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Werckismic chords: Difference between revisions

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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]].
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 [[Triad|triads]] consist of three pairs of inversely related chords:  
The werckismic [[triad]]s consist of three pairs of inversely related chords:  
* 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;
* 1-11/9-7/5 with [[step]]s 11/9-8/7-10/7, and its inverse
* 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
* 1-11/9-7/4 with steps 11/9-10/7-8/7;
* 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.
* 1-11/7-7/4 with steps 11/7-10/9-8/7, and its inverse
* 1-11/7-9/5 with steps 11/7-8/7-10/9;  
* 1-7/5-11/7 with steps 7/5-9/8-14/11, and its inverse
* 1-14/11-10/7 with steps 14/11-9/8-7/5.


The werckismic [[Tetrad|tetrads]] consist of three palindromic (self-inversive) chords and five pairs of chords in an inverse relationship. The palindromic chords are  
The werckismic [[tetrad]]s 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;  
* 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:  
The pairs of chords are:  
* 1-11/9-11/7-7/4 with steps 11/9-9/7-10/9-8/7, and its inverse
* 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 its inverse
* 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 its inverse
* 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 its inverse
* 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 its inverse
* 1-10/9-5/4-7/4 with steps 10/9-9/8-7/5-8/7;
* 1-9/8-9/7-10/7 with steps 9/8-8/7-10/9-7/5, and its inverse
* 1-9/8-11/7-7/4 with steps 9/8-7/5-10/9-8/7.


* 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 werckismic [[pentad]]s consist of three pairs of chords in an inverse relationship:
* 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;
* 1-9/8-5/4-10/7-7/4 with steps 9/8-10/9-8/7-11/9-8/7, and its inverse
* 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;  
* 1-11/9-7/5-14/9-7/4 with steps 11/9-8/7-10/9-9/8-8/7;  
* 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;
* 1-11/9-11/8-11/7-7/4 with steps 11/9-9/8-8/7-10/9-8/7, and its inverse
* 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
* 1-9/8-11/8-11/7-7/4 with steps 9/8-11/9-8/7-10/9-8/7;  
* 1-9/8-9/7-10/7 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.
* 1-9/8-9/7-10/7-11/7 with steps 9/8-8/7-10/9-11/10-14/11, and its inverse
* 1-9/8-10/7-11/7-7/4 with steps 9/8-14/11-11/10-10/9-8/7.


The werckismic [[Pentad|pentads]] consist of three pairs of chords in an inverse relationship:  
The count of chords is therefore triads: 6, tetrads: 15, pentads: 6, for a total of 27.
* 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 temperament]]s 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]].
 
[[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]].


[[Category:11-odd-limit]]
[[Category:11-odd-limit]]
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