Scala analysis: Bendeler
Bendeler, as quoted by Toepfer, compiled by A.Sparschuh
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: 32768/29403 187.589111 3: 32/27 294.134997 Pythagorean minor third 4: 4096/3267 391.499113 5: 4/3 498.044999 perfect fourth 6: 1024/729 588.269995 Pythagorean diminished fifth 7: 65536/43923 692.773230 8: 128/81 792.179997 Pythagorean minor sixth 9: 16384/9801 889.544112 10: 16/9 996.089998 Pythagorean minor seventh 11: 2048/1089 1093.454114 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 1089/1024 106.5459 two (undecimal comma, al-Farabi's 1/4-tone) 1: 256/243 90.2250 limma, Pythagorean minor second 2: 128/121 97.3641 undecimal semitone 3: 1089/1024 106.5459 two (undecimal comma, al-Farabi's 1/4-tone) 4: 128/121 97.3641 undecimal semitone 5: 1089/1024 106.5459 two (undecimal comma, al-Farabi's 1/4-tone) 6: 256/243 90.2250 limma, Pythagorean minor second 7: 15552/14641 104.5032 8: 14641/13824 99.4068 9: 128/121 97.3641 undecimal semitone 10: 1089/1024 106.5459 two (undecimal comma, al-Farabi's 1/4-tone) 11: 128/121 97.3641 undecimal semitone 12: 1089/1024 106.5459 two (undecimal comma, al-Farabi's 1/4-tone)
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 256/243 90.225 cents limma, Pythagorean minor second 1: 4 128/121 97.364 cents undecimal semitone 1: 1 14641/13824 99.407 cents 1: 1 15552/14641 104.503 cents 1: 4 1089/1024 106.546 cents two (undecimal comma, al-Farabi's 1/4-tone) 2: 1 32768/29403 187.589 cents 2: 1 16384/14641 194.728 cents two (undecimal semitone) 2: 3 121/108 196.771 cents 2: 7 9/8 203.910 cents major whole tone 3: 5 32/27 294.135 cents Pythagorean minor third 3: 4 144/121 301.274 cents two (undecimal neutral second, 3/4-tone) 3: 1 14641/12288 303.317 cents 3: 2 9801/8192 310.456 cents 4: 3 4096/3267 391.499 cents 4: 1 18432/14641 398.638 cents 4: 5 121/96 400.681 cents 4: 3 81/64 407.820 cents Pythagorean major third 5: 9 4/3 498.045 cents perfect fourth 5: 2 162/121 505.184 cents 5: 1 43923/32768 507.227 cents 6: 1 1024/729 588.270 cents Pythagorean diminished fifth 6: 4 512/363 595.409 cents 6: 1 14641/10368 597.452 cents 6: 1 20736/14641 602.548 cents 6: 4 363/256 604.591 cents 6: 1 729/512 611.730 cents Pythagorean tritone 7: 1 65536/43923 692.773 cents 7: 2 121/81 694.816 cents two (undecimal neutral third) 7: 9 3/2 701.955 cents perfect fifth 8: 3 128/81 792.180 cents Pythagorean minor sixth 8: 5 192/121 799.319 cents 8: 1 14641/9216 801.362 cents 8: 3 3267/2048 808.501 cents 9: 2 16384/9801 889.544 cents 9: 1 24576/14641 896.683 cents 9: 4 121/72 898.726 cents 9: 5 27/16 905.865 cents Pythagorean major sixth 10: 7 16/9 996.090 cents Pythagorean minor seventh 10: 3 216/121 1003.229 cents 10: 1 14641/8192 1005.272 cents 10: 1 29403/16384 1012.411 cents 11: 4 2048/1089 1093.454 cents 11: 1 14641/7776 1095.497 cents 11: 1 27648/14641 1100.593 cents 11: 4 121/64 1102.636 cents two (undecimal semi-augmented fourth) 11: 2 243/128 1109.775 cents Pythagorean major seventh Highest number of different intervals for one interval class: 6 Average number of different intervals per interval class: 4.18182 = 46/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 90.2 187.6 294.1 391.5 498.0 588.3 692.8 792.2 889.5 996.1 1093.5 1200.0 90.2 : 97.4 203.9 301.3 407.8 498.0 602.5 702.0 799.3 905.9 1003.2 1109.8 1200.0 187.6 : 106.5 203.9 310.5 400.7 505.2 604.6 702.0 808.5 905.9 1012.4 1102.6 1200.0 294.1 : 97.4 