Werckismic chords: Difference between revisions
Jump to navigation
Jump to search
m Categories |
m Added links (as requested by the to:do notice) |
||
| Line 1: | Line 1: | ||
A '''werckismic chord''' is an [[essentially tempered dyadic chord]] in werckismic (441/440) tempering in the 2.9.5.7.11 subgroup in the 11-odd-limit. | A '''werckismic chord''' is an [[essentially tempered dyadic chord]] in [[Werckismic temperaments|werckismic]] ([[441/440]]) tempering in the 2.9.5.7.11 [[Just intonation subgroup|subgroup]] in the [[11-odd-limit]]. | ||
The werckismic triads consist of three pairs of inversely related chords: | The werckismic [[Triad|triads]] consist of three pairs of inversely related chords: | ||
* 1-11/9-7/5 with steps 11/9-8/7-10/7 and 1-11/9-7/4 with steps 11/9-10/7-8/7; | * 1-11/9-7/5 with [[Step|steps]] 11/9-8/7-10/7 and 1-11/9-7/4 with steps 11/9-10/7-8/7; | ||
* 1-11/7-7/4 with steps 11/7-10/9-8/7 and 1-11/7-9/5 with steps 11/7-8/7-10/9; and | * 1-11/7-7/4 with steps 11/7-10/9-8/7 and 1-11/7-9/5 with steps 11/7-8/7-10/9; and | ||
* 1-7/5-11/7 with steps 7/5-9/8-14/11 and 1-14/11-10/7 with steps 14/11-9/8-7/5. | * 1-7/5-11/7 with steps 7/5-9/8-14/11 and 1-14/11-10/7 with steps 14/11-9/8-7/5. | ||
The werckismic tetrads consist of three palindromic (self-inversive) chords and five pairs of chords in an inverse relationship. The palindromic chords are | The werckismic [[Tetrad|tetrads]] consist of three palindromic (self-inversive) chords and five pairs of chords in an inverse relationship. The palindromic chords are | ||
* 1-5/4-10/7-7/4 with steps 5/4-8/7-11/9-8/7; | * 1-5/4-10/7-7/4 with steps 5/4-8/7-11/9-8/7; | ||
* 1-9/8-10/7-11/7 with steps 9/8-14/11-11/10-14/11; and | * 1-9/8-10/7-11/7 with steps 9/8-14/11-11/10-14/11; and | ||
* 1-14/11-7/5-16/9 with steps 14/11-11/10-14/11-9/8. | * 1-14/11-7/5-16/9 with steps 14/11-11/10-14/11-9/8. | ||
The pairs of chords are 1-11/9-11/7-7/4 with steps 11/9-9/7-10/9-8/7 and 1-9/7-11/7-9/5 with steps 9/7-11/9-8/7-10/9; | The pairs of chords are: | ||
* 1-11/9-11/7-7/4 with steps 11/9-9/7-10/9-8/7 and 1-9/7-11/7-9/5 with steps 9/7-11/9-8/7-10/9; | |||
* 1-10/7-11/7-7/4 with steps 10/7-11/10-10/9-8/7 and 1-8/7-14/11-7/5 with steps 8/7-10/9-11/10-10/7; | * 1-10/7-11/7-7/4 with steps 10/7-11/10-10/9-8/7 and 1-8/7-14/11-7/5 with steps 8/7-10/9-11/10-10/7; | ||
* 1-9/8-10/7-7/4 with steps 9/8-14/11-11/9-8/7 and 1-11/9-14/9-7/4 with steps 11/9-14/11-9/8-8/7; | * 1-9/8-10/7-7/4 with steps 9/8-14/11-11/9-8/7 and 1-11/9-14/9-7/4 with steps 11/9-14/11-9/8-8/7; | ||
| Line 18: | Line 20: | ||
* 1-9/8-9/7-10/7 with steps 9/8-8/7-10/9-7/5 and 1-9/8-11/7-7/4 with steps 9/8-7/5-10/9-8/7. | * 1-9/8-9/7-10/7 with steps 9/8-8/7-10/9-7/5 and 1-9/8-11/7-7/4 with steps 9/8-7/5-10/9-8/7. | ||
The werckismic pentads consist of three pairs of chords in an inverse relationship: | The werckismic [[Pentad|pentads]] consist of three pairs of chords in an inverse relationship: | ||
