814edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
814edo is [[consistency|distinctly consistent]] to the [[17-odd-limit]] and is a strong 17-limit system. The equal temperament is [[enfactoring|enfactored]] in the 5-limit, tempering out the [[schisma]] as does 407et. In the 7-limit it tempers out [[2401/2400]] 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]]. | 814edo is [[consistency|distinctly consistent]] to the [[17-odd-limit]] and is a strong 17-limit system. The equal temperament is [[enfactoring|enfactored]] in the 5-limit, tempering out the [[schisma]] as does 407et. In the 7-limit it tempers out [[2401/2400]] 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 {{nowrap|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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== 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]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 70: | Line 71: | ||
{| 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> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Sesquiquartififths]] | | [[Sesquiquartififths]] | ||
|} | |} | ||
<nowiki>* | <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 | ||
[[Category:Sesquiquartififths]] | [[Category:Sesquiquartififths]] | ||
Latest revision as of 13:31, 13 March 2026
| ← 813edo | 814edo | 815edo → |
814 equal divisions of the octave (abbreviated 814edo or 814ed2), also called 814-tone equal temperament (814tet) or 814 equal temperament (814et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 814 equal parts of about 1.47 ¢ each. Each step represents a frequency ratio of 21/814, or the 814th root of 2.
Theory
814edo is distinctly consistent to the 17-odd-limit and is a strong 17-limit system. The equal temperament is enfactored in the 5-limit, tempering out the schisma as does 407et. In the 7-limit it tempers out 2401/2400 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
| 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) | |
Subsets and supersets
Since 814 factors into 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 |
| 2.3.5.7.11.13.17.19 | 1445/1444, 1521/1520, 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2601/2600 | [⟨814 1290 1890 2285 2816 3012 3327 3458]] | +0.0421 | 0.0629 | 4.27 |
| 2.3.5.7.11.13.17.19.23 | 1445/1444, 1521/1520, 1701/1700, 1863/1862, 2058/2057, 2376/2375, 2401/2400, 2601/2600 | [⟨814 1290 1890 2285 2816 3012 3327 3682]] | +0.0439 | 0.0595 | 4.04 |
- 814et is notable in the 17- and 23-limit with lower absolute errors than any previous equal temperaments, beating 764 in the 17-limit and 742i in the 23-limit, and is only bettered by 935 in either subgroup.
Rank-2 temperaments
Note: 5-limit temperaments supported by 407edo are not included.
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 119\814 | 175.43 | 448/405 | Sesquiquartififths |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct