718edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|718}}
{{ED intro}}


== Theory ==
== Theory ==
718edo is [[consistency|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 [[5/1|5]], [[13/1|13]], [[17/1|17]] and [[31/1|31]].  
718edo is [[consistency|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 [[5/1|5]], [[13/1|13]], [[17/1|17]] and [[31/1|31]].  


As does 359et, 718et [[Tempering out|tempers out]] the 359-comma in the 3-limit, rendering a very accurate perfect fifth. In the 5-limit it tempers out the gammic comma, {{monzo| -29 -11 20 }}, and the [[monzisma]], {{monzo| 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]].  
As does 359et, 718et [[Tempering out|tempers out]] the 359-comma in the 3-limit, rendering a very accurate [[harmonic]] [[3/1|3]]. In the 5-limit it [[tempering out|tempers out]] the gammic comma, {{monzo| -29 -11 20 }}, and the [[monzisma]], {{monzo| 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 [[support]]s [[gammic]], [[monzismic]] and [[abigail]].  


=== Prime harmonics ===
=== Prime harmonics ===
Line 11: Line 11:


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 718 factors into 2 × 359, 718edo contains [[2edo]] and [[359edo]] as subsets.
Since 718 factors into {{factorization|718}}, 718edo contains [[2edo]] and [[359edo]] as subsets.


== 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|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 4375/4374, 40500000/40353607, {{monzo| 31 -6 -2 -6 }}
| 4375/4374, 40500000/40353607, {{monzo| 31 -6 -2 -6 }}
| {{mapping| 718 1138 1667 2016 }}
| {{mapping| 718 1138 1667 2016 }}
| -0.0207
| −0.0207
| 0.1063
| 0.1063
| 6.36
| 6.36
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| 3025/3024, 4375/4374, 131072/130977, 40500000/40353607
| 3025/3024, 4375/4374, 131072/130977, 40500000/40353607
| {{mapping| 718 1138 1667 2016 2484 }}
| {{mapping| 718 1138 1667 2016 2484 }}
| -0.0290
| −0.0290
| 0.0965
| 0.0965
| 5.77
| 5.77
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| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 7031250/7014007
| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 7031250/7014007
| {{mapping| 718 1138 1667 2016 2484 2657 }}
| {{mapping| 718 1138 1667 2016 2484 2657 }}
| -0.0305
| −0.0305
| 0.0881
| 0.0881
| 5.27
| 5.27
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| 1275/1274, 1716/1715, 2025/2023, 2080/2079, 3025/3024, 4096/4095
| 1275/1274, 1716/1715, 2025/2023, 2080/2079, 3025/3024, 4096/4095
| {{mapping| 718 1138 1667 2016 2484 2657 2935 }}
| {{mapping| 718 1138 1667 2016 2484 2657 2935 }}
| -0.0379
| −0.0379
| 0.0836
| 0.0836
| 5.00
| 5.00
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| 1275/1274, 1716/1715, 2025/2023, 2080/2079, 2432/2431, 3025/3024, 3250/3249
| 1275/1274, 1716/1715, 2025/2023, 2080/2079, 2432/2431, 3025/3024, 3250/3249
| {{mapping| 718 1138 1667 2016 2484 2657 2935 3050 }}
| {{mapping| 718 1138 1667 2016 2484 2657 2935 3050 }}
| -0.0326
| −0.0326
| 0.0795
| 0.0795
| 4.76
| 4.76
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| 1275/1274, 1716/1715, 2024/2023, 2025/2023, 2080/2079, 2185/2184, 2432/2431, 3025/3024
| 1275/1274, 1716/1715, 2024/2023, 2025/2023, 2080/2079, 2185/2184, 2432/2431, 3025/3024
| {{mapping| 718 1138 1667 2016 2484 2657 2935 3050 3248 }}
| {{mapping| 718 1138 1667 2016 2484 2657 2935 3050 3248 }}
| -0.0323
| −0.0323
| 0.0749
| 0.0749
| 4.48
| 4.48
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| 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>Ratio
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Abigail]]
| [[Abigail]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

Latest revision as of 06:32, 21 February 2025

← 717edo 718edo 719edo →
Prime factorization 2 × 359
Step size 1.67131 ¢ 
Fifth 420\718 (701.95 ¢) (→ 210\359)
Semitones (A1:m2) 68:54 (113.6 ¢ : 90.25 ¢)
Consistency limit 23
Distinct consistency limit 23

718 equal divisions of the octave (abbreviated 718edo or 718ed2), also called 718-tone equal temperament (718tet) or 718 equal temperament (718et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 718 equal parts of about 1.67 ¢ each. Each step represents a frequency ratio of 21/718, or the 718th root of 2.

Theory

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.

Prime harmonics

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
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)

Subsets and supersets

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
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.

Rank-2 temperaments

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 distinct

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