412edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
Line 32: Line 24:
| {{monzo| 32 -7 -9 }}, {{monzo| -5 31 -19 }}
| {{monzo| 32 -7 -9 }}, {{monzo| -5 31 -19 }}
| {{mapping| 412 653 957 }}
| {{mapping| 412 653 957 }}
| -0.1501
| &minus;0.1501
| 0.2182
| 0.2182
| 7.49
| 7.49
Line 39: Line 31:
| 6144/6125, 2460375/2458624, 49009212/48828125
| 6144/6125, 2460375/2458624, 49009212/48828125
| {{mapping| 412 653 957 1157 }}
| {{mapping| 412 653 957 1157 }}
| -0.2085
| &minus;0.2085
| 0.2143
| 0.2143
| 7.36
| 7.36
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 76: Line 62:
| 16875/16384
| 16875/16384
| [[Septisuperfourth]] (7-limit)
| [[Septisuperfourth]] (7-limit)
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 01:57, 16 November 2024

← 411edo 412edo 413edo →
Prime factorization 22 × 103
Step size 2.91262 ¢ 
Fifth 241\412 (701.942 ¢)
Semitones (A1:m2) 39:31 (113.6 ¢ : 90.29 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

Theory

412edo has a very accurate perfect fifth, but it is not quite accurate beyond that. The equal temperament tempers out [32 -7 -9⟩ (escapade comma) and [-69 45 -1⟩ (counterschisma) in the 5-limit; 6144/6125, 118098/117649, 2460375/2458624, 49009212/48828125, and notably the nanisma in the 7-limit. It supports nanic and counterschismic.

Prime harmonics

Approximation of prime harmonics in 412edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.01 +1.06 +1.08 -0.83 +1.22 -0.10 -0.43 +0.85 -1.42 -0.38
Relative (%) +0.0 -0.5 +36.6 +37.0 -28.6 +41.9 -3.5 -14.6 +29.2 -48.8 -12.9
Steps
(reduced) 		412
(0) 		653
(241) 		957
(133) 		1157
(333) 		1425
(189) 		1525
(289) 		1684
(36) 		1750
(102) 		1864
(216) 		2001
(353) 		2041
(393)

Subsets and supersets

412 factors into 22 × 103, with subset edos 2, 4, 103, and 206. 1236edo, which triples it, gives a good correction to harmonics 5, 7, and 11.

Regular temperament properties

Template:Comma basis begin |- | 2.3 | [-653 412⟩ | [⟨412 653]] | +0.0042 | 0.0042 | 0.14 |- | 2.3.5 | [32 -7 -9⟩, [-5 31 -19⟩ | [⟨412 653 957]] | −0.1501 | 0.2182 | 7.49 |- | 2.3.5.7 | 6144/6125, 2460375/2458624, 49009212/48828125 | [⟨412 653 957 1157]] | −0.2085 | 0.2143 | 7.36 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 9\412 | 26.21 | 49/48 | Sfourth (5-limit) |- | 1 | 19\412 | 55.34 | 16875/16384 | Escapade (5-limit) |- | 1 | 171\412 | 498.06 | 4/3 | Counterschismic
Nanic |- | 2 | 19\412 | 55.34 | 16875/16384 | Septisuperfourth (7-limit) Template:Rank-2 end Template:Orf

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