Scala analysis: grailrms
RMS optimized Holy Grail
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 84.048 cents 84.048251 2: 196.736 cents 196.735887 3: 309.424 cents 309.423522 4: 393.472 cents 393.471773 5: 506.897 cents 506.897045 6: 580.913 cents 580.913009 7: 699.794 cents 699.794113 8: 796.736 cents 796.735887 9: 893.678 cents 893.677661 10: 1012.559 cents 1012.558765 11: 1086.575 cents 1086.574729 12: 1200.000 cents 1200.000000
SHOW/INTERVAL
0: 113.4253 cents 113.4253 1: 84.0483 cents 84.0483 2: 112.6876 cents 112.6876 3: 112.6876 cents 112.6876 4: 84.0483 cents 84.0483 5: 113.4253 cents 113.4253 6: 74.0160 cents 74.0160 7: 118.8811 cents 118.8811 8: 96.9418 cents 96.9418 9: 96.9418 cents 96.9418 10: 118.8811 cents 118.8811 11: 74.0160 cents 74.0160 12: 113.4253 cents 113.4253
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 74.01596 cents 1: 2 84.04825 cents 1: 2 96.94177 cents 1: 2 112.68764 cents 1: 2 113.42527 cents 1: 2 118.88110 cents 2: 2 187.44124 cents 2: 2 192.89707 cents 2: 1 193.88355 cents 2: 2 196.73589 cents 2: 2 197.47352 cents 2: 2 215.82288 cents 2: 1 225.37527 cents 3: 2 271.48949 cents 3: 2 289.83884 cents 3: 2 306.32234 cents 3: 2 309.42352 cents 3: 2 310.16116 cents 3: 2 312.76465 cents 4: 2 384.17712 cents 4: 2 386.78062 cents 4: 2 390.37059 cents 4: 1 393.47177 cents 4: 2 403.26411 cents 4: 2 422.84879 cents 4: 1 431.64576 cents 5: 2 487.31236 cents 5: 2 496.86476 cents 5: 2 500.20589 cents 5: 2 503.05823 cents 5: 2 505.66172 cents 5: 2 506.89705 cents 6: 1 579.67768 cents 6: 2 580.91301 cents 6: 2 584.25414 cents 6: 2 600.00000 cents 6: 2 615.74586 cents 6: 2 619.08699 cents 6: 1 620.32232 cents 7: 2 693.10296 cents 7: 2 694.33828 cents 7: 2 696.94177 cents 7: 2 699.79411 cents 7: 2 703.13524 cents 7: 2 712.68764 cents 8: 1 768.35424 cents 8: 2 777.15121 cents 8: 2 796.73589 cents 8: 1 806.52823 cents 8: 2 809.62941 cents 8: 2 813.21938 cents 8: 2 815.82288 cents 9: 2 887.23535 cents 9: 2 889.83884 cents 9: 2 890.57648 cents 9: 2 893.67766 cents 9: 2 910.16116 cents 9: 2 928.51051 cents 10: 1 974.62473 cents 10: 2 984.17712 cents 10: 2 1002.52648 cents 10: 2 1003.26411 cents 10: 1 1006.11645 cents 10: 2 1007.10293 cents 10: 2 1012.55876 cents 11: 2 1081.11890 cents 11: 2 1086.57473 cents 11: 2 1087.31236 cents 11: 2 1103.05823 cents 11: 2 1115.95175 cents 11: 2 1125.98404 cents Highest number of different intervals for one interval class: 7 Average number of different intervals per interval class: 6.45455 = 71/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 84.0 196.7 309.4 393.5 506.9 580.9 699.8 796.7 893.7 1012.6 1086.6 1200.0 84.0 : 112.7 225.4 309.4 422.8 496.9 615.7 712.7 809.6 928.5 1002.5 1116.0 1200.0 196.7 : 112.7 196.7 310.2 384.2 503.1 600.0 696.9 815.8 889.8 1003.3 1087.3 1200.0 309.4 : 84.0 197.5 271.5 390.4 487.3 584.3 703.1 777.2 890.6 974.6 1087.3 1200.0 393.5 : 113.4 187.4 306.3 403.3 500.2 619.1 693.1 806.5 890.6 1003.3 1116.0 1200.0 506.9 : 74.0 