Scala analysis: xen17-bohlen-harmonic-2
12-tone non-tempered scale based on 4:7:10 triad
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 11/10 165.004228 Ptolemy's second, 4/5-tone 2: 6/5 315.641287 minor third 3: 30/23 459.994367 4: 10/7 617.487807 Euler's tritone 5: 11/7 782.492036 undecimal augmented fifth 6: 7/4 968.825906 harmonic seventh 7: 21/11 1119.462965 undecimal major seventh 8: 21/10 1284.467193 minor semitone +1 octave 9: 23/10 1441.960633 10: 5/2 1586.313714 major 10th 11: 11/4 1751.317942 undecimal semi-augmented fourth +1 octave 12: 3/1 1901.955001 perfect 12th
SHOW/INTERVAL
0: 12/11 150.6371 undecimal neutral second, 3/4-tone 1: 11/10 165.0042 Ptolemy's second, 4/5-tone 2: 12/11 150.6371 undecimal neutral second, 3/4-tone 3: 25/23 144.3531 4: 23/21 157.4934 5: 11/10 165.0042 Ptolemy's second, 4/5-tone 6: 49/44 186.3339 undecimal minor whole tone 7: 12/11 150.6371 undecimal neutral second, 3/4-tone 8: 11/10 165.0042 Ptolemy's second, 4/5-tone 9: 23/21 157.4934 10: 25/23 144.3531 11: 11/10 165.0042 Ptolemy's second, 4/5-tone 12: 12/11 150.6371 undecimal neutral second, 3/4-tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 25/23 144.353 cents 1: 3 12/11 150.637 cents undecimal neutral second, 3/4-tone 1: 2 23/21 157.493 cents 1: 4 11/10 165.004 cents Ptolemy's second, 4/5-tone 1: 1 49/44 186.334 cents undecimal minor whole tone 2: 1 300/253 294.990 cents 2: 2 25/21 301.847 cents BP second, quasi-equal minor third 2: 1 55/46 309.357 cents 2: 4 6/5 315.641 cents minor third 2: 2 253/210 322.498 cents 2: 1 147/121 336.971 cents 2: 1 49/40 351.338 cents larger approximation to neutral third 3: 1 100/77 452.484 cents 3: 2 30/23 459.994 cents 3: 1 72/55 466.278 cents undecimal semi-diminished fourth 3: 3 55/42 466.851 cents 3: 1 46/35 473.135 cents 3: 1 33/25 480.646 cents 2 pentatones 3: 2 147/110 501.975 cents 3: 1 161/120 508.832 cents 4: 1 360/253 610.631 cents 4: 4 10/7 617.488 cents Euler's tritone 4: 1 33/23 624.999 cents 4: 1 36/25 631.283 cents classic diminished fifth 4: 1 121/84 631.855 cents 4: 1 35/24 653.185 cents septimal semi-diminished fifth 4: 2 161/110 659.469 cents 4: 1 147/100 666.979 cents 5: 1 120/77 768.125 cents 5: 2 36/23 775.636 cents 5: 4 11/7 782.492 cents undecimal augmented fifth 5: 3 35/22 803.822 cents 5: 2 161/100 824.473 cents 6: 1 900/529 919.989 cents 6: 4 12/7 933.129 cents septimal major sixth 6: 1 121/70 947.496 cents 6: 1 210/121 954.459 cents 6: 4 7/4 968.826 cents harmonic seventh 6: 1 529/300 981.966 cents 7: 2 300/161 1077.482 cents 7: 3 66/35 1098.133 cents 7: 4 21/11 1119.463 cents undecimal major seventh 7: 2 23/12 1126.319 cents vicesimotertial major seventh 7: 1 77/40 1133.830 cents 8: 1 100/49 1234.976 cents Erlich's decatonic comma, tritonic diesis +1 octave 8: 2 330/161 1242.486 cents 8: 1 72/35 1248.770 cents septimal diesis, 1/4-tone +1 octave 8: 1 252/121 1270.100 cents 8: 1 25/12 1270.672 cents classic augmented octave 8: 1 23/11 1276.956 cents 8: 4 21/10 1284.467 cents minor