2129edo: Difference between revisions

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Til I find out which val supports hemischis
Rework on theory, mentioning the most important comma -- schisma, and remove the obscure, complex commas
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== 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 the [[3/1|harmonic 3]] commends itself to a 2.9.7.11.13.… [[subgroup]] interpretation.  


=== Odd harmonics ===
=== Odd harmonics ===
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|-
|-
| 2.9
| 2.9
| {{monzo|-6749 2129}}
| {{monzo| -6749 2129 }}
| {{val|2129 6749}}
| {{mapping| 2129 6749 }}
| -0.0204
| -0.0204
| 0.0204
| 0.0204
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|-
|-
| 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
Line 38: Line 38:
| 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
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| 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
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| 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
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| 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

Revision as of 12:32, 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

2129edo is only consistent to the 5-odd-limit, where it tempers out the schisma. Otherwise its poor approximation to the harmonic 3 commends itself to a 2.9.7.11.13.… subgroup interpretation.

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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