405edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
Newer edit →
VisualWikitext
Francium (talk | contribs)
4,827 edits
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..."
 
Review
Line 3: Line 3:


== 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 3<sup>4</sup> × 5, 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" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3.5
|{{monzo|214 -135}}
| 15625/15552, {{monzo| 110 -65 -3 }}
|{{mapping|405 642}}
| {{mapping| 405 642 940 }}
| -0.0843
| 0.0843
| 2.85
|-
|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
Line 60: Line 53:
! 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>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is 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

Revision as of 14:48, 19 January 2024

← 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

Template:EDO intro

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 it is distinct

Retrieved from "https://en.xen.wiki/index.php?title=405edo&oldid=132099"