203.9 294.1 398.6 498.0 595.4 702.0 799.3 905.9 996.1 1093.5 1200.0 391.5 : 106.5 196.8 301.3 400.7 498.0 604.6 702.0 808.5 898.7 996.1 1102.6 1200.0 498.0 : 90.2 194.7 294.1 391.5 498.0 595.4 702.0 792.2 889.5 996.1 1093.5 1200.0 588.3 : 104.5 203.9 301.3 407.8 505.2 611.7 702.0 799.3 905.9 1003.2 1109.8 1200.0 692.8 : 99.4 196.8 303.3 400.7 507.2 597.5 694.8 801.4 898.7 1005.3 1095.5 1200.0 792.2 : 97.4 203.9 301.3 407.8 498.0 595.4 702.0 799.3 905.9 996.1 1100.6 1200.0 889.5 : 106.5 203.9 310.5 400.7 498.0 604.6 702.0 808.5 898.7 1003.2 1102.6 1200.0 996.1 : 97.4 203.9 294.1 391.5 498.0 595.4 702.0 792.2 896.7 996.1 1093.5 1200.0 1093.5: 106.5 196.8 294.1 400.7 498.0 604.6 694.8 799.3 898.7 996.1 1102.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 106.5459 cents 264627/262144, 16.3209 cents 256/243, 90.2250 cents 2: 187.5891 .. 203.9100 cents 264627/262144, 16.3209 cents 33554432/32019867, 81.0432 cents 3: 294.1350 .. 310.4559 cents 264627/262144, 16.3209 cents 256/243, 90.2250 cents 4: 391.4991 .. 407.8200 cents 264627/262144, 16.3209 cents 33554432/32019867, 81.0432 cents 5: 498.0450 .. 507.2268 cents 131769/131072, 9.1818 cents 256/243, 90.2250 cents 6: 588.2700 .. 611.7300 cents 531441/524288, 23.4600 cents 33554432/32019867, 81.0432 cents 7: 692.7732 .. 701.9550 cents 131769/131072, 9.1818 cents 33554432/32019867, 81.0432 cents 8: 792.1800 .. 808.5009 cents 264627/262144, 16.3209 cents 256/243, 90.2250 cents 9: 889.5441 .. 905.8650 cents 264627/262144, 16.3209 cents 33554432/32019867, 81.0432 cents 10: 996.0900 .. 1012.4109 cents 264627/262144, 16.3209 cents 256/243, 90.2250 cents 11: 1093.4541 .. 1109.7750 cents 264627/262144, 16.3209 cents 33554432/32019867, 81.0432 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 : 1089/1024, 106.5459 cents Average / Smallest step : 1.108340 Largest / Average step : 1.065459 Largest / Smallest step : 1.180891 Median interval of one step : 98.38544 cents, amount: 0 Least squares average step : 99.25430 cents, oct.: 1191.05159 cents Scale is strictly proper Step pattern alph. order: ABCBCADEBCBC Step pattern size order : ABEBEADCBEBE Scale is a Constant Structure, by a margin of 81.04322 cents Scale diversity : 1.118891 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.856341 Rothenberg efficiency : 0.381355 redundancy: 0.618645 Efficiency x scale size : 4.576263 Number of different interval sizes : 46 = 4.18182 / class Number of one step interval sizes : 5 Highest interval variety : 6 Mean interval variety : 4.18182 = 46/11 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 1771561/1769472, 2.0427 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 9 Most common triad is 0.0 498.045 701.955 cents, amount: 7 Number of recognisable fifths : 12, average 700.0000 cents Scale is a well-temperament. It has 3 different sizes of fifth Scale has monotonic third-sizes over circle of fifths Best fifths form a closed circle Formal octave complements present : 1 = 8.3333% Limited transpositions with margin 23.4600 cents: 1 2 3 4 5 7 8 9 10 11 Limited inverse transpositions with margin 16.3209 cents: 1 2 3 4 5 6 7 8 9 10 11 -- Rational properties -- Prime limit : 11 Odd number limit : 43923 (O: 43923 U: 43923) Highest odd numerator or denominator: 43923 Scale harmonicity : 0.009549 Average absolute harmonicity : 0.134982 Specific harmonicity : 0.058634 Fundamental : 1/10673289, -23.3475 octaves, 0.0000 Hz. Guide tone : 65536, 16.0000 octaves, 17145893.048 Hz. Exponens Consonantiae : 6.994847E+11, 39.34750 octaves Euler's gradus suavitatis : 69 Sum of Mann's harmonic distance : 105435.0, average 8786.25000 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 29.67308, average 2.47276 Sum of Tenney's harmonic distance : 61.28123, average 5.10677 Vogel's harmonic complexity : 50.75000 Wille's k value : 5336644 Wilson's harmonic complexity : 62 Rectangular lattice diameter : 10 Triangular lattice diameter : 7 Lattice compactness : 570.65538, average 7.31609 Lattice compactness (without 2's) : 236.85176, average 3.03656 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: 1 .. 16 8.75000 105 1 3: -6 .. 0 -3.00000 36 6 11: -4 .. 0 -1.00000 12 5 Average exponent except 2's :-48 / 12 =-4.00000 Average absolute exponent except 2's: 48 / 12 = 4.00000
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 8.0545 c. M:-12.4109 c. Twelve-tone Chromatic (1/11-comma) 43: 3 4 4 3 4 3 4 3 4 4 3 4 SP M ME SD: 6.7261 c. M:-12.8417 c. Meantone Chromatic (1/5-comma) 45: 3 4 4 4 4 3 4 4 3 4 4 4 SP D3 SD: 6.4329 c. M: 10.2250 c. 49: 4 4 4 4 4 4 4 4 4 5 4 4 SP M ME SD: 6.5832 c. M:-8.5867 c. 53: 4 4 5 4 5 4 4 5 4 5 4 5 SP M ME SD: 5.4433 c. M: 13.5279 c. 57: 4 5 5 5 5 4 5 5 4 5 5 5 SP D3 SD: 5.0397 c. M:-8.5009 c. 64: 5 5 6 5 6 4 6 5 5 6 5 6 SP D3 SD: 5.0090 c. M: 8.2941 c. 65: 5 5 6 5 6 5 6 5 5 6 5 6 SP M ME SD: 3.4795 c. M:-8.7652 c. 70: 5 6 6 6 6 5 6 6 6 6 6 6 SP D3 S SD: 3.5329 c. M: 7.0589 c. 77: 6 6 7 6 7 6 6 7 6 7 6 7 SP M ME SD: 2.9000 c. M: 7.0589 c. 79: 6 6 7 7 7 6 7 6 7 7 6 7 SP D3 SD: 4.3360 c. M:-6.6584 c. 81: 6 7 7 6 8 6 7 6 7 7 7 7 SP D3 SD: 4.1245 c. M: 6.9948 c. 82: 6 7 7 7 7 6 7 7 7 7 7 7 SP D3 S SD: 2.7823 c. M: 4.9684 c. 89: 7 7 8 7 8 7 7 8 7 8 7 8 SP M ME SD: 2.7798 c. M: 5.1328 c. 94: 7 8 8 8 8 7 8 8 8 8 8 8 SP D3 S SD: 2.6410 c. M:-4.4182 c. 111: 8 9 10 9 10 8 10 9 9 10 9 10 SP SD: 2.6340 c. M: 4.4862 c. 116: 9 9 10 10 10 9 10 10 9 10 10 10 SP D3 S SD: 2.4956 c. M: 4.4798 c. 120: 9 10 10 10 11 9 10 10 10 11 9 11 SP T3 SD: 2.4477 c. M: 4.1350 c. 123: 9 10 11 10 11 9 11 10 10 11 10 11 SP SD: 1.6193 c. M: 2.9041 c. 135: 10 11 12 11 12 10 12 11 11 12 11 12 SP SD: 0.8295 c. M: 1.6033 c. 147: 11 12 13 12 13 11 13 12 12 13 12 13 SP SD: 0.4353 c. M:-1.1043 c. 282: 21 23 25 23 25 21 25 23 23 25 23 25 SP SD: 0.5485 c. M: 1.0360 c. 306: 23 25 27 25 27 23 27 25 25 27 25 27 SP SD: 0.5421 c. M:-1.3444 c. 347: 26 28 31 28 31 26 30 29 28 31 28 31 SP SD: 0.5711 c. M: 1.1306 c. 359: 27 29 32 29 32 27 31 30 29 32 29 32 SP SD: 0.3416 c. M: 0.8512 c. 371: 28 30 33 30 33 28 32 31 30 33 30 33 SP SD: 0.2626 c. M: 0.5899 c. 494: 37 40 44 40 44 37 43 41 40 44 40 44 SP SD: 0.3481 c. M: 0.5446 c. 506: 38 41 45 41 45 38 44 42 41 45 41 45 SP SD: 0.1561 c. M: 0.2831 c. 518: 39 42 46 42 46 39 45 43 42 46 42 46 SP SD: 0.0769 c. M:-0.1470 c. 665: 50 54 59 54 59 50 58 55 54 59 54 59 SP SD: 0.0651 c. M:-0.1591 c. 1171: 88 95 104 95 104 88 102 97 95 104 95 104 SP SD: 0.0371 c. M: 0.0571 c. 1689: 127 137 150 137 150 127 147 140 137 150 137 150 SP SD: 0.0216 c. M: 0.0556 c. 1836: 138 149 163 149 163 138 160 152 149 163 149 163 SP SD: 0.0196 c. M:-0.0372 c. 2354: 177 191 209 191 209 177 205 195 191 209 191 209 SP SD: 0.0041 c. M:-0.0065 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: 4.5909 cents 3 x x x x 4 x x x 5 x x 6 S SD: 1.7285 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 7.3081 