718edo: Difference between revisions

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== 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, with [[28/25]] and its octave complement [[25/14]] being the only inconsistencies in the [[35-odd-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 [[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]].

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

| {{mapping| 718 1138 1667 2016 }}

| ~~−0~~.0207

| −0.0207

| 0.1063

| 6.36

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

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

| ~~−0~~.0290

| −0.0290

| 0.0965

| 5.77

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

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

| ~~−0~~.0305

| −0.0305

| 0.0881

| 5.27

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

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

| ~~−0~~.0379

| −0.0379

| 0.0836

| 5.00

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

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

| ~~−0~~.0326

| −0.0326

| 0.0795

| 4.76

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| 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 }}

| ~~−0~~.0323

| −0.0323

| 0.0749

| 4.48

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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 ~~it is~~ distinct

<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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|718}}
+{{ED intro}}
 == Theory ==
-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]].
+edo is [[consistency|distinctly consistent]] in the [[23-odd-limit]], and does well enough in the 31-limit, with [[28/25]] and its octave complement [[25/14]] being the only inconsistencies in the [[35-odd-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 [[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]].
@@ Line 35: / Line 35: @@
 | 4375/4374, 40500000/40353607, {{monzo| 31 -6 -2 -6 }}
 | {{mapping| 718 1138 1667 2016 }}
-| &minus;0.0207
+| −0.0207
 | 0.1063
 | 6.36
@@ Line 42: / Line 42: @@
 | 3025/3024, 4375/4374, 131072/130977, 40500000/40353607
 | {{mapping| 718 1138 1667 2016 2484 }}
-| &minus;0.0290
+| −0.0290
 | 0.0965
 | 5.77
@@ Line 49: / Line 49: @@
 | 1716/1715, 2080/2079, 3025/3024, 4096/4095, 7031250/7014007
 | {{mapping| 718 1138 1667 2016 2484 2657 }}
-| &minus;0.0305
+| −0.0305
 | 0.0881
 | 5.27
@@ Line 56: / Line 56: @@
 | 1275/1274, 1716/1715, 2025/2023, 2080/2079, 3025/3024, 4096/4095
 | {{mapping| 718 1138 1667 2016 2484 2657 2935 }}
-| &minus;0.0379
+| −0.0379
 | 0.0836
 | 5.00
@@ Line 63: / Line 63: @@
 | 1275/1274, 1716/1715, 2025/2023, 2080/2079, 2432/2431, 3025/3024, 3250/3249
 | {{mapping| 718 1138 1667 2016 2484 2657 2935 3050 }}
-| &minus;0.0326
+| −0.0326
 | 0.0795
 | 4.76
@@ Line 70: / Line 70: @@
 | 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 }}
-| &minus;0.0323
+| −0.0323
 | 0.0749
 | 4.48
@@ Line 104: / Line 104: @@
 | [[Abigail]]
 |}
-<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 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