Scala analysis: lumma
Carl Lumma, 7-limit, 6 tetrads + 4 triads within 2c of Just, TL 19-2-99
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 115.587 cents 115.587000 2: 200.054 cents 200.054200 3: 268.799 cents 268.798800 4: 384.386 cents 384.385800 5: 499.973 cents 499.972900 6: 584.440 cents 584.440100 7: 700.027 cents 700.027100 8: 815.614 cents 815.614200 9: 5/3 884.358713 major sixth, BP sixth 10: 7/4 968.825906 harmonic seventh 11: 1084.413 cents 1084.413000 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 115.5870 cents 115.5870 1: 115.5870 cents 115.5870 2: 84.4672 cents 84.4672 3: 68.7446 cents 68.7446 4: 115.5870 cents 115.5870 5: 115.5871 cents 115.5871 6: 84.4672 cents 84.4672 7: 115.5870 cents 115.5870 8: 115.5871 cents 115.5871 9: 68.7445 cents 68.7445 10: 21/20 84.4672 minor semitone 11: 115.5871 cents 115.5871 12: 115.5870 cents 115.5870
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 68.74460 cents 1: 3 84.46720 cents 1: 7 115.58700 cents 2: 2 153.21180 cents 2: 2 184.33160 cents 2: 4 200.05420 cents 2: 4 231.17410 cents 3: 4 268.79880 cents 3: 2 299.91870 cents 3: 5 315.64130 cents 3: 1 346.76109 cents 4: 8 384.38580 cents 4: 4 431.22830 cents 5: 2 468.85310 cents 5: 8 499.97290 cents 5: 1 515.69549 cents 5: 1 546.81540 cents 6: 6 584.44010 cents 6: 6 615.56000 cents 7: 1 653.18460 cents 7: 1 684.30451 cents 7: 8 700.02710 cents 7: 2 731.14690 cents 8: 4 768.77171 cents 8: 8 815.61420 cents 9: 1 853.23891 cents 9: 5 5/3 884.359 cents major sixth, BP sixth 9: 2 900.08130 cents 9: 4 931.20120 cents 10: 4 7/4 968.826 cents harmonic seventh 10: 4 999.94580 cents 10: 2 1015.66840 cents 10: 2 1046.78820 cents 11: 7 1084.41300 cents 11: 3 1115.53280 cents 11: 2 1131.25540 cents Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.27273 = 36/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 115.6 200.1 268.8 384.4 500.0 584.4 700.0 815.6 884.4 968.8 1084.4 1200.0 115.6 : 84.5 153.2 268.8 384.4 468.9 584.4 700.0 768.8 853.2 968.8 1084.4 1200.0 200.1 : 68.7 184.3 299.9 384.4 500.0 615.6 684.3 768.8 884.4 999.9 1115.5 1200.0 268.8 : 115.6 231.2 315.6 431.2 546.8 615.6 700.0 815.6 931.2 1046.8 1131.3 1200.0 384.4 : 115.6 200.1 315.6 431.2 500.0 584.4 700.0 815.6 931.2 1015.7 1084.4 1200.0 500.0 : 84.5 200.1 315.6 384.4 468.9 584.4 700.0 815.6 900.1 968.8 1084.4 1200.0 584.4 : 115.6 231.2 299.9 384.4 500.0 615.6 731.1 815.6 884.4 999.9 1115.5 1200.0 700.0 : 115.6 184.3 268.8 384.4 500.0 615.6 700.0 768.8 884.4 999.9 1084.4 1200.0 815.6 : 68.7 153.2 268.8 384.4 500.0 584.4 653.2 768.8 884.4 968.8 1084.4 1200.0 884.4 : 84.5 200.1 315.6 431.2 515.7 584.4 700.0 815.6 900.1 1015.7 1131.3 1200.0 968.8 : 115.6 231.2 346.8 431.2 500.0 615.6 731.1 815.6 931.2 1046.8 1115.5 1200.0 1084.4: 115.6 231.2 315.6 384.4 500.0 615.6 700.0 815.6 931.2 999.9 1084.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 68.7446 .. 115.5870 cents 46.84240 cents 68.74460 cents 2: 153.2118 .. 231.1741 cents 77.96230 cents 37.62480 cents 3: 268.7988 .. 346.7611 cents 77.96229 cents 37.62470 cents 4: 384.3858 .. 431.2283 cents 46.84250 cents 37.62471 cents 5: 468.8531 .. 546.8154 cents 77.96230 cents 37.62480 cents 6: 584.4401 .. 615.5600 cents 31.11990 cents 37.62470 cents 7: 653.1846 .. 731.1469 cents 77.96230 cents 37.62460 cents 8: 768.7717 .. 