814edo: Difference between revisions

Revision as of 12:30, 16 November 2024

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.

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
Error	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)

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

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	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.

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

Table of rank-2 temperaments by generator
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 it is distinct

@@ Line 12: / Line 12: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| 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
@@ Line 55: / Line 64: @@
 | 0.0595
 | 4.04
-{{comma basis end}}
+|}
 * 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.
@@ Line 61: / Line 70: @@
 Note: 5-limit temperaments supported by 407edo are not included.
-{{rank-2 begin}}
+{| 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
@@ Line 68: / Line 84: @@
 | 448/405
 | [[Sesquiquartififths]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 [[Category:Sesquiquartififths]]