semitone +1 octave 8: 1 253/120 1291.324 cents 9: 1 360/161 1393.123 cents 9: 2 110/49 1399.980 cents quasi-equal major second +1 octave 9: 1 25/11 1421.309 cents undecimal acute whole tone +1 octave 9: 1 105/46 1428.820 cents 9: 3 126/55 1435.104 cents 9: 1 55/24 1435.677 cents undecimal semi-augmented whole tone +1 octave 9: 2 23/10 1441.961 cents 9: 1 231/100 1449.471 cents 5/4-tone +1 octave 10: 1 120/49 1550.617 cents smaller approximation to neutral third +1 octave 10: 1 121/49 1564.984 cents 10: 2 630/253 1579.457 cents 10: 4 5/2 1586.314 cents major 10th 10: 1 138/55 1592.598 cents 10: 2 63/25 1600.108 cents quasi-equal major tenth, BP eleventh 10: 1 253/100 1606.965 cents 11: 1 132/49 1715.621 cents 11: 4 30/11 1736.951 cents undecimal augmented fourth +1 octave 11: 2 63/23 1744.462 cents 11: 3 11/4 1751.318 cents undecimal semi-augmented fourth +1 octave 11: 2 69/25 1757.602 cents Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 6.54545 = 72/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 165.0 315.6 460.0 617.5 782.5 968.8 1119.5 1284.5 1442.0 1586.3 1751.3 1902.0 165.0 : 150.6 295.0 452.5 617.5 803.8 954.5 1119.5 1277.0 1421.3 1586.3 1737.0 1902.0 315.6 : 144.4 301.8 466.9 653.2 803.8 968.8 1126.3 1270.7 1435.7 1586.3 1751.3 1902.0 460.0 : 157.5 322.5 508.8 659.5 824.5 982.0 1126.3 1291.3 1442.0 1607.0 1757.6 1902.0 617.5 : 165.0 351.3 502.0 667.0 824.5 968.8 1133.8 1284.5 1449.5 1600.1 1744.5 1902.0 782.5 : 186.3 337.0 502.0 659.5 803.8 968.8 1119.5 1284.5 1435.1 1579.5 1737.0 1902.0 968.8 : 150.6 315.6 473.1 617.5 782.5 933.1 1098.1 1248.8 1393.1 1550.6 1715.6 1902.0 1119.5: 165.0 322.5 466.9 631.9 782.5 947.5 1098.1 1242.5 1400.0 1565.0 1751.3 1902.0 1284.5: 157.5 301.8 466.9 617.5 782.5 933.1 1077.5 1235.0 1400.0 1586.3 1737.0 1902.0 1442.0: 144.4 309.4 460.0 625.0 775.6 920.0 1077.5 1242.5 1428.8 1579.5 1744.5 1902.0 1586.3: 165.0 315.6 480.6 631.3 775.6 933.1 1098.1 1284.5 1435.1 1600.1 1757.6 1902.0 1751.3: 150.6 315.6 466.3 610.6 768.1 933.1 1119.5 1270.1 1435.1 1592.6 1737.0 1902.0 1902.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 144.3531 .. 186.3339 cents 1127/1100, 41.9808 cents 25/23, 144.3531 cents 2: 294.9901 .. 351.3381 cents 12397/12000, 56.3480 cents 1200/1127, 108.6563 cents 3: 452.4836 .. 508.8315 cents 12397/12000, 56.3480 cents 4000/3773, 101.1455 cents 4: 610.6314 .. 666.9794 cents 12397/12000, 56.3480 cents 43200/40733, 101.7999 cents 5: 768.1249 .. 824.4728 cents 12397/12000, 56.3480 cents 4000/3773, 101.1455 cents 6: 919.9887 .. 981.9663 cents 279841/270000, 61.9775 cents 90000/85169, 95.5159 cents 7: 1077.4822 .. 1133.8301 cents 12397/12000, 56.3480 cents 90000/85169, 95.5159 cents 8: 1234.9756 .. 1291.3236 cents 12397/12000, 56.3480 cents 4000/3773, 101.1455 cents 9: 1393.1235 .. 1449.4714 cents 12397/12000, 56.3480 cents 43200/40733, 101.7999 cents 10: 1550.6169 .. 1606.9649 cents 12397/12000, 56.3480 cents 4000/3773, 101.1455 cents 11: 1715.6211 .. 