403edo: Difference between revisions

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== Theory ==

403edo is only [[consistent]] to the [[5-odd-limit]], since the error of [[harmonic]] [[7/1|7]] is quite large. To start with, the 403def [[val]] {{val| 403 639 936 '''1132''' '''1395''' '''1492''' }}, the 403df [[val]] {{val| 403 639 936 '''1132''' '''1394''' '''1492''' }}, and the [[patent val]] {{val| 403 639 936 '''1131''' '''1394''' '''1491''' }} are worth considering.

~~The equal temperament~~ [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 70 0 -31 }} (31-5, or birds comma) in the 5-limit.

In any case, it [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 70 0 -31 }} (31-5, or birds comma) in the 5-limit.

Using the 403d val, it tempers out [[4375/4374]], [[5120/5103]], and [[6144/6125]] in the 7-limit, so that it [[support]]s [[amity|septimal amity]], the 152 & 251 temperament. Extending it by the 403def val, it tempers out [[540/539]], [[5632/5625]], [[6250/6237]], and [[19712/19683]] in the 11-limit, supporting 11-limit amity; and [[1575/1573]], [[1716/1715]], [[2200/2197]], 3584/3575 in the 13-limit. Extending it by the alternative 403df val, 1375/1372, [[14641/14580]] in the 11-limit; [[352/351]], [[847/845]], and [[2080/2079]] in the 13-limit.

Using the 403d val, it tempers out [[4375/4374]], [[5120/5103]], and [[6144/6125]] in the 7-limit, so that it [[support]]s [[amity|septimal amity]], the {{nowrap|152 & 251}} temperament. Extending it by the 403def val, it tempers out [[540/539]], [[5632/5625]], [[6250/6237]], and [[19712/19683]] in the 11-limit, supporting 11-limit amity; and [[1575/1573]], [[1716/1715]], [[2200/2197]], 3584/3575 in the 13-limit. Extending it by the alternative 403df val, [[1375/1372]], [[14641/14580]] in the 11-limit; [[352/351]], [[847/845]], and [[2080/2079]] in the 13-limit.

Using the patent val, it tempers out [[3136/3125]], [[2100875/2097152]], and [[78125000/78121827]] in the 7-limit; [[3025/3024]], [[3388/3375]], 12005/11979, 14641/14580, 42875/42768, and [[131072/130977]] in the 11-limit; [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]], and [[10648/10647]] in the 13-limit.

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=== Subsets and supersets ===

Since 403 factors into {{factorization|403}}, 403edo contains [[13edo]] and [[31edo]] as subsets.

== 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

| {{monzo| 639 -403 }}

| {{mapping| 403 639 }}

| ~~−0~~.2443

| −0.2443

| 0.2443

| 8.20

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| 1600000/1594323, {{monzo| 81 -13 -26 }}

| {{mapping| 403 639 936 }}

| ~~−0~~.2753

| −0.2753

| 0.2042

| 6.86

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| 3136/3125, 1600000/1594323, 2100875/2097152

| {{mapping| 403 639 936 1131 }} (403)

| ~~−0~~.3751

| −0.3751

| 0.2473

| 8.31

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| 3025/3024, 3136/3125, 12005/11979, 131072/130977

| {{mapping| 403 639 936 1131 1394 }} (403)

| ~~−0~~.0621

| −0.0621

| 0.3160

| 10.61

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| 2080/2079, 3025/3024, 3136/3125, 4096/4095, 12005/11979

| {{mapping| 403 639 936 1131 1394 1491 }} (403)

| ~~−0~~.0146

| −0.0146

| 0.3074

| 10.32

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| 0.2927

| 9.83

~~{{comma basis end}~~}

|}

=== 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

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| 243/200

| [[Amity]] (403defff)

