68ed7/3

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← 67ed7/3 68ed7/3 69ed7/3 →
Prime factorization 22 × 17
Step size 21.5716 ¢ 
Octave 56\68ed7/3 (1208.01 ¢) (→ 14\17ed7/3)
Twelfth 88\68ed7/3 (1898.3 ¢) (→ 22\17ed7/3)
Consistency limit 2
Distinct consistency limit 2

68 equal divisions of 7/3 (abbreviated 68ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 68 equal parts of about 21.6 ¢ each. Each step represents a frequency ratio of (7/3)1/68, or the 68th root of 7/3.

Intervals

Degrees Enneatonic Pentadecatonic Enneadecatonic ed11\9~ed7/3
1 1+ G+ G' Qq Q+ 21.5686 21.5716
Jv, Av, Jbv, Abv, W\\v Wbv
2 1# G# G^ Qp/W\\\ Q#/Wb 43.13725 43.1433
Jv Av Jbv Abv
3 1#+ G#+ G^' Qpq Q#+ 64.7059 64.7149
J, A, Jb, Ab, W\ Wd
4 1x/2bb Gx J A Jb Ab W 86.2745 86.2865
Jbb Abb
5 2bv Jbv Abv J'/A\\v, A'/B\\v, Jb' Ab' Wq W+ W+/Ebd 107.8431 107.8582
G#v,
6 2b Jb Ab J^/A\\v A^/B\\v Jb^ Ab^ Wp W# W#/Eb 129.4118 129.4298
G#v
7 2d Jd Ad J^'/A\\, A^'/B\\, Jb^' Ab^' Wpq/E\\\ W#+/E(b)d 150.9804 151.0014
G#,
8 2 J A A\\ B\\ G# E\\ Eb E 172.549 172.57305
9 2+ J+ A+ A\\'\/Jpv, B\\'/Apv, G#' E\ Ed E+/Rd 194.11765 194.1447
Jv, Av,
10 2# J# A# A\\^/Jpv B\\^/Apv G#^ E R 215.6863 215.7163
Jv Av
11 2#+ J#+ A#+ Jp, Ap, G#^' Epq/R\\\ E+/Rbd R+/Tbd 237.2549 237.2879
J, A,
12 2x/3bb Jx/Abb Ax/Bbb Jp Ap J A Ep/R\\ E#/Rb R#/Tb 258.8235 258.8596
13 3bd Abd Bbd Jp'/Av, Ap^/Bv J'/A, A'/Bv, Epq/R\ E#+/Rd R#+/Td 280.3922 280.4312
14 3b Ab Bb Jp^/Av Ap^/Bv J^/Av A^/Bv R T 301.9608 302.0028
15 3d Ad Bd Jp^'/A, Ap^'/B, J^'/A, A^'/Bv, Rq/T\\\ R+/Tbd T+/Ybd 323.5294 323.5745
16 3 A B A B A B Rp/T\\ R#/Tb T#/Yb 345.098 345.1461
17 3+ A+ B+ A'/Bv, B'/Cv, A'/Bbv, B^'/Cbv, Rpq/T\ R+/Tbd T#+/Yd 366.6 366.7177
18 3# A# B# A^/Bv B^/Cv A^/Bbv B^/Cbv T Y 388.2353 388.2894
19 3#+ A#+ B#+ A^'/B, B^'/Cv, A^'/Bb, B^'/Cbv, Tq\A\\\ T#+/Ad Y+/Ubd 409.8039 409.861
20 3x/4b Ax/Bbb Bx/Cbb B C Bb Cb Tp\A\\ T#/Ab Y#/Ub 431.37255 431.4326
21 4bd Bbd Cbd B^'C\\v, C'/Q\\v, Bb'/A#v, Cb'/B#v, Tpq\A\ T#+/Ad Y#+/Ud 452.9412 453.00425
