Catadictmic chords: Difference between revisions

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For triads, there are three pairs of chords in inverse relationship:  
For triads, there are three pairs of chords in inverse relationship:  
* 1-16/13-11/8 with steps of 16/13-10/9-16/11 and its inverse 1-16/11-13/8 with steps of 16/11-10/9-16/13;
* 1–16/13–11/8 with steps of 16/13, 10/9, 16/11, and its inverse  
* 1-13/10-16/11 with steps of 13/10-9/8-11/8 and its inverse 1-11/8-20/13 with steps of 11/8-9/8-13/10;
* 1–16/11–13/8 with steps of 16/11, 10/9, 16/13;
* 1-11/10-16/13 with steps of 11/10-9/8-13/8 and its inverse 1-9/8-16/13 with steps of 9/8-11/10-13/8.
* 1–13/10–16/11 with steps of 13/10, 9/8, 11/8, and its inverse  
* 1–11/8–20/13 with steps of 11/8, 9/8, 13/10;
* 1–11/10–16/13 with steps of 11/10, 9/8, 13/8, and its inverse  
* 1–9/8–16/13 with steps of 9/8, 11/10, 13/8.


For tetrads, there are three palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are:
For tetrads, there are three palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are  
* 1-16/13-16/11-9/5 chord with steps of 16/13-13/11-16/13-10/9;
* 1–16/13–16/11–9/5 with steps of 16/13, 13/11, 16/13, 10/9;
* 1-9/8-11/8-20/13 chord with steps of 9/8-11/9-9/8-13/10;
* 1–9/8–11/8–20/13 with steps of 9/8, 11/9, 9/8, 13/10;
* 1-9/8-16/13-18/13 chord with steps of 9/8-11/10-9/8-13/9.
* 1–9/8–16/13–18/13 with steps of 9/8, 11/10, 9/8, 13/9.


The inversely related pairs of chords are:
The inversely related pairs of chords are  
* 1-13/10-16/11-9/5 with steps of 13/10-9/8-16/13-10/9 and its inverse 1-16/13-18/13-9/5 with steps of 16/13-9/8-13/10-10/9;
* 1–13/10–16/11–9/5 with steps of 13/10, 9/8, 16/13, 10/9, and its inverse  
* 1-9/8-16/11-9/5 with steps of 9/8-13/10-16/13-10/9 and its inverse 1-16/13-8/5-9/5 with steps of 16/13-13/10-9/8-10/9;
* 1–16/13–18/13–9/5 with steps of 16/13, 9/8, 13/10, 10/9;
* 1-13/11-13/10-16/11 with steps of 13/11-11/10-9/8-11/8 and its inverse 1-9/8-16/13-16/11 with steps of 9/8-11/10-13/11-11/8;
* 1–9/8–16/11–9/5 with steps of 9/8, 13/10, 16/13, 10/9, and its inverse  
* 1-13/10-16/11-8/5 with steps of 13/10-9/8-11/10-5/4 and its inverse 1-11/10-16/13-8/5 with steps of 11/10-9/8-13/10-5/4;
* 1–16/13–8/5–9/5 with steps of 16/13, 13/10, 9/8, 10/9;
* 1-11/10-16/13-11/8 with steps of 11/10-9/8-10/9-16/11 and its inverse 1-10/9-5/4-11/8 with steps of 10/9-9/8-11/10-16/11;
* 1–13/11–13/10–16/11 with steps of 13/11, 11/10, 9/8, 11/8, and its inverse  
* 1-11/10-16/13-9/5 with steps of 11/10-9/8-16/11-10/9 and its inverse 1-16/11-18/11-9/5 with steps of 16/11-9/8-11/10-10/9.
* 1–9/8–16/13–16/11 with steps of 9/8, 11/10, 13/11, 11/8;
* 1–13/10–16/11–8/5 with steps of 13/10, 9/8, 11/10, 5/4, and its inverse  
* 1–11/10–16/13–8/5 with steps of 11/10, 9/8, 13/10, 5/4;
* 1–11/10–16/13–11/8 with steps of 11/10, 9/8, 10/9, 16/11, and its inverse  
* 1–10/9–5/4–11/8 with steps of 10/9, 9/8, 11/10, 16/11;
* 1–11/10–16/13–9/5 with steps of 11/10, 9/8, 16/11, 10/9, and its inverse  
* 1–16/11–18/11–9/5 with steps of 16/11, 9/8, 11/10, 10/9.