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<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|403}}
+{{ED intro}}
 == Theory ==
 edo is only [[consistent]] to the [[5-odd-limit]], since the error of [[harmonic]] [[7/1|7]] is quite large. To start with, the 403def [[val]] {{val| 403 639 936 '''1132''' '''1395''' '''1492''' }}, the 403df [[val]] {{val| 403 639 936 '''1132''' '''1394''' '''1492''' }}, and the [[patent val]] {{val| 403 639 936 '''1131''' '''1394''' '''1491''' }} are worth considering.
-The equal temperament [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 70 0 -31 }} (31-5, or birds comma) in the 5-limit.
+In any case, it [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 70 0 -31 }} (31-5, or birds comma) in the 5-limit.
-Using the 403d val, it tempers out [[4375/4374]], [[5120/5103]], and [[6144/6125]] in the 7-limit, so that it [[support]]s [[amity|septimal amity]], the 152 & 251 temperament. Extending it by the 403def val, it tempers out [[540/539]], [[5632/5625]], [[6250/6237]], and [[19712/19683]] in the 11-limit, supporting 11-limit amity; and [[1575/1573]], [[1716/1715]], [[2200/2197]], 3584/3575 in the 13-limit. Extending it by the alternative 403df val, 1375/1372, [[14641/14580]] in the 11-limit; [[352/351]], [[847/845]], and [[2080/2079]] in the 13-limit.
+Using the 403d val, it tempers out [[4375/4374]], [[5120/5103]], and [[6144/6125]] in the 7-limit, so that it [[support]]s [[amity|septimal amity]], the {{nowrap|152 & 251}} temperament. Extending it by the 403def val, it tempers out [[540/539]], [[5632/5625]], [[6250/6237]], and [[19712/19683]] in the 11-limit, supporting 11-limit amity; and [[1575/1573]], [[1716/1715]], [[2200/2197]], 3584/3575 in the 13-limit. Extending it by the alternative 403df val, [[1375/1372]], [[14641/14580]] in the 11-limit; [[352/351]], [[847/845]], and [[2080/2079]] in the 13-limit.
 Using the patent val, it tempers out [[3136/3125]], [[2100875/2097152]], and [[78125000/78121827]] in the 7-limit; [[3025/3024]], [[3388/3375]], 12005/11979, 14641/14580, 42875/42768, and [[131072/130977]] in the 11-limit; [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]], and [[10648/10647]] in the 13-limit.
@@ Line 15: / Line 15: @@
 === Subsets and supersets ===
 Since 403 factors into {{factorization|403}}, 403edo contains [[13edo]] and [[31edo]] as subsets.
 == 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
 | {{monzo| 639 -403 }}
 | {{mapping| 403 639 }}
-| &minus;0.2443
+| −0.2443
 | 0.2443
 | 8.20
@@ Line 30: / Line 39: @@
 | 1600000/1594323, {{monzo| 81 -13 -26 }}
 | {{mapping| 403 639 936 }}
-| &minus;0.2753
+| −0.2753
 | 0.2042
 | 6.86
@@ Line 37: / Line 46: @@
 | 3136/3125, 1600000/1594323, 2100875/2097152
 | {{mapping| 403 639 936 1131 }} (403)
-| &minus;0.3751
+| −0.3751
 | 0.2473
 | 8.31
@@ Line 44: / Line 53: @@
 | 3025/3024, 3136/3125, 12005/11979, 131072/130977
 | {{mapping| 403 639 936 1131 1394 }} (403)
-| &minus;0.0621
+| −0.0621
 | 0.3160
 | 10.61
@@ Line 51: / Line 60: @@
 | 2080/2079, 3025/3024, 3136/3125, 4096/4095, 12005/11979
 | {{mapping| 403 639 936 1131 1394 1491 }} (403)
-| &minus;0.0146
+| −0.0146
 | 0.3074
 | 10.32
@@ Line 61: / Line 70: @@
 | 0.2927
 | 9.83
-{{comma basis end}}
+|}
 === 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 83: / Line 99: @@
 | 243/200
 | [[Amity]] (403defff)
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct