Swetismic chords: Difference between revisions

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A '''swetismic chord''' is a [[Swetismic temperaments|swetismic]] ([[540/539]]) [[11-odd-limit]], [[essentially tempered chord]].
A '''swetismic chord''' is a [[Swetismic temperaments|swetismic]] ([[540/539]]) [[11-odd-limit]], [[essentially tempered chord]].


There are six swetismic [[Triad|triads]], consisting of three pairs of chords in inverse relationship:
There are six swetismic [[triad]]s, consisting of three pairs of chords in inverse relationship:
* 1-7/5-12/7 with [[step]]s 7/5-11/9-7/6, and its inverse
* 1-7/6-10/7 with steps 7/6-11/9-7/5;
* 1-7/6-9/7 with steps 7/6-11/10-14/9, and its inverse
* 1-14/9-12/7 with steps 14/9-11/10-7/6;
* 1-9/7-7/5 with steps 9/7-12/11-10/7, and its inverse
* 1-10/7-14/9 with steps 10/7-12/11-9/7.


* 1-7/5-12/7 with [[Step|steps]] 7/5-11/9-7/6 and 1-7/6-10/7 with steps 7/6-11/9-7/5;
There are fifteen swetismic [[tetrad]]s, consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. The palindromic tetrads are:
* 1-7/6-9/7 with steps 7/6-11/10-14/9 and 1-14/9-12/7 with steps 14/9-11/10-7/6;
* and 1-9/7-7/5 with steps 9/7-12/11-10/7 and 1-10/7-14/9 with steps 10/7-12/11-9/7.
 
There are fifteen swetismic [[Tetrad|tetrads]], consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. The palