# 451edo

 ← 450edo 451edo 452edo →
Prime factorization 11 × 41
Step size 2.66075¢
Fifth 264\451 (702.439¢) (→24\41)
Semitones (A1:m2) 44:33 (117.1¢ : 87.8¢)
Consistency limit 7
Distinct consistency limit 7

451 equal divisions of the octave (abbreviated 451edo), or 451-tone equal temperament (451tet), 451 equal temperament (451et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 451 equal parts of about 2.66 ¢ each. Each step of 451edo represents a frequency ratio of 21/451, or the 451st root of 2.

## Theory

451 = 11 × 41, and 451edo shares its fifth with 41edo. Unlike 41, however, 451 is only consistent to the 7-odd-limit, though it has a reasonable approximation up to the 13-limit using the patent val. The equal temperament tempers out 2401/2400, 65625/65536, 703125/702464, 2100875/2097152, and 390625000/387420489 in the 7-limit; 6250/6237, 42592/42525, 42875/42768, 43923/43904 in the 11-limit; and 625/624, 2080/2079, 2200/2197, 4096/4095, 4225/4224, 4459/4455, and 17303/17280 in the 13-limit. It supports tertiaseptal, tertiseptisix, and hemermacomp.

### Prime harmonics

Approximation of prime harmonics in 451edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.00 +0.48 -0.50 -0.31 -0.54 +0.27 -1.19 +0.49 -0.34 +0.13 -0.91
relative (%) +0 +18 -19 -12 -20 +10 -45 +18 -13 +5 -34
Steps
(reduced)
451
(0)
715
(264)
1047
(145)
1266
(364)
1560
(207)
1669
(316)
1843
(39)
1916
(112)
2040
(236)
2191
(387)
2234
(430)

### Subsets and supersets

Since 451 factors into 11 × 41, 451edo has 11edo and 41edo as its subsets.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5 [3 -18 11, [-59 5 22 [451 715 1047]] -0.0294 0.2144 8.06
2.3.5.7 2401/2400, 65625/65536, 390625000/387420489 [451 715 1047 126 6]] +0.0057 0.1953 7.34
2.3.5.7.11 2401/2400, 6250/6237, 42592/42525, 43923/43904 [451 715 1047 1266 1560]] +0.0359 0.1849 6.95
2.3.5.7.11.13 625/624, 2080/2079, 2200/2197, 2401/2400, 17303/17280 [451 715 1047 1266 1560 1669]] +0.0177 0.1736 6.52

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 29\451 77.16 256/245 Tertiaseptal

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