192.9 289.8 386.8 505.7 579.7 693.1 777.2 889.8 1002.5 1086.6 1200.0 580.9 : 118.9 215.8 312.8 431.6 505.7 619.1 703.1 815.8 928.5 1012.6 1126.0 1200.0 699.8 : 96.9 193.9 312.8 386.8 500.2 584.3 696.9 809.6 893.7 1007.1 1081.1 1200.0 796.7 : 96.9 215.8 289.8 403.3 487.3 600.0 712.7 796.7 910.2 984.2 1103.1 1200.0 893.7 : 118.9 192.9 306.3 390.4 503.1 615.7 699.8 813.2 887.2 1006.1 1103.1 1200.0 1012.6: 74.0 187.4 271.5 384.2 496.9 580.9 694.3 768.4 887.2 984.2 1081.1 1200.0 1086.6: 113.4 197.5 310.2 422.8 506.9 620.3 694.3 813.2 910.2 1007.1 1126.0 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 74.0160 .. 118.8811 cents 44.86514 cents 74.01596 cents 2: 187.4412 .. 225.3753 cents 37.93403 cents 68.56013 cents 3: 271.4895 .. 312.7647 cents 41.27516 cents 46.11422 cents 4: 384.1771 .. 431.6458 cents 47.46863 cents 71.41247 cents 5: 487.3124 .. 506.8970 cents 19.58468 cents 55.66661 cents 6: 579.6777 .. 620.3223 cents 40.64463 cents 72.78064 cents 7: 693.1030 .. 712.6876 cents 19.58468 cents 72.78064 cents 8: 768.3542 .. 815.8229 cents 47.46863 cents 55.66661 cents 9: 887.2353 .. 928.5105 cents 41.27517 cents 71.41247 cents 10: 974.6247 .. 1012.5588 cents 37.93404 cents 46.11421 cents 11: 1081.1189 .. 1125.9840 cents 44.86514 cents 68.56013 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 74.01596 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 118.88110 cents Average / Smallest step : 1.351060 Largest / Average step : 1.188811 Largest / Smallest step : 1.606155 Median interval of one step : 104.81470 cents, amount: 0 Least squares average step : 99.68169 cents, oct.: 1196.18022 cents Scale is strictly proper Step pattern alph. order: ABBACDEFFEDC Step pattern size order : BDDBEAFCCFAE Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 46.11421 cents Scale diversity : 1.300938 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.647583 Rothenberg efficiency : 0.243687 redundancy: 0.756313 Efficiency x scale size : 2.924242 Number of different interval sizes : 71 = 6.45455 / class Number of one step interval sizes : 6 Highest interval variety : 7 Mean interval variety : 6.45455 = 71/11 Median interval variety : 6 Lowest interval variety : 6 Smallest interval difference : 0.73764 cents Number of recognisable fifths : 12, average 700.0000 cents Best fifths form a closed circle Formal octave complements present : 1 = 8.3333% Limited transpositions with margin 39.4093 cents: 7 Limited inverse transpositions : 4 Inversional symmetry on degrees : 2 8 -- Rational properties --
