Scala analysis: xen17-bohlen-harmonic-1
13-tone non-tempered scale
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 27/25 133.237575 large limma, BP small semitone, Zarlino semitone 2: 25/21 301.846520 BP second, quasi-equal minor third 3: 9/7 435.084095 septimal major third, BP third 4: 7/5 582.512193 septimal or Huygens' tritone, BP fourth 5: 75/49 736.930616 BP fifth 6: 5/3 884.358713 major sixth, BP sixth 7: 9/5 1017.596288 just minor seventh, BP seventh 8: 49/25 1165.024385 BP eighth 9: 15/7 1319.442808 septimal minor ninth, BP ninth 10: 7/3 1466.870906 minimal tenth, BP tenth 11: 63/25 1600.108480 quasi-equal major tenth, BP eleventh 12: 25/9 1768.717426 classic augmented eleventh, BP twelfth 13: 3/1 1901.955001 perfect 12th
SHOW/INTERVAL
0: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 1: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 2: 625/567 168.6089 BP great semitone, major BP chroma 3: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 4: 49/45 147.4281 BP minor semitone 5: 375/343 154.4184 BP major semitone, minor BP chroma 6: 49/45 147.4281 BP minor semitone 7: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 8: 49/45 147.4281 BP minor semitone 9: 375/343 154.4184 BP major semitone, minor BP chroma 10: 49/45 147.4281 BP minor semitone 11: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 12: 625/567 168.6089 BP great semitone, major BP chroma 13: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 5 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 1: 4 49/45 147.428 cents BP minor semitone 1: 2 375/343 154.418 cents BP major semitone, minor BP chroma 1: 2 625/567 168.609 cents BP great semitone, major BP chroma 2: 1 729/625 266.475 cents two (large limma, BP small semitone, Zarlino semitone) 2: 4 147/125 280.666 cents 2: 8 25/21 301.847 cents BP second, quasi-equal minor third 3: 1 2401/1875 428.094 cents 3: 8 9/7 435.084 cents septimal major third, BP third 3: 4 35/27 449.275 cents septimal semi-diminished fourth, 9/4-tone 4: 2 243/175 568.322 cents 4: 8 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 4: 3 625/441 603.693 cents two (BP second, quasi-equal minor third) 5: 4 189/125 715.750 cents 5: 2 343/225 729.940 cents 5: 5 75/49 736.931 cents BP fifth 5: 2 125/81 751.121 cents 6: 2 1029/625 863.178 cents 6: 3 81/49 870.168 cents two (septimal major third, BP third) 6: 8 5/3 884.359 cents major sixth, BP sixth 7: 8 9/5 1017.596 cents just minor seventh, BP seventh 7: 3 49/27 1031.787 cents 7: 2 625/343 1038.777 cents 8: 2 243/125 1150.834 cents octave - maximal diesis 8: 5 49/25 1165.024 cents BP eighth 8: 2 675/343 1172.015 cents 8: 4 125/63 1186.205 cents 9: 3 1323/625 1298.262 cents 9: 8 15/7 1319.443 cents septimal minor ninth, BP ninth 9: 2 175/81 1333.633 cents 10: 4 81/35 1452.680 cents Al-Hwarizmi's lute middle finger +1 octave 10: 8 7/3 1466.871 cents minimal tenth, BP tenth 10: 1 5625/2401 1473.861 cents 11: 8 63/25 1600.108 cents quasi-equal major tenth, BP eleventh 11: 4 125/49 1621.289 cents 11: 1 625/243 1635.480 cents 12: 2 1701/625 1733.346 cents 12: 2 343/125 1747.537 cents 12: 4 135/49 1754.527 cents 12: 5 25/9 1768.717 cents classic augmented eleventh, BP twelfth Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.33333 = 10/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 13 0.0 : 133.2 301.8 