Dakotismic chords - Revision history

FloraC: One chord per line

2025-12-02T08:34:54Z

One chord per line

← Older revision		Revision as of 08:34, 2 December 2025
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	For triads, there are seven pairs of chords in inverse relationship:		For triads, there are seven pairs of chords in inverse relationship:
	* 1–6/5–17/11 with steps 6/5, 9/7, 22/17 and its inverse 1–22/17–5/3 with steps 22/17, 9/7, 6/5;		* 1–6/5–17/11 with steps 6/5, 9/7, 22/17, and its inverse
	* 1–7/6–22/17 with steps 7/6, 10/9, 17/11 and its inverse 1–17/11–12/7 with steps 17/11, 10/9, 7/6;		* 1–22/17–5/3 with steps 22/17, 9/7, 6/5;
	* 1–11/10–17/12 with steps 11/10, 9/7, 24/17 and its inverse 1–9/7–17/12 with steps 9/7, 11/10, 24/17;		* 1–7/6–22/17 with steps 7/6, 10/9, 17/11, and its inverse
	* 1–14/11–24/17 with steps 14/11, 10/9, 17/12 and its inverse 1–10/9–24/17 with steps 10/9, 14/11, 17/12;		* 1–17/11–12/7 with steps 17/11, 10/9, 7/6;
	* 1–9/7–15/11 with steps 9/7, 18/17, 22/15 and its inverse 1–9/7–17/9 with steps 9/7, 22/15, 18/17;		* 1–11/10–17/12 with steps 11/10, 9/7, 24/17, and its inverse
	* 1–11/7–5/3 with steps 11/7, 18/17, 6/5 and its inverse 1–11/7–17/9 with steps 11/7, 6/5, 18/17;		* 1–9/7–17/12 with steps 9/7, 11/10, 24/17;
	* 1–11/10–7/6 with steps 11/10, 18/17, 12/7 and its inverse 1–11/10–17/9 with steps 11/10, 12/7, 18/17.		* 1–14/11–24/17 with steps 14/11, 10/9, 17/12, and its inverse
			* 1–10/9–24/17 with steps 10/9, 14/11, 17/12;
			* 1–9/7–15/11 with steps 9/7, 18/17, 22/15, and its inverse
			* 1–9/7–17/9 with steps 9/7, 22/15, 18/17;
			* 1–11/7–5/3 with steps 11/7, 18/17, 6/5, and its inverse
			* 1–11/7–17/9 with steps 11/7, 6/5, 18/17;
			* 1–11/10–7/6 with steps 11/10, 18/17, 12/7, and its inverse
			* 1–11/10–17/9 with steps 11/10, 12/7, 18/17.

