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403edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|403}}
{{ED intro}}


== Theory ==
== 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.  
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.  
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 ===
=== Subsets and supersets ===
Since 403 factors into {{factorization|403}}, 403edo contains [[13edo]] and [[31edo]] as subsets.  
Since 403 factors into {{factorization|403}}, 403edo contains [[13edo]] and [[31edo]] 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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| {{monzo| 639 -403 }}
| {{monzo| 639 -403 }}
| {{mapping| 403 639 }}
| {{mapping| 403 639 }}
| -0.2443
| −0.2443
| 0.2443
| 0.2443
| 8.20
| 8.20
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| 1600000/1594323, {{monzo| 81 -13 -26 }}
| 1600000/1594323, {{monzo| 81 -13 -26 }}
| {{mapping| 403 639 936 }}
| {{mapping| 403 639 936 }}
| -0.2753
| −0.2753
| 0.2042
| 0.2042
| 6.86
| 6.86
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| 3136/3125, 1600000/1594323, 2100875/2097152
| 3136/3125, 1600000/1594323, 2100875/2097152
| {{mapping| 403 639 936 1131 }} (403)
| {{mapping| 403 639 936 1131 }} (403)
| -0.3751
| −0.3751
| 0.2473
| 0.2473
| 8.31
| 8.31
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| 3025/3024, 3136/3125, 12005/11979, 131072/130977
| 3025/3024, 3136/3125, 12005/11979, 131072/130977
| {{mapping| 403 639 936 1131 1394 }} (403)
| {{mapping| 403 639 936 1131 1394 }} (403)
| -0.0621
| −0.0621
| 0.3160
| 0.3160
| 10.61
| 10.61
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| 2080/2079, 3025/3024, 3136/3125, 4096/4095, 12005/11979
| 2080/2079, 3025/3024, 3136/3125, 4096/4095, 12005/11979
| {{mapping| 403 639 936 1131 1394 1491 }} (403)
| {{mapping| 403 639 936 1131 1394 1491 }} (403)
| -0.0146
| −0.0146
| 0.3074
| 0.3074
| 10.32
| 10.32
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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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| [[Amity]] (403defff)
| [[Amity]] (403defff)
|}
|}
<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
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