2129edo: Difference between revisions

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Rework on theory, mentioning the most important comma -- schisma, and remove the obscure, complex commas
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|2129}}
{{ED intro}}


== Theory ==
== Theory ==
2129edo is only [[consistent]] to the [[5-odd-limit]], where it tempers out the [[schisma]]. Otherwise its poor approximation to the [[3/1|harmonic 3]] commends itself to a 2.9.7.11.13.… [[subgroup]] interpretation.  
2129edo is only [[consistent]] to the [[5-odd-limit]], where it tempers out the [[schisma]]. Otherwise its poor approximation to both [[harmonic]]s [[3/1|3]] and [[5/1|5]] commends itself to a 2.9.15.7.11.13.… [[subgroup]] interpretation. However, its representation of [[5/3]] and its octave complement [[6/5]] are extremely accurate, due to being a continued fraction convergent to their logarithms.


=== Odd harmonics ===
=== Odd 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]] (¢)
! [[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
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| {{monzo| -6749 2129 }}
| {{monzo| -6749 2129 }}
| {{mapping| 2129 6749 }}
| {{mapping| 2129 6749 }}
| -0.0204
| −0.0204
| 0.0204
| 0.0204
| 3.62
| 3.62
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| {{monzo| 37 29 -33 }}, {{monzo| 209 -61 -4 }}
| {{monzo| 37 29 -33 }}, {{monzo| 209 -61 -4 }}
| {{mapping| 2129 6749 8318 }}
| {{mapping| 2129 6749 8318 }}
| -0.0247
| −0.0247
| 0.0177
| 0.0177
| 3.14
| 3.14
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| 24414062500/24407490807, 13841287201/13839609375, 2199023255552/2197176384375
| 24414062500/24407490807, 13841287201/13839609375, 2199023255552/2197176384375
| {{mapping| 2129 6749 8318 5977 }}
| {{mapping| 2129 6749 8318 5977 }}
| -0.0256
| −0.0256
| 0.0154
| 0.0154
| 2.73
| 2.73
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| 9800/9801, 5767168/5764801, 104857600/104825259, 13841287201/13839609375
| 9800/9801, 5767168/5764801, 104857600/104825259, 13841287201/13839609375
| {{mapping| 2129 6749 8318 5977 7365 }}
| {{mapping| 2129 6749 8318 5977 7365 }}
| -0.0162
| −0.0162
| 0.0232
| 0.0232
| 4.12
| 4.12
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| 10648/10647, 9801/9800, 196625/196608, 36924979/36905625, 304117528/303807105
| 10648/10647, 9801/9800, 196625/196608, 36924979/36905625, 304117528/303807105
| {{mapping| 2129 6749 8318 5977 7365 7878 }}
| {{mapping| 2129 6749 8318 5977 7365 7878 }}
| -0.0075
| −0.0075
| 0.0288
| 0.0288
| 5.11
| 5.11
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| 2431/2430, 10648/10647, 9801/9800, 845325/845152, 297440/297381, 11275335/11275264, 15980544/15978655
| 2431/2430, 10648/10647, 9801/9800, 845325/845152, 297440/297381, 11275335/11275264, 15980544/15978655
| {{mapping| 2129 6749 8318 5977 7365 7878 8702 }}
| {{mapping| 2129 6749 8318 5977 7365 7878 8702 }}
| -0.0024
| −0.0024
| 0.0295
| 0.0295
| 5.2
| 5.2
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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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| 498.262
| 498.262
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
|}
|}
<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 ==
* [[Hemischis29]]


== Music ==
== Music ==
; [[User:Francium|Francium]]
; [[User:Francium|Francium]]
* [https://www.youtube.com/watch?v=xF_1MKMlsjY ''Brid Dance''] (2023) – [[hemischis]] in 2129edo tuning
* "Brid Dance" from ''HemischisMatic EP'' (2023) – [https://open.spotify.com/track/6yXO4yHFM0EWk3pCjop8Dp Spotify] | [https://francium223.bandcamp.com/track/brid-dance Bandcamp] | [https://youtu.be/xF_1MKMlsjY?si=8Db0YVe6GQ9ZgPET YouTube] – [[hemischis]] in 2129edo tuning
 
[[Category:Listen]]

Latest revision as of 02:30, 17 April 2025

← 2128edo 2129edo 2130edo →
Prime factorization 2129 (prime)
Step size 0.563645 ¢ 
Fifth 1245\2129 (701.738 ¢)
Semitones (A1:m2) 199:162 (112.2 ¢ : 91.31 ¢)
Dual sharp fifth 1246\2129 (702.302 ¢)
Dual flat fifth 1245\2129 (701.738 ¢)
Dual major 2nd 362\2129 (204.039 ¢)
Consistency limit 5
Distinct consistency limit 5

2129 equal divisions of the octave (abbreviated 2129edo or 2129ed2), also called 2129-tone equal temperament (2129tet) or 2129 equal temperament (2129et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2129 equal parts of about 0.564 ¢ each. Each step represents a frequency ratio of 21/2129, or the 2129th root of 2.

Theory

2129edo is only consistent to the 5-odd-limit, where it tempers out the schisma. Otherwise its poor approximation to both harmonics 3 and 5 commends itself to a 2.9.15.7.11.13.… subgroup interpretation. However, its representation of 5/3 and its octave complement 6/5 are extremely accurate, due to being a continued fraction convergent to their logarithms.

Odd harmonics

Approximation of odd harmonics in 2129edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.217 -0.217 +0.080 +0.129 -0.073 -0.133 +0.130 -0.117 +0.091 -0.137 +0.190
Relative (%) -38.5 -38.5 +14.1 +23.0 -13.0 -23.6 +23.0 -20.8 +16.2 -24.4 +33.7
Steps
(reduced) 		3374
(1245) 		4943
(685) 		5977
(1719) 		6749
(362) 		7365
(978) 		7878
(1491) 		8318
(1931) 		8702
(186) 		9044
(528) 		9351
(835) 		9631
(1115)

Subsets and supersets

2129edo is the 320th prime edo. 4258edo, which doubles it, gives a good correction to the harmonics 3 and 5.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.9 [-6749 2129⟩ [⟨2129 6749]] −0.0204 0.0204 3.62
2.9.15 [37 29 -33⟩, [209 -61 -4⟩ [⟨2129 6749 8318]] −0.0247 0.0177 3.14
2.9.15.7 24414062500/24407490807, 13841287201/13839609375, 2199023255552/2197176384375 [⟨2129 6749 8318 5977]] −0.0256 0.0154 2.73
2.9.15.7.11 9800/9801, 5767168/5764801, 104857600/104825259, 13841287201/13839609375 [⟨2129 6749 8318 5977 7365]] −0.0162 0.0232 4.12
2.9.15.7.11.13 10648/10647, 9801/9800, 196625/196608, 36924979/36905625, 304117528/303807105 [⟨2129 6749 8318 5977 7365 7878]] −0.0075 0.0288 5.11
2.9.15.7.11.13.17 2431/2430, 10648/10647, 9801/9800, 845325/845152, 297440/297381, 11275335/11275264, 15980544/15978655 [⟨2129 6749 8318 5977 7365 7878 8702]] −0.0024 0.0295 5.2

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 884\2129 498.262 4/3 Helmholtz

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

Scales

Music

Francium
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