815.6142 cents 46.84249 cents 37.62481 cents 9: 853.2389 .. 931.2012 cents 77.96229 cents 37.62471 cents 10: 968.8259 .. 1046.7882 cents 77.96229 cents 37.62471 cents 11: 1084.4130 .. 1131.2554 cents 46.84240 cents 37.62480 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 68.74451 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 115.58700 cents Average / Smallest step : 1.454662 Largest / Average step : 1.155870 Largest / Smallest step : 1.681400 Median interval of one step : 115.58700 cents, amount: 7 Most common interval of one step : 115.58700 cents, amount: 7 Least squares average step : 98.87273 cents, oct.: 1186.47275 cents Scale is strictly proper Step pattern alph. order: ABCAABAACBAA Step pattern size order : LMSLLMLLSMLL Scale is a Constant Structure, by a margin of 37.62460 cents Scale diversity : 1.399206 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.428114 Rothenberg efficiency : 0.395893 redundancy: 0.604107 Efficiency x scale size : 4.750722 Number of different interval sizes : 36 = 3.27273 / class Number of one step interval sizes : 3 Highest interval variety : 4 Mean interval variety : 3.27273 = 36/11 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 15.72259 cents Number of recognisable fifths : 8, average 700.0271 cents Best fifths form a closed circle Formal octave complements present : 7 = 58.3333% Limited inverse transpositions : 11 Inversional symmetry on intervals : 5-6 11-12 -- Rational properties -- Prime limit : 7 (not all pitches rational)
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 16.8464 c. M:-31.2012 c. Twelve-tone Chromatic (1/11-comma) 19: 2 1 1 2 2 1 2 2 1 1 2 2 P D3 SD: 10.9382 c. M: 21.4575 c. 22: 2 2 1 2 2 2 2 2 1 2 2 2 SP B ME SD: 9.7985 c. M:-18.1276 c. Jubilee-12 31: 3 2 2 3 3 2 3 3 2 2 3 3 SP D3 SD: 3.3494 c. M: 6.5058 c. Fokker 12-tone 41: 4 3 2 4 4 3 4 4 2 3 4 4 SP SD: 3.5143 c. M: 6.3099 c. 53: 5 4 3 5 5 4 5 5 3 4 5 5 SP SD: 2.6368 c. M:-4.7590 c. 72: 7 5 4 7 7 5 7 7 4 5 7 7 SP SD: 1.1560 c. M: 2.1592 c. Smith's Prism 103: 10 7 6 10 10 7 10 10 6 7 10 10 SP SD: 1.1754 c. M: 1.9959 c. 125: 12 9 7 12 12 9 12 12 7 9 12 12 SP SD: 0.7889 c. M:-1.5458 c. 156: 15 11 9 15 15 11 15 15 9 11 15 15 SP SD: 0.2307 c. M:-0.4320 c. 228: 22 16 13 22 22 16 22 22 13 16 22 22 SP SD: 0.2100 c. M: 0.4049 c. 384: 37 27 22 37 37 27 37 37 22 27 37 37 SP SD: 0.0410 c. M: 0.0759 c. 924: 89 65 53 89 89 65 89 89 53 65 89 89 SP SD: 0.0304 c. M:-0.0569 c. 1308: 126 92 75 126 126 92 126 126 75 92 126 126 SP SD: 0.0274 c. M: 0.0542 c. 1973: 190 139 113 190 190 139 190 190 113 139 190 190 SP SD: 0.0311 c. M:-0.0540 c. 2045: 197 144 117 197 197 144 197 197 117 144 197 197 SP SD: 0.0306 c. M: 0.0555 c. 2076: 200 146 119 200 200 146 200 200 119 146 200 200 SP SD: 0.0295 c. M: 0.0542 c. 2201: 212 155 126 212 212 155 212 212 126 155 212 212 SP SD: 0.0188 c. M:-0.0367 c. 2357: 227 166 135 227 227 166 227 227 135 166 227 227 SP SD: 0.0194 c. M:-0.0328 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: 0.9640 cents 3 x x x x 4 x x x 5 x x 6 S SD: 0.6426 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 0.6816 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 13.5354 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 5.8681 cents 7 x 8 x 9 x 10 x 11 x 12:13:14 S SD: 12.0913 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 5.2365 