412edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
VisualWikitext
Francium (talk | contribs)
4,827 edits
Created page with "{{Infobox ET}} {{EDO intro|412}} == Theory == 412et tempers out 2460375/2458624, 6144/6125, 102760448/102515625, 1640558367/1638400000 and 200120949/200000000 in the..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
(6 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|412}}
{{ED intro}}


== Theory ==
== Theory ==
412et tempers out 2460375/2458624, [[6144/6125]], 102760448/102515625, [[1640558367/1638400000]] and 200120949/200000000 in the 7-limit. It supports [[nanic]] and [[counterschismic]].
412edo has a very accurate [[3/2|perfect fifth]], but it is not quite accurate beyond that. The equal temperament [[tempering out|tempers out]] {{monzo| 32 -7 -9 }} ([[escapade comma]]) and {{monzo| -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 ===
=== Prime harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
412 factors into 2<sup>2</sup> × 103, with subset edos {{EDOs|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to the harmonic 11.
412 factors into {{factorisation|412}}, with subset edos {{EDOs|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to harmonics 5, 7, and 11.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! 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
! rowspan="2" | [[Subgroup]]
|{{monzo|-653 412}}
! rowspan="2" | [[Comma list]]
|{{mapping|412 653}}
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo|-653 412}}
| {{mapping| 412 653 }}
| +0.0042
| +0.0042
| 0.0042
| 0.0042
| 0.14
| 0.14
|-
|-
|2.3.5
| 2.3.5
|{{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
| −0.1501
| 0.2182
| 0.2182
| 7.49
| 7.49
|-
|-
|2.3.5.7
| 2.3.5.7
|6144/6125, 2460375/2458624, 100442349/100000000
| 6144/6125, 2460375/2458624, 49009212/48828125
|{{mapping|412 653 957 1157}}
| {{mapping| 412 653 957 1157 }}
| -0.2085
| −0.2085
| 0.2143
| 0.2143
| 7.36
| 7.36
Line 46: Line 47:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)*
! Periods<br />per 8ve
! Cents<br>(reduced)*
! Generator*
! Associated<br>Ratio*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|9\412
| 9\412
|26.21
| 26.21
|49/48
| 49/48
|[[Sfourth]]
| [[Sfourth]] (5-limit)
|-
|-
|1
| 1
|19\412
| 19\412
|55.34
| 55.34
|16875/16384
| 16875/16384
|[[Escapade]]
| [[Escapade]] (5-limit)
|-
|-
|1
| 1
|171\412
| 171\412
|498.06
| 498.06
|4/3
| 4/3
|[[Counterschismic]]
| [[Counterschismic]]<br>[[Nanic]]
|-
|-
|2
| 2
|19\412
| 19\412
|55.34
| 55.34
|16875/16384
| 16875/16384
|[[Semisuperfourth]]
| [[Septisuperfourth]] (7-limit)
|}
|}
 
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Latest revision as of 13:31, 13 March 2026

← 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

412 equal divisions of the octave (abbreviated 412edo or 412ed2), also called 412-tone equal temperament (412tet) or 412 equal temperament (412et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 412 equal parts of about 2.91 ¢ each. Each step represents a frequency ratio of 21/412, or the 412th root of 2.

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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
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

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
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)

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

Retrieved from "https://en.xen.wiki/index.php?title=412edo&oldid=225759"