	For tetrads, there are three palindromic chords and twenty-four pairs of chords in inverse relationship. The palindromic chords are		For tetrads, there are three palindromic chords and twenty-four pairs of chords in inverse relationship. The palindromic chords are
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	The inversely related pairs of chords are		The inversely related pairs of chords are
	* 1–14/11–17/11–9/5 with steps 14/11, 17/14, 7/6, 10/9 and its inverse 1–7/6–17/12–9/5 with steps 7/6, 17/14, 14/11, 10/9;		* 1–14/11–17/11–9/5 with steps 14/11, 17/14, 7/6, 10/9, and its inverse
	* 1–6/5–17/11–12/7 with steps 6/5, 9/7, 10/9, 7/6 and its inverse 1–7/6–22/17–5/3 with steps 7/6, 10/9, 9/7, 6/5;		* 1–7/6–17/12–9/5 with steps 7/6, 17/14, 14/11, 10/9;
	* 1–6/5–17/11–9/5 with steps 6/5, 9/7, 7/6, 10/9 and its inverse 1–10/9–22/17–5/3 with steps 10/9, 7/6, 9/7, 6/5;		* 1–6/5–17/11–12/7 with steps 6/5, 9/7, 10/9, 7/6, and its inverse
	* 1–9/7–17/12–11/7 with steps 9/7, 11/10, 10/9, 14/11 and its inverse 1–14/11–24/17–14/9 with steps 14/11, 10/9, 11/10, 9/7;		* 1–7/6–22/17–5/3 with steps 7/6, 10/9, 9/7, 6/5;
	* 1–11/10–17/12–11/7 with steps 11/10, 9/7, 10/9, 14/11 and its inverse 1–14/11–24/17–20/11 with steps 14/11, 10/9, 9/7, 11/10;		* 1–6/5–17/11–9/5 with steps 6/5, 9/7, 7/6, 10/9, and its inverse
	* 1–11/10–17/12–17/10 with steps 11/10, 9/7, 6/5, 20/17 and its inverse 1–6/5–17/11–17/10 with steps 6/5, 9/7, 11/10, 20/17;		* 1–10/9–22/17–5/3 with steps 10/9, 7/6, 9/7, 6/5;
	* 1–17/12–11/7–11/6 with steps 17/12, 10/9, 7/6, 12/11 and its inverse 1–17/12–17/11–9/5 with steps 17/12, 12/11, 7/6, 10/9;		* 1–9/7–17/12–11/7 with steps 9/7, 11/10, 10/9, 14/11, and its inverse
	* 1–22/17–5/3–11/6 with steps 22/17, 9/7, 11/10, 12/11 and its inverse 1–11/10–17/12–11/6 with steps 11/10, 9/7, 22/17, 12/11;		* 1–14/11–24/17–14/9 with steps 14/11, 10/9, 11/10, 9/7;
	* 1–17/12–3/2–9/5 with steps 17/12, 18/17, 6/5, 10/9 and its inverse 1–18/17–3/2–5/3 with steps 18/17, 17/12, 10/9, 6/5;		* 1–11/10–17/12–11/7 with steps 11/10, 9/7, 10/9, 14/11, and its inverse
	* 1–9/7–15/11–3/2 with steps 9/7, 18/17, 11/10, 4/3 and its inverse 1–11/10–7/6–3/2 with steps 11/10, 18/17, 9/7, 4/3;		* 1–14/11–24/17–20/11 with steps 14/11, 10/9, 9/7, 11/10;
	* 1–9/7–17/12–3/2 with steps 9/7, 11/10, 18/17, 4/3 and its inverse 1–18/17–7/6–3/2 with steps 18/17, 11/10, 9/7, 4/3;		* 1–11/10–17/12–17/10 with steps 11/10, 9/7, 6/5, 20/17, and its inverse
	* 1–11/10–17/12–3/2 with steps 11/10, 9/7, 18/17, 4/3 and its inverse 1–18/17–15/11–3/2 with steps 18/17, 9/7, 11/10, 4/3;		* 1–6/5–17/11–17/10 with steps 6/5, 9/7, 11/10, 20/17;
	* 1–9/7–15/11–5/3 with steps 9/7, 18/17, 11/9, 6/5 and its inverse 1–9/7–17/11–17/9 with steps 9/7, 6/5, 11/9, 18/17;		* 1–17/12–11/7–11/6 with steps 17/12, 10/9, 7/6, 12/11, and its inverse
	* 1–9/7–15/11–18/11 with steps 9/7, 18/17, 6/5, 11/9 and its inverse 1–9/7–11/7–17/9 with steps 9/7, 11/9, 6/5, 18/17;		* 1–17/12–17/11–9/5 with steps 17/12, 12/11, 7/6, 10/9;
	* 1–9/7–17/11–18/11 with steps 9/7, 6/5, 18/17, 11/9 and its inverse 1–6/5–17/11–17/9 with steps 6/5, 9/7, 11/9, 18/17;		* 1–22/17–5/3–11/6 with steps 22/17, 9/7, 11/10, 12/11, and its inverse
