2129edo: Difference between revisions
2.9.15.7.... seems like a good interpretation |
m Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|" Tags: Mobile edit Mobile web edit |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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. | 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 === | ||
| Line 13: | Line 13: | ||
== 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 | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 25: | Line 26: | ||
| {{monzo| -6749 2129 }} | | {{monzo| -6749 2129 }} | ||
| {{mapping| 2129 6749 }} | | {{mapping| 2129 6749 }} | ||
| | | −0.0204 | ||
| 0.0204 | | 0.0204 | ||
| 3.62 | | 3.62 | ||
| Line 32: | 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 | ||
| 0.0177 | | 0.0177 | ||
| 3.14 | | 3.14 | ||
| Line 39: | 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 | ||
| 0.0154 | | 0.0154 | ||
| 2.73 | | 2.73 | ||
| Line 46: | 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 | ||
| 0.0232 | | 0.0232 | ||
| 4.12 | | 4.12 | ||
| Line 53: | 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 | ||
| 0.0288 | | 0.0288 | ||
| 5.11 | | 5.11 | ||
| Line 60: | 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 | ||
| 0.0295 | | 0.0295 | ||
| 5.2 | | 5.2 | ||
| Line 67: | Line 68: | ||
=== 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 | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 78: | Line 80: | ||
| 498.262 | | 498.262 | ||
| 4/3 | | 4/3 | ||
| [[Helmholtz]] | | [[Helmholtz (temperament)|Helmholtz]] | ||
|} | |} | ||
<nowiki>* | <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 == | ||
; [[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) – [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]] | |||