435.1 582.5 736.9 884.4 1017.6 1165.0 1319.4 1466.9 1600.1 1768.7 1902.0 133.2 : 168.6 301.8 449.3 603.7 751.1 884.4 1031.8 1186.2 1333.6 1466.9 1635.5 1768.7 1902.0 301.8 : 133.2 280.7 435.1 582.5 715.7 863.2 1017.6 1165.0 1298.3 1466.9 1600.1 1733.3 1902.0 435.1 : 147.4 301.8 449.3 582.5 729.9 884.4 1031.8 1165.0 1333.6 1466.9 1600.1 1768.7 1902.0 582.5 : 154.4 301.8 435.1 582.5 736.9 884.4 1017.6 1186.2 1319.4 1452.7 1621.3 1754.5 1902.0 736.9 : 147.4 280.7 428.1 582.5 729.9 863.2 1031.8 1165.0 1298.3 1466.9 1600.1 1747.5 1902.0 884.4 : 133.2 280.7 435.1 582.5 715.7 884.4 1017.6 1150.8 1319.4 1452.7 1600.1 1754.5 1902.0 1017.6: 147.4 301.8 449.3 582.5 751.1 884.4 1017.6 1186.2 1319.4 1466.9 1621.3 1768.7 1902.0 1165.0: 154.4 301.8 435.1 603.7 736.9 870.2 1038.8 1172.0 1319.4 1473.9 1621.3 1754.5 1902.0 1319.4: 147.4 280.7 449.3 582.5 715.7 884.4 1017.6 1165.0 1319.4 1466.9 1600.1 1747.5 1902.0 1466.9: 133.2 301.8 435.1 568.3 736.9 870.2 1017.6 1172.0 1319.4 1452.7 1600.1 1754.5 1902.0 1600.1: 168.6 301.8 435.1 603.7 736.9 884.4 1038.8 1186.2 1319.4 1466.9 1621.3 1768.7 1902.0 1768.7: 133.2 266.5 435.1 568.3 715.7 870.2 1017.6 1150.8 1298.3 1452.7 1600.1 1733.3 1902.0 1902.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 133.2376 .. 168.6089 cents 15625/15309, 35.3714 cents 27/25, 133.2376 cents 2: 266.4751 .. 301.8465 cents 15625/15309, 35.3714 cents 413343/390625, 97.8662 cents 3: 428.0938 .. 449.2746 cents 3125/3087, 21.1808 cents 16807/15625, 126.2472 cents 4: 568.3217 .. 603.6930 cents 15625/15309, 35.3714 cents 6561/6125, 119.0471 cents 5: 715.7498 .. 751.1211 cents 15625/15309, 35.3714 cents 83349/78125, 112.0567 cents 6: 863.1779 .. 884.3587 cents 3125/3087, 21.1808 cents 83349/78125, 112.0567 cents 7: 1017.5963 .. 1038.7771 cents 3125/3087, 21.1808 cents 27/25, 133.2376 cents 8: 1150.8339 .. 1186.2052 cents 15625/15309, 35.3714 cents 83349/78125, 112.0567 cents 9: 1298.2620 .. 1333.6333 cents 15625/15309, 35.3714 cents 83349/78125, 112.0567 cents 10: 1452.6804 .. 1473.8612 cents 3125/3087, 21.1808 cents 6561/6125, 119.0471 cents 11: 1600.1085 .. 1635.4799 cents 15625/15309, 35.3714 cents 16807/15625, 126.2472 cents 12: 1733.3461 .. 1768.7174 cents 15625/15309, 35.3714 cents 413343/390625, 97.8662 cents
SHOW DATA
Number of notes : 13 -- Interval properties -- Smallest interval : 27/25, 133.2376 cents, class 1 Average step (divided formal octave): 146.3042 cents Largest one step interval : 625/567, 168.6089 cents Average / Smallest step : 1.098070 Largest / Average step : 1.152454 Largest / Smallest step : 1.265476 Median interval of one step : 49/45, 147.4281 cents, amount: 4 Most common interval of one step : 27/25, 133.2376 cents, amount: 5 Least squares average step : 146.39966 cents, oct.: 1903.19552 cents Scale is strictly proper Step pattern alph. order: ABACDCACDCABA Step pattern size order : ADABCBABCBADA Scale is a Constant Structure, by a margin of 97.86620 cents Scale diversity : 0.289267 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.806676 Number of different interval sizes : 40 = 3.33333 / class Number of one step interval sizes : 4 Highest interval variety : 4 Mean interval variety : 3.33333 = 10/3 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 16875/16807, 6.9903 cents Number of recognisable fifths : 4, average 715.7498 cents