	* 1–6/5–14/9–17/9 with steps 6/5, 22/17, 17/14, 18/17 and its inverse 1–17/14–11/7–17/9 with steps 17/14, 22/17, 6/5, 18/17;		* 1–11/10–17/12–11/6 with steps 11/10, 9/7, 22/17, 12/11;
	* 1–9/7–5/3–17/9 with steps 9/7, 22/17, 17/15, 18/17 and its inverse 1–17/15–22/15–17/9 with steps 17/15, 22/17, 9/7, 18/17;		* 1–17/12–3/2–9/5 with steps 17/12, 18/17, 6/5, 10/9, and its inverse
	* 1–6/5–17/10–17/9 with steps 6/5, 17/12, 10/9, 18/17 and its inverse 1–10/9–11/7–17/9 with steps 10/9, 17/12, 6/5, 18/17;		* 1–18/17–3/2–5/3 with steps 18/17, 17/12, 10/9, 6/5;
	* 1–17/14–9/7–17/12 with steps 17/14, 18/17, 11/10, 24/17 and its inverse 1–11/10–7/6–17/12 with steps 11/10, 18/17, 17/14, 24/17;		* 1–9/7–15/11–3/2 with steps 9/7, 18/17, 11/10, 4/3, and its inverse
	* 1–10/7–11/7–5/3 with steps 10/7, 11/10, 18/17, 6/5 and its inverse 1–10/7–12/7–20/11 with steps 10/7, 6/5, 18/17, 11/10;		* 1–11/10–7/6–3/2 with steps 11/10, 18/17, 9/7, 4/3;
	* 1–11/10–11/9–17/9 with steps 11/10, 10/9, 17/11, 18/17 and its inverse 1–17/11–12/7–17/9 with steps 17/11, 10/9, 11/10, 18/17;		* 1–9/7–17/12–3/2 with steps 9/7, 11/10, 18/17, 4/3, and its inverse
	* 1–10/9–12/7–17/9 with steps 10/9, 17/11, 11/10, 18/17 and its inverse 1–11/10–17/10–17/9 with steps 11/10, 17/11, 10/9, 18/17;		* 1–18/17–7/6–3/2 with steps 18/17, 11/10, 9/7, 4/3;
	* 1–11/10–6/5–17/9 with steps 11/10, 12/11, 11/7, 18/17 and its inverse 1–11/7–12/7–17/9 with steps 11/7, 12/11, 11/10, 18/17;		* 1–11/10–17/12–3/2 with steps 11/10, 9/7, 18/17, 4/3, and its inverse
	* 1–15/14–9/7–15/11 with steps 15/14, 6/5, 18/17, 22/15 and its inverse 1–18/17–14/11–15/11 with steps 18/17, 6/5, 15/14, 22/15.		* 1–18/17–15/11–3/2 with steps 18/17, 9/7, 11/10, 4/3;
			* 1–9/7–15/11–5/3 with steps 9/7, 18/17, 11/9, 6/5, and its inverse
			* 1–9/7–17/11–17/9 with steps 9/7, 6/5, 11/9, 18/17;
			* 1–9/7–15/11–18/11 with steps 9/7, 18/17, 6/5, 11/9, and its inverse
			* 1–9/7–11/7–17/9 with steps 9/7, 11/9, 6/5, 18/17;
			* 1–9/7–17/11–18/11 with steps 9/7, 6/5, 18/17, 11/9, and its inverse
			* 1–6/5–17/11–17/9 with steps 6/5, 9/7, 11/9, 18/17;
			* 1–6/5–14/9–17/9 with steps 6/5, 22/17, 17/14, 18/17, and its inverse
			* 1–17/14–11/7–17/9 with steps 17/14, 22/17, 6/5, 18/17;
			* 1–9/7–5/3–17/9 with steps 9/7, 22/17, 17/15, 18/17, and its inverse
			* 1–17/15–22/15–17/9 with steps 17/15, 22/17, 9/7, 18/17;
			* 1–6/5–17/10–17/9 with steps 6/5, 17/12, 10/9, 18/17, and its inverse
			* 1–10/9–11/7–17/9 with steps 10/9, 17/12, 6/5, 18/17;
			* 1–17/14–9/7–17/12 with steps 17/14, 18/17, 11/10, 24/17, and its inverse
			* 1–11/10–7/6–17/12 with steps 11/10, 18/17, 17/14, 24/17;
			* 1–10/7–11/7–5/3 with steps 10/7, 11/10, 18/17, 6/5, and its inverse
			* 1–10/7–12/7–20/11 with steps 10/7, 6/5, 18/17, 11/10;
			* 1–11/10–11/9–17/9 with steps 11/10, 10/9, 17/11, 18/17, and its inverse
			* 1–17/11–12/7–17/9 with steps 17/11, 10/9, 11/10, 18/17;
			* 1–10/9–12/7–17/9 with steps 10/9, 17/11, 11/10, 18/17, and its inverse
			* 1–11/10–17/10–17/9 with steps 11/10, 17/11, 10/9, 18/17;
			* 1–11/10–6/5–17/9 with steps 11/10, 12/11, 11/7, 18/17, and its inverse
			* 1–11/7–12/7–17/9 with steps 11/7, 12/11, 11/10, 18/17;
			* 1–15/14–9/7–15/11 with steps 15/14, 6/5, 18/17, 22/15, and its inverse
			* 1–18/17–14/11–15/11 with steps 18/17, 6/5, 15/14, 22/15.