1757.6019 cents 1127/1100, 41.9808 cents 1200/1127, 108.6563 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 25/23, 144.3531 cents, class 1 Average step (divided formal octave): 158.4963 cents Largest one step interval : 49/44, 186.3339 cents Average / Smallest step : 1.097976 Largest / Average step : 1.175636 Largest / Smallest step : 1.290820 Median interval of one step : 23/21, 157.4934 cents, amount: 2 Most common interval of one step : 11/10, 165.0042 cents, amount: 4 Least squares average step : 159.09593 cents, oct.: 1909.15117 cents Scale is strictly proper Step pattern alph. order: ABCDAEBADCAB Step pattern size order : DBACDEBDCADB Scale is a Constant Structure, by a margin of 95.51591 cents Scale diversity : 0.301019 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.686258 Rothenberg efficiency : 0.242845 redundancy: 0.757155 Efficiency x scale size : 2.914141 Number of different interval sizes : 72 = 6.54545 / class Number of one step interval sizes : 5 Highest interval variety : 8 Mean interval variety : 6.54545 = 72/11 Median interval variety : 5 Lowest interval variety : 5 Smallest interval difference : 3025/3024, 0.5724 cents Number of recognisable fifths : 0 Number of recognisable fourths : 4, average 498.3568 cents Number of appreciable fourths : 3, average 504.2606 cents Length of longest harmonic part : 2 at degree 0 Length of longest subharmonic part : 2 at degree 6 Formal octave complements present : 9 = 75.0000% 2/1 octave complements present : 0 = 0.0000% Limited transpositions with margin 49.4916 cents: 1 Limited inverse transpositions with margin 49.4916 cents: 1 10 11 -- Rational properties -- Prime limit : 23 Odd number limit : 529 (O: 529 U: 529) Highest odd numerator or denominator: 23 Scale harmonicity : 0.006267 Average absolute harmonicity : 0.082651 Specific harmonicity : 0.057646 Fundamental : 1/35420, -15.1123 octaves, 0.0074 Hz. Guide tone : 53130, 15.6972 octaves, 13900166.284 Hz. Exponens Consonantiae : 1.881865E+09, 30.80952 octaves Euler's gradus suavitatis : 90 Sum of Mann's harmonic distance : 126.5, average 10.54167 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 12.05733, average 1.00478 Sum of Tenney's harmonic distance : 21.70424, average 1.80869 Vogel's harmonic complexity : 50.83333 Wille's k value : 48702 Wilson's harmonic complexity : 95 Rectangular lattice diameter : 9 Triangular lattice diameter : 3 Lattice compactness : 177.64737, average 2.27753 Lattice compactness (without 2's) : 138.87231, average 1.78041 Number of different primes : 6 Prime exponents' range, average, count, tones@limit: 2: -2 .. 1 -0.41667 11 3: 0 .. 1 0.41667 5 1 5: -1 .. 1 -0.08333 7 2 7: -1 .. 1 0.08333 5 3 11: -1 .. 1 0.16667 4 4 23: -1 .. 