22 4b Bb Cb B^/C\\v C^/Q\\v Bb^/A#v Cb^/B#v A U 474.5098 474.5759
23 4d B Cd B^'/C\\v, C^'/Q\\, Bb^'/A#, Cb^'/B#, Apq A+ U+/Abd 496.0784 496.1475
24 4 B C C\\ Q\\ A# B# Ap A# U#/Ab 517.6471 517.7191
25 4+ B+ C+ C\\'/Bpv, Q\\'/Cpv, A#'/Bv, B#'/Cv, Apq/S\\\ A#+/Sbd U#+/Ad 539.2157 539.2908
26 4# B# C# C\\^/Bpv Q\\^/Cpv A#^/Bv B#^/Cv S\\ Sb A 560.7843 560.8624
27 4#+/5bd B#+/Cbd C#/Dbd C\\^'/Bp, Q\\^'/Cpv, A#^'/B, B#^'/C, S\ Sd A+/Sd 582.3529 582.434
28 5b Cb Qb Bp Cp B C S 603.9216 604.0057
29 5d Cd Qd Bp'/Cv, Cp'/Qv, B'/Cv, C'/Qv, Sq/D\\\ S+/Dbd 625.4902 625.5773
30 5 C Q Bp^/Cv Cp^/Qv B^/Cv C^/Qv Sp/D\\ S#/Db 647.0588 647.1489
31 5+ C+ Q+ Bp^'/C, Cp^/'Q, B^'/C, C^'/Q, Spq/D\ S#+/Dbd 668.62745 668.7206
32 5# C# Q# C Q C Q D 690.1961 690.2922
33 5#+ C#+ Q#+ C'/Qv, Q'/Dv, C'/Qbv, Q'/Dbv, Dq/F\\\ D+/Fbd 711.7647 711.8638
34 5x/6bb Cx/Qbb Qx/Dbb C^/Qv Q^/Dv C^/Qbv Q^/Dbv Dp/F\\ D#/Fb 733.3 733.43545
35 6bd Qbd Dbd C^/Qv Q^/Dv C^/Qbv Q^/Dbv Dp/F\\ D#/Fb 754.902 755.0071
36 6b Qb Db Q D Qb Db F 776.4706 776.5787
37 6d Qd Dd Q'/D\\v, D'/E\\v, Qb'/C#v, Db'/Q#v, Fq/G\\\ F+/Gbd 798.0392 798.15035
38 6 Q D Q^/D\\v D^/E\\v Qb^/C#v Db^/Q#v Fp/G\\ F#/Gb 819.6078 819.722
39 6+ Q+ D+ Q^'/D\\, D^'/E\\, Qb^'/C#, Db^'/Q#, Fpq/G\ F+/Gd 841.1765 841.2936
40 6# Q# D# D\\ E\\ C# Q# G 862.7451 862.8982
41 6#+ Q#+ D#+ D\\'/Qpv, E\\'/Dpv, C#'/Qv, Q#'/Dv, Gq G+ G+\Hd 884.3137 884.4369
42 6x/7bb Qx/Dbb Dx/Sbb D\\^/Qpv E\\^/Dpv C#^/Qv Q#^/Dv Gp G# H 905.88235 906.0085
43 7bd Dbd Sbd D\\^'/Qp, E\\^'/Dp, C#^'/Qv, Q#^'/Dv, Gpq/Z\\\ G#+/Zbd Hd 927.451 927.5801
44 7b Db Sb Qp Dp Q D Z\\ Zb H# 949.0196 949.1518
45 7d Dd Sd Qp'/Dv, Dp'/Ev, Q'/Dv, D'/Sv, Z\ Zd H#+/Jbd 970.5882 970.7234
46 7 D S Qp^/Dv Dp^/Ev Q^/Dv D^/Sv Z Jb 992.1569 992.295
47 7+ D+ S+ Qp^'/D, Dp^'/E, Q^'/D, D^'/S, Zq/X\\\ Z+/Xbd Jd 1013.7255 1013.8667
48 7# D# S# D E D S Zp/X\\ Z#/Xb J 1035.2941 1035.4383
49 7#+ D#+ S#+ D'/Ev, E'/Sv, D' S' Zpq/X\ Z#+/Xd J+/Zbd 1056.86275 1057.0099
