2129edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|2129}}
{{EDO intro}}


== Theory ==
== Theory ==
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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
Line 33: Line 33:
| {{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
Line 40: Line 40:
| 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
Line 47: Line 47:
| 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
Line 54: Line 54:
| 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
Line 61: Line 61:
| 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
Line 82: Line 82:
| [[Helmholtz]]
| [[Helmholtz]]
|}
|}
<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 ==
Line 89: Line 89:
== Music ==
== Music ==
; [[User:Francium|Francium]]
; [[User:Francium|Francium]]
* "Brid Dance" from ''HemischisMatic EP'' (2023) &ndash; [https://open.spotify.com/track/6yXO4yHFM0EWk3pCjop8Dp Spotify] | [https://francium223.bandcamp.com/track/brid-dance Bandcamp] | [https://youtu.be/xF_1MKMlsjY?si=8Db0YVe6GQ9ZgPET YouTube] &ndash; [[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]]
[[Category:Listen]]

Revision as of 18:07, 15 January 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

Template:EDO intro

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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