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Biyatismic chords: Difference between revisions

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'''Biyatismic chords''' are [[Dyadic chord|essentially tempered dyadic chords]] tempered by the biyatisma, [[121/120]].
'''Biyatismic chords''' are [[Dyadic chord|essentially tempered dyadic chords]] tempered by the biyatisma, [[121/120]].


Biyatismic temperament contains many possibilities for essentially tempered dyadic chords including five triads and 33 tetrads in the 2.3.5.11 subgroup.
There are two triads, nine tetrads, seven pentads and one hexad as 2.3.5.11 subgroup [[11-odd-limit]] essentially tempered chords. In this article, voicing with the [[perfect fifth]] is prioritized.  


The biyatismic triads consist of a palindromic chord and two pairs of chords in inverse relationship. The palindromic chord is
The two biyatismic triads are in inverse relationship:
* 1-[[11/8]]-[[16/11]] chord with steps of 11/8-[[16/15]]-11/8.  
* 1–11/8–5/3 with steps 11/8, 11/9, 6/5;
* 1–11/8–18/11 with steps 11/8, 6/5, 11/9.


The inversely related pairs of chords are  
For tetrads, there are three palindromic chords and three pairs of chords in inverse relationship. The palindromic chords are  
* 1-11/8-[[11/6]] chord with steps of 11/8-[[4/3]]-[[11/10]] and 1-11/10-16/11 chord with steps of 11/10-4/3-11/8; and
* 1–9/8–11/8–18/11 with steps 9/8, 11/9, 6/5, 11/9;
* 1-11/8-[[5/3]] chord with steps of 11/8-[[11/9]]-[[6/5]] and 1-6/5-16/11 chord with steps of 6/5-11/9-11/8.
* 1–11/10–11/8–3/2 with steps 11/10, 5/4, 11/10, 4/3;
* 1–3/2–18/11–11/6 with steps 3/2, 11/10, 10/9, 11/10.  


The biyatismic tetrads consist of three palindromic chords and fifteen pairs of chords in inverse relationship. The palindromic chords are  
The inversely related pairs of chords are
* 1-9/8-11/8-18/11 chord with steps of 9/8-11/9-6/5-11/9,  
* 1–11/8–3/2–18/11 with steps 11/8, 11/10, 11/10, 11/9, and its inverse
* 1-5/4-11/8-11/6 chord with steps of 5/4-11/10-4/3-11/10, and  
* 1–11/10–3/2–11/6 with steps 11/10, 11/8, 11/9, 11/10;
* 1-11/10-3/2-18/11 chord with steps of 11/10-11/8-11/10-11/9.  
* 1–5/4–3/2–11/6 with steps 5/4, 6/5, 11/9, 11/10, and its inverse
* 1–6/5–3/2–18/11 with steps 6/5, 5/4, 11/10, 11/9;
* 1–11/8–3/2–5/3 with steps 11/8, 11/10, 10/9, 6/5, and its inverse
* 1–11/10–3/2–9/5 with steps 11/10, 11/8, 6/5, 10/9.  


