405edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|405}} == Theory == 405et is consistent to the 7-odd-limit. Using the patent val, it tempers out 4096000/4084101, 2460375/2458624, 3355443..."
 
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"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|405}}
{{ED intro}}


== Theory ==
== Theory ==
405et is consistent to the [[7-odd-limit]]. Using the patent val, it tempers out [[4096000/4084101]], 2460375/2458624, [[33554432/33480783]], [[2100875/2097152]] and [[40353607/40310784]] in the 7-limit; 56723625/56689952, 55296000/55240493, [[117440512/117406179]], 820125/819896, [[4000/3993]], 78675968/78594219, 7168000/7144929, 496125/495616, 117649/117612, 1296000/1294139, [[131072/130977]], 1265625/1261568, 42875/42768, 5767168/5764801, 1375/1372, [[19712/19683]], [[41503/41472]], 199297406/199290375, 1362944/1361367, 166375/165888 and 67110351/67108864 in the 11-limit. It supports [[marthirds]], [[novemkleismic]] and [[kleirtismic]]. It provides the [[optimal patent val]] for novemkleismic.
405edo is [[enfactoring|enfactored]] in the 3-limit, with the same tuning as [[135edo]]. Like 135edo, it is [[consistent]] to the [[7-odd-limit]] with a poor approximation to the [[harmonic]] [[5/1|5]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[2100875/2097152]], and 2460375/2458624 in the 7-limit; 1375/1372, [[4000/3993]], [[19712/19683]], and [[41503/41472]] in the 11-limit. It [[support]]s [[marthirds]], [[novemkleismic]] and [[kleirtismic]]. It provides the [[optimal patent val]] for 7- and 11-limit novemkleismic.


=== Prime harmonics ===
=== Prime harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
405 factors into 3<sup>4</sup> × 5, with subset edos {{EDOs|3, 5, 9, 15, 27, 45, 81, and 135}}.
Since 405 factors into {{factorisation|405}}, 405edo has subset edos {{EDOs| 3, 5, 9, 15, 27, 45, 81, and 135 }}.


== 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|214 -135}}
! rowspan="2" | [[Comma list]]
|{{mapping|405 642}}
! rowspan="2" | [[Mapping]]
| -0.0843
! rowspan="2" | Optimal<br />8ve stretch (¢)
| 0.0843
! colspan="2" | Tuning error
| 2.85
|-
|-
|2.3.5
! [[TE error|Absolute]] (¢)
|15625/15552, {{monzo|110 -65 -3}}
! [[TE simple badness|Relative]] (%)
|{{mapping|405 642 940}}
|-
| 2.3.5
| 15625/15552, {{monzo| 110 -65 -3 }}
| {{mapping| 405 642 940 }}
| +0.1058
| +0.1058
| 0.2776
| 0.2776
| 9.37
| 9.37
|-
|-
|2.3.5.7
| 2.3.5.7
|15625/15552, 2460375/2458624, 4096000/4084101
| 15625/15552, 2100875/2097152, 2460375/2458624
|{{mapping|405 642 940 1137}}
| {{mapping| 405 642 940 1137 }}
| +0.0737
| +0.0737
| 0.2467
| 0.2467
| 8.33
| 8.33
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|1375/1372, 42875/42768, 19712/19683, 41503/41472
| 1375/1372, 4000/3993, 19712/19683, 41503/41472
|{{mapping|405 642 940 1137 1401}}
| {{mapping| 405 642 940 1137 1401 }}
| +0.0709
| +0.0709
| 0.2207
| 0.2207
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=== 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
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|56\405
| 56\405
|165.93
| 165.93
|8575/7776
| 11/10
|[[Satin]]
| [[Satin]]
|-
|-
|1
| 1
|107\405
| 107\405
|317.04
| 317.04
|6/5
| 6/5
|[[Hanson]]
| [[Hanson]]
|-
|-
|9
| 9
|107\405<br>(17\405)
| 107\405<br />(17\405)
|317.04<br>(50.37)
| 317.04<br />(50.37)
|6/5<br>(36/35)
| 6/5<br />(36/35)
|[[Novemkleismic]]
| [[Novemkleismic]]
|}
|}
 
<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 13:33, 13 March 2026

← 404edo 405edo 406edo →
Prime factorization 34 × 5
Step size 2.96296 ¢ 
Fifth 237\405 (702.222 ¢) (→ 79\135)
Semitones (A1:m2) 39:30 (115.6 ¢ : 88.89 ¢)
Consistency limit 7
Distinct consistency limit 7

405 equal divisions of the octave (abbreviated 405edo or 405ed2), also called 405-tone equal temperament (405tet) or 405 equal temperament (405et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 405 equal parts of about 2.96 ¢ each. Each step represents a frequency ratio of 21/405, or the 405th root of 2.

Theory

405edo is enfactored in the 3-limit, with the same tuning as 135edo. Like 135edo, it is consistent to the 7-odd-limit with a poor approximation to the harmonic 5. Using the patent val, the equal temperament tempers out 15625/15552 in the 5-limit; 2100875/2097152, and 2460375/2458624 in the 7-limit; 1375/1372, 4000/3993, 19712/19683, and 41503/41472 in the 11-limit. It supports marthirds, novemkleismic and kleirtismic. It provides the optimal patent val for 7- and 11-limit novemkleismic.

Prime harmonics

Approximation of prime harmonics in 405edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.27 -1.13 +0.06 -0.21 +0.95 -1.25 -1.22 -0.13 -1.43 -1.33
Relative (%) +0.0 +9.0 -38.1 +2.1 -7.0 +32.2 -42.2 -41.1 -4.3 -48.2 -45.0
Steps
(reduced) 		405
(0) 		642
(237) 		940
(130) 		1137
(327) 		1401
(186) 		1499
(284) 		1655
(35) 		1720
(100) 		1832
(212) 		1967
(347) 		2006
(386)

Subsets and supersets

Since 405 factors into 34 × 5, 405edo has subset edos 3, 5, 9, 15, 27, 45, 81, and 135.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 15625/15552, [110 -65 -3⟩ [⟨405 642 940]] +0.1058 0.2776 9.37
2.3.5.7 15625/15552, 2100875/2097152, 2460375/2458624 [⟨405 642 940 1137]] +0.0737 0.2467 8.33
2.3.5.7.11 1375/1372, 4000/3993, 19712/19683, 41503/41472 [⟨405 642 940 1137 1401]] +0.0709 0.2207 7.45

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 56\405 165.93 11/10 Satin
1 107\405 317.04 6/5 Hanson
9 107\405
(17\405) 		317.04
(50.37) 		6/5
(36/35) 		Novemkleismic

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

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