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]]

Latest revision as of 15:16, 16 March 2025

← 1428edo 1429edo 1430edo →
Prime factorization 1429 (prime)
Step size 0.839748 ¢ 
Fifth 836\1429 (702.029 ¢)
Semitones (A1:m2) 136:107 (114.2 ¢ : 89.85 ¢)
Consistency limit 9
Distinct consistency limit 9

1429 equal divisions of the octave (abbreviated 1429edo or 1429ed2), also called 1429-tone equal temperament (1429tet) or 1429 equal temperament (1429et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1429 equal parts of about 0.84 ¢ each. Each step represents a frequency ratio of 21/1429, or the 1429th root of 2.

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 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.

As an equal temperament, it 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 alphatrillium temperament.

Prime harmonics

Approximation of prime harmonics in 1429edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.074 -0.030 +0.243 +0.397 +0.060 +0.013 -0.242 -0.143 -0.046 +0.381
Relative (%) +0.0 +8.9 -3.5 +29.0 +47.2 +7.2 +1.6 -28.8 -17.0 -5.5 +45.3
Steps
(reduced) 		1429
(0) 		2265
(836) 		3318
(460) 		4012
(1154) 		4944
(657) 		5288
(1001) 		5841
(125) 		6070
(354) 		6464
(748) 		6942
(1226) 		7080
(1364)

Subsets and supersets

1429edo is the 226th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [2265 -1429⟩ [⟨1429 2265]] −0.0235 0.0234 2.80
2.3.5 [39 -29 3⟩, [-66 -36 53⟩ [⟨1429 2265 3318]] −0.0114 0.0257 3.06
2.3.5.7 4375/4374, [26 4 -3 -14⟩, [40 -22 -1 -1⟩ [⟨1429 2265 3318 4012]] −0.0302 0.0395 4.70
2.3.5.7.11 4375/4374, 131072/130977, 759375/758912, 3294225/3294172 [⟨1429 2265 3318 4012 4944]] −0.0471 0.0488 5.81
2.3.5.7.11.13 2080/2079, 4096/4095, 4375/4374, 78125/78078, 3294225/3294172 [⟨1429 2265 3318 4012 4944 5288]] −0.0420 0.0460 5.48
2.3.5.7.11.13.17 2080/2079, 2500/2499, 4096/4095, 4375/4374, 11016/11011, 108086/108045 [⟨1429 2265 3318 4012 4944 5288 5841]] −0.0364 0.0447 5.32

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 109\1429 91.533 [144 -22 -47⟩ Gross
1 674\1429 565.990 104/75 Alphatrillium

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

Scales

Music

Francium
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