70edo

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← 69edo 70edo 71edo →
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 perfect fifth, which is the 4th number in the convergent sequence to the silver ratio, following 29edo, 12edo, and 5edo and preceding 169edo. It is the last edo to have exactly one diatonic perfect fifth, and this perfect 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 17.1 ^D, ^3E♭♭
2 34.3 ^^D, v3E♭
3 51.4 32/31, 33/32, 34/33 ^3D, vvE♭
4 68.6 27/26 v3D♯, vE♭
5 85.7 21/20 vvD♯, E♭
6 102.9 17/16 vD♯, ^E♭
7 120 15/14, 29/27 D♯, ^^E♭
8 137.1 13/12 ^D♯, ^3E♭
9 154.3 12/11, 23/21 ^^D♯, v3E
10 171.4 32/29 ^3D♯, vvE
11 188.6 29/26 v3D𝄪, vE
12 205.7 9/8 E
13 222.9 33/29 ^E, ^3F♭
14 240 23/20, 31/27 ^^E, v3F
15 257.1 22/19, 36/31 ^3E, vvF
16 274.3 27/23, 34/29 v3E♯, vF
17 291.4 13/11, 32/27 F
18 308.6 37/31 ^F, ^3G♭♭
19 325.7 29/24 ^^F, v3G♭
20 342.9 28/23 ^3F, vvG♭
21 360 16/13, 37/30 v3F♯, vG♭
22 377.1 36/29 vvF♯, G♭
23 394.3 vF♯, ^G♭
24 411.4 33/26 F♯, ^^G♭
25 428.6 ^F♯, ^3G♭
26 445.7 22/17, 31/24 ^^F♯, v3G
27 462.9 17/13, 30/23 ^3F♯, vvG
28 480 29/22, 37/28 v3F𝄪, vG
29 497.1 4/3 G
30 514.3 31/23 ^G, ^3A♭♭
31 531.4 ^^G, v3A♭
32 548.6 11/8, 37/27 ^3G, vvA♭
33 565.7 18/13 v3G♯, vA♭
34 582.9 7/5 vvG♯, A♭
35 600 17/12, 24/17 vG♯, ^A♭
36 617.1 10/7 G♯, ^^A♭
37 634.3 13/9 ^G♯, ^3A♭
38 651.4 16/11 ^^G♯, v3A
39 668.6 ^3G♯, vvA
40 685.7 v3G𝄪, vA
41 702.9 3/2 A
42 720 ^A, ^3B♭♭
43 737.1 23/15, 26/17 ^^A, v3B♭
44 754.3 17/11 ^3A, vvB♭
45 771.4 v3A♯, vB♭
46 788.6 vvA♯, B♭
47 805.7 vA♯, ^B♭
48 822.9 29/18, 37/23 A♯, ^^B♭
49 840 13/8 ^A♯, ^3B♭
50 857.1 23/14 ^^A♯, v3B
51 874.3 ^3A♯, vvB
52 891.4 v3A𝄪, vB
53 908.6 22/13, 27/16 B
54 925.7 29/17 ^B, ^3C♭
55 942.9 19/11, 31/18 ^^B, v3C
56 960 ^3B, vvC
57 977.1 37/21 v3B♯, vC
58 994.3 16/9 C
59 1011.4 ^C, ^3D♭♭
60 1028.6 29/16 ^^C, v3D♭
61 1045.7 11/6 ^3C, vvD♭
62 1062.9 24/13, 37/20 v3C♯, vD♭
63 1080 28/15 vvC♯, D♭
64 1097.1 32/17 vC♯, ^D♭
65 1114.3 C♯, ^^D♭
66 1131.4 ^C♯, ^3D♭
67 1148.6 31/16, 33/17 ^^C♯, v3D
68 1165.7 ^3C♯, vvD
69 1182.9 v3C𝄪, vD
70 1200 2/1 D

Notation

Ups and downs notation

70edo can be notated using ups and downs. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol                
                 
