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1429edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1429}}
{{ED intro}}


== Theory ==
== Theory ==
1429edo has a reasonable approximation of the full 17-limit. It is [[consistent]] to the [[9-odd-limit]] with only [[11/10]] barely missing the line. The 11-limit [[TE tuning|optimal tuning]] of the equal temperament is consistent to the 18-integer-limit; however, the 13- and 17-limit optimal tunings, which have less of octave compression, are not, so one might want to keep the compression tight.  
1429edo has a reasonable approximation of the full [[17-limit]]. It is [[consistent]] to the [[9-odd-limit]] with only [[11/10]] barely missing the line. The 11-limit [[TE tuning|optimal tuning]] of the equal temperament is consistent to the 18-integer-limit; however, the 13- and 17-limit optimal tunings, which have less of octave compression, are not, so one might want to keep the compression tight.  


The equal temperament [[Tempering out|tempers out]] [[4375/4374]] in the 7-limit; [[131072/130977]], 759375/758912, 1953125/1951488, 2359296/2358125, 2657205/2656192, and 3294225/3294172 in the 11-limit; [[2080/2079]], [[4096/4095]], [[4225/4224]], 78125/78078, and [[123201/123200]] in the 13-limit; [[2500/2499]], [[5832/5831]], [[11016/11011]], and [[12376/12375]] in the 17-limit. It supports the [[gross]] temperament and provides the [[optimal patent val]] for the 11- and 13-limit [[trillium]] temperament.  
As an equal temperament, it [[tempering out|tempers out]] [[4375/4374]] in the 7-limit; [[131072/130977]], 759375/758912, 1953125/1951488, 2359296/2358125, 2657205/2656192, and 3294225/3294172 in the 11-limit; [[2080/2079]], [[4096/4095]], [[4225/4224]], 78125/78078, and [[123201/123200]] in the 13-limit; [[2500/2499]], [[5832/5831]], [[11016/11011]], and [[12376/12375]] in the 17-limit. It [[support]]s the [[gross]] temperament and provides the [[optimal patent val]] for the 11- and 13-limit [[alphatrillium]] temperament.  


=== Prime harmonics ===
=== Prime harmonics ===
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== 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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|-
|-
| 2.3
| 2.3
| {{monzo| 2265 -1429 }}
| {{Monzo| 2265 -1429 }}
| {{mapping| 1429 2265 }}
| {{Mapping| 1429 2265 }}
| -0.0235
| −0.0235
| 0.0234
| 0.0234
| 2.80
| 2.80
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 39 -29 3 }}, {{monzo| -66 -36 53 }}
| {{Monzo| 39 -29 3 }}, {{monzo| -66 -36 53 }}
| {{mapping| 1429 2265 3318 }}
| {{Mapping| 1429 2265 3318 }}
| -0.0114
| −0.0114
| 0.0257
| 0.0257
| 3.06
| 3.06
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| 2.3.5.7
| 2.3.5.7
| 4375/4374, {{monzo| 26 4 -3 -14 }}, {{monzo| 40 -22 -1 -1 }}
| 4375/4374, {{monzo| 26 4 -3 -14 }}, {{monzo| 40 -22 -1 -1 }}
| {{mapping| 1429 2265 3318 4012 }}
| {{Mapping| 1429 2265 3318 4012 }}
| -0.0302
| −0.0302
| 0.0395
| 0.0395
| 4.70
| 4.70
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| 2.3.5.7.11
| 2.3.5.7.11
| 4375/4374, 131072/130977, 759375/758912, 3294225/3294172
| 4375/4374, 131072/130977, 759375/758912, 3294225/3294172
| {{mapping| 1429 2265 3318 4012 4944 }}
| {{Mapping| 1429 2265 3318 4012 4944 }}
| -0.0471
| −0.0471
| 0.0488
| 0.0488
| 5.81
| 5.81
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2080/2079, 4096/4095, 4375/4374, 78125/78078, 3294225/3294172
| 2080/2079, 4096/4095, 4375/4374, 78125/78078, 3294225/3294172
| {{mapping| 1429 2265 3318 4012 4944 5288 }}
| {{Mapping| 1429 2265 3318 4012 4944 5288 }}
| -0.0420
| −0.0420
| 0.0460
| 0.0460
| 5.48
| 5.48
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 2080/2079, 2500/2499, 4096/4095, 4375/4374, 11016/11011, 108086/108045
| 2080/2079, 2500/2499, 4096/4095, 4375/4374, 11016/11011, 108086/108045
| {{mapping| 1429 2265 3318 4012 4944 5288 5841 }}
| {{Mapping| 1429 2265 3318 4012 4944 5288 5841 }}
| -0.0364
| −0.0364
| 0.0447
| 0.0447
| 5.32
| 5.32
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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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| 674\1429
| 674\1429
| 565.990
| 565.990
| 25/18
| 104/75
| [[Trillium]]
| [[Alphatrillium]]
|}
|}
<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


== Scales ==
== Scales ==
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== Music ==
== Music ==
; [[User:Francium|Francium]]
; [[Francium]]
* "Gross Pattern" from ''Gross temperament EP'' (2023) [https://open.spotify.com/track/1BwMVnnrzfug6pSLUJ0jSG Spotify] | [https://francium223.bandcamp.com/track/gross-pattern Bandcamp] | [https://youtu.be/ttQVdzSy96M?si=Puvjpvzud8yQ2TAj YouTube] – gross in 1429edo tuning
* "Gross Pattern" from ''Gross temperament EP'' (2023) [https://open.spotify.com/track/1BwMVnnrzfug6pSLUJ0jSG Spotify] | [https://francium223.bandcamp.com/track/gross-pattern Bandcamp] | [https://youtu.be/ttQVdzSy96M YouTube] – gross in 1429edo tuning
 
[[Category:Listen]]
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