414edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The '''414 equal divisions of the octave''' ('''414edo'''), or the '''414(-tone) equal temperament''' ('''414tet''', '''414et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 414 parts of about 2.90 [[cent]]s each.
{{EDO intro|414}}


== Theory ==
== Theory ==
414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for 5. It is [[consistent]] to the [[17-odd-limit]], tempering out {{monzo| -36 11 8 }} (submajor comma) and {{monzo|1 -27 18}} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], 1275/1274, and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].
414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1|5]]. It is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the harmonics, making for a good full 17-limit system. The equal temperament [[tempering out|tempers out]] {{monzo| -36 11 8 }} (submajor comma) and {{monzo| 1 -27 18 }} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], [[1275/1274]], and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|414}}
{{Harmonics in equal|414}}
=== Subsets and supersets ===
Since 414 factors into {{factorization|414}}, 414edo has subset edos {{EDOs| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}.


== 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]]
! 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]] (¢)
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| 2.3.5
| 2.3.5
| {{monzo| -36 11 8 }}, {{monzo| 1 -27 18 }}
| {{monzo| -36 11 8 }}, {{monzo| 1 -27 18 }}
| [{{val| 414 656 961 }}]
| {{mapping| 414 656 961 }}
| +0.2222
| +0.2222
| 0.1575
| 0.1575
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 4375/4374, {{monzo| -36 11 8 }}
| 2401/2400, 4375/4374, {{monzo| -36 11 8 }}
| [{{val| 414 656 961 1162 }}]
| {{mapping| 414 656 961 1162 }}
| +0.2299
| +0.2299
| 0.1371
| 0.1371
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 4375/4374, 1366875/1362944
| 2401/2400, 3025/3024, 4375/4374, 1366875/1362944
| [{{val| 414 656 961 1162 1432 }}]
| {{mapping| 414 656 961 1162 1432 }}
| +0.2182
| +0.2182
| 0.1248
| 0.1248
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 625/624, 729/728, 1575/1573, 2200/2197, 2401/2400
| 625/624, 729/728, 1575/1573, 2200/2197, 2401/2400
| [{{val| 414 656 961 1162 1432 1532 }}]
| {{mapping| 414 656 961 1162 1432 1532 }}
| +0.1795
| +0.1795
| 0.1431
| 0.1431
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 625/624, 729/728, 833/832, 1089/1088, 1225/1224, 2200/2197
| 625/624, 729/728, 833/832, 1089/1088, 1225/1224, 2200/2197
| [{{val| 414 656 961 1162 1432 1532 1692 }}]
| {{mapping| 414 656 961 1162 1432 1532 1692 }}
| +0.1751
| +0.1751
| 0.1329
| 0.1329
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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Semihemiennealimmal]]
| [[Semihemiennealimmal]]
|}
|}
 
<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:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 14:43, 6 November 2023

← 413edo 414edo 415edo →
Prime factorization 2 × 32 × 23
Step size 2.89855 ¢ 
Fifth 242\414 (701.449 ¢) (→ 121\207)
Semitones (A1:m2) 38:32 (110.1 ¢ : 92.75 ¢)
Consistency limit 17
Distinct consistency limit 17

Template:EDO intro

Theory

414edo is closely related to 207edo, but the patent vals differ on the mapping for harmonic 5. It is consistent to the 17-odd-limit with a flat tendency for most of the harmonics, making for a good full 17-limit system. The equal temperament tempers out [-36 11 8⟩ (submajor comma) and [1 -27 18⟩ (ennealimma) in the 5-limit; 2401/2400, 4375/4374, and [-37 4 12 1⟩ in the 7-limit; 3025/3024, 9801/9800, 41503/41472, and 1265625/1261568 in the 11-limit; 625/624, 729/728, 1575/1573, 2200/2197, and 26411/26364 in the 13-limit; 833/832, 1089/1088, 1225/1224, 1275/1274, and 1701/1700 in the 17-limit. It supports the 11-limit hemiennealimmal and the 13-limit quatracot.

Prime harmonics

Approximation of prime harmonics in 414edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.51 -0.81 -0.71 -0.59 +0.05 -0.61 +1.04 +0.71 -0.59 -0.11
Relative (%) +0.0 -17.4 -27.8 -24.5 -20.5 +1.8 -21.0 +35.8 +24.5 -20.4 -3.7
Steps
(reduced) 		414
(0) 		656
(242) 		961
(133) 		1162
(334) 		1432
(190) 		1532
(290) 		1692
(36) 		1759
(103) 		1873
(217) 		2011
(355) 		2051
(395)

Subsets and supersets

Since 414 factors into 2 × 32 × 23, 414edo has subset edos 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207.

Regular temperament properties

[Subgroup]] Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3.5 [-36 11 8⟩, [1 -27 18⟩ [⟨414 656 961]] +0.2222 0.1575 5.43
2.3.5.7 2401/2400, 4375/4374, [-36 11 8⟩ [⟨414 656 961 1162]] +0.2299 0.1371 4.73
2.3.5.7.11 2401/2400, 3025/3024, 4375/4374, 1366875/1362944 [⟨414 656 961 1162 1432]] +0.2182 0.1248 4.30
2.3.5.7.11.13 625/624, 729/728, 1575/1573, 2200/2197, 2401/2400 [⟨414 656 961 1162 1432 1532]] +0.1795 0.1431 4.94
2.3.5.7.11.13.17 625/624, 729/728, 833/832, 1089/1088, 1225/1224, 2200/2197 [⟨414 656 961 1162 1432 1532 1692]] +0.1751 0.1329 4.58

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 125\414 362.31 10125/8192 Submajor (5-limit)
2 61\414 176.81 195/176 Quatracot
9 109\414
(17\414) 		315.94
(49.28) 		6/5
(36/35) 		Ennealimmal
18 86\414
(6\414) 		249.28
(17.39) 		231/200
(99/98) 		Hemiennealimmal
18 164\414
(3\414) 		475.36
(8.70) 		1053/800
(1287/1280) 		Semihemiennealimmal

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

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