	Equal temperaments with dakotismic chords include {{Optimal ET sequence\| 12, 14c, 15g, 19eg, 22, 26, 27eg, 31g, 41, 46, 50, 58, 72, 94, 118, 121, 140, 239 and 311 }}.		Equal temperaments with dakotismic chords include {{Optimal ET sequence\| 12, 14c, 15g, 19eg, 22, 26, 27eg, 31g, 41, 46, 50, 58, 72, 94, 118, 121, 140, 239 and 311 }}.

FloraC: Cleanup

2025-12-02T08:31:36Z

Cleanup

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Xenllium: Update the article including recategorize and adding supporting ET

2023-08-20T05:53:54Z

Update the article including recategorize and adding supporting ET

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FloraC: Recategorize

2023-05-23T11:50:35Z

Recategorize

← Older revision		Revision as of 11:50, 23 May 2023
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	* 1-18/17-7/6-21/17-3/2 with steps 18/17-11/10-18/17-17/14-4/3;		* 1-18/17-7/6-21/17-3/2 with steps 18/17-11/10-18/17-17/14-4/3;

			[[Category:21-odd-limit]]
	[[Category:Essentially tempered chords]]		[[Category:Essentially tempered chords]]
			[[Category:Triads]]
			[[Category:Tetrads]]
			[[Category:Pentads]]
	[[Category:Dakotismic]]		[[Category:Dakotismic]]

FloraC: Created page with "'''Dakotismic chords''' are essentially tempered dyadic chords tempered by the dakotisma, 595/594. Dakotismic chords are numerous. If we disallow intervals that are..."

2023-02-18T15:58:48Z

Created page with "'''Dakotismic chords''' are essentially tempered dyadic chords tempered by the dakotisma, 595/594. Dakotismic chords are numerous. If we disallow intervals that are..."

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'''Dakotismic chords''' are [[essentially tempered dyadic chord]]s tempered by the dakotisma, [[595/594]].

Dakotismic chords are numerous. If we disallow intervals that are typically too small for harmonic use, in this case the semitones [[15/14]], [[18/17]], [[21/20]], and [[22/21]], there are four pairs of triads in inverse relationship in the 21-odd-limit:

* 1-6/5-17/11 with steps 6/5-9/7-22/17, and
* 1-22/17-5/3 with steps 22/17-9/7-6/5;
* 1-7/6-22/17 with steps 7/6-10/9-17/11, and
* 1-17/11-12/7 with steps 17/11-10/9-7/6;
* 1-11/10-17/12 with steps 11/10-9/7-24/17, and
* 1-9/7-17/12 with steps 9/7-11/10-24/17;
* 1-14/11-24/17 with steps 14/11-10/9-17/12, and
* 1-10/9-24/17 with steps 10/9-14/11-17/12.

If we allow the semitones, there are four additional pairs of triads in inverse relationship:
* 1-9/5-21/11 with steps 9/5-18/17-22/21, and its inverse
* 1-9/5-17/9 with steps 9/5-22/21-18/17;
* 1-9/7-15/11 with steps 9/7-18/17-22/15, and its inverse
* 1-9/7-17/9 with steps 9/7-22/15-18/17;
* 1-11/7-5/3 with steps 11/7-18/17-6/5, and its inverse
* 1-11/7-17/9 with steps 11/7-6/5-18/17;
* 1-11/10-7/6 with steps 11/10-18/17-12/7, and its inverse
* 1-11/10-17/9 with steps 11/10-12/7-18/17.