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 9.9352 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 9.8991 cents 11:12 x 13:14:15 x 16 x 18:19:21:22 SD: 10.8273 cents 12:13 x 14:15:16:17:18:19:20:21:22:24 S SD: 4.5622 cents 13:14 x 15:16:17:18:19:21:22:23:24:26 SD: 7.3768 cents 14:15:16 x 17:19:20:21:22:23:25:26:28 SD: 8.2996 cents 15:16:17:18:19:20:21:22:24:25:26:28:30 S SD: 5.7563 cents 16:17:18:19:20:21:22:24:26:27:28:30:32 S SD: 5.2752 cents 17:18:19:20:21:23:24:25:27:28:30:32:34 SD: 4.9607 cents 18:19:20:21:22:24:25:27:29:30:32:34:36 SD: 4.9593 cents 23:25:26:27:29:31:32:34:37:38:40:43:46 SD: 4.3543 cents 24:26:27:28:30:32:34:36:38:40:42:45:48 S SD: 3.0273 cents 30:32:34:35:37:40:42:45:48:50:53:56:60 SD: 2.6982 cents 32:34:36:37:40:43:45:48:51:53:56:60:64 SD: 2.3793 cents 40:43:45:47:50:53:56:60:64:67:70:75:80 SD: 1.8316 cents 48:51:54:56:60:64:67:72:77:80:84:90:96 SD: 1.2205 cents 60:64:67:70:75:80:84:90:96:100:105:112:120 SD: 0.9622 cents 72:77:81:84:90:96:101:108:115:120:126:135:144 SD: 0.7054 cents 113:121:127:132:141:151:158:169:181:188:198:211:226 SD: 0.6966 cents 120:128:135:140:150:160:168:180:192:200:210:225:240 SD: 0.6816 cents 129:138:145:151:161:172:181:193:207:215:226:241:258 SD: 0.6182 cents 132:141:148:154:165:176:185:198:211:220:231:247:264 SD: 0.4832 cents 156:167:175:182:195:208:219:234:250:260:273:292:312 SD: 0.4652 cents 161:172:181:188:201:215:226:241:258:268:282:301:322 SD: 0.4415 cents 173:185:194:202:216:231:242:259:277:288:303:324:346 SD: 0.4330 cents 185:198:208:216:231:247:259:277:296:308:324:346:370 SD: 0.4314 cents 189:202:212:221:236:252:265:283:303:315:331:354:378 SD: 0.3654 cents 221:236:248:258:276:295:310:331:354:368:387:413:442 SD: 0.3169 cents 245:262:275:286:306:327:343:367:392:408:429:458:490 SD: 0.3103 cents 261:279:293:305:326:348:366:391:418:435:457:488:522 SD: 0.2338 cents 317:339:356:370:396:423:444:475:508:528:555:593:634 SD: 0.2225 cents 334:357:375:390:417:446:468:500:535:557:585:625:668 SD: 0.2185 cents 359:384:403:419:448:479:503:538:575:598:628:672:718 SD: 0.2153 cents 391:418:439:457:488:522:548:586:626:652:684:731:782 SD: 0.1995 cents 409:437:459:478:511:546:573:613:655:682:716:765:818 SD: 0.1967 cents 423:452:475:494:528:565:593:634:678:705:740:791:846 SD: 0.1941 cents 433:463:486:506:541:578:607:649:694:722:758:810:866 SD: 0.1847 cents 434:464:487:507:542:579:608:650:695:723:760:812:868 SD: 0.1817 cents 435:465:488:508:543:581:610:652:697:725:761:814:870 SD: 0.1751 cents 448:479:503:523:559:598:628:671:718:747:784:838:896 SD: 0.1744 cents 451:482:506:527:563:602:632:676:722:752:789:844:902 SD: 0.1741 cents 463:495:520:541:578:618:649:694:742:772:810:866:926 SD: 0.1630 cents 475:508:533:555:593:634:666:712:761:792:831:889:950 SD: 0.1613 cents 495:529:556:578:618:661:694:742:793:825:866:926:990 SD: 0.1570 cents 507:542:569:592:633:677:711:760:812:845:887:949:1014 SD: 0.1545 cents 520:556:584:607:649:694:729:779:833:867:910:973:1040 SD: 0.1514 cents 523:559:587:611:653:698:733:784:838:872:915:978:1046 SD: 0.1366 cents 540:577:606:631:674:721:757:809:865:900:945:1010:1080 SD: 0.1336 cents 580:620:651:677:724:774:813:869:929:967:1015:1085:1160 SD: 0.1183 cents 588:629:660:687:734:785:824:881:942:980:1029:1100:1176 SD: 0.1175 cents 595:636:668:695:743:794:834:892:953:992:1041:1113:1190 SD: 0.1157 cents