* 1-9/8-5/4-10/7-7/4 with steps 9/8-10/9-8/7-11/9-8/7 and 1-11/9-7/5-14/9-7/4 with steps 11/9-8/7-10/9-9/8-8/7; | * 1-9/8-5/4-10/7-7/4 with steps 9/8-10/9-8/7-11/9-8/7 and 1-11/9-7/5-14/9-7/4 with steps 11/9-8/7-10/9-9/8-8/7; | ||
* 1-11/9-11/8-11/7-7/4 with steps 11/9-9/8-8/7-10/9-8/7 and 1-9/8-11/8-11/7-7/4 with steps 9/8-11/9-8/7-10/9-8/7; | * 1-11/9-11/8-11/7-7/4 with steps 11/9-9/8-8/7-10/9-8/7 and 1-9/8-11/8-11/7-7/4 with steps 9/8-11/9-8/7-10/9-8/7; | ||
| Line 24: | Line 26: | ||
The count of chords is therefore triads: 3, tetrads: 13, pentads: 6, for a total of 22. | The count of chords is therefore triads: 3, tetrads: 13, pentads: 6, for a total of 22. | ||
[[Category:11-odd-limit]] | [[Category:11-odd-limit]] | ||
| Line 34: | Line 34: | ||
[[Category:Werckismic]] | [[Category:Werckismic]] | ||
{{ | [[EDO|Equal temperaments]] with werckismic chords include {{EDOs| 31, 41, 43, 46, 58, 72, 77, 84, 103, 118, 130, 159, 171, 190, 248, 289, and 320 }}, with [[320edo]] giving the [[optimal patent val]]. | ||
Revision as of 08:07, 20 April 2023
A werckismic chord is an essentially tempered dyadic chord in werckismic (441/440) tempering in the 2.9.5.7.11 subgroup in the 11-odd-limit.
The werckismic triads consist of three pairs of inversely related chords:
- 1-11/9-7/5 with steps 11/9-8/7-10/7 and 1-11/9-7/4 with steps 11/9-10/7-8/7;
- 1-11/7-7/4 with steps 11/7-10/9-8/7 and 1-11/7-9/5 with steps 11/7-8/7-10/9; and
- 1-7/5-11/7 with steps 7/5-9/8-14/11 and 1-14/11-10/7 with steps 14/11-9/8-7/5.
The werckismic tetrads consist of three palindromic (self-inversive) chords and five pairs of chords in an inverse relationship. The palindromic chords are
- 1-5/4-10/7-7/4 with steps 5/4-8/7-11/9-8/7;
- 1-9/8-10/7-11/7 with steps 9/8-14/11-11/10-14/11; and
- 1-14/11-7/5-16/9 with steps 14/11-11/10-14/11-9/8.
The pairs of chords are:
- 1-11/9-11/7-7/4 with steps 11/9-9/7-10/9-8/7 and 1-9/7-11/7-9/5 with steps 9/7-11/9-8/7-10/9;
- 1-10/7-11/7-7/4 with steps 10/7-11/10-10/9-8/7 and 1-8/7-14/11-7/5 with steps 8/7-10/9-11/10-10/7;
- 1-9/8-10/7-7/4 with steps 9/8-14/11-11/9-8/7 and 1-11/9-14/9-7/4 with steps 11/9-14/11-9/8-8/7;
- 1-11/9-11/8-7/4 with steps 11/9-9/8-14/11-8/7 and 1-14/11-10/7-7/4 with steps 14/11-9/8-11/9-8/7;
- 1-9/8-5/4-10/7 with steps 9/8-10/9-8/7-7/5 and 1-10/9-5/4-7/4 with steps 10/9-9/8-7/5-8/7; and
- 1-9/8-9/7-10/7 with steps 9/8-8/7-10/9-7/5 and 1-9/8-11/7-7/4 with steps 9/8-7/5-10/9-8/7.
The werckismic pentads consist of three pairs of chords in an inverse relationship:
- 1-9/8-5/4-10/7-7/4 with steps 9/8-10/9-8/7-11/9-8/7 and 1-11/9-7/5-14/9-7/4 with steps 11/9-8/7-10/9-9/8-8/7;
- 1-11/9-11/8-11/7-7/4 with steps 11/9-9/8-8/7-10/9-8/7 and 1-9/8-11/8-11/7-7/4 with steps 9/8-11/9-8/7-10/9-8/7;
- 1-9/8-9/7-10/7-11/7 with steps 9/8-8/7-10/9-11/10-14/11 and 1-9/8-10/7-11/7-7/4 with steps 9/8-14/11-11/10-10/9-8/7.
The count of chords is therefore triads: 3, tetrads: 13, pentads: 6, for a total of 22.
Equal temperaments with werckismic chords include 31, 41, 43, 46, 58, 72, 77, 84, 103, 118, 130, 159, 171, 190, 248, 289, and 320, with 320edo giving the optimal patent val.