2129edo: Difference between revisions

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Til I find out which val supports hemischis
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|2129}}
{{EDO intro|2129}}
== 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.
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.
===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" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|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]] (%)
|-
|-
|2.9
| 2.9
|{{monzo|-6749 2129}}
| {{monzo|-6749 2129}}
|{{val|2129 6749}}
| {{val|2129 6749}}
| -0.0204
| -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}}
| {{val|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}}
| {{val|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}}
| {{val|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}}
| {{val|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}}
| {{val|2129 6749 8318 5977 7365 7878 8702}}
| -0.0024
| -0.0024
| 0.0295
| 0.0295
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
|-
|-
|1
| 1
|442\2129
| 884\2129
|249.131
| 498.262
|81/70
| 4/3
|[[Hemischis]] (5-limit)
| [[Helmholtz]]
|-
|1
|884\2129
|498.262
|4/3
|[[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


== Music ==
== Music ==
*[https://www.youtube.com/watch?v=xF_1MKMlsjY Brid Dance] by [[User:Francium|Francium]]
; [[User:Francium|Francium]]
* [https://www.youtube.com/watch?v=xF_1MKMlsjY ''Brid Dance''] (2023) – [[hemischis]] in 2129edo tuning

Revision as of 12:26, 15 October 2023

← 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

Template:EDO intro

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.

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 it is distinct

Music

Francium
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