407edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|407}} == Theory == 407et tempers out 32805/32768 in the 5-limit; 4096000/4084101, 134217728/133984375, 26873856/26796875, 78125000/781218..."
 
Review
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== Theory ==
== Theory ==
407et tempers out [[32805/32768]] in the 5-limit; [[4096000/4084101]], 134217728/133984375, 26873856/26796875, [[78125000/78121827]] and 48828125/48771072 in the 7-limit. It supports the pinkanberry chords and the temperament [[subsemifourth]].
407edo is a strong 5-limit system and 2.3.5.11.13.19.23 [[subgroup]] system. The equal temperament [[tempering out|tempers out]] [[32805/32768]] in the 5-limit; using the [[patent val]], [[16875/16807]], [[4096000/4084101]], and 26873856/26796875 in the 7-limit. It [[support]]s and provides the [[optimal patent val]] for the [[subsemifourth]] temperament in the 7- and 11-limit. [[Essentially tempered chord]]s available in 407et include [[pinkanberry chords]].  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
407 factors into 11 × 37, with [[11edo]] and [[37edo]] as its subset edos.
407 factors into 11 × 37, with [[11edo]] and [[37edo]] as its subset edos. [[814edo]], which doubles it, gives a good correction to harmonics 7 and 17, and is a notable full 23-limit temperament.  


== 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
|{{monzo|-645 407}}
| {{monzo| -645 407 }}
|{{mapping|407 645}}
| {{mapping| 407 645 }}
| 0.0742
| +0.0742
| 0.0742
| 0.0742
| 2.52
| 2.52
|-
|-
|2.3.5
| 2.3.5
|32805/32768, {{monzo|30 47 -45}}
| 32805/32768, {{monzo| 30 47 -45 }}
|{{mapping|407 645 945}}
| {{mapping| 407 645 945 }}
| 0.0599
| +0.0599
| 0.0638
| 0.0638
| 2.16
| 2.16
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)*
! Generator*
! Cents<br>(reduced)*
! Cents*
! Associated<br>Ratio*
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|63\407
| 63\407
|185.75
| 185.75
|{{monzo|24 4 -13}}
| {{monzo| 24 4 -13 }}
|[[Pirate]]
| [[Pirate]]
|-
| 1
| 83\407
| 244.72
| 15/13
| [[Subsemifourth]] (407f)
|-
|-
|1
| 1
|169\407
| 169\407
|498.28
| 498.28
|4/3
| 4/3
|[[Helmholtz]]
| [[Helmholtz]]
|}
|}
<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
[[Category:Subsemifourth]]

Revision as of 11:12, 1 January 2024

← 406edo 407edo 408edo →
Prime factorization 11 × 37
Step size 2.9484 ¢ 
Fifth 238\407 (701.72 ¢)
Semitones (A1:m2) 38:31 (112 ¢ : 91.4 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

407edo is a strong 5-limit system and 2.3.5.11.13.19.23 subgroup system. The equal temperament tempers out 32805/32768 in the 5-limit; using the patent val, 16875/16807, 4096000/4084101, and 26873856/26796875 in the 7-limit. It supports and provides the optimal patent val for the subsemifourth temperament in the 7- and 11-limit. Essentially tempered chords available in 407et include pinkanberry chords.

Prime harmonics

Approximation of prime harmonics in 407edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.24 -0.07 +1.20 +0.03 -0.23 +1.19 +0.28 -0.26 -0.58 -1.06
Relative (%) +0.0 -8.0 -2.5 +40.7 +1.1 -7.9 +40.3 +9.4 -9.0 -19.8 -35.8
Steps
(reduced) 		407
(0) 		645
(238) 		945
(131) 		1143
(329) 		1408
(187) 		1506
(285) 		1664
(36) 		1729
(101) 		1841
(213) 		1977
(349) 		2016
(388)

Subsets and supersets

407 factors into 11 × 37, with 11edo and 37edo as its subset edos. 814edo, which doubles it, gives a good correction to harmonics 7 and 17, and is a notable full 23-limit temperament.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-645 407⟩ [⟨407 645]] +0.0742 0.0742 2.52
2.3.5 32805/32768, [30 47 -45⟩ [⟨407 645 945]] +0.0599 0.0638 2.16

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 63\407 185.75 [24 4 -13⟩ Pirate
1 83\407 244.72 15/13 Subsemifourth (407f)
1 169\407 498.28 4/3 Helmholtz

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

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