Scala analysis: marpurg
Marpurg, Versuch ueber die musikalische Temperatur (1776), p. 153
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 101.955 cents 101.955000 2: 200.978 cents 200.978000 3: 300.000 cents 300.000000 4: 401.955 cents 401.955000 5: 500.978 cents 500.978000 6: 600.000 cents 600.000000 7: 3/2 701.955001 perfect fifth 8: 800.978 cents 800.978000 9: 900.000 cents 900.000000 10: 1001.955 cents 1001.955000 11: 1100.978 cents 1100.978000 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 99.0220 cents 99.0220 1: 101.9550 cents 101.9550 2: 99.0230 cents 99.0230 3: 99.0220 cents 99.0220 4: 101.9550 cents 101.9550 5: 99.0230 cents 99.0230 6: 99.0220 cents 99.0220 7: 101.9550 cents 101.9550 8: 99.0230 cents 99.0230 9: 99.0220 cents 99.0220 10: 101.9550 cents 101.9550 11: 99.0230 cents 99.0230 12: 99.0220 cents 99.0220
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 4 99.02200 cents 1: 4 99.02300 cents 1: 4 101.95500 cents 2: 4 198.04500 cents 2: 4 200.97700 cents 2: 4 200.97800 cents 3: 12 300.00000 cents 4: 4 399.02200 cents 4: 4 399.02300 cents 4: 4 401.95500 cents 5: 4 498.04500 cents 5: 4 500.97700 cents 5: 4 500.97800 cents 6: 12 600.00000 cents 7: 4 699.02200 cents 7: 4 699.02300 cents 7: 4 3/2 701.955 cents perfect fifth 8: 4 798.04500 cents 8: 4 800.97700 cents 8: 4 800.97800 cents 9: 12 900.00000 cents 10: 4 999.02200 cents 10: 4 999.02300 cents 10: 4 1001.95500 cents 11: 4 1098.04500 cents 11: 4 1100.97700 cents 11: 4 1100.97800 cents Highest number of different intervals for one interval class: 3 Average number of different intervals per interval class: 2.45455 = 27/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 102.0 201.0 300.0 402.0 501.0 600.0 702.0 801.0 900.0 1002.0 1101.0 1200.0 102.0 : 99.0 198.0 300.0 399.0 498.0 600.0 699.0 798.0 900.0 999.0 1098.0 1200.0 201.0 : 99.0 201.0 300.0 399.0 501.0 600.0 699.0 801.0 900.0 999.0 1101.0 1200.0 300.0 : 102.0 201.0 300.0 402.0 501.0 600.0 702.0 801.0 900.0 1002.0 1101.0 1200.0 402.0 : 99.0 198.0 300.0 399.0 498.0 600.0 699.0 798.0 900.0 999.0 1098.0 1200.0 501.0 : 99.0 201.0 300.0 399.0 501.0 600.0 699.0 801.0 900.0 999.0 1101.0 1200.0 600.0 : 102.0 201.0 300.0 402.0 501.0 600.0 702.0 801.0 900.0 1002.0 1101.0 1200.0 702.0 : 99.0 198.0 300.0 399.0 498.0 600.0 699.0 798.0 900.0 999.0 1098.0 1200.0 801.0 : 99.0 201.0 300.0 399.0 501.0 600.0 699.0 801.0 900.0 999.0 1101.0 1200.0 900.0 : 102.0 201.0 300.0 402.0 501.0 600.0 702.0 801.0 900.0 1002.0 1101.0 1200.0 1002.0: 99.0 198.0 300.0 399.0 498.0 600.0 699.0 798.0 900.0 999.0 1098.0 1200.0 1101.0: 99.0 201.0 300.0 399.0 501.0 600.0 699.0 801.0 900.0 999.0 1101.0 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 99.0220 .. 101.9550 cents 2.93300 cents 99.02200 cents 2: 198.0450 .. 200.9780 cents 2.93300 cents 96.09000 cents 3: 300.0000 .. 300.0000 cents 1/1, 0.0000 cents 99.02200 cents 4: 399.0220 .. 401.9550 cents 2.93300 cents 99.02200 cents 5: 498.0450 .. 500.9780 cents 2.93300 cents 96.09000 cents 6: 600.0000 .. 600.0000 cents 1/1, 0.0000 cents 99.02200 cents 7: 699.0220 .. 701.9550 cents 2.93300 cents 99.02200 cents 8: 798.0450 .. 800.9780 cents 2.93300 cents 96.09000 cents 9: 900.0000 .. 900.0000 cents 1/1, 0.0000 cents 99.02200 cents 10: 999.0220 .. 1001.9550 cents 2.93300 cents 99.02200 cents 11: 1098.0450 .. 1100.9780 cents 2.93300 cents 96.09000 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 99.02200 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 101.95500 cents Average / Smallest step : 1.009877 Largest / Average step : 1.019550 Largest / Smallest step : 1.029620 Median interval of one step : 99.02300 cents, amount: 4 Least squares average step : 100.10529 cents, oct.: 1201.26347 cents Scale is strictly proper Scale has 4 repeating blocks of 3 notes Scale is 3-distributional even Scale is a mode of a 1224-tone equal temperament with octave 2/1 degrees: 104 205 306 410 511 612 716 817 918 1022 1123 1224 Least number of segments generator : 613 of 600.980 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 713 of 699.020 cents and inv. generated superset size: 921 = 909 more = 7675.000% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABCABCABCABC Step pattern size order : LMSLMSLMSLMS Scale is a Constant Structure, by a margin of 96.09000 cents Scale diversity : 1.015574 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.980447 Rothenberg efficiency (ambiguous key): 0.194444 redundancy: 0.805556 Efficiency x scale size : 2.333333 Number of different interval sizes : 27 = 2.45455 / class Number of one step interval sizes : 3 Highest interval variety : 3 Mean interval variety : 2.45455 = 27/11 Median interval variety : 3 Lowest interval variety : 1 Smallest interval difference : 0.00100 cents Most common triad is 0.0 300.000 600.000 cents, amount: 12 Number of recognisable fifths : 12, average 700.0000 cents Scale contains two identical disjunct hexachords Scale is a well-temperament. It has 3 different sizes of fifth Best fifths form a closed circle Formal octave complements present : 4 = 33.3333% Limited transpositions : 3 6 9 Limited inverse transpositions : 1 4 7 10 Inversional symmetry on degrees : 2 5 8 11 Inversional symmetry on intervals : 0-1 3-4 6-7 9-10 -- 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: 1.2621 c. M: 1.9550 c. Twelve-tone Chromatic (1/11-comma) 208: 18 17 17 18 17 17 18 17 17 18 17 17 SP B ME S SD: 1.2206 c. M:-1.8912 c. 220: 19 18 18 19 18 18 19 18 18 19 18 18 SP B ME S SD: 1.0852 c. M:-1.6814 c. 232: 20 19 19 20 19 19 20 19 19 20 19 19 SP B ME S SD: 0.9638 c. M:-1.4933 c. 244: 21 20 20 21 20 20 21 20 20 21 20 20 SP B ME S SD: 0.8543 c. M:-1.3237 c. 256: 22 21 21 22 21 21 22 21 21 22 21 21 SP B ME S SD: 0.7551 c. M:-1.1700 c. 268: 23 22 22 23 22 22 23 22 22 23 22 22 SP B ME S SD: 0.6648 c. M:-1.0301 c. 280: 24 23 23 24 23 23 24 23 23 24 23 23 SP B ME S SD: 0.5822 c. M:-0.9021 c. 292: 25 24 24 25 24 24 25 24 24 25 24 24 SP B ME S SD: 0.5064 c. M:-0.7847 c. 304: 26 25 25 26 25 25 26 25 25 26 25 25 SP B ME S SD: 0.4366 c. M:-0.6766 c. 316: 27 26 26 27 26 26 27 26 26 27 26 26 SP B ME S SD: 0.3721 c. M:-0.5766 c. 328: 28 27 27 28 27 27 28 27 27 28 27 27 SP B ME S SD: 0.3123 c. M:-0.4840 c. 340: 29 28 28 29 28 28 29 28 28 29 28 28 SP B ME S SD: 0.2567 c. M:-0.3979 c. 352: 30 29 29 30 29 29 30 29 29 30 29 29 SP B ME S SD: 0.2050 c. M:-0.3177 c. 364: 31 30 30 31 30 30 31 30 30 31 30 30 SP B ME S SD: 0.1566 c. M:-0.2428 c. 376: 32 31 31 32 31 31 32 31 31 32 31 31 SP B ME S SD: 0.1113 c. M:-0.1727 