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: 8.8169 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 12.0540 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.7059 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.9631 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.1399 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 10.2786 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.6000 cents 13:14 x 15:16:17:18:19:21:22:23:24:26 SD: 9.3836 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 8.2604 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 5.3499 cents 17:18:19:20:21:23:24:25:27:28:30:32:34 SD: 4.6915 cents 18:19:20:21:23:24:25:27:28:30:32:34:36 SD: 4.6817 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 4.5915 cents 23:24:26:27:29:31:32:34:36:38:41:43:46 SD: 4.5240 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.6958 cents 27:28:30:32:34:36:38:40:43:45:48:51:54 SD: 2.9547 cents 33:35:37:39:41:44:46:49:52:55:59:62:66 SD: 2.5183 cents 36:38:40:43:45:48:51:54:57:60:64:68:72 SD: 2.1146 cents 42:44:47:50:53:56:59:63:66:70:75:79:84 SD: 2.0325 cents 43:45:48:51:54:57:60:64:68:72:76:81:86 SD: 1.9061 cents 51:54:57:60:64:68:72:76:81:85:91:96:102 SD: 1.8514 cents 52:55:58:62:65:69:73:78:82:87:92:98:104 SD: 1.7428 cents 54:57:60:64:68:72:76:81:85:90:96:102:108 SD: 1.4955 cents 55:58:61:65:69:73:77:82:87:92:98:103:110 SD: 1.3636 cents 59:62:66:70:74:79:83:88:93:99:105:111:118 SD: 1.1524 cents 71:75:79:84:89:95:100:106:112:119:126:134:142 SD: 1.1044 cents 76:80:85:90:95:101:107:113:120:127:135:143:152 SD: 1.0522 cents 86:91:96:102:108:115:121:128:136:144:153:162:172 SD: 1.0100 cents 91:96:101:108:114:121:128:136:144:152:162:171:182 SD: 0.8934 cents 98:103:109:116:123:131:138:146:155:164:174:184:196 SD: 0.8347 cents 107:113:119:127:134:143:150:160:169:179:190:201:214 SD: 0.8064 cents 112:118:125:133:140:149:157:167:177:187:199:211:224 SD: 0.7825 cents 114:120:127:135:143:152:160:170:180:191:203:214:228 SD: 0.5500 cents 150:158:167:178:188:200:211:224:237:251:267:282:300 SD: 0.4114 cents 183:193:204:217:229:244:257:273:289:306:325:344:366 SD: 0.3806 cents 185:195:206:219:232:247:260:276:292:309:329:348:370 SD: 0.3768 cents 201:212:224:238:252:268:282:300:318:336:357:378:402 SD: 0.3531 cents 210:221:234:249:263:280:295:313:332:351:373:395:420 SD: 0.3136 cents 228:240:254:270:286:304:320:340:360:381:405:429:456 SD: 0.3055 cents 243:256:271:288:305:324:341:363:384:406:432:457:486 SD: 0.3011 cents 244:257:272:289:306:325:343:364:386:408:434:459:488 SD: 0.2840 cents 260:274:290:308:326:347:365:388:411:435:462:489:520 SD: 0.2662 cents 264:278:294:313:331:352:371:394:417:441:469:496:528 SD: 0.2619 cents 279:294:311:331:350:372:392:416:441:466:496:525:558 SD: 0.2551 cents 319:336:356:378:400:425:448:476:504:533:567:600:638 SD: 0.2500 cents 324:341:361:384:406:432:455:483:512:542:576:609:648 SD: 0.2459 cents 334:352:372:396:419:445:469:498:528:558:594:628:668 SD: 0.2434 cents 339:357:378:402:425:452:476:506:536:567:603:638:678 SD: 0.2267 cents 351:370:391:416:440:468:493:524:555:587:624:660:702 SD: 0.1726 cents 378:398:421:448:474:504:531:564:597:632:672:711:756 SD: 0.1513 cents 410:432:457:486:514:547:576:612:648:685:729:771:820 SD: 0.1451 cents 453:477:505:537:568:604:636:676:716:757:805:852:906 SD: 0.1439 cents 524:552:584:621:657:699:736:782:828:876:932:985:1048 SD: 0.1228 cents 579:610:645:686:726:772:813:864:915:968:1029:1089:1158 SD: 0.1083 cents