Scala analysis: stanhope
Well temperament of Charles, third earl of Stanhope (1806)
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 256/243 90.224996 limma, Pythagorean minor second 2: 196.090 cents 196.090000 3: 32/27 294.134997 Pythagorean minor third 4: 8192/6561 384.359993 Pythagorean diminished fourth 5: 4/3 498.044999 perfect fourth 6: 1024/729 588.269995 Pythagorean diminished fifth 7: 3/2 701.955001 perfect fifth 8: 128/81 792.179997 Pythagorean minor sixth 9: 890.225 cents 890.225000 10: 16/9 996.089998 Pythagorean minor seventh 11: 4096/2187 1086.314994 Pythagorean diminished octave 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 2187/2048 113.6850 apotome 1: 256/243 90.2250 limma, Pythagorean minor second 2: 105.8650 cents 105.8650 3: 98.0450 cents 98.0450 4: 256/243 90.2250 limma, Pythagorean minor second 5: 2187/2048 113.6850 apotome 6: 256/243 90.2250 limma, Pythagorean minor second 7: 2187/2048 113.6850 apotome 8: 256/243 90.2250 limma, Pythagorean minor second 9: 98.0450 cents 98.0450 10: 105.8650 cents 105.8650 11: 256/243 90.2250 limma, Pythagorean minor second 12: 2187/2048 113.6850 apotome
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 5 256/243 90.225 cents limma, Pythagorean minor second 1: 2 98.04500 cents 1: 2 105.86500 cents 1: 3 2187/2048 113.685 cents apotome 2: 2 188.26999 cents 2: 2 196.09000 cents 2: 8 9/8 203.910 cents major whole tone 3: 7 32/27 294.135 cents Pythagorean minor third 3: 2 301.95500 cents 3: 2 309.77500 cents 3: 1 19683/16384 317.595 cents Pythagorean augmented second 4: 2 8192/6561 384.360 cents Pythagorean diminished fourth 4: 2 392.17999 cents 4: 2 400.00000 cents 4: 6 81/64 407.820 cents Pythagorean major third 5: 9 4/3 498.045 cents perfect fourth 5: 3 505.86500 cents 6: 4 1024/729 588.270 cents Pythagorean diminished fifth 6: 2 596.09000 cents 6: 2 603.91000 cents 6: 4 729/512 611.730 cents Pythagorean tritone 7: 3 694.13500 cents 7: 9 3/2 701.955 cents perfect fifth 8: 6 128/81 792.180 cents Pythagorean minor sixth 8: 2 800.00000 cents 8: 2 807.82001 cents 8: 2 6561/4096 815.640 cents Pythagorean augmented fifth 9: 1 32768/19683 882.405 cents Pythagorean diminished seventh 9: 2 890.22500 cents 9: 2 898.04500 cents 9: 7 27/16 905.865 cents Pythagorean major sixth 10: 8 16/9 996.090 cents Pythagorean minor seventh 10: 2 1003.91000 cents 10: 2 1011.73001 cents 11: 3 4096/2187 1086.315 cents Pythagorean diminished octave 11: 2 1094.13500 cents 11: 2 1101.95500 cents 11: 5 243/128 1109.775 cents Pythagorean major seventh Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.45455 = 38/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 90.2 196.1 294.1 384.4 498.0 588.3 702.0 792.2 890.2 996.1 1086.3 1200.0 90.2 : 105.9 203.9 294.1 407.8 498.0 611.7 702.0 800.0 905.9 996.1 1109.8 1200.0 196.1 : 98.0 188.3 302.0 392.2 505.9 596.1 694.1 800.0 890.2 1003.9 1094.1 1200.0 294.1 : 90.2 203.9 294.1 407.8 498.0 596.1 702.0 792.2 905.9 996.1 1102.0 1200.0 384.4 : 113.7 203.9 317.6 407.8 505.9 611.7 702.0 815.6 905.9 1011.7 1109.8 1200.0 498.0 : 90.2 203.9 294.1 392.2 498.0 588.3 702.0 792.2 898.0 996.1 1086.3 1200.0 588.3 : 113.7 203.9 302.0 407.8 498.0 611.7 702.0 807.8 905.9 996.1 1109.8 1200.0 702.0 : 90.2 188.3 294.1 384.4 498.0 588.3 694.1 792.2 882.4 996.1 1086.3 1200.0 792.2 : 98.0 203.9 294.1 407.8 498.0 603.9 702.0 792.2 905.9 996.1 1109.8 1200.0 890.2 : 105.9 196.1 309.8 400.0 505.9 603.9 694.1 807.8 898.0 1011.7 1102.0 1200.0 996.1 : 90.2 203.9 294.1 400.0 498.0 588.3 702.0 792.2 905.9 996.1 1094.1 1200.0 1086.3: 113.7 203.9 309.8 407.8 498.0 611.7 702.0 815.6 905.9 1003.9 1109.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 113.6850 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 2: 188.2700 .. 