Flat symbol              
                 

Alternatively, sharps and flats with arrows borrowed from Helmholtz–Ellis notation can be used:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Sharp symbol  
Flat symbol

Sagittal notation

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8055/5433/32

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8055/5433/32

Approximation to JI

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 70edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 70edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/5, 10/7 0.345 2.0
13/8, 16/13 0.528 3.1
15/14, 28/15 0.557 3.3
3/2, 4/3 0.902 5.3
13/12, 24/13 1.430 8.3
9/8, 16/9 1.804 10.5
13/11, 22/13 2.219 12.9
13/9, 18/13 2.332 13.6
11/8, 16/11 2.747 16.0
11/6, 12/11 3.649 21.3
11/9, 18/11 4.551 26.5
15/11, 22/15 5.522 32.2
11/7, 14/11 6.079 35.5
9/5, 10/9 6.168 36.0
11/10, 20/11 6.424 37.5
9/7, 14/9 6.513 38.0
5/3, 6/5 7.070 41.2
7/6, 12/7 7.415 43.3
15/13, 26/15 7.741 45.2
5/4, 8/5 7.972 46.5
15/8, 16/15 8.269 48.2
13/7, 14/13 8.298 48.4
7/4, 8/7 8.317 48.5
13/10, 20/13 8.500 49.6
15-odd-limit intervals in 70edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/5, 10/7 0.345 2.0
13/8, 16/13 0.528 3.1
15/14, 28/15 0.557 3.3
3/2, 4/3 0.902 5.3
13/12, 24/13 1.430 8.3
9/8, 16/9 1.804 10.5
13/11, 22/13 2.219 12.9
13/9, 18/13 2.332 13.6
11/8, 16/11 2.747 16.0
11/6, 12/11 3.649 21.3
11/9, 18/11 4.551 26.5
9/5, 10/9 6.168 36.0
9/7, 14/9 6.513 38.0
5/3, 6/5 7.070 41.2
7/6, 12/7 7.415 43.3
5/4, 8/5 7.972 46.5
7/4, 8/7 8.317 48.5
13/10, 20/13 8.500 49.6
13/7, 14/13 8.845 51.6
15/8, 16/15 8.874 51.8
15/13, 26/15 9.402 54.8
11/10, 20/11 10.719 62.5
11/7, 14/11 11.063 64.5
15/11, 22/15 11.621 67.8
15-odd-limit intervals by 70cd val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/5, 10/7 0.345 2.0
13/8, 16/13 0.528 3.1
15/14, 28/15 0.557 3.3
3/2, 4/3 0.902 5.3
13/12, 24/13 1.430 8.3
9/8, 16/9 1.804 10.5
13/11, 22/13 2.219 12.9
13/9, 18/13 2.332 13.6
11/8, 16/11 2.747 16.0
11/6, 12/11 3.649 21.3
11/9, 18/11 4.551 26.5
15/11, 22/15 5.522 32.2
11/7, 14/11 6.079 35.5
11/10, 20/11 6.424 37.5
15/13, 26/15 7.741 45.2
15/8, 16/15 8.269 48.2
13/7, 14/13 8.298 48.4
13/10, 20/13 8.643 50.4
7/4, 8/7 8.826 51.5
5/4, 8/5 9.171 53.5
7/6, 12/7 9.728 56.7
5/3, 6/5 10.073 58.8
9/7, 14/9 10.630 62.0
9/5, 10/9 10.975 64.0

Scales

Kukula's 2.3.13 70edo MOS

In July 2025, composer and theorist James Kukula created a 17-tone MOS scale for his piece in the Monthly Tunings project. The scale is generated by stacking the interval 33\70, 17 times, then octave reducing the result. It is designed to approximate the 2.3.13 subgroup very accurately. He discusses it in his blog post titled Stepping Outside.

Subsets

Instruments

A Lumatone mapping for 70edo is available.

Music

Bryan Deister
James Kukula
Budjarn Lambeth
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