829edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|829}}
{{ED intro}}


== Theory ==
== Theory ==
829edo is [[consistent]] to the [[11-odd-limit]]. The equal temperament [[tempering out|tempers out]] [[4375/4374]], {{monzo| -25 6 -3 8 }}, and {{monzo| 16 -9 -8 6 }} in the 7-limit; [[41503/41472]], [[200704/200475]], and 3750705/3748096 in the 11-limit. It [[support]]s [[squarschmidt]], [[senior]] and [[acrokleismic]].
829edo is [[consistent]] to the [[11-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] [[4375/4374]], {{monzo| -25 6 -3 8 }}, and {{monzo| 16 -9 -8 6 }} in the 7-limit; [[41503/41472]], [[200704/200475]], and 3750705/3748096 in the 11-limit. It [[support]]s [[squarschmidt]], [[senior]] and [[acrokleismic]].


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
829edo is the 145th [[prime edo]].  
829edo is the 145th [[prime edo]].


== 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]] (¢)
! [[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 1314 -829 }}
| {{Monzo| 1314 -829 }}
| {{mapping| 829 1314 }}
| {{Mapping| 829 1314 }}
| -0.0302
| –0.0302
| 0.0302
| 0.0302
| 2.09
| 2.09
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 39 -29 3 }}, {{monzo| 61 4 -29 }}
| {{Monzo| 39 -29 3 }}, {{monzo| 61 4 -29 }}
| {{mapping| 829 1314 1925 }}
| {{Mapping| 829 1314 1925 }}
| -0.0454
| −0.0454
| 0.0327
| 0.0327
| 2.26
| 2.26
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| 2.3.5.7
| 2.3.5.7
| 4375/4374, {{monzo| -25 6 -3 8 }}, {{monzo| 16 -9 -8 6 }}
| 4375/4374, {{monzo| -25 6 -3 8 }}, {{monzo| 16 -9 -8 6 }}
| {{mapping| 829 1314 1925 2327 }}
| {{Mapping| 829 1314 1925 2327 }}
| +0.0043
| +0.0043
| 0.0906
| 0.0906
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| 2.3.5.7.11
| 2.3.5.7.11
| 4375/4374, 41503/41472, 200704/200475, 3750705/3748096
| 4375/4374, 41503/41472, 200704/200475, 3750705/3748096
| {{mapping| 829 1314 1925 2327 2868 }}
| {{Mapping| 829 1314 1925 2327 2868 }}
| -0.0076
| −0.0076
| 0.0844
| 0.0844
| 5.83
| 5.83
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 4096/4095, 4375/4374, 4459/4455, 47432/47385, 59535/59488
| 4096/4095, 4375/4374, 4459/4455, 47432/47385, 59535/59488
| {{mapping| 829 1314 1925 2327 2868 3068 }}
| {{Mapping| 829 1314 1925 2327 2868 3068 }}
| -0.0282
| −0.0282
| 0.0898
| 0.0898
| 6.20
| 6.20
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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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| 565.983
| 565.983
| 59049/40960
| 59049/40960
| [[Tricot]]
| [[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 forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Music ==
== Music ==
; [[Francium]]
; [[Francium]]
* "confluencelessness" from ''albumwithoutspaces'' (2024) – [https://francium223.bandcamp.com/track/confluencelessness Bandcamp] | [https://www.youtube.com/watch?v=eKDAs6aHF1s YouTube] – acrokleismic[19] in 829edo tuning
* "confluencelessness" from ''albumwithoutspaces'' (2024) – [https://open.spotify.com/track/7luOwK0nu7Cq9hmJy6W7L9 Spotify] | [https://francium223.bandcamp.com/track/confluencelessness Bandcamp] | [https://www.youtube.com/watch?v=eKDAs6aHF1s YouTube] – Acrokleismic[19] in 829edo tuning

Latest revision as of 13:38, 13 March 2026

← 828edo 829edo 830edo →
Prime factorization 829 (prime)
Step size 1.44753 ¢ 
Fifth 485\829 (702.051 ¢)
Semitones (A1:m2) 79:62 (114.4 ¢ : 89.75 ¢)
Consistency limit 11
Distinct consistency limit 11

829 equal divisions of the octave (abbreviated 829edo or 829ed2), also called 829-tone equal temperament (829tet) or 829 equal temperament (829et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 829 equal parts of about 1.45 ¢ each. Each step represents a frequency ratio of 21/829, or the 829th root of 2.

Theory

829edo is consistent to the 11-odd-limit. As an equal temperament, it tempers out 4375/4374, [-25 6 -3 8⟩, and [16 -9 -8 6⟩ in the 7-limit; 41503/41472, 200704/200475, and 3750705/3748096 in the 11-limit. It supports squarschmidt, senior and acrokleismic.

Prime harmonics

Approximation of prime harmonics in 829edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.096 +0.176 -0.430 +0.190 +0.486 +0.714 +0.678 -0.048 -0.385 -0.042
Relative (%) +0.0 +6.6 +12.2 -29.7 +13.1 +33.5 +49.3 +46.8 -3.3 -26.6 -2.9
Steps
(reduced) 		829
(0) 		1314
(485) 		1925
(267) 		2327
(669) 		2868
(381) 		3068
(581) 		3389
(73) 		3522
(206) 		3750
(434) 		4027
(711) 		4107
(791)

Subsets and supersets

829edo is the 145th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [1314 -829⟩ [⟨829 1314]] –0.0302 0.0302 2.09
2.3.5 [39 -29 3⟩, [61 4 -29⟩ [⟨829 1314 1925]] −0.0454 0.0327 2.26
2.3.5.7 4375/4374, [-25 6 -3 8⟩, [16 -9 -8 6⟩ [⟨829 1314 1925 2327]] +0.0043 0.0906 6.25
2.3.5.7.11 4375/4374, 41503/41472, 200704/200475, 3750705/3748096 [⟨829 1314 1925 2327 2868]] −0.0076 0.0844 5.83
2.3.5.7.11.13 4096/4095, 4375/4374, 4459/4455, 47432/47385, 59535/59488 [⟨829 1314 1925 2327 2868 3068]] −0.0282 0.0898 6.20

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 218\829 315.561 6/5 Acrokleismic
1 223\829 322.799 3087/2560 Senior / seniority
1 274\829 396.622 98304/78125 Squarschmidt
1 391\829 565.983 59049/40960 Alphatrillium

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

Music

Francium
  • "confluencelessness" from albumwithoutspaces (2024) – Spotify | Bandcamp | YouTube – Acrokleismic[19] in 829edo tuning
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