718edo: Difference between revisions

718edo is distinctly consistent in the 23-odd-limit, and does well enough in the 31-limit. It is closely related to 359edo, but the mapping differs for 5, 13, 17 and 31.

As does 359et, 718et tempers out the 359-comma in the 3-limit, rendering a very accurate harmonic 3. In the 5-limit it tempers out the gammic comma, [-29 -11 20⟩, and the monzisma, [54 -37 2⟩. In the 7-limit it tempers out 4375/4374; in the 11-limit 3025/3024, 9801/9800 and 131072/130977; in the 13-limit 1716/1715, 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647; in the 17-limit 1275/1274, 2025/2023; in the 19-limit 2432/2431, 3250/3249, 4200/4199 and 5985/5984; and in the 23-limit 2024/2023, 2025/2024, 2185/2184, 3060/3059. It supports gammic, monzismic and abigail.

Approximation of prime harmonics in 718edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	-0.005	-0.241	+0.533	+0.214	+0.141	+0.337	-0.020	+0.138	-0.051	-0.189
Error	Relative (%)	+0.0	-0.3	-14.4	+31.9	+12.8	+8.4	+20.2	-1.2	+8.3	-3.0	-11.3
Steps (reduced)		718 (0)	1138 (420)	1667 (231)	2016 (580)	2484 (330)	2657 (503)	2935 (63)	3050 (178)	3248 (376)	3488 (616)	3557 (685)

Since 718 factors into 2 × 359, 718edo contains 2edo and 359edo as subsets.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3.5	[-29 -11 20⟩, [54 -37 2⟩	[⟨718 1138 1667]]	+0.0357	0.0482	2.89
2.3.5.7	4375/4374, 40500000/40353607, [31 -6 -2 -6⟩	[⟨718 1138 1667 2016]]	−0.0207	0.1063	6.36
2.3.5.7.11	3025/3024, 4375/4374, 131072/130977, 40500000/40353607	[⟨718 1138 1667 2016 2484]]	−0.0290	0.0965	5.77
2.3.5.7.11.13	1716/1715, 2080/2079, 3025/3024, 4096/4095, 7031250/7014007	[⟨718 1138 1667 2016 2484 2657]]	−0.0305	0.0881	5.27
2.3.5.7.11.13.17	1275/1274, 1716/1715, 2025/2023, 2080/2079, 3025/3024, 4096/4095	[⟨718 1138 1667 2016 2484 2657 2935]]	−0.0379	0.0836	5.00
2.3.5.7.11.13.17.19	1275/1274, 1716/1715, 2025/2023, 2080/2079, 2432/2431, 3025/3024, 3250/3249	[⟨718 1138 1667 2016 2484 2657 2935 3050]]	−0.0326	0.0795	4.76
2.3.5.7.11.13.17.19.23	1275/1274, 1716/1715, 2024/2023, 2025/2023, 2080/2079, 2185/2184, 2432/2431, 3025/3024	[⟨718 1138 1667 2016 2484 2657 2935 3050 3248]]	−0.0323	0.0749	4.48

718et has a lower absolute error in the 23-limit than any previous equal temperaments, past 581 and followed by 742i.

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	21\718	35.10	234375/229376	Gammic
1	249\718	249.03	[-27 11 3 1⟩	Monzismic
2	125\718	208.91	44/39	Abigail

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

@@ Line 14: / Line 14: @@
 == 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
@@ Line 64: / Line 73: @@
 | 0.0749
 | 4.48
-{{comma basis end}}
+|}
 * 718et has a lower [[Tenney-Euclidean temperament measures #TE error|absolute error]] in the 23-limit than any previous equal temperaments, past [[581edo|581]] and followed by [[742edo|742i]].
 === Rank-2 temperaments ===
-{{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 87: / Line 103: @@
 | 44/39
 | [[Abigail]]
-{{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