203.9100 cents 15.64001 cents 74.58499 cents 3: 294.1350 .. 317.5950 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 4: 384.3600 .. 407.8200 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents 5: 498.0450 .. 505.8650 cents 7.82000 cents 256/243, 90.2250 cents 6: 588.2700 .. 611.7300 cents 531441/524288, 23.4600 cents 82.40499 cents 7: 694.1350 .. 701.9550 cents 7.82000 cents 82.40499 cents 8: 792.1800 .. 815.6400 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 9: 882.4050 .. 905.8650 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents 10: 996.0900 .. 1011.7300 cents 15.64001 cents 256/243, 90.2250 cents 11: 1086.3150 .. 1109.7750 cents 531441/524288, 23.4600 cents 74.58499 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 256/243, 90.2250 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 2187/2048, 113.6850 cents Average / Smallest step : 1.108340 Largest / Average step : 1.136850 Largest / Smallest step : 1.260017 Median interval of one step : 98.04500 cents, amount: 2 Most common interval of one step : 256/243, 90.2250 cents, amount: 5 Least squares average step : 99.22402 cents, oct.: 1190.68818 cents Scale is strictly proper Scale is a mode of a 1995-tone equal temperament with octave 2/1 degrees: 150 326 489 639 828 978 1167 1317 1480 1656 1806 1995 Least number of segments generator : 998 of 600.301 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 1682 of 1011.729 cents and inv. generated superset size: 1000 = 988 more = 8333.333% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABCADADACBAD Step pattern size order : ACBADADABCAD Scale is a Constant Structure, by a margin of 66.76499 cents Scale diversity : 1.142979 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.824050 Rothenberg efficiency : 0.366162 redundancy: 0.633838 Efficiency x scale size : 4.393939 Number of different interval sizes : 38 = 3.45455 / class Number of one step interval sizes : 4 Highest interval variety : 4 Mean interval variety : 3.45455 = 38/11 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 7.82000 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 9 Most common triad is 0.0 498.045 701.955 cents, amount: 8 Number of recognisable fifths : 12, average 700.0000 cents Best fifths form a closed circle Scale contains a complete diamond : 1 3 Formal octave complements present : 3 = 25.0000% Limited transpositions with margin 23.4600 cents: 2 5 7 9 10 Limited inverse transpositions : 11 Inversional symmetry on intervals : 5-6 11-12 -- Rational properties -- Prime limit : 3 (not all pitches rational)
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 8.6330 c. M:-15.6400 c. Twelve-tone Chromatic (1/11-comma) 41: 3 4 3 3 4 3 4 3 3 4 3 4 SP M ME SD: 4.7742 c. M: 12.1762 c. Schismic-12 45: 3 4 4 3 5 3 4 4 3 4 4 4 SP D3 SD: 8.0134 c. M: 11.0267 c. 49: 4 4 4 4 4 4 5 3 4 5 3 5 SP D3 SD: 6.7022 c. M: 8.7640 c. 50: 4 4 4 4 5 4 4 4 4 5 3 5 SP D3 SD: 6.3448 c. M:-11.9100 c. 51: 4 4 4 4 5 4 5 4 4 4 4 5 SP D3 SD: 6.2079 c. M: 11.7821 c. 53: 4 5 4 4 5 4 5 4 4 5 4 5 SP M ME SD: 3.0540 c. M:-7.6836 c. Schismic-12 65: 5 6 5 5 6 5 6 5 5 6 5 6 SP M ME SD: 2.9006 c. M:-6.9869 c. 77: 6 7 6 6 7 6 7 6 6 7 6 7 SP M ME SD: 3.3455 c. M:-6.5074 c. 82: 6 7 7 6 8 6 8 6 7 7 6 8 SP D3 SD: 2.7118 c. M: 5.8461 c. 93: 7 8 8 7 9 7 8 7 8 8 7 9 SP SD: 3.4248 c. M:-5.2784 c. 94: 7 8 8 7 9 7 9 7 8 8 7 9 SP D3 SD: 1.7978 c. M: 4.6006 c. 102: 8 9 8 8 9 8 10 7 9 9 7 10 SP S SD: 3.3913 c. M: 3.9621 c. 103: 8 9 8 8 10 7 10 8 8 9 8 10 SP SD: 3.3622 c. M: 5.7987 c. 104: 8 9 8 8 10 8 10 8 8 9 8 10 SP SD: 2.7795 c. M: 5.6735 c. 106: 8 9 9 8 10 8 10 8 9 9 8 10 SP D3 SD: 1.6102 c. M:-4.1146 c. 123: 9 11 10 9 12 9 12 9 10 11 9 12 SP SD: 2.1374 c. M: 3.8722 c. 135: 10 12 11 10 13 10 13 10 11 12 10 13 SP SD: 1.1800 c. M: 2.1378 c. 147: 11 13 12 11 14 11 14 11 12 13 11 14 SP SD: 0.3790 c. M: 0.6865 c. 159: 12 14 13 12 15 12 15 12 13 14 12 15 SP SD: 0.3012 c. M:-0.5457 c. 306: 23 27 25 23 29 23 29 23 25 27 23 29 SP SD: 0.0255 c. M: 0.0463 c. 1077: 81 95 88 81 102 81 102 81 88 95 81 102 SP SD: 0.0227 c. M:-0.0411 c. 1383: 104 122 113 104 131 104 131 104 113 122 104 131 SP SD: 0.0120 c. M:-0.0218 c. 1689: 127 149 138 127 160 127 160 127 138 149 127 160 SP SD: 0.0052 c. M:-0.0095 c. 1995: 150 176 163 150 189 150 189 150 163 176 150 189 SP SD: 0.0005 c. M:-0.0009 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: 0.0000 cents 3 x x x x 4 x x x 5 x x 6 S SD: 1.9554 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 6.8335 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.5385 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 7.7158 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 11.7852 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.3993 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.7659 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.5729 cents 11 x 12:13:14:15 x x 17:18:20:21:22 SD: 11.2310 cents 12:13 x 14:15:16:17:18:19:20:21:22:24 S SD: 6.7151 cents 13:14 x 15:16:17:18:20:21:22:23:24:26 S SD: 9.2948 cents 14:15:16:17 x 19:20:21:22:23:25:26:28 SD: 7.5814 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.5541 cents 16:17:18:19:20:21:22:24:25:27:28:30:32 SD: 5.1209 cents 17:18:19:20:21:23:24:26:27:28:30:32:34 SD: 4.8953 cents 22:23:25:26:27:29:31:33:35:37:39:41:44 SD: 4.1463 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.3734 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.5760 cents 33:35:37:39:41:44:46:50:52:55:59:62:66 SD: 2.5007 cents 36:38:40:43:45:48:51:54:57:60:64:67:72 SD: 2.3065 cents 40:42:45:47:50:53:56:60:63:67:71:75:80 SD: 1.9082 cents 48:51:54:57:60:64:67:72:76:80:85:90:96 SD: 1.8785 cents 52:55:58:62:65:69:73:78:82:87:92:97:104 SD: 1.6888 cents 57:60:64:68:71:76:80:86:90:95:101:107:114 SD: 1.5733 cents 60:63:67:71:75:80:84:90:95:100:107:112:120 SD: 1.2108 cents 72:76:81:85:90:96:101:108:114:120:128:135:144 SD: 1.1098 cents 76:80:85:90:95:101:107:114:120:127:135:142:152 SD: 0.7662 cents 108:114:121:128:135:144:152:162:171:181:192:202:216 SD: 0.6336 cents 126:133:141:149:157:168:177:189:199:211:224:236:252 SD: 0.5736 cents 134:141:150:159:167:179:188:201:212:224:238:251:268 SD: 0.5358 cents 150:158:168:178:187:200:211:225:237:251:267:281:300 SD: 0.4081 cents 190:200:213:225:237:253:267:285:300:318:338:356:380 SD: 0.3867 cents 210:221:235:249:262:280:295:315:332:351:373:393:420 SD: 0.3042 cents 242:255:271:287:302:323:340:363:382:405:430:453:484 SD: 0.2937 cents 260:274:291:308:325:347:365:390:411:435:462:487:520 SD: 0.2820 cents 274:289:307:325:342:365:385:411:433:458:487:513:548 SD: 0.2765 cents 282:297:316:334:352:376:396:423:446:472:501:528:564 SD: 0.2488 cents 302:318:338:358:377:403:424:453:477:505:537:566:604 SD: 0.2250 cents 334:352:374:396:417:445:469:501:528:559:594:626:668 SD: 0.2246 cents 351:370:393:416:438:468:493:527:555:587:624:657:702 SD: 0.2219 cents 374:394:419:443:467:499:525:561:591:625:665:700:748 SD: 0.2197 cents 378:398:423:448:472:504:531:567:597:632:672:708:756 SD: 0.1649 cents 410:432:459:486:512:547:576:615:648:686:729:768:820 SD: 0.1391 cents 486:512:544:576:607:648:683:729:768:813:864:910:972 SD: 0.1259 cents 558:588:625:661:697:744:784:837:882:933:992:1045:1116 SD: 0.1174 cents 576:607:645:683:719:768:809:864:910:963:1024:1079:1152 SD: 0.1116 cents 600:632:672:711:749:800:843:900:948:1003:1067:1124:1200 SD: 0.1006 cents