814edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The '''814 equal division''' divides the [[octave]] into 814 [[equal]] parts of 1.474 [[cent]]s each. It is uniquely [[consistent]] to the [[17-odd-limit]] and is a strong 17-limit system. It tempers out [[32805/32768]] in the 5-limit and [[2401/2400]] in the 7-limit, so that it [[support]]s and gives a good tuning for [[sesquiquartififths]]. In the 11-limit it tempers out [[9801/9800]], in the 13-limit [[4225/4224]] and [[6656/6655]], and in the 17-limit [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[5832/5831]]. The 171&643 temperament gives an extension of sesquiquartififths to the 17-limit for which 814edo provides the [[optimal patent val]].
{{EDO intro|814}}
 
== Theory ==
814edo is uniquely [[consistent]] to the [[17-odd-limit]] and is a strong 17-limit system. It tempers out [[32805/32768]] in the 5-limit and [[2401/2400]] in the 7-limit, so that it [[support]]s and gives a good tuning for [[sesquiquartififths]]. In the 11-limit it tempers out [[9801/9800]], in the 13-limit [[4225/4224]] and [[6656/6655]], and in the 17-limit [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[5832/5831]]. The 171&643 temperament gives an extension of sesquiquartififths to the 17-limit for which 814edo provides the [[optimal patent val]].


=== Prime harmonics ===
=== Prime harmonics ===
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=== Miscellany ===
=== Miscellany ===
Since 814 = 2 × 11 × 37, 814edo has subset edos {{EDOs| 2, 11, 22, 37, 74, and 407 }}.  
Since 814 = 2 × 11 × 37, 814edo has subset edos {{EDOs| 2, 11, 22, 37, 74, and 407 }}.  
== Regular temperament properties ==
{| 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]] (%)
|-
| 2.3.5.7
| 2401/2400, 32805/32768, {{monzo| 25 20 -22 -2 }}
| [{{val| 814 1290 1890 2285 }}]
| +0.0695
| 0.0577
| 3.91
|-
| 2.3.5.7.11
| 2401/2400, 9801/9800, 32805/32768, 20155392/20131375
| [{{val| 814 1290 1890 2285 2816 }}]
| +0.0536
| 0.0605
| 4.11
|-
| 2.3.5.7.11.13
| 2401/2400, 4225/4224, 6656/6655, 9801/9800, 32805/32768
| [{{val| 814 1290 1890 2285 2816 3012 }}]
| +0.0552
| 0.0554
| 3.76
|-
| 2.3.5.7.11.13.17
| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4225/4224, 6656/6655
| [{{val| 814 1290 1890 2285 2816 3012 3327 }}]
| +0.0573
| 0.0528
| 3.50
|}
* 814et is notable in the 17- and 23-limit, having lower absolute errors than any previous equal temperaments, and is only bettered by [[935edo|935]] in either subgroup.
=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by 407edo are not included.
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per Octave
! Generator<br>(Reduced)
! Cents<br>(Reduced)
! Associated<br>Ratio
! Temperaments
|-
| 1
| 119\814
| 175.43
| 448/405
| [[Sesquiquartififths]]
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Sesquiquartififths]]
[[Category:Sesquiquartififths]]

Revision as of 11:40, 17 December 2022

← 813edo 814edo 815edo →
Prime factorization 2 × 11 × 37
Step size 1.4742 ¢ 
Fifth 476\814 (701.72 ¢) (→ 238\407)
Semitones (A1:m2) 76:62 (112 ¢ : 91.4 ¢)
Consistency limit 17
Distinct consistency limit 17

Template:EDO intro

Theory

814edo is uniquely consistent to the 17-odd-limit and is a strong 17-limit system. It tempers out 32805/32768 in the 5-limit and 2401/2400 in the 7-limit, so that it supports and gives a good tuning for sesquiquartififths. In the 11-limit it tempers out 9801/9800, in the 13-limit 4225/4224 and 6656/6655, and in the 17-limit 1701/1700, 2058/2057, 2601/2600, 4914/4913 and 5832/5831. The 171&643 temperament gives an extension of sesquiquartififths to the 17-limit for which 814edo provides the optimal patent val.

Prime harmonics

Approximation of prime harmonics in 814edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.235 -0.073 -0.276 +0.033 -0.233 -0.287 +0.276 -0.265 -0.585 +0.419
Relative (%) +0.0 -15.9 -4.9 -18.7 +2.3 -15.8 -19.5 +18.7 -17.9 -39.7 +28.4
Steps
(reduced) 		814
(0) 		1290
(476) 		1890
(262) 		2285
(657) 		2816
(374) 		3012
(570) 		3327
(71) 		3458
(202) 		3682
(426) 		3954
(698) 		4033
(777)

Miscellany

Since 814 = 2 × 11 × 37, 814edo has subset edos 2, 11, 22, 37, 74, and 407.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3.5.7 2401/2400, 32805/32768, [25 20 -22 -2⟩ [⟨814 1290 1890 2285]] +0.0695 0.0577 3.91
2.3.5.7.11 2401/2400, 9801/9800, 32805/32768, 20155392/20131375 [⟨814 1290 1890 2285 2816]] +0.0536 0.0605 4.11
2.3.5.7.11.13 2401/2400, 4225/4224, 6656/6655, 9801/9800, 32805/32768 [⟨814 1290 1890 2285 2816 3012]] +0.0552 0.0554 3.76
2.3.5.7.11.13.17 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4225/4224, 6656/6655 [⟨814 1290 1890 2285 2816 3012 3327]] +0.0573 0.0528 3.50
  • 814et is notable in the 17- and 23-limit, having lower absolute errors than any previous equal temperaments, and is only bettered by 935 in either subgroup.

Rank-2 temperaments

Note: 5-limit temperaments supported by 407edo are not included.

Table of rank-2 temperaments by generator
Periods
per Octave 		Generator
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
1 119\814 175.43 448/405 Sesquiquartififths
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