Ebv,
50 7x/8bb Dx/Ebb Sx/Ebb D^/Ev E^/Sv D^ S^ X J#/Zb 1078.4314 1078.58155
Ebv
51 8bd Ebd D^'/E, E^'/Sv, D^' S^' Xq/C\\\ X+/Cbd J#+/Zd 1100 1100.1532
Eb,
52 8b Eb E S Eb Xp/C\\ X#/Cb Z 1121.5686 1121.7248
53 8d Ed E' S' Eb' Xpq/C\ X#+/Cbd Z+/Xd 1143.13725 1143.2964
Fv, D#v, S#v,
54 8 E E^ S^ Eb^ C X 1164.7059 1164.8681
Fv D#v S#v
55 8+ E+ E^' S^' Eb^' Cq/V\\\ C+/Vbd X+/Cbd 1186.2745 1186.4397
F, D#, S#,
56 8# E# F D# S# Cp/V\\ C#/Vb X#/Cb 1207.8731 1208.0113
57 8#+ E#+ F'/G\\v, D#' S#' Cpq/V\ C#+/Vd X#+/Cd 1229.4118 1229.583
Ev,
58 8x/9bb Ex/Fbb F^/G\\v D#^ S#^ V C 1250.9804 1251.1546
Ev
59 9bd Fbd F^'/G\\, D#^' S#^' Vq V+ C+/Vd 1272.2549 1272.7262
E,
60 9b Fb G\\ E Vp V# V 1294.11765 1294.2979
61 9d Fd G\\'/Fpv, E'/Fv, Vpq/B\\\ V#+/Bbd V+/Bbd 1315.6863 1315.8695
62 9 F G\\^/Fpv E^/Fv B\\ Bb V#/Bb 1337.2549 1337.4411
63 9+ F+ G\\^/'Fp, E^'/F, B\ Bd V#+/Bd 1358.8235 1359.01275
64 9# F# Fp F B 1380.3922 1380.5844
65 9#+/1bd F#+/Gbd Fp' F' Bpq/Q\\ B+/Qbd 1401.9608 1402.156
Gv
66 1b Gb Fp^ F^ Bp/Q\\ B#/Qb 1423.5294 1423.7276
Gv
67 1d Gd Fp^' F^' Bpq/Q\ B#+/Qd 1445.098 1445.993
Gv,
68 1 G Q 1466.6 1466.8709

Harmonics

Approximation of harmonics in 135ed7/3
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.77 -0.46 +1.32 -4.69 -5.23 -0.46 -3.45 -0.91 +1.40 -0.62 +0.87
Relative (%) -43.9 -4.2 +12.2 -43.2 -48.1 -4.2 -31.7 -8.4 +12.9 -5.7 +8.0
Steps
(reduced) 		110
(110) 		175
(40) 		221
(86) 		256
(121) 		285
(15) 		310
(40) 		331
(61) 		350
(80) 		367
(97) 		382
(112) 		396
(126)
Approximation of harmonics in 135ed7/3
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +3.55 -5.23 -5.15 +2.64 -4.52 +5.18 -1.49 -3.37 -0.91 -5.39 +4.58
Relative (%) +32.7 -48.1 -47.4 +24.3 -41.6 +47.7 -13.8 -31.0 -8.4 -49.6 +42.2
Steps
(reduced) 		409
(4) 		420
(15) 		431
(26) 		442
(37) 		451
(46) 		461
(56) 		469
(64) 		477
(72) 		485
(80) 		492
(87) 		500
(95)
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