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]] tempered by the werckisma, [[441/440]].  


The werckismic [[Triad|triads]] consist of three pairs of inversely related chords:
Werckismic chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 2]] in the [[11-odd-limit]], meaning that there are 6 [[triad]]s, 15 [[tetrad]]s and 6 [[pentad]]s, for a total of 27 distinct chord structures.  
* 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
* 1-7/5-11/7 with steps 7/5-9/8-14/11 and 1-14/11-10/7 with steps 14/11-9/8-7/5.


The werckismic [[Tetrad|tetrads]] consist of three palindromic (self-inversive) chords and five pairs of chords in an inverse relationship. The palindromic chords are  
The werckismic triads consist of three pairs of inversely related chords:
* 1-5/4-10/7-7/4 with steps 5/4-8/7-11/9-8/7;  
* 1–11/9–7/5 with steps 11/9, 8/7, 10/7, and its inverse
* 1-9/8-10/7-11/7 with steps 9/8-14/11-11/10-14/11; and
* 1–11/9–7/4 with steps 11/9, 10/7, 8/7;
* 1-14/11-7/5-16/9 with steps 14/11-11/10-14/11-9/8.  
* 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 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;  
* 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 pentads 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:  
* 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.


[[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]].
[[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]].


[[Category:11-odd-limit]]
[[Category:11-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Triads]]

Latest revision as of 13:35, 11 October 2024

A werckismic chord is an essentially tempered dyadic chord tempered by the werckisma, 441/440.

Werckismic chords are of pattern 2 in the 11-odd-limit, meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.

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 its inverse
  • 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 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 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;
  • 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 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.

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 its inverse
  • 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 its inverse
  • 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 its inverse
  • 1–9/8–10/7–11/7–7/4 with steps 9/8, 14/11, 11/10, 10/9, 8/7.

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.

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