70edo

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← 69edo70edo71edo →
Prime factorization 2 × 5 × 7
Step size 17.1429¢ 
Fifth 41\70 (702.857¢)
Semitones (A1:m2) 7:5 (120¢ : 85.71¢)
Consistency limit 9
Distinct consistency limit 9

70 equal divisions of the octave (abbreviated 70edo or 70ed2), also called 70-tone equal temperament (70tet) or 70 equal temperament (70et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 70 equal parts of about 17.1 ¢ each. Each step represents a frequency ratio of 21/70, or the 70th root of 2.

Theory

This tuning was singled out by William Stoney in his article "Theoretical Possibilities for Equally Tempered Systems" (in the book The Computer and Music) as one of the six best systems of size 72 or smaller, along with 72, 65, 58, 53, and 41. These other systems have had notice paid to them, but the same does not seem to be true of 70, which seems to have been ignored ever since, despite its excellent fifth, which is the 4th number in the convergent sequence to the silver ratio, following 29edo, 12edo & 5edo and preceding 169edo. The fifth 41\70 is the true center of the diatonic tuning spectrum, as it is the geometric mean of 3\5edo and 4\7edo.

The patent val for 70edo tempers out 2048/2025, making it a diaschismic system. An alternative mapping is 70c, with a flat rather than a sharp major third, tempering out 32805/32768. In the 7-limit, the patent val tempers out 126/125, 2430/2401 and 5120/5103, and provides the optimum patent val for the kumonga temperament. The 70c val tempers out 50/49, making it a tuning for doublewide even better than the optimal patent val. The 70cd val tempers out 225/224 and 3125/3087 instead. The alternative mapping begins to make more sense in the 11-limit and higher, where the patent val tempers out 99/98 and 121/120 in the 11-limit, 169/168 and 352/351 in the 13-limit, and 221/220 in the 17-limit. 70cd on the other hand, with flat 5 and 7, tempers out 100/99 and 245/242 in the 11-limit, 105/104 and 196/195 in the 13-limit, and 154/153 and 170/169 in the 17-limit. 70 also makes sense as a no-5 or -7 system, tempering out 131769/131072 in the 11-limit, 352/351 and 2197/2187 in the 13-limit, and 289/288 and 1089/1088 in the 17-limit.

The 17-limit 2*70 subgroup, on which 70 is tuned like 140edo, is 2.3.25.35.11.13.17.

Prime harmonics

Approximation of prime harmonics in 70edo
Harmonic 2 3 5 7 11 13 17 19 23
Error Absolute (¢) +0.00 +0.90 +7.97 +8.32 -2.75 -0.53 -2.10 -6.08 +6.01
Relative (%) +0.0 +5.3 +46.5 +48.5 -16.0 -3.1 -12.2 -35.5 +35.1
Steps
(reduced)
70
(0)
111
(41)
163
(23)
197
(57)
242
(32)
259
(49)
286
(6)
297
(17)
317
(37)
Approximation of prime harmonics in 70edo (continued)
Harmonic 29 31 37 41 43 47 53 59 61
Error Absolute (¢) -1.01 +3.54 +5.80 -0.49 +2.77 +3.06 +0.78 +3.69 -2.60
Relative (%) -5.9 +20.6 +33.8 -2.9 +16.1 +17.9 +4.6 +21.5 -15.2
Steps
(reduced)
340
(60)
347
(67)
365
(15)
375
(25)
380
(30)
389
(39)
401
(51)
412
(62)
415
(65)

Subsets and supersets

Since 70 factors into 2 × 5 × 7, 70edo has subset edos 2, 5, 7, 10, 14, and 35. 140edo, which doubles it, provides good correction for its approximation to harmonics 5 and 7.

Intervals

Steps Cents Approximate Ratios Ups and Downs Notation
0 0 1/1 D
1 17.143 ^D, v4E♭
2 34.286 ^^D, v3E♭
3 51.429 32/31, 33/32, 34/33 ^3D, vvE♭
4 68.571 27/26 ^4D, vE♭
5 85.714 21/20 ^5D, E♭
6 102.857 17/16 ^6D, v6E
7 120 15/14, 29/27 D♯, v5E
8 137.143 13/12 ^D♯, v4E
9 154.286 12/11, 23/21 ^^D♯, v3E
10 171.429 32/29 ^3D♯, vvE
11 188.571 29/26 ^4D♯, vE
12 205.714 9/8 E
13 222.857 33/29 ^E, v4F
14 240 23/20, 31/27 ^^E, v3F
15 257.143 22/19, 36/31 ^3E, vvF
16 274.286 27/23, 34/29 ^4E, vF
17 291.429 13/11, 32/27 F
18 308.571 37/31 ^F, v4G♭
19 325.714 29/24 ^^F, v3G♭
20 342.857 28/23 ^3F, vvG♭
21 360 16/13, 37/30 ^4F, vG♭
22 377.143 36/29 ^5F, G♭
23 394.286 ^6F, v6G
24 411.429 33/26 F♯, v5G
25 428.571 ^F♯, v4G
26 445.714 22/17, 31/24 ^^F♯, v3G
27 462.857 17/13, 30/23 ^3F♯, vvG
28 480 29/22, 37/28 ^4F♯, vG
29 497.143 4/3 G
30 514.286 31/23 ^G, v4A♭
31 531.429 ^^G, v3A♭
32 548.571 11/8, 37/27 ^3G, vvA♭
33 565.714 18/13 ^4G, vA♭
34 582.857 7/5 ^5G, A♭
35 600 17/12, 24/17 ^6G, v6A
36 617.143 10/7 G♯, v5A
37 634.286 13/9 ^G♯, v4A
38 651.429 16/11 ^^G♯, v3A
39 668.571 ^3G♯, vvA
40 685.714 ^4G♯, vA
41 702.857 3/2 A
42 720 ^A, v4B♭
43 737.143 23/15, 26/17 ^^A, v3B♭
44 754.286 17/11 ^3A, vvB♭
45 771.429 ^4A, vB♭
46 788.571 ^5A, B♭
47 805.714 ^6A, v6B
48 822.857 29/18, 37/23 A♯, v5B
49 840 13/8 ^A♯, v4B
50 857.143 23/14 ^^A♯, v3B
51 874.286 ^3A♯, vvB
52 891.429 ^4A♯, vB
53 908.571 22/13, 27/16 B
54 925.714 29/17 ^B, v4C
55 942.857 19/11, 31/18 ^^B, v3C
56 960 ^3B, vvC
57 977.143 37/21 ^4B, vC
58 994.286 16/9 C
59 1011.429 ^C, v4D♭
60 1028.571 29/16 ^^C, v3D♭
61 1045.714 11/6 ^3C, vvD♭
62 1062.857 24/13, 37/20 ^4C, vD♭
63 1080 28/15 ^5C, D♭
64 1097.143 32/17 ^6C, v6D
65 1114.286 C♯, v5D
66 1131.429 ^C♯, v4D
67 1148.571 31/16, 33/17 ^^C♯, v3D
68 1165.714 ^3C♯, vvD
69 1182.857 ^4C♯, vD
70 1200 2/1 D