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 9.9877 c. M:-19.0870 c. Twelve-tone Chromatic (1/11-comma) 31: 2 3 3 2 3 2 3 2 3 3 2 3 SP M ME SD: 7.5360 c. M: 22.5423 c. Meantone Chromatic (53/220-comma) 33: 2 3 4 2 3 2 3 3 3 3 2 3 P D3 SD: 9.5530 c. M:-17.8492 c. 35: 2 4 3 2 4 2 3 3 3 4 2 3 P SD: 10.3438 c. M: 16.3289 c. 36: 3 3 3 3 3 2 4 3 3 3 3 3 SP T3 SD: 9.2902 c. M:-15.9517 c. 39: 3 3 4 3 3 3 4 3 3 4 2 4 SP D3 SD: 7.4350 c. M: 14.5894 c. 40: 3 4 3 3 4 2 4 4 3 4 2 4 SP SD: 8.4230 c. M:-13.2641 c. 42: 3 4 4 3 4 2 4 4 3 4 3 4 SP D3 SD: 7.3881 c. M: 14.0798 c. 43: 3 4 4 3 4 3 4 4 3 4 3 4 SP M ME SD: 4.9292 c. M:-12.5664 c. 47: 3 5 4 3 5 3 4 4 4 5 3 4 SP SD: 7.2191 c. M:-11.2976 c. 50: 3 5 5 3 5 3 5 4 4 5 3 5 SP D3 S SD: 5.9829 c. M: 12.0483 c. 51: 4 4 5 4 5 3 5 4 4 5 3 5 SP SD: 6.2206 c. M:-10.7500 c. 54: 4 5 5 4 5 3 6 4 4 6 3 5 SP S SD: 5.5343 c. M:-11.3170 c. 55: 4 5 5 4 5 4 5 5 4 5 4 5 SP M ME SD: 5.2622 c. M:-10.5368 c. 57: 4 5 6 4 5 4 5 5 4 6 4 5 SP SD: 5.8343 c. M: 9.4671 c. 60: 4 6 5 5 5 4 6 5 5 6 3 6 SP SD: 5.4477 c. M: 9.4235 c. 62: 4 6 6 4 6 4 6 5 5 6 4 6 SP D3 S SD: 3.9105 c. M: 6.6289 c. 74: 5 7 7 5 7 5 7 6 6 7 5 7 SP D3 S SD: 3.2208 c. M: 7.1534 c. 75: 5 7 7 6 7 4 8 6 6 7 5 7 SP SD: 4.2581 c. M:-6.5282 c. 82: 6 7 8 6 8 5 8 6 7 8 5 8 SP SD: 3.9009 c. M: 6.4920 c. 85: 6 8 8 6 8 5 9 6 7 9 5 8 SP SD: 3.1714 c. M: 6.1477 c. 86: 6 8 8 6 8 6 8 7 7 9 5 8 SP SD: 3.0220 c. M:-6.0459 c. 89: 6 9 8 6 9 5 9 7 7 9 6 8 SP SD: 3.2694 c. M:-5.5601 c. 97: 7 9 9 7 9 6 10 7 8 10 6 9 SP SD: 2.6505 c. M:-5.3605 c. 104: 7 10 10 7 10 6 11 8 8 11 6 10 SP S SD: 2.7273 c. M: 5.2161 c. 106: 7 10 10 8 10 6 11 8 9 10 7 10 SP SD: 3.2985 c. M: 5.0116 c. 109: 8 10 10 8 10 7 11 8 9 11 7 10 SP SD: 2.7255 c. M:-4.7930 c. 110: 8 10 10 8 10 7 11 9 9 11 7 10 SP S SD: 2.6821 c. M: 5.0789 c. 113: 8 11 10 8 11 7 11 9 9 11 7 11 SP SD: 2.5137 c. M:-5.0340 c. 115: 8 11 11 8 11 7 11 9 10 11 7 11 SP SD: 2.6792 c. M:-4.4073 c. 116: 8 11 11 8 11 7 12 9 9 12 7 11 SP S SD: 1.7435 c. M: 4.0225 c. 128: 9 12 12 9 12 8 13 10 10 13 8 12 SP S SD: 1.3549 c. M:-3.3309 c. 140: 10 13 13 10 13 9 14 11 11 14 9 13 SP S SD: 1.5647 c. M:-3.0630 c. 146: 10 14 14 10 14 9 14 12 12 14 9 14 SP S SD: 1.8099 c. M:-2.9052 c. 158: 11 15 15 11 15 9 16 13 13 15 10 15 SP SD: 1.7922 c. M: 3.6978 c. 159: 11 15 15 11 15 10 16 13 12 16 10 15 SP SD: 1.5872 c. M:-3.2641 c. 173: 12 16 17 12 16 11 17 14 14 17 11 16 SP SD: 1.5836 c. M:-2.7152 c. 182: 13 17 17 13 17 11 18 15 15 18 11 17 SP S SD: 1.5983 c. M:-3.0256 c. 190: 13 18 18 13 18 12 19 15 15 19 12 18 SP S SD: 1.5153 c. M: 3.1513 c. 197: 14 18 19 14 18 12 20 16 16 19 12 19 SP SD: 1.6204 c. M:-2.4673 c. 199: 14 18 19 14 19 12 20 16 16 20 12 19 SP SD: 1.5308 c. M: 3.7711 c. 200: 14 19 19 14 19 12 20 16 16 20 12 19 SP S SD: 1.6935 c. M:-3.1030 c. 201: 14 19 19 14 19 12 20 16 17 20 12 19 SP SD: 1.3913 c. M: 2.7060 c. 211: 15 19 20 15 20 13 21 17 17 21 13 20 SP SD: 1.3465 c. M: 3.3710 c. 213: 15 20 20 15 20 13 21 17 18 21 13 20 SP SD: 1.1607 c. M: 2.3697 c. 225: 16 21 21 16 21 14 22 18 19 22 14 21 SP SD: 1.1990 c. M:-2.3223 c. 231: 16 22 22 16 22 14 23 18 19 23 14 22 SP SD: 1.3164 c. M:-2.2648 c. 232: 16 22 22 16 22 14 23 19 19 23 14 22 SP S SD: 0.9385 c. M: 1.6027 c. 244: 17 23 23 17 23 15 24 20 20 24 15 23 SP S SD: 0.6619 c. M: 1.4335 c. 303: 21 29 28 21 29 19 30 24 25 30 18 29 SP SD: 1.0706 c. M: 1.4262 c. 305: 21 29 29 21 29 19 30 25 24 30 19 