Number of appreciable fifths : 0 Number of recognisable fourths : 0 Scale contains two identical disjunct tetrachords Best fifths form a closed circle Best major thirds form a closed circle Best minor thirds form a closed circle Formal octave complements present : 13 = 100.0000% 2/1 octave complements present : 0 = 0.0000% Limited transpositions with margin 35.3714 cents: 3 4 5 6 7 8 9 10 Limited inverse transpositions with margin 35.3714 cents: 3 4 5 6 7 8 9 10 Inversional symmetry on degrees : 0 Inversional symmetry on intervals : 6-7 -- Rational properties -- Prime limit : 7 Odd number limit : 5625 (O: 5625 U: 2401) Highest odd numerator or denominator: 75 Scale harmonicity : 0.012488 Average absolute harmonicity : 0.081408 Specific harmonicity : 0.078336 Fundamental : 1/11025, -13.4285 octaves, 0.0237 Hz. Guide tone : 33075, 15.0135 octaves, 8653265.572 Hz. Exponens Consonantiae : 3.646519E+08, 28.44194 octaves Euler's gradus suavitatis : 51 Sum of Mann's harmonic distance : 252.0, average 19.38462 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 15.54271, average 1.19559 Sum of Tenney's harmonic distance : 27.74556, average 2.13427 Vogel's harmonic complexity : 31.61538 Wille's k value : 16537 Wilson's harmonic complexity : 63 Rectangular lattice diameter : 13 Triangular lattice diameter : 5 Lattice compactness : 274.27289, average 3.01399 Lattice compactness (without 2's) : 274.27289, average 3.01399 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 3: -2 .. 3 0.53846 17 1 5: -2 .. 2 0.00000 16 4 7: -2 .. 2 0.00000 10 8 Average exponent except 2's : 7 / 13 = 0.53846 Average absolute exponent except 2's: 43 / 13 = 3.30769 Scale is weakly epimorphic with val: <13 17 23| Scale is weakly epimorphic with val: <13 17 24| Scale is JI-epimorphic with val: <13 19 23| = patent Scale is weakly epimorphic with val: <13 20 24| Scale is weakly epimorphic with val: <13 21 22|
FIT/MODE
13: 1 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 7.3380 c. M: 13.0667 c. 56: 4 5 4 4 5 4 4 4 5 4 4 5 4 SP M ME I SD: 5.4970 c. M:-10.2660 c. 62: 4 6 4 5 5 5 4 5 5 5 4 6 4 SP D3 I SD: 5.4759 c. M: 10.5308 c. 69: 5 6 5 5 6 5 5 5 6 5 5 6 5 SP M ME I SD: 4.4786 c. M:-7.3126 c. 75: 5 7 5 6 6 6 5 6 6 6 5 7 5 SP D3 I SD: 3.4322 c. M: 6.4406 c. 82: 6 7 6 6 7 6 6 6 7 6 6 7 6 SP M ME I SD: 4.1252 c. M: 5.9299 c. 88: 6 8 6 7 7 7 6 7 7 7 6 8 6 SP D3 I SD: 2.1400 c. M:-3.5588 c. 101: 7 9 7 8 8 8 7 8 8 8 7 9 7 SP D3 I SD: 1.4971 c. M:-2.5123 c. 114: 8 10 8 9 9 9 8 9 9 9 8 10 8 SP D3 I SD: 1.4800 c. M: 2.8427 c. 157: 11 14 11 12 13 12 11 12 13 12 11 14 11 SP I SD: 1.0614 c. M: 2.0455 c. 170: 12 15 12 13 14 13 12 13 14 13 12 15 12 SP I SD: 0.9301 c. M:-1.4754 c. 245: 17 22 17 19 20 19 17 19 20 19 17 22 17 SP I SD: 0.7184 c. M: 1.2652 c. 258: 18 23 18 20 21 20 18 20 21 20 18 23 18 SP I SD: 0.3127 c. M:-0.5430 c. 271: 19 24 19 21 22 21 19 21 22 21 19 24 19 SP I SD: 0.0573 c. M: 0.1098 c. 800: 56 71 56 62 65 62 56 62 65 62 56 71 56 SP I SD: 0.0658 c. M: 0.1007 c. 1071: 75 95 75 83 87 83 75 83 87 83 75 95 75 SP I SD: 0.0376 c. M:-0.0547 c. 1342: 94 119 94 104 109 104 94 104 109 104 94 119 94 SP I SD: 0.0230 c. M:-0.0415 c. 1613: 113 143 113 125 131 125 113 125 131 125 113 143 113 SP I SD: 0.0172 c. M:-0.0327 c. 1884: 132 167 132 146 153 146 132 146 153 146 132 167 132 SP I SD: 0.0173 c. M: 0.0265 c.
