718edo: Difference between revisions

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== 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 [[5/1|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 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 ===

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=== Subsets and supersets ===

718 contains [[2edo]] and [[359edo]] as subsets.

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

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| 2.3.5

| {{monzo| -29 -11 20 }}, {{monzo| 54 -37 2 }}

| [{{~~val~~| 718 1138 1667 }}]

| {{mapping| 718 1138 1667 }}

| +0.0357

| 0.0482

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| 2.3.5.7

| 4375/4374, 40500000/40353607, {{monzo| 31 -6 -2 -6 }}

| [{{~~val~~| 718 1138 1667 2016 }}]

| {{mapping| 718 1138 1667 2016 }}

| -0.0207

| −0.0207

| 0.1063

| 6.36

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| 2.3.5.7.11

| 3025/3024, 4375/4374, 131072/130977, 40500000/40353607

| [{{~~val~~| 718 1138 1667 2016 2484 }}]

| {{mapping| 718 1138 1667 2016 2484 }}

| -0.0290

| −0.0290

| 0.0965

| 5.77

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| 2.3.5.7.11.13

| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 7031250/7014007

| [{{~~val~~| 718 1138 1667 2016 2484 2657 }}]

| {{mapping| 718 1138 1667 2016 2484 2657 }}

| -0.0305

| −0.0305

| 0.0881

| 5.27

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| 2.3.5.7.11.13.17

| 1275/1274, 1716/1715, 2025/2023, 2080/2079, 3025/3024, 4096/4095

| [{{~~val~~| 718 1138 1667 2016 2484 2657 2935 }}]

| {{mapping| 718 1138 1667 2016 2484 2657 2935 }}

| -0.0379

| −0.0379

| 0.0836

| 5.00

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| 2.3.5.7.11.13.17.19

| 1275/1274, 1716/1715, 2025/2023, 2080/2079, 2432/2431, 3025/3024, 3250/3249

| [{{~~val~~| 718 1138 1667 2016 2484 2657 2935 3050 }}]

| {{mapping| 718 1138 1667 2016 2484 2657 2935 3050 }}

| -0.0326

| −0.0326

| 0.0795

| 4.76

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| 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

| [{{~~val~~| 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

| 4.48

|}

* 718et has a lower [[Tenney-Euclidean temperament measures #TE ~~simple badness~~|~~relative~~ error]] in the 23-limit than any previous equal temperaments, past [[581edo|581]] and followed by [[742edo|742i]].

* 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 ===

{| 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 per ~~Octave~~

|-

! Generator~~ (Reduced)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! Associated ratio*

! Temperaments

|-

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| [[Abigail]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|718}}
+{{ED intro}}
 == Theory ==
-edo 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 [[5/1|5]], [[13/1|13]], [[17/1|17]] and [[31/1|31]].
+edo 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 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 ===
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-contains [[2edo]] and [[359edo]] as subsets.
+Since 718 factors into {{factorization|718}}, 718edo contains [[2edo]] and [[359edo]] as subsets.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 26: / Line 27: @@
 | 2.3.5
 | {{monzo| -29 -11 20 }}, {{monzo| 54 -37 2 }}
-| [{{val| 718 1138 1667 }}]
+| {{mapping| 718 1138 1667 }}
 | +0.0357
 | 0.0482
@@ Line 33: / Line 34: @@
 | 2.3.5.7
 | 4375/4374, 40500000/40353607, {{monzo| 31 -6 -2 -6 }}
-| [{{val| 718 1138 1667 2016 }}]
+| {{mapping| 718 1138 1667 2016 }}
-| -0.0207
+| −0.0207
 | 0.1063
 | 6.36
@@ Line 40: / Line 41: @@
 | 2.3.5.7.11
 | 3025/3024, 4375/4374, 131072/130977, 40500000/40353607
-| [{{val| 718 1138 1667 2016 2484 }}]
+| {{mapping| 718 1138 1667 2016 2484 }}
-| -0.0290
+| −0.0290
 | 0.0965
 | 5.77
@@ Line 47: / Line 48: @@
 | 2.3.5.7.11.13
 | 1716/1715, 2080/2079, 3025/3024, 4096/4095, 7031250/7014007
-| [{{val| 718 1138 1667 2016 2484 2657 }}]
+| {{mapping| 718 1138 1667 2016 2484 2657 }}
-| -0.0305
+| −0.0305
 | 0.0881
 | 5.27
@@ Line 54: / Line 55: @@
 | 2.3.5.7.11.13.17
 | 1275/1274, 1716/1715, 2025/2023, 2080/2079, 3025/3024, 4096/4095
-| [{{val| 718 1138 1667 2016 2484 2657 2935 }}]
+| {{mapping| 718 1138 1667 2016 2484 2657 2935 }}
-| -0.0379
+| −0.0379
 | 0.0836
 | 5.00
@@ Line 61: / Line 62: @@
 | 2.3.5.7.11.13.17.19
 | 1275/1274, 1716/1715, 2025/2023, 2080/2079, 2432/2431, 3025/3024, 3250/3249
-| [{{val| 718 1138 1667 2016 2484 2657 2935 3050 }}]
+| {{mapping| 718 1138 1667 2016 2484 2657 2935 3050 }}
-| -0.0326
+| −0.0326
 | 0.0795
 | 4.76
@@ Line 68: / Line 69: @@
 | 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
-| [{{val| 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
 | 4.48
 |}
-* 718et has a lower [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] in the 23-limit than any previous equal temperaments, past [[581edo|581]] and followed by [[742edo|742i]].
+* 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 ===
 {| 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 Octave
+|-
-! Generator<br>(Reduced)
+! Periods<br />per 8ve
-! Cents<br>(Reduced)
+! Generator*
-! Associated<br>Ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 102: / Line 104: @@
 | [[Abigail]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct