# 742edo

 ← 741edo 742edo 743edo →
Prime factorization 2 × 7 × 53
Step size 1.61725¢
Fifth 434\742 (701.887¢) (→31\53)
Semitones (A1:m2) 70:56 (113.2¢ : 90.57¢)
Consistency limit 21
Distinct consistency limit 21
Special properties

742 equal divisions of the octave (abbreviated 742edo or 742ed2), also called 742-tone equal temperament (742tet) or 742 equal temperament (742et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 742 equal parts of about 1.62 ¢ each. Each step represents a frequency ratio of 21/742, or the 742nd root of 2.

## Theory

742edo is a very strong 19-limit system and a zeta peak edo, and is distinctly consistent in the 21-odd-limit. The equal temperament tempers out the vishnuzma and the fortune comma in the 5-limit, supporting vishnu and fortune; 2401/2400 in the 7-limit, 9801/9800 in the 11-limit, 4096/4095, 6656/6655, 10648/10647 in the 13-limit, 1701/1700, 2058/2057, 2601/2600, 4914/4913, 5832/5831 in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.

### Prime harmonics

Approximation of prime harmonics in 742edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.068 +0.209 -0.093 +0.165 +0.443 +0.166 +0.061 -0.781 +0.611 -0.022
Relative (%) +0.0 -4.2 +12.9 -5.7 +10.2 +27.4 +10.3 +3.8 -48.3 +37.8 -1.4
Steps
(reduced)
742
(0)
1176
(434)
1723
(239)
2083
(599)
2567
(341)
2746
(520)
3033
(65)
3152
(184)
3356
(388)
3605
(637)
3676
(708)

### Subsets and supersets

Since 742 factors into 2 × 7 × 53, 742edo has subset edos 2, 7, 14, 53, 106, and 371, of which 7edo, 14edo and 53edo are very notable. It supports silicon (224 & 518) with period 14 in the 13-limit, and iodine (159 & 583f) with period 53 in the 17-limit.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5 [23 6 -14, [-84 53 [742 1176 1723]] -0.0157 0.0555 3.43
2.3.5.7 2401/2400, 14348907/14336000, [23 6 -14 [742 1176 1723 2083]] -0.0035 0.0525 3.24
2.3.5.7.11 2401/2400, 9801/9800, 172032/171875, 1240029/1239040 [742 1176 1723 2083 2567]] -0.0123 0.0501 3.10
2.3.5.7.11.13 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325 [742 1176 1723 2083 2567 2746]] -0.0302 0.0608 3.76
2.3.5.7.11.13.17 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655 [742 1176 1723 2083 2567 2746 3033]] -0.0317 0.0564 3.49
2.3.5.7.11.13.17.19 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211 [742 1176 1723 2083 2567 2746 3033 3152]] -0.0295 0.0531 3.28
2.3.5.7.11.13.17.19.23 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211 [742 1176 1723 2083 2567 2746 3033 3152 3357]] (742i) -0.0468 0.0699 4.32
• 742et has a lower 19-limit relative error than any previous equal temperaments. It is only bettered by 935 in terms of absolute error, and by 1178 in terms of relative error.
• 742et (742i val) is also notable in the 17- and 23-limit, where it has lower absolute errors than any previous equal temperaments. In the 17-limit it beats 581 and is bettered by 764; in the 23-limit it beats 718 and is bettered by 814.

### Rank-2 temperaments

Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 137\742 221.563 8388608/7381125 Fortune
1 303\742 490.026 896/675 Surmarvelpyth
2 44\742 71.159 25/24 Vishnu
14 434\742
(10\742)
701.886
(16.173)
3/2
(105/104)
Silicon
53 239\742
(1\742)
386.523
(1.617)
5/4
(32805/32768)
Mercator
53 565\742
(5\742)
913.746
(8.086)
441/260
(196/195)
Iodine

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