FIT/HARMONIC
1 x x x x x x x 2 x x x x 3 S SD: 17.4878 cents 2 x x x x 3 x x 4 x x 5 x 6 S SD: 12.8376 cents 3 x x 4 x x 5 x 6 x 7 x 8:9 S SD: 16.8176 cents 4 x x 5 x 6 x 7:8 x 9:10:11:12 S SD: 13.4981 cents 5 x 6 x 7 x 8:9:10:11:12:13:14:15 S SD: 11.8238 cents 6 x 7:8 x 9:10:11:12:13:14:15:17:18 SD: 9.2596 cents 7 x 8:9:10:11:12:13:14:15:16:18:19:21 SD: 11.4634 cents 8:9 x 10:11:12:13:14:16:17:19:20:22:24 SD: 10.7801 cents 9:10:11:12:13:14:15:16:18:19:21:23:25:27 SD: 9.4555 cents 10:11:12:13:14:15:17:18:20:21:23:25:28:30 SD: 6.6224 cents 14:15:17:18:20:21:23:25:27:30:33:35:39:42 SD: 5.9235 cents 15:16:18:19:21:23:25:27:29:32:35:38:42:45 SD: 3.6290 cents 21:23:25:27:29:32:35:38:41:45:49:53:58:63 SD: 2.9235 cents 25:27:30:32:35:38:42:45:49:54:58:63:69:75 SD: 2.4294 cents 27:29:32:35:38:41:45:49:53:58:63:68:75:81 SD: 2.2441 cents 35:38:42:45:49:54:58:63:69:75:82:88:97:105 SD: 2.0991 cents 36:39:43:46:50:55:60:65:71:77:84:91:100:108 SD: 1.8207 cents 42:45:50:54:59:64:70:76:82:90:98:106:117:126 SD: 1.6218 cents 48:52:57:62:67:73:80:86:94:103:112:121:133:144 SD: 1.4443 cents 49:53:58:63:69:75:82:88:96:105:114:123:136:147 SD: 1.4294 cents 52:56:62:67:73:80:87:94:102:111:121:131:144:156 SD: 1.4237 cents 53:57:63:68:74:81:88:95:104:114:124:134:147:159 SD: 1.3431 cents 60:65:71:77:84:92:100:108:118:129:140:151:167:180 SD: 1.1960 cents 62:67:74:80:87:95:103:112:122:133:145:156:172:186 SD: 1.1573 cents 63:68:75:81:88:96:105:113:123:135:147:159:175:189 SD: 0.9901 cents 73:79:87:94:102:112:122:131:143:156:170:184:203:219 SD: 0.9109 cents 75:81:89:96:105:115:125:135:147:161:175:189:208:225 SD: 0.8315 cents 85:92:101:109:119:130:142:153:167:182:198:214:236:255 SD: 0.7482 cents 90:97:107:116:126:138:150:162:176:193:210:227:250:270 SD: 0.6209 cents 100:108:119:129:140:153:167:180:196:214:233:252:278:300 SD: 0.5933 cents 115:124:137:148:161:176:192:207:225:246:268:290:319:345 SD: 0.5521 cents 125:135:149:161:175:191:208:225:245:268:292:315:347:375 SD: 0.4665 cents 126:136:150:162:176:193:210:227:247:270:294:318:350:378 SD: 0.4024 cents 147:159:175:189:206:225:245:265:288:315:343:370:408:441 SD: 0.3710 cents 175:189:208:225:245:268:292:315:343:375:408:441:486:525 SD: 0.2940 cents 200:216:238:257:280:306:333:360:392:429:467:504:556:600 SD: 0.2589 cents 225:243:268:289:315:344:375:405:441:482:525:567:625:675 SD: 0.2155 cents 252:272:300:324:353:386:420:454:494:540:588:635:700:756 SD: 0.1894 cents 273:295:325:351:382:418:455:491:535:585:637:688:758:819 SD: 0.1667 cents 300:324:357:386:420:459:500:540:588:643:700:756:833:900 SD: 0.1383 cents 315:340:375:405:441:482:525:567:617:675:735:794:875:945 SD: 0.1275 cents 462:499:550:594:647:707:770:832:906:990:1078:1164:1283:1386 SD: 0.1165 cents 477:515:568:613:668:730:795:859:935:1022:1113:1202:1325:1431 SD: 0.1139 cents 525:567:625:675:735:804:875:945:1029:1125:1225:1323:1458:1575 SD: 0.0773 cents