1 0.00000 2 2 Average exponent except 2's : 7 / 12 = 0.58333 Average absolute exponent except 2's: 23 / 12 = 1.91667 Scale is JI-epimorphic with val: <8 12 18 22 27 35|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 11.7274 c. M: 17.8484 c. Twelve-tone Chromatic (1/11-comma) 37: 3 3 3 3 3 4 3 3 3 3 3 3 SP M ME SD: 6.8987 c. M: 11.4292 c. 49: 4 4 4 4 4 5 4 4 4 4 4 4 SP M ME SD: 5.3229 c. M: 9.7426 c. 61: 5 5 5 5 5 6 5 5 5 5 5 5 SP M ME SD: 5.4497 c. M: 9.1063 c. 71: 6 6 5 6 6 7 6 6 6 5 6 6 SP SD: 5.1408 c. M: 10.0915 c. 73: 6 6 6 6 6 7 6 6 6 6 6 6 SP M ME SD: 6.0118 c. M:-8.9808 c. 78: 7 6 6 6 7 8 6 7 6 6 7 6 SP SD: 4.5950 c. M: 7.8868 c. 83: 7 7 6 7 7 8 7 7 7 6 7 7 SP SD: 4.4945 c. M: 9.7688 c. 90: 8 7 7 7 8 9 7 8 7 7 8 7 SP SD: 3.2971 c. M: 4.9280 c. 102: 9 8 8 8 9 10 8 9 8 8 9 8 SP SD: 2.8961 c. M:-6.1711 c. 104: 9 8 8 9 9 10 8 9 9 8 9 8 SP SD: 3.5018 c. M: 4.7448 c. 114: 10 9 9 9 10 11 9 10 9 9 10 9 SP SD: 3.1147 c. M:-7.1525 c. 127: 11 10 10 10 11 13 10 11 10 10 11 10 SP SD: 3.0707 c. M:-4.6156 c. 138: 12 11 10 12 12 13 11 12 12 10 12 11 SP SD: 3.0118 c. M: 5.1790 c. 139: 12 11 11 11 12 14 11 12 11 11 12 11 SP SD: 2.6371 c. M: 5.2320 c. 151: 13 12 11 13 13 15 12 13 12 12 13 12 SP SD: 2.7162 c. M: 6.5481 c. 161: 14 13 12 13 14 16 13 14 13 12 14 13 SP SD: 2.3813 c. M: 3.3201 c. 163: 14 13 12 14 14 16 13 14 14 12 14 13 SP SD: 2.1588 c. M:-4.9254 c. 173: 15 14 13 14 15 17 14 15 14 13 15 14 SP SD: 2.1027 c. M: 3.2784 c. 175: 15 14 13 15 15 17 14 15 15 13 15 14 SP SD: 1.8696 c. M: 3.5252 c. 182: 16 14 14 15 16 18 14 16 15 14 16 14 SP SD: 1.9804 c. M:-4.3328 c. 187: 16 15 14 16 16 18 15 16 16 14 16 15 SP SD: 1.9272 c. M: 2.9360 c. 194: 17 15 15 16 17 19 15 17 16 15 17 15 SP SD: 1.6814 c. M:-3.5787 c. 206: 18 16 16 17 18 20 16 18 17 16 18 16 SP SD: 1.6990 c. M:-2.9124 c. 216: 19 17 16 18 19 21 17 19 18 16 19 17 SP SD: 1.4178 c. M:-2.2974 c. 228: 20 18 17 19 20 22 18 20 19 17 20 18 SP SD: 1.1878 c. M:-1.8339 c. 253: 22 20 19 21 22 25 20 22 21 19 22 20 SP SD: 0.8281 c. M: 1.4202 c. 265: 23 21 20 22 23 26 21 23 22 20 23 21 SP SD: 0.3046 c. M: 0.6543 c. 542: 47 43 41 45 47 53 43 47 45 41 47 43 SP SD: 0.1912 c. M: 0.3027 c. 1174: 102 93 89 97 102 115 93 102 97 89 102 93 SP SD: 0.1702 c. M: 0.2712 c. 1439: 125 114 109 119 125 141 114 125 119 109 125 114 SP SD: 0.1569 c. M: 0.2498 c. 1451: 126 115 110 120 126 142 115 126 120 110 126 115 SP SD: 0.1405 c. M: 0.2589 c. 1488: 129 118 113 123 129 146 118 129 123 113 129 118 SP SD: 0.1448 c. M: 0.2364 c. 1592: 138 126 121 132 138 156 126 138 132 121 138 126 SP SD: 0.1336 c. M:-0.2417 c. 1716: 149 136 130 142 149 168 136 149 142 130 149 136 SP SD: 0.1277 c. M: 0.2428 c. 1753: 152 139 133 145 152 172 139 152 145 133 152 139 SP SD: 0.1291 c. M: 0.2276 c. 1765: 153 140 134 146 153 173 140 153 146 134 153 140 SP SD: 0.1227 c. M: 0.2262 c. 1857: 161 147 141 154 161 182 147 161 154 141 161 147 SP SD: 0.1107 c. M: 0.1851 c. 1959: 170 155 149 162 170 192 155 170 162 149 170 155 SP SD: 0.1099 c. M:-0.2030 c. 2030: 176 161 154 168 176 199 161 176 168 154 176 161 SP SD: 0.1071 c. M: 0.2076 c. 2134: 185 169 162 177 185 209 169 185 177 162 185 169 SP SD: 0.1019 c. M: 0.1574 c. 2212: 192 175 168 183 192 217 175 192 183 168 192 175 SP SD: 0.1030 c. M:-0.2081 c. 2224: 193 176 169 184 193 218 176 193 184 169 193 176 SP SD: 0.0730 c. M:-0.1226 c.