The inversely related pairs of chords are  
For pentads, there are one palindromic chord and three pairs of chords in inverse relationship. The palindromic chord is
* 1-4/3-5/3-11/6 chord with steps of 4/3-5/4-11/10-11/10 and 1-11/10-6/5-3/2 chord with steps of 11/10-11/10-5/4-4/3;
* 1–9/8–11/8–3/2–18/11 chord with steps of 9/8, 11/9, 11/10, 11/10, 11/9.  
* 1-11/9-11/8-18/11 chord with steps of 11/9-9/8-6/5-11/9 and 1-11/9-16/11-18/11 chord with steps of 11/9-6/5-9/8-11/9;
* 1-11/8-5/3-11/6 chord with steps of 11/8-11/9-11/10-11/10 and 1-11/10-6/5-16/11 chord with steps of 11/10-11/10-11/9-11/8;
* 1-4/3-3/2-11/6 chord with steps of 4/3-9/8-11/9-11/10 and 1-11/10-4/3-3/2 chord with steps of 11/10-11/9-9/8-4/3;
* 1-9/8-11/8-11/6 chord with steps of 9/8-11/9-4/3-11/10 and 1-11/10-16/11-16/9 chord with steps of 11/10-4/3-11/9-9/8;
* 1-4/3-3/2-18/11 chord with steps of 4/3-9/8-11/10-11/9 and 1-11/9-4/3-3/2 chord with steps of 11/9-11/10-9/8-4/3;
* 1-5/4-3/2-11/6 chord with steps of 5/4-6/5-11/9-11/10 and 1-11/10-4/3-8/5 chord with steps of 11/10-11/9-6/5-5/4;
* 1-6/5-3/2-11/6 chord with steps of 6/5-5/4-11/9-11/10 and 1-11/10-4/3-5/3 chord with steps of 11/10-11/9-5/4-6/5;
* 1-6/5-16/11-11/6 chord with steps of 6/5-11/9-5/4-11/10 and 1-11/10-11/8-5/3 chord with steps of 11/10-5/4-11/9-6/5;
* 1-5/4-11/8-16/11 chord with steps of 5/4-11/10-16/15-11/8 and 1-11/8-16/11-8/5 chord with steps of 11/8-16/15-11/10-5/4;
* 1-5/4-4/3-16/11 chord with steps of 5/4-16/15-11/10-11/8 and 1-11/8-3/2-8/5 chord with steps of 11/8-11/10-16/15-5/4;
* 1-5/4-4/3-11/6 chord with steps of 5/4-16/15-11/8-11/10 and 1-11/10-3/2-8/5 chord with steps of 11/10-11/8-16/15-5/4;
* 1-11/9-11/8-16/11 chord with steps of 11/9-9/8-16/15-11/8 and 1-11/8-16/11-18/11 chord with steps of 11/8-16/15-9/8-11/9;
* 1-11/8-5/3-16/9 chord with steps of 11/8-11/9-16/15-9/8 and 1-9/8-6/5-16/11 chord with steps of 9/8-16/15-11/9-11/8; and
* 1-11/8-16/11-16/9 chord with steps of 11/8-16/15-11/9-9/8 and 1-9/8-11/8-16/11 chord with steps of 9/8-11/9-16/15-11/8.


Equal temperaments with biyatismic chords include {{Vals| 7-, 9-, 15-, 16-, 22-, 24-, 29-, 31-, 38-, 46-, and 53et }}.
The inversely related pairs of chords are
* 1–11/8–3/2–5/3–11/6 with steps 11/8, 11/10, 10/9, 11/10, 11/10 and its inverse
* 1–11/10–3/2–18/11–9/5 with steps 11/10, 11/8, 11/10, 11/10, 10/9;
* 1–5/4–3/2–5/3–11/6 with steps 5/4, 6/5, 10/9, 11/10, 11/10 and its inverse
* 1–6/5–3/2–18/11–11/6 with steps 6/5, 5/4, 11/10, 11/10, 10/9;
* 1–5/4–11/8–3/2–11/6 with steps 5/4, 11/10, 11/10, 11/9, 11/10 and its inverse
* 1–11/10–6/5–3/2–18/11 with steps 11/10, 11/10, 5/4, 11/10, 11/9.
 
Finally, there is a palindromic hexad,
* 1–5/4–11/8–3/2–5/3–11/6 with steps 5/4, 11/10, 11/10, 10/9, 11/10, 11/10.
 
[[Equal temperament]]s with biyatismic chords include {{Optimal ET sequence| 7, 9, 15, 16, 22, 24, 29, 31, 38, 46 and 53 }}.


== See also ==
== See also ==
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* [[Chords of porcupine]]
* [[Chords of porcupine]]


[[Category:Chords]]
[[Category:11-odd-limit chords]]
[[Category:Dyadic]]
[[Category:Essentially tempered chords]]
[[Category:Triad]]
[[Category:Triads]]
[[Category:Tetrad]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]
[[Category:Biyatismic]]
[[Category:Biyatismic]]
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