29 SP SD: 1.0474 c. M:-1.9526 c. 306: 21 29 29 21 29 19 30 25 25 30 19 29 SP S SD: 0.9534 c. M: 1.7549 c. 310: 22 29 29 22 29 19 31 25 25 31 19 29 SP S SD: 0.8776 c. M:-1.6348 c. 318: 22 30 30 22 30 20 31 26 26 31 20 30 SP S SD: 0.8679 c. M: 1.6809 c. 326: 23 30 31 23 31 20 32 26 27 32 20 31 SP SD: 0.8648 c. M: 1.6439 c. 329: 23 31 31 23 31 20 33 26 27 33 20 31 SP SD: 0.7402 c. M: 1.5991 c. 341: 24 32 32 24 32 21 34 27 28 34 21 32 SP SD: 0.6274 c. M: 1.4280 c. 353: 25 33 33 25 33 22 35 28 29 35 22 33 SP SD: 0.6994 c. M: 1.2685 c. 357: 25 34 33 25 34 22 35 29 29 35 22 34 SP SD: 0.6682 c. M:-1.5834 c. 360: 25 34 34 25 34 22 36 29 29 36 22 34 SP S SD: 0.4590 c. M: 0.9130 c. 372: 26 35 35 26 35 23 37 30 30 37 23 35 SP S SD: 0.2622 c. M:-0.5220 c. 616: 43 58 58 43 58 38 61 50 50 61 38 58 SP S SD: 0.3292 c. M:-0.4782 c. 717: 50 68 67 50 68 44 71 58 58 71 44 68 SP SD: 0.2947 c. M:-0.7537 c. 729: 51 69 68 51 69 45 72 59 59 72 45 69 SP SD: 0.2607 c. M:-0.7950 c. 857: 60 81 80 60 81 53 85 69 69 85 53 81 SP SD: 0.2533 c. M:-0.6970 c. 866: 61 81 81 61 82 53 86 70 70 86 53 82 SP S SD: 0.2522 c. M:-0.4783 c. 902: 63 85 85 63 85 56 89 73 73 89 56 85 SP S SD: 0.2858 c. M:-0.5543 c. 909: 64 85 85 64 86 56 90 74 73 90 56 86 SP SD: 0.2717 c. M:-0.6238 c. 916: 64 86 86 64 87 56 91 74 74 91 56 87 SP S SD: 0.3116 c. M: 0.5637 c. 928: 65 87 87 65 88 57 92 75 75 92 57 88 SP S SD: 0.2323 c. M: 0.3718 c. 940: 66 88 88 66 89 58 93 76 76 93 58 89 SP S SD: 0.2134 c. M: 0.4874 c. 1089: 76 103 102 76 103 67 108 88 88 108 67 103 SP SD: 0.1977 c. M:-0.5093 c. 1101: 77 104 103 77 104 68 109 89 89 109 68 104 SP SD: 0.1684 c. M:-0.5393 c. 1214: 85 114 114 85 115 75 120 98 98 120 75 115 SP S SD: 0.1697 c. M: 0.3677 c. 1238: 87 116 116 87 117 76 123 100 100 123 76 117 SP S SD: 0.1732 c. M:-0.3653 c. 1298: 91 122 122 91 122 80 129 105 105 128 80 123 SP SD: 0.2424 c. M:-0.3649 c. 1300: 91 122 122 91 123 80 129 105 105 129 80 123 SP S SD: 0.1520 c. M: 0.2976 c. 1312: 92 123 123 92 124 81 130 106 106 130 81 124 SP S SD: 0.1306 c. M: 0.2772 c. 1446: 101 136 136 101 137 89 143 117 117 143 89 137 SP S SD: 0.1437 c. M: 0.2677 c. 1458: 102 137 137 102 138 90 144 118 118 144 90 138 SP S SD: 0.1240 c. M: 0.2131 c. 1512: 106 142 142 106 143 93 150 122 122 150 93 143 SP S SD: 0.1269 c. M:-0.2458 c. 1524: 107 143 143 107 144 94 151 123 123 151 94 144 SP S SD: 0.1410 c. M:-0.2290 c. 1543: 108 145 145 108 146 95 153 124 125 153 95 146 SP SD: 0.1361 c. M: 0.3652 c. 1555: 109 146 146 109 147 96 154 125 126 154 96 147 SP SD: 0.1308 c. M: 0.3372 c. 1574: 110 148 148 110 149 97 156 127 127 156 97 149 SP S SD: 0.1064 c. M: 0.1855 c. 1586: 111 149 149 111 150 98 157 128 128 157 98 150 SP S SD: 0.0915 c. M: 0.2006 c. 1830: 128 172 172 128 173 113 181 148 148 181 113 173 SP S SD: 0.0709 c. M: 0.1220 c. 1958: 137 184 184 137 185 121 194 158 158 194 121 185 SP S SD: 0.0699 c. M: 0.1128 c. 2202: 154 207 207 154 208 136 218 178 178 218 136 208 SP S SD: 0.0642 c. M: 0.1245 c.