For tetrads without the semitones, there are two pairs in inverse relationship:
* 1-6/5-17/11-12/7 with steps 6/5-9/7-10/9-7/6, and its inverse
* 1-7/6-22/17-5/3 with steps 7/6-10/9-9/7-6/5;
* 1-11/10-17/12-11/7 with steps 11/10-9/7-10/9-14/11, and its inverse
* 1-14/11-24/17-20/11 with steps 14/11-10/9-9/7-11/10;

And if we use the semitones, there are many more. Some of these involve the perfect fifth:
* 1-3/2-9/5-21/11 with steps 3/2-6/5-18/17-22/21, and its inverse
* 1-3/2-11/7-5/3 with steps 3/2-22/21-18/17-6/5;
* 1-9/7-15/11-3/2 with steps 9/7-18/17-11/10-4/3, and its inverse
* 1-11/10-7/6-3/2 with steps 11/10-18/17-9/7-4/3;
* 1-9/7-17/12-3/2 with steps 9/7-11/10-18/17-4/3, and its inverse
* 1-18/17-7/6-3/2 with steps 18/17-11/10-9/7-4/3.
* 1-11/10-17/12-3/2 with steps 11/10-9/7-18/17-4/3, and its inverse
* 1-18/17-15/11-3/2 with steps 18/17-9/7-11/10-4/3.

Others do not:
* 1-7/5-9/5-21/11 with steps 7/5-9/7-18/17-22/21, and its inverse
* 1-7/5-22/15-14/9 with steps 7/5-22/21-18/17-9/7;
* 1-10/7-11/7-5/3 with steps 10/7-11/10-18/17-6/5, and its inverse
* 1-10/7-12/7-20/11 with steps 10/7-6/5-18/17-11/10;
* 1-18/11-9/5-21/11 with steps 18/11-11/10-18/17-22/21, and its inverse
* 1-18/11-12/7-20/11 with steps 18/11-22/21-18/17-11/10;
* 1-9/7-15/11-18/11 with steps 9/7-18/17-6/5-11/9, and its inverse
* 1-6/5-14/11-18/11 with steps 6/5-18/17-9/7-11/9;
* 1-17/14-9/7-17/12 with steps 17/14-18/17-11/10-24/17, and its inverse
* 1-11/10-7/6-17/12 with steps 11/10-18/17-17/14-24/17;
* 1-11/10-6/5-17/9 with steps 11/10-12/11-11/7-18/17, and its inverse
* 1-11/7-12/7-17/9 with steps 11/7-12/11-11/10-18/17;

The following two tetrads are palindromic:
* 1-18/17-7/6-21/17 with steps 18/17-11/10-18/17-34/21;
* 1-18/17-5/3-30/17 with steps 18/17-11/7-18/17-17/15.

For pentads, there are at least six pairs in inverse relationship, all featuring the fifth and using the semitones:
* 1-3/2-18/11-9/5-21/11 with steps 3/2-12/11-11/10-18/17-22/21, and its inverse
* 1-3/2-11/7-5/3-11/6 with steps 3/2-22/21-18/17-11/10-12/11;
* 1-7/5-3/2-9/5-21/11 with steps 7/5-15/14-6/5-18/17-22/21, and its inverse
* 1-15/14-3/2-11/7-5/3 with steps 15/14-7/5-22/21-18/17-6/5;
* 1-9/7-15/11-3/2-18/11 with steps 9/7-18/17-11/10-12/11-11/9, and its inverse
* 1-11/10-7/6-3/2-11/6 with steps 11/10-18/17-9/7-11/9-12/11;
* 1-11/10-17/12-3/2-11/7 with steps 11/10-9/7-18/17-22/21-14/11, and its inverse
* 1-18/17-15/11-3/2-21/11 with steps 18/17-9/7-11/10-14/11-22/21;
* 1-9/7-15/11-10/7-3/2 with steps 9/7-18/17-22/21-21/20-4/3, and its inverse
* 1-21/20-11/10-7/6-3/2 with steps 21/20-22/21-18/17-9/7-4/3;
* 1-17/14-9/7-17/12-3/2 with steps 17/14-18/17-11/10-18/17-4/3, and its inverse
* 1-18/17-7/6-21/17-3/2 with steps 18/17-11/10-18/17-17/14-4/3;

[[Category:Essentially tempered chords]]
[[Category:Dakotismic]]