# 718edo

 ← 717edo 718edo 719edo →
Prime factorization 2 × 359
Step size 1.67131¢
Fifth 420\718 (701.95¢) (→210\359)
Semitones (A1:m2) 68:54 (113.6¢ : 90.25¢)
Consistency limit 23
Distinct consistency limit 23

718 equal divisions of the octave (abbreviated 718edo or 718ed2), also called 718-tone equal temperament (718tet) or 718 equal temperament (718et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 718 equal parts of about 1.67 ¢ each. Each step represents a frequency ratio of 21/718, or the 718th root of 2.

## Theory

718edo is distinctly consistent in the 23-odd-limit, and does well enough in the 31-limit. It is closely related to 359edo, but the mapping differs for 5, 13, 17 and 31.

As does 359et, 718et tempers out the 359-comma in the 3-limit, rendering a very accurate harmonic 3. In the 5-limit it tempers out the gammic comma, [-29 -11 20, and the monzisma, [54 -37 2. In the 7-limit it tempers out 4375/4374; in the 11-limit 3025/3024, 9801/9800 and 131072/130977; in the 13-limit 1716/1715, 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647; in the 17-limit 1275/1274, 2025/2023; in the 19-limit 2432/2431, 3250/3249, 4200/4199 and 5985/5984; and in the 23-limit 2024/2023, 2025/2024, 2185/2184, 3060/3059. It supports gammic, monzismic and abigail.

### Prime harmonics

Approximation of prime harmonics in 718edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.005 -0.241 +0.533 +0.214 +0.141 +0.337 -0.020 +0.138 -0.051 -0.189
Relative (%) +0.0 -0.3 -14.4 +31.9 +12.8 +8.4 +20.2 -1.2 +8.3 -3.0 -11.3
Steps
(reduced)
718
(0)
1138
(420)
1667
(231)
2016
(580)
2484
(330)
2657
(503)
2935
(63)
3050
(178)
3248
(376)
3488
(616)
3557
(685)

### Subsets and supersets

Since 718 factors into 2 × 359, 718edo contains 2edo and 359edo as subsets.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5 [-29 -11 20, [54 -37 2 [718 1138 1667]] +0.0357 0.0482 2.89
2.3.5.7 4375/4374, 40500000/40353607, [31 -6 -2 -6 [718 1138 1667 2016]] -0.0207 0.1063 6.36
2.3.5.7.11 3025/3024, 4375/4374, 131072/130977, 40500000/40353607 [718 1138 1667 2016 2484]] -0.0290 0.0965 5.77
2.3.5.7.11.13 1716/1715, 2080/2079, 3025/3024, 4096/4095, 7031250/7014007 [718 1138 1667 2016 2484 2657]] -0.0305 0.0881 5.27
2.3.5.7.11.13.17 1275/1274, 1716/1715, 2025/2023, 2080/2079, 3025/3024, 4096/4095 [718 1138 1667 2016 2484 2657 2935]] -0.0379 0.0836 5.00
2.3.5.7.11.13.17.19 1275/1274, 1716/1715, 2025/2023, 2080/2079, 2432/2431, 3025/3024, 3250/3249 [718 1138 1667 2016 2484 2657 2935 3050]] -0.0326 0.0795 4.76
2.3.5.7.11.13.17.19.23 1275/1274, 1716/1715, 2024/2023, 2025/2023, 2080/2079, 2185/2184, 2432/2431, 3025/3024 [718 1138 1667 2016 2484 2657 2935 3050 3248]] -0.0323 0.0749 4.48
• 718et has a lower absolute error in the 23-limit than any previous equal temperaments, past 581 and followed by 742i.

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 21\718 35.10 234375/229376 Gammic
1 249\718 249.03 [-27 11 3 1 Monzismic
2 125\718 208.91 44/39 Abigail

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct