2129edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! 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
| 2.9
| {{monzo| -6749 2129 }}
| {{monzo| -6749 2129 }}
| {{mapping| 2129 6749 }}
| {{mapping| 2129 6749 }}
| -0.0204
| &minus;0.0204
| 0.0204
| 0.0204
| 3.62
| 3.62
Line 32: Line 24:
| {{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
| &minus;0.0247
| 0.0177
| 0.0177
| 3.14
| 3.14
Line 39: Line 31:
| 24414062500/24407490807, 13841287201/13839609375, 2199023255552/2197176384375
| 24414062500/24407490807, 13841287201/13839609375, 2199023255552/2197176384375
| {{mapping| 2129 6749 8318 5977 }}
| {{mapping| 2129 6749 8318 5977 }}
| -0.0256
| &minus;0.0256
| 0.0154
| 0.0154
| 2.73
| 2.73
Line 46: Line 38:
| 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
| &minus;0.0162
| 0.0232
| 0.0232
| 4.12
| 4.12
Line 53: Line 45:
| 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
| &minus;0.0075
| 0.0288
| 0.0288
| 5.11
| 5.11
Line 60: Line 52:
| 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
| &minus;0.0024
| 0.0295
| 0.0295
| 5.2
| 5.2
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 79: Line 65:
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


== Scales ==
== Scales ==
Line 87: Line 73:
== Music ==
== Music ==
; [[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
* "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


[[Category:Listen]]
[[Category:Listen]]

Revision as of 02:35, 16 November 2024

← 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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 884\2129 | 498.262 | 4/3 | Helmholtz Template:Rank-2 end Template:Orf

Scales

Music

Francium
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