FIT/HARMONIC
1 x x x x x x x x x x x 2 S SD: 0.0000 cents 2 x x x x x x 3 x x x x 4 S SD: 1.0804 cents 3 x x x x 4 x x x 5 x x 6 S SD: 4.2844 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 11.0919 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 3.7625 cents 6 x x 7 x 8 x 9 x 10:11 x 12 S SD: 9.6251 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 11.2810 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.6750 cents 9 x 10:11 x 12:13 x 14:15:16:17:18 S SD: 8.8955 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.3284 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 10.8433 cents 12:13 x 14:15:16:17:18:19:20:22 x 24 S SD: 8.2756 cents 13:14:15 x 16:17:18:19:21:22:23:24:26 SD: 10.0919 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 7.6254 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 4.8007 cents 20:21:22:24:25:27:28:30:32:34:36:37:40 SD: 4.3015 cents 24:25:27:29:30:32:34:36:38:40:43:45:48 SD: 2.9858 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.9115 cents 35:37:39:42:44:47:49:52:55:59:63:66:70 SD: 2.5997 cents 36:38:40:43:45:48:50:54:57:60:65:67:72 SD: 2.5052 cents 40:42:45:48:50:54:56:60:63:67:72:75:80 SD: 1.7559 cents 55:58:62:66:69:74:77:82:87:92:99:103:110 SD: 1.6188 cents 56:59:63:67:70:75:78:84:89:94:101:105:112 SD: 1.4767 cents 58:61:65:69:73:78:81:87:92:97:104:109:116 SD: 1.2021 cents 77:81:86:92:97:103:108:115:122:129:138:144:154 SD: 1.0521 cents 82:86:92:98:103:110:115:123:130:137:147:154:164 SD: 0.7958 cents 98:103:110:117:123:131:137:147:155:164:176:184:196 SD: 0.7413 cents 102:107:114:122:128:137:143:153:162:171:183:191:204 SD: 0.7154 cents 117:123:131:140:147:157:164:175:185:196:210:219:234 SD: 0.6046 cents 118:124:132:141:148:158:165:177:187:198:212:221:236 SD: 0.4525 cents 180:189:202:215:226:241:252:270:285:302:323:337:360 SD: 0.4495 cents 183:192:205:219:230:245:256:274:290:307:328:343:366 SD: 0.4009 cents 185:194:207:221:232:248:259:277:293:310:332:347:370 SD: 0.3971 cents 199:209:223:238:250:267:278:298:315:333:357:373:398 SD: 0.3955 cents 200:210:224:239:251:268:280:300:317:335:359:375:400 SD: 0.2961 cents 259:272:290:310:325:347:362:388:410:434:465:485:518 SD: 0.2600 cents 301:316:337:360:378:403:421:451:477:504:540:564:602 SD: 0.2325 cents 306:321:343:366:384:410:428:458:485:513:549:573:612 SD: 0.2236 cents 317:333:355:379:398:425:443:475:502:531:569:594:634 SD: 0.2153 cents 341:358:382:408:428:457:477:511:540:571:612:639:682 SD: 0.1740 cents 423:444:474:506:531:567:592:634:670:709:759:792:846 SD: 0.1623 cents 424:445:475:507:532:568:593:635:672:710:761:794:848 SD: 0.1512 cents 459:482:514:549:576:615:642:688:727:769:824:860:918 SD: 0.1494 cents 482:506:540:576:605:646:674:722:764:808:865:903:964 SD: 0.1275 cents 506:531:567:605:635:678:708:758:802:848:908:948:1012 SD: 0.1074 cents