814edo: Difference between revisions
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{{Infobox ET}} | |||
[[Category: | {{ED intro}} | ||
== 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 {{nowrap|171 & 643}} temperament gives an extension of sesquiquartififths to the 17-limit for which 814edo provides the [[optimal patent val]]. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|814}} | |||
=== Subsets and supersets === | |||
Since 814 factors into {{factorization|814}}, 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]] | |||
! 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 }} | |||
| {{mapping| 814 1290 1890 2285 }} | |||
| +0.0695 | |||
| 0.0577 | |||
| 3.91 | |||
|- | |||
| 2.3.5.7.11 | |||
| 2401/2400, 9801/9800, 32805/32768, 20155392/20131375 | |||
| {{mapping| 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 | |||
| {{mapping| 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 | |||
| {{mapping| 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 | |||
| {{mapping| 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 | |||
| {{mapping| 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 [[764edo|764]] in the 17-limit and [[742edo|742i]] in the 23-limit, 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" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 119\814 | |||
| 175.43 | |||
| 448/405 | |||
| [[Sesquiquartififths]] | |||
|} | |||
<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]] | |||