For pentads, there are three pairs of chords in inverse relationship:
For pentads, there are three pairs of chords in inverse relationship:  
* 1-16/13-16/11-8/5-9/5 with steps of 16/13-13/11-11/10-9/8-10/9 and its inverse 1-9/8-16/13-16/11-9/5 with steps of 9/8-11/10-13/11-16/13-10/9;
* 1–16/13–16/11–8/5–9/5 with steps of 16/13, 13/11, 11/10, 9/8, 10/9, and its inverse  
* 1-9/8-16/11-18/11-9/5 with steps of 9/8-13/10-9/8-11/10-10/9 and its inverse 1-11/10-16/13-8/5-9/5 with steps of 11/10-9/8-13/10-9/8-10/9;
* 1–9/8–16/13–16/11–9/5 with steps of 9/8, 11/10, 13/11, 16/13, 10/9;
* 1-13/10-16/11-8/5-9/5 with steps of 13/10-9/8-11/10-9/8-10/9 and its inverse 1-9/8-16/13-18/13-9/5 with steps of 9/8-11/10-9/8-13/10-10/9.
* 1–9/8–16/11–18/11–9/5 with steps of 9/8, 13/10, 9/8, 11/10, 10/9, and its inverse  
* 1–11/10–16/13–8/5–9/5 with steps of 11/10, 9/8, 13/10, 9/8, 10/9;
* 1–13/10–16/11–8/5–9/5 with steps of 13/10, 9/8, 11/10, 9/8, 10/9, and its inverse  
* 1–9/8–16/13–18/13–9/5 with steps of 9/8, 11/10, 9/8, 13/10, 10/9.


Equal temperaments with catadictmic chords include {{Optimal ET sequence| 7, 12, 19, 22, 24, 26, 29, 31, 43, 46, 50, 53, 72 and 96 }}.
Equal temperaments with catadictmic chords include {{Optimal ET sequence| 7, 12, 19, 22, 24, 26, 29, 31, 43, 46, 50, 53, 72 and 96 }}.


[[Category:13-odd-limit]]
[[Category:13-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Triads]]

Latest revision as of 13:47, 19 March 2025

Catadictmic chords are essentially tempered dyadic chords tempered by the catadictma, 1287/1280.

Catadictmic chords are of pattern 2 in the 13-odd-limit, meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.

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

  • 1–16/13–11/8 with steps of 16/13, 10/9, 16/11, and its inverse
  • 1–16/11–13/8 with steps of 16/11, 10/9, 16/13;
  • 1–13/10–16/11 with steps of 13/10, 9/8, 11/8, and its inverse
  • 1–11/8–20/13 with steps of 11/8, 9/8, 13/10;
  • 1–11/10–16/13 with steps of 11/10, 9/8, 13/8, and its inverse
  • 1–9/8–16/13 with steps of 9/8, 11/10, 13/8.

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

  • 1–16/13–16/11–9/5 with steps of 16/13, 13/11, 16/13, 10/9;
  • 1–9/8–11/8–20/13 with steps of 9/8, 11/9, 9/8, 13/10;
  • 1–9/8–16/13–18/13 with steps of 9/8, 11/10, 9/8, 13/9.

The inversely related pairs of chords are

  • 1–13/10–16/11–9/5 with steps of 13/10, 9/8, 16/13, 10/9, and its inverse
  • 1–16/13–18/13–9/5 with steps of 16/13, 9/8, 13/10, 10/9;
  • 1–9/8–16/11–9/5 with steps of 9/8, 13/10, 16/13, 10/9, and its inverse
  • 1–16/13–8/5–9/5 with steps of 16/13, 13/10, 9/8, 10/9;
  • 1–13/11–13/10–16/11 with steps of 13/11, 11/10, 9/8, 11/8, and its inverse
  • 1–9/8–16/13–16/11 with steps of 9/8, 11/10, 13/11, 11/8;
  • 1–13/10–16/11–8/5 with steps of 13/10, 9/8, 11/10, 5/4, and its inverse
  • 1–11/10–16/13–8/5 with steps of 11/10, 9/8, 13/10, 5/4;
  • 1–11/10–16/13–11/8 with steps of 11/10, 9/8, 10/9, 16/11, and its inverse
  • 1–10/9–5/4–11/8 with steps of 10/9, 9/8, 11/10, 16/11;
  • 1–11/10–16/13–9/5 with steps of 11/10, 9/8, 16/11, 10/9, and its inverse
  • 1–16/11–18/11–9/5 with steps of 16/11, 9/8, 11/10, 10/9.

For pentads, there are three pairs of chords in inverse relationship:

  • 1–16/13–16/11–8/5–9/5 with steps of 16/13, 13/11, 11/10, 9/8, 10/9, and its inverse
  • 1–9/8–16/13–16/11–9/5 with steps of 9/8, 11/10, 13/11, 16/13, 10/9;
  • 1–9/8–16/11–18/11–9/5 with steps of 9/8, 13/10, 9/8, 11/10, 10/9, and its inverse
  • 1–11/10–16/13–8/5–9/5 with steps of 11/10, 9/8, 13/10, 9/8, 10/9;
  • 1–13/10–16/11–8/5–9/5 with steps of 13/10, 9/8, 11/10, 9/8, 10/9, and its inverse
  • 1–9/8–16/13–18/13–9/5 with steps of 9/8, 11/10, 9/8, 13/10, 10/9.

Equal temperaments with catadictmic chords include 7, 12, 19, 22, 24, 26, 29, 31, 43, 46, 50, 53, 72 and 96.

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