Biyatismic chords: Difference between revisions

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Corrections. 1-11/8-11/6 is clearly utonal; 1-9/8-11/8-11/6, 1-4/3-3/2-18/11, and 1-6/5-3/2-11/6 aren't 11-odd-limit concordances; all 15-odd-limit chords here are either essentially just or aren't concordances
m Fix a ratio
 
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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 3 triads and 8 tetrads in the [[11-odd-limit]] of 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 3 biyatismic triads are a palindromic chord and a pair 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-[[5/3]] chord with steps of 11/8-[[11/9]]-[[6/5]];  
* 1–9/8–11/8–18/11 with steps 9/8, 11/9, 6/5, 11/9;  
* 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.  


For tetrads, there are 2 palindromic chords and 3 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.  
* 1–11/10–3/2–11/6 with steps 11/10, 11/8, 11/9, 11/10;  
* 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
* 1-11/9-11/8-18/11 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;
* 1–9/8–11/8–3/2–18/11 chord with steps of 9/8, 11/9, 11/10, 11/10, 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;
* 1-5/4-3/2-11/6 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.  


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]]

Latest revision as of 10:25, 10 October 2024

Biyatismic chords are essentially tempered dyadic chords tempered by the biyatisma, 121/120.

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 two biyatismic triads are in inverse relationship:

  • 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.

For tetrads, there are three palindromic chords and three pairs of chords in inverse relationship. The palindromic chords are

  • 1–9/8–11/8–18/11 with steps 9/8, 11/9, 6/5, 11/9;
  • 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 inversely related pairs of chords are

  • 1–11/8–3/2–18/11 with steps 11/8, 11/10, 11/10, 11/9, and its inverse
  • 1–11/10–3/2–11/6 with steps 11/10, 11/8, 11/9, 11/10;
  • 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.

For pentads, there are one palindromic chord and three pairs of chords in inverse relationship. The palindromic chord is

  • 1–9/8–11/8–3/2–18/11 chord with steps of 9/8, 11/9, 11/10, 11/10, 11/9.

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 temperaments with biyatismic chords include 7, 9, 15, 16, 22, 24, 29, 31, 38, 46 and 53.

See also

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