Ainismic chords: Difference between revisions

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An '''ainic chord''' is an [[Dyadic chord|essentially tempered dyadic chord]] tempered by [[936/935]], the ainos comma.  Assuming we stick to the 17-odd-limit, sixteen ainic triads and eight ainic tetrads are known, however, other triads and tetrads, as well as pentads and so forth, are likely to exist.  However, if we are willing to go as high as the 39-odd-limit, many more options become available.
'''Ainismic chords''' are [[Dyadic chord|essentially tempered chords]] tempered by [[936/935]], the ainisma.  


The known ainic triads are:
Ainismic chords are numerous, including 22 triads and 99 tetrads as 2.3.5.11.13.17 subgroup [[17-odd-limit]] essentially tempered chords.
* 1-17/16-13/11 with steps 17/16-10/9-22/13, and its inversion, 1-10/9-13/11 with steps 10/9-17/16-22/13.
* 1-18/17-5/4 with steps 18/17-13/11-8/5, and its inversion, 1-13/11-5/4 with steps 13/11-18/17-8/5.  
* 1-13/11-17/12 with steps 13/11-6/5-24/17, and its inversion, 1-6/5-17/12 with steps 6/5-13/11-24/17.  
* 1-12/11-17/12 with steps 12/11-13/10-24/17, and its inversion, 1-13/10-17/12 with steps 13/10-12/11-24/17.
* 1-11/9-24/17 with steps 11/9-15/13-17/12, and its inversion, 1-15/13-24/17 with steps 15/13-11/9-17/12.
* 1-6/5-17/13 with steps 6/5-12/11-26/17 and its inversion, 1-12/11-17/13 with steps 12/11-6/5-26/17.  
* 1-5/4-26/17 with steps 5/4-11/9-17/13, dubbed the "ainic major triad".  
* 1-11/9-26/17 with steps 11/9-5/4-17/13, dubbed the "ainic neutral triad".  
* 1-11/8-26/17 with steps 11/8-10/9-17/13, dubbed the "ainic suspended demisharp four chord".
* 1-10/9-26/17 with steps 10/9-11/8-17/13, dubbed the "ainic suspended two chord".  


The known ainic tetrads are:
For triads, there are eleven pairs of chords in inverse relationship:
* 1-12/11-17/12-3/2 with steps 12/11-13/10-18/17-4/3, and its inversion, 1-12/11-15/13-3/2 with steps 12/11-18/17-13/10-4/3.
* 1-5/4-26/17 with steps of 5/4-11/9-17/13 (ainismic major triad), and its inversion
* 1-13/10-17/12-3/2 with steps 13/10-12/11-18/17-4/3, and its inversion, 1-13/10-11/8-3/2 with steps 13/10-18/17-12/11-4/3.
* 1-11/9-26/17 with steps of 11/9-5/4-17/13 (ainismic neutral triad);
* 1-18/17-15/13-3/2 with steps 18/17-12/11-13/10-4/3, and its inversion, 1-18/17-11/8-3/2 with steps 18/17-13/10-12/11-4/3.
* 1-11/8-26/17 with steps of 11/8-10/9-17/13 (ainismic suspended demisharp four), and its inversion
* 1-15/13-24/17-3/2 with steps 15/13-11/9-17/16-4/3, and its inversion, 1-17/16-13/10-3/2 with steps 17/16-11/9-15/13-4/3.  
* 1-10/9-26/17 with steps of 10/9-11/8-17/13 (ainismic suspended two);
* 1-6/5-17/12 with steps of 6/5-13/11-24/17, and its inversion
* 1-13/11-17/12 with steps of 13/11-6/5-24/17;
* 1-13/10-17/12 with steps of 13/10-12/11-24/17, and its inversion
* 1-12/11-17/12 with steps of 12/11-13/10-24/17;
* 1-11/9-24/17 with steps of 11/9-15/13-17/12, and its inversion
* 1-15/13-24/17 with steps of 15/13-11/9-17/12;
* 1-13/10-11/8 with steps of 13/10-18/17-16/11, and its inversion
* 1-18/17-11/8 with steps of 18/17-13/10-16/11;
* 1-6/5-17/13 with steps of 6/5-12/11-26/17, and its inversion
* 1-12/11-17/13 with steps of 12/11-6/5-26/17;
* 1-11/9-13/10 with steps of 11/9-17/16-20/13, and its inversion
* 1-17/16-13/10 with steps of 17/16-11/9-20/13;
* 1-13/11-5/4 with steps of 13/11-18/17-8/5, and its inversion
* 1-18/17-5/4 with steps of 18/17-13/11-8/5;
* 1-10/9-13/11 with steps of 10/9-17/16-22/13, and its inversion
* 1-17/16-13/11 with steps of 17/16-10/9-22/13;
* 1-12/11-15/13 with steps of 12/11-18/17-26/15, and its inversion
* 1-18/17-15/13 with steps of 18/17-12/11-26/15.
 
For tetrads, there are three palindromic chords and forty-eight pairs of chords in inverse relationship. The palindromic chords are:
* 1-12/11-6/5-17/13 with steps of 12/11-11/10-12/11-26/17;
* 1-13/11-17/12-5/3 with steps of 13/11-6/5-20/17-6/5;
* 1-12/11-17/12-17/11 with steps of 12/11-13/10-12/11-22/17.
 
Inversely related pairs of tetrads include:  
* 1-12/11-17/12-3/2 with steps of 12/11-13/10-18/17-4/3, and its inversion  
* 1-12/11-15/13-3/2 with steps of 12/11-18/17-13/10-4/3;
* 1-13/10-17/12-3/2 with steps of 13/10-12/11-18/17-4/3, and its inversion  
* 1-13/10-11/8-3/2 with steps of 13/10-18/17-12/11-4/3;
* 1-18/17-11/8-3/2 with steps of 18/17-13/10-12/11-4/3, and its inversion  
* 1-18/17-15/13-3/2 with steps of 18/17-12/11-13/10-4/3;
* 1-15/13-24/17-3/2 with steps of 15/13-11/9-17/16-4/3, and its inversion  
* 1-17/16-13/10-3/2 with steps of 17/16-11/9-15/13-4/3.
 
Equal temperaments with ainismic chords include {{Optimal ET sequence|22, 24, 26, 31, 34, 46, 53, 65, 72, 87, 96, 111, 137, 145, 159, 183, 217 and 270}}.


[[Category:17-odd-limit]]
[[Category:17-odd-limit]]
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Dyadic]]
[[Category:Triads]]
[[Category:Triad]]
[[Category:Tetrads]]
[[Category:Tetrad]]
[[Category:Ainismic]]
[[Category:Ainic]]

Latest revision as of 12:24, 9 December 2023

Ainismic chords are essentially tempered chords tempered by 936/935, the ainisma.

Ainismic chords are numerous, including 22 triads and 99 tetrads as 2.3.5.11.13.17 subgroup 17-odd-limit essentially tempered chords.

For triads, there are eleven pairs of chords in inverse relationship:

  • 1-5/4-26/17 with steps of 5/4-11/9-17/13 (ainismic major triad), and its inversion
  • 1-11/9-26/17 with steps of 11/9-5/4-17/13 (ainismic neutral triad);
  • 1-11/8-26/17 with steps of 11/8-10/9-17/13 (ainismic suspended demisharp four), and its inversion
  • 1-10/9-26/17 with steps of 10/9-11/8-17/13 (ainismic suspended two);
  • 1-6/5-17/12 with steps of 6/5-13/11-24/17, and its inversion
  • 1-13/11-17/12 with steps of 13/11-6/5-24/17;
  • 1-13/10-17/12 with steps of 13/10-12/11-24/17, and its inversion
  • 1-12/11-17/12 with steps of 12/11-13/10-24/17;
  • 1-11/9-24/17 with steps of 11/9-15/13-17/12, and its inversion
  • 1-15/13-24/17 with steps of 15/13-11/9-17/12;
  • 1-13/10-11/8 with steps of 13/10-18/17-16/11, and its inversion
  • 1-18/17-11/8 with steps of 18/17-13/10-16/11;
  • 1-6/5-17/13 with steps of 6/5-12/11-26/17, and its inversion
  • 1-12/11-17/13 with steps of 12/11-6/5-26/17;
  • 1-11/9-13/10 with steps of 11/9-17/16-20/13, and its inversion
  • 1-17/16-13/10 with steps of 17/16-11/9-20/13;
  • 1-13/11-5/4 with steps of 13/11-18/17-8/5, and its inversion
  • 1-18/17-5/4 with steps of 18/17-13/11-8/5;
  • 1-10/9-13/11 with steps of 10/9-17/16-22/13, and its inversion
  • 1-17/16-13/11 with steps of 17/16-10/9-22/13;
  • 1-12/11-15/13 with steps of 12/11-18/17-26/15, and its inversion
  • 1-18/17-15/13 with steps of 18/17-12/11-26/15.

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

  • 1-12/11-6/5-17/13 with steps of 12/11-11/10-12/11-26/17;
  • 1-13/11-17/12-5/3 with steps of 13/11-6/5-20/17-6/5;
  • 1-12/11-17/12-17/11 with steps of 12/11-13/10-12/11-22/17.

Inversely related pairs of tetrads include:

  • 1-12/11-17/12-3/2 with steps of 12/11-13/10-18/17-4/3, and its inversion
  • 1-12/11-15/13-3/2 with steps of 12/11-18/17-13/10-4/3;
  • 1-13/10-17/12-3/2 with steps of 13/10-12/11-18/17-4/3, and its inversion
  • 1-13/10-11/8-3/2 with steps of 13/10-18/17-12/11-4/3;
  • 1-18/17-11/8-3/2 with steps of 18/17-13/10-12/11-4/3, and its inversion
  • 1-18/17-15/13-3/2 with steps of 18/17-12/11-13/10-4/3;
  • 1-15/13-24/17-3/2 with steps of 15/13-11/9-17/16-4/3, and its inversion
  • 1-17/16-13/10-3/2 with steps of 17/16-11/9-15/13-4/3.

Equal temperaments with ainismic chords include 22, 24, 26, 31, 34, 46, 53, 65, 72, 87, 96, 111, 137, 145, 159, 183, 217 and 270.

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