c. 388: 33 32 32 33 32 32 33 32 32 33 32 32 SP B ME S SD: 0.0688 c. M:-0.1069 c. 400: 34 33 33 34 33 33 34 33 33 34 33 33 SP B ME S SD: 0.0289 c. M:-0.0450 c. 412: 35 34 34 35 34 34 35 34 34 35 34 34 SP B ME S SD: 0.0087 c. M: 0.0133 c. 1224: 104 101 101 104 101 101 104 101 101 104 101 101 SP B DE S SD: 0.0036 c. M:-0.0058 c. 1636: 139 135 135 139 135 135 139 135 135 139 135 135 SP B DE S SD: 0.0006 c. M:-0.0010 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: 5.3046 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 9.1590 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 6.1859 cents 6 x x 7 x 8 x 9 x 10:11 x 12 S SD: 9.9976 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 9.7012 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 9.2250 cents 9 x 10:11 x 12:13 x 14:15:16:17:18 S SD: 8.2563 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 6.9926 cents 11:12 x 13:14:15:16 x 17:19:20:21:22 SD: 11.0457 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.2419 cents 13:14:15 x 16:17:18:20:21:22:23:25:26 SD: 9.7181 cents 14:15:16:17:18:19:20:21:22:24:25:26:28 S SD: 6.9691 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 4.9090 cents 22:23:25:26:28:29:31:33:35:37:39:42:44 SD: 4.0981 cents 26:28:29:31:33:35:37:39:41:44:46:49:52 SD: 3.4999 cents 27:29:30:32:34:36:38:41:43:45:48:51:54 SD: 3.4273 cents 31:33:35:37:39:41:44:47:49:52:55:59:62 SD: 2.8909 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.2242 cents 42:45:47:50:53:56:59:63:67:71:75:79:84 SD: 2.1800 cents 46:49:52:55:58:61:65:69:73:77:82:87:92 SD: 1.8634 cents 48:51:54:57:61:64:68:72:76:81:86:91:96 SD: 1.6086 cents 54:57:61:64:68:72:76:81:86:91:96:102:108 SD: 1.5811 cents 63:67:71:75:79:84:89:95:100:106:112:119:126 SD: 1.4320 cents 65:69:73:77:82:87:92:98:103:109:116:123:130 SD: 1.1590 cents 80:85:90:95:101:107:113:120:127:135:143:151:160 SD: 0.7737 cents 114:121:128:136:144:152:161:171:181:192:203:215:228 SD: 0.7203 cents 116:123:130:138:146:155:164:174:184:195:207:219:232 SD: 0.5019 cents 164:174:184:195:207:219:232:246:260:276:293:310:328 SD: 0.4226 cents 179:190:201:213:226:239:253:269:284:301:319:338:358 SD: 0.4066 cents 180:191:202:214:227:240:255:270:286:303:321:340:360 SD: 0.4033 cents 196:208:220:233:247:262:277:294:311:330:350:370:392 SD: 0.3654 cents 218:231:245:259:275:291:308:327:346:367:389:412:436 SD: 0.3391 cents 226:240:254:269:285:302:320:339:359:380:403:427:452 SD: 0.3170 cents 244:259:274:290:308:326:345:366:388:410:435:461:488 SD: 0.2961 cents 260:276:292:309:328:347:368:390:413:437:464:491:520 SD: 0.2476 cents 268:284:301:319:338:358:379:402:426:451:478:506:536 SD: 0.2452 cents 318:337:357:378:401:425:450:477:505:535:567:601:636 SD: 0.2370 cents 333:353:374:396:420:445:471:500:529:560:594:629:666 SD: 0.1878 cents 348:369:391:414:439:465:492:522:553:585:621:657:696 SD: 0.1780 cents 406:431:456:483:512:542:574:609:645:683:724:767:812 SD: 0.1768 cents 430:456:483:511:542:574:608:645:683:723:767:812:860 SD: 0.1747 cents 432:458:485:514:545:577:611:648:686:727:771:816:864 SD: 0.1681 cents 440:467:494:523:555:588:622:660:699:740:785:831:880 SD: 0.1680 cents 444:471:499:528:560:593:628:666:705:747:792:839:888 SD: 0.1401 cents 512:543:575:609:646:684:724:768:813:861:913:967:1024 SD: 0.0932 cents