FIT/HARMONIC
1 x x x x x x 2 x x x x 3 S SD: 40.2685 cents 2 x x x x 3 x 4 x x 5 x 6 S SD: 28.4741 cents 3 x x 4 x x 5:6 x 7 x 8:9 S SD: 22.6858 cents 4 x 5 x x 6:7:8 x 9:10:11:12 S SD: 17.4172 cents 5 x 6 x 7:8:9:10 x x 13:14:15 SD: 16.1268 cents 6 x 7:8 x 9 x 11:13:14:15:17:18 SD: 16.2715 cents 7:8 x 9:10:11:12:13:15:16:18:19:21 SD: 10.2790 cents 8:9:10 x 11:13:14:15:17:18:20:22:24 SD: 11.8864 cents 9:10:11:12:13:14:16:17:19:21:23:25:27 SD: 6.9581 cents 10:11:12:13:14:16:18:19:21:23:25:28:30 SD: 6.2637 cents 16:18:19:21:23:25:28:31:34:37:40:44:48 SD: 4.8118 cents 19:21:23:25:27:30:33:36:40:44:48:52:57 SD: 3.3076 cents 20:22:24:26:29:31:35:38:42:46:50:55:60 SD: 3.0404 cents 29:32:35:38:41:46:51:55:61:67:73:80:87 SD: 2.9101 cents 30:33:36:39:43:47:53:57:63:69:75:83:90 SD: 1.9387 cents 39:43:47:51:56:61:68:74:82:90:98:107:117 SD: 1.8996 cents 40:44:48:52:57:63:70:76:84:92:100:110:120 SD: 0.9716 cents 70:77:84:91:100:110:123:134:147:161:175:193:210 SD: 0.9333 cents 79:87:95:103:113:124:138:151:166:182:198:217:237 SD: 0.7126 cents 89:98:107:116:127:140:156:170:187:205:223:245:267 SD: 0.6252 cents 100:110:120:130:143:157:175:191:210:230:250:275:300 SD: 0.5242 cents 119:131:143:155:170:187:208:227:250:274:298:327:357 SD: 0.4861 cents 121:133:145:158:173:190:212:231:254:278:303:333:363 SD: 0.4769 cents 131:144:157:171:187:206:229:250:275:301:328:360:393 SD: 0.4329 cents 140:154:168:183:200:220:245:267:294:322:350:385:420 SD: 0.3421 cents 180:198:216:235:257:283:315:344:378:414:450:495:540 SD: 0.2299 cents 200:220:240:261:286:314:350:382:420:460:500:550:600 SD: 0.2189 cents 220:242:264:287:314:346:385:420:462:506:550:605:660 SD: 0.1786 cents 240:264:288:313:343:377:420:458:504:552:600:660:720 SD: 0.1014 cents 420:462:504:548:600:660:735:802:882:966:1050:1155:1260 SD: 0.0563 cents 460:506:552:600:657:723:805:878:966:1058:1150:1265:1380 SD: 0.0519 cents