451edo

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← 450edo451edo452edo →
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