1429edo: Difference between revisions
m →Theory: cleanup |
m Cleanup and update |
||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
1429edo has a reasonable approximation of the full 17-limit. It is [[consistent]] to the [[9-odd-limit]] with only [[11/10]] barely missing the line. The 11-limit [[TE tuning|optimal tuning]] of the equal temperament is consistent to the 18-integer-limit; however, the 13- and 17-limit optimal tunings, which have less of octave compression, are not, so one might want to keep the compression tight. | 1429edo has a reasonable approximation of the full [[17-limit]]. It is [[consistent]] to the [[9-odd-limit]] with only [[11/10]] barely missing the line. The 11-limit [[TE tuning|optimal tuning]] of the equal temperament is consistent to the 18-integer-limit; however, the 13- and 17-limit optimal tunings, which have less of octave compression, are not, so one might want to keep the compression tight. | ||
As an equal temperament, it [[tempering out|tempers out]] [[4375/4374]] in the 7-limit; [[131072/130977]], 759375/758912, 1953125/1951488, 2359296/2358125, 2657205/2656192, and 3294225/3294172 in the 11-limit; [[2080/2079]], [[4096/4095]], [[4225/4224]], 78125/78078, and [[123201/123200]] in the 13-limit; [[2500/2499]], [[5832/5831]], [[11016/11011]], and [[12376/12375]] in the 17-limit. It | As an equal temperament, it [[tempering out|tempers out]] [[4375/4374]] in the 7-limit; [[131072/130977]], 759375/758912, 1953125/1951488, 2359296/2358125, 2657205/2656192, and 3294225/3294172 in the 11-limit; [[2080/2079]], [[4096/4095]], [[4225/4224]], 78125/78078, and [[123201/123200]] in the 13-limit; [[2500/2499]], [[5832/5831]], [[11016/11011]], and [[12376/12375]] in the 17-limit. It [[support]]s the [[gross]] temperament and provides the [[optimal patent val]] for the 11- and 13-limit [[alphatrillium]] temperament. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 19: | Line 19: | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 26: | Line 26: | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| 2265 -1429 }} | ||
| {{ | | {{Mapping| 1429 2265 }} | ||
| −0.0235 | | −0.0235 | ||
| 0.0234 | | 0.0234 | ||
| Line 33: | Line 33: | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{ | | {{Monzo| 39 -29 3 }}, {{monzo| -66 -36 53 }} | ||
| {{ | | {{Mapping| 1429 2265 3318 }} | ||
| −0.0114 | | −0.0114 | ||
| 0.0257 | | 0.0257 | ||
| Line 41: | Line 41: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 4375/4374, {{monzo| 26 4 -3 -14 }}, {{monzo| 40 -22 -1 -1 }} | | 4375/4374, {{monzo| 26 4 -3 -14 }}, {{monzo| 40 -22 -1 -1 }} | ||
| {{ | | {{Mapping| 1429 2265 3318 4012 }} | ||
| −0.0302 | | −0.0302 | ||
| 0.0395 | | 0.0395 | ||
| Line 48: | Line 48: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 4375/4374, 131072/130977, 759375/758912, 3294225/3294172 | | 4375/4374, 131072/130977, 759375/758912, 3294225/3294172 | ||
| {{ | | {{Mapping| 1429 2265 3318 4012 4944 }} | ||
| −0.0471 | | −0.0471 | ||
| 0.0488 | | 0.0488 | ||
| Line 55: | Line 55: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 2080/2079, 4096/4095, 4375/4374, 78125/78078, 3294225/3294172 | | 2080/2079, 4096/4095, 4375/4374, 78125/78078, 3294225/3294172 | ||
| {{ | | {{Mapping| 1429 2265 3318 4012 4944 5288 }} | ||
| −0.0420 | | −0.0420 | ||
| 0.0460 | | 0.0460 | ||
| Line 62: | Line 62: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 2080/2079, 2500/2499, 4096/4095, 4375/4374, 11016/11011, 108086/108045 | | 2080/2079, 2500/2499, 4096/4095, 4375/4374, 11016/11011, 108086/108045 | ||
| {{ | | {{Mapping| 1429 2265 3318 4012 4944 5288 5841 }} | ||
| −0.0364 | | −0.0364 | ||
| 0.0447 | | 0.0447 | ||
| Line 72: | Line 72: | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 87: | Line 87: | ||
| 674\1429 | | 674\1429 | ||
| 565.990 | | 565.990 | ||
| | | 104/75 | ||
| [[ | | [[Alphatrillium]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if 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 97: | Line 97: | ||
== Music == | == Music == | ||
; [[Francium]] | ; [[Francium]] | ||
* "Gross Pattern" from ''Gross temperament EP'' (2023) [https://open.spotify.com/track/1BwMVnnrzfug6pSLUJ0jSG Spotify] | [https://francium223.bandcamp.com/track/gross-pattern Bandcamp] | [https://youtu.be/ttQVdzSy96M | * "Gross Pattern" from ''Gross temperament EP'' (2023) [https://open.spotify.com/track/1BwMVnnrzfug6pSLUJ0jSG Spotify] | [https://francium223.bandcamp.com/track/gross-pattern Bandcamp] | [https://youtu.be/ttQVdzSy96M YouTube] – gross in 1429edo tuning | ||
[[Category:Listen]] | [[Category:Listen]] | ||