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

Latest revision as of 06:31, 21 February 2025

← 402edo 403edo 404edo →
Prime factorization 13 × 31
Step size 2.97767 ¢ 
Fifth 236\403 (702.73 ¢)
Semitones (A1:m2) 40:29 (119.1 ¢ : 86.35 ¢)
Consistency limit 5
Distinct consistency limit 5

403 equal divisions of the octave (abbreviated 403edo or 403ed2), also called 403-tone equal temperament (403tet) or 403 equal temperament (403et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 403 equal parts of about 2.98 ¢ each. Each step represents a frequency ratio of 21/403, or the 403rd root of 2.

Theory

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

In any case, it tempers out 1600000/1594323 (amity comma) and [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 supports 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 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.

Odd harmonics

Approximation of odd harmonics in 403edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.77 +0.78 -1.08 -1.43 -0.45 -0.83 -1.42 -0.74 +0.25 -0.31 +0.01
Relative (%) +26.0 +26.3 -36.4 -48.0 -15.1 -27.7 -47.7 -24.8 +8.5 -10.4 +0.5
Steps
(reduced) 		639
(236) 		936
(130) 		1131
(325) 		1277
(68) 		1394
(185) 		1491
(282) 		1574
(365) 		1647
(35) 		1712
(100) 		1770
(158) 		1823
(211)

Subsets and supersets

Since 403 factors into 13 × 31, 403edo contains 13edo and 31edo as subsets.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [639 -403⟩ [⟨403 639]] −0.2443 0.2443 8.20
2.3.5 1600000/1594323, [81 -13 -26⟩ [⟨403 639 936]] −0.2753 0.2042 6.86
2.3.5.7 3136/3125, 1600000/1594323, 2100875/2097152 [⟨403 639 936 1131]] (403) −0.3751 0.2473 8.31
2.3.5.7.11 3025/3024, 3136/3125, 12005/11979, 131072/130977 [⟨403 639 936 1131 1394]] (403) −0.0621 0.3160 10.61
2.3.5.7.11.13 2080/2079, 3025/3024, 3136/3125, 4096/4095, 12005/11979 [⟨403 639 936 1131 1394 1491]] (403) −0.0146 0.3074 10.32
2.3.5.7.11.13.17 595/594, 833/832, 1225/1224, 3025/3024, 3136/3125, 4096/4095 [⟨403 639 936 1131 1394 1491 1647]] (403) +0.0133 0.2927 9.83

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 53\403 157.82 36756909/33554432 Hemiegads
1 106\403 315.63 6/5 Egads
1 114\403 339.45 243/200 Amity (403defff)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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