Catadictmic chords - Revision history

FloraC: Cleanup

2025-03-19T13:47:54Z

Cleanup

← Older revision		Revision as of 13:47, 19 March 2025
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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		* 1–16/13–11/8 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;		* 1–16/11–13/8 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		* 1–13/10–16/11 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;		* 1–11/8–20/13 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		* 1–11/10–16/13 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.		* 1–9/8–16/13 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 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 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 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		* 1–13/10–16/11–9/5 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;		* 1–16/13–18/13–9/5 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		* 1–9/8–16/11–9/5 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;		* 1–16/13–8/5–9/5 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		* 1–13/11–13/10–16/11 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;		* 1–9/8–16/13–16/11 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		* 1–13/10–16/11–8/5 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;		* 1–11/10–16/13–8/5 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		* 1–11/10–16/13–11/8 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;		* 1–10/9–5/4–11/8 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		* 1–11/10–16/13–9/5 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.		* 1–16/11–18/11–9/5 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		* 1–16/13–16/11–8/5–9/5 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;		* 1–9/8–16/13–16/11–9/5 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		* 1–9/8–16/11–18/11–9/5 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;		* 1–11/10–16/13–8/5–9/5 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		* 1–13/10–16/11–8/5–9/5 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.		* 1–9/8–16/13–18/13–9/5 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]]

Xenllium at 05:15, 25 November 2023

2023-11-25T05:15:44Z

← Older revision		Revision as of 05:15, 25 November 2023
Line 4:		Line 4:

	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 }}.

Xenllium: Created page with "'''Catadictmic chords''' are essentially tempered dyadic chords tempered by the catadictma, 1287/1280. Catadictmic chords are of Dyadic chord/Pattern o..."

2023-09-11T11:07:50Z

Created page with "'''Catadictmic chords''' are essentially tempered dyadic chords tempered by the catadictma, 1287/1280. Catadictmic chords are of Dyadic chord/Pattern o..."

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'''Catadictmic chords''' are [[Dyadic chord|essentially tempered dyadic chords]] tempered by the catadictma, [[1287/1280]].

Catadictmic chords are of [[Dyadic chord/Pattern of essentially tempered chords|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 chord 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;
* 1-9/8-16/13-18/13 chord 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 {{Optimal ET sequence| 7, 12, 19, 22, 24, 26, 29, 31, 43, 46, 50, 53, 72 and 96 }}.

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