cents 318:343:395:407:424:458:477:636 SD: 0.1264 cents 468:505:581:599:624:674:702:936 SD: 0.1078 cents 582:628:723:745:776:838:873:1164 SD: 0.0677 cents