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2129edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|2129}}
{{ED intro}}
 
== Theory ==
== Theory ==
2129et tempers out 95703125/95664294, 5767168/5764801, 47265625/47258883, 67110351/67108864 and 43923/43904 in the 11-limit; 33792000/33787663, 200000/199927, 34034175/34027136, 2250423/2249390, 78125/78078, 1449459/1449175, 1990656/1990625, 67392/67375, [[4225/4224]], 8858304/8857805, 59319/59290 and 4084223/4084101 in the 13-limit.
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 ===
{{Harmonics in equal|2129}}
{{Harmonics in equal|2129}}
===Subsets and supersets===
 
=== Subsets and supersets ===
2129edo is the 320th [[prime edo]]. 4258edo, which doubles it, gives a good correction to the harmonics 3 and 5.
2129edo is the 320th [[prime edo]]. 4258edo, which doubles it, gives a good correction to the harmonics 3 and 5.
==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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.9
! rowspan="2" | [[Subgroup]]
|{{monzo|-6749 2129}}
! rowspan="2" | [[Comma list]]
|{{val|2129 6749}}
! rowspan="2" | [[Mapping]]
| -0.0204
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.9
| {{monzo| -6749 2129 }}
| {{mapping| 2129 6749 }}
| −0.0204
| 0.0204
| 0.0204
| 3.62
| 3.62
|-
|-
|2.9.15
| 2.9.15
|{{monzo|37 29 -33}}, {{monzo|209 -61 -4}}
| {{monzo| 37 29 -33 }}, {{monzo| 209 -61 -4 }}
|{{val|2129 6749 8318}}
| {{mapping| 2129 6749 8318 }}
| -0.0247
| −0.0247
| 0.0177
| 0.0177
| 3.14
| 3.14
|-
|-
|2.9.15.7
| 2.9.15.7
|24414062500/24407490807, 13841287201/13839609375, 2199023255552/2197176384375
| 24414062500/24407490807, 13841287201/13839609375, 2199023255552/2197176384375
|{{val|2129 6749 8318 5977}}
| {{mapping| 2129 6749 8318 5977 }}
| -0.0256
| −0.0256
| 0.0154
| 0.0154
| 2.73
| 2.73
|-
|-
|2.9.15.7.11
| 2.9.15.7.11
|9800/9801, 5767168/5764801, 104857600/104825259, 13841287201/13839609375
| 9800/9801, 5767168/5764801, 104857600/104825259, 13841287201/13839609375
|{{val|2129 6749 8318 5977 7365}}
| {{mapping| 2129 6749 8318 5977 7365 }}
| -0.0162
| −0.0162
| 0.0232
| 0.0232
| 4.12
| 4.12
|-
|-
|2.9.15.7.11.13
| 2.9.15.7.11.13
|10648/10647, 9801/9800, 196625/196608, 36924979/36905625, 304117528/303807105
| 10648/10647, 9801/9800, 196625/196608, 36924979/36905625, 304117528/303807105
|{{val|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
|-
|-
|2.9.15.7.11.13.17
| 2.9.15.7.11.13.17
|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
|{{val|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
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|442\2129
| 884\2129
|249.131
| 498.262
|81/70
| 4/3
|[[Hemischis]] (5-limit)
| [[Helmholtz (temperament)|Helmholtz]]
|-
|1
|884\2129
|498.262
|4/3
|[[Helmholtz]]
|}
|}
<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 ==
*[https://www.youtube.com/watch?v=xF_1MKMlsjY Brid Dance] by [[User:Francium|Francium]]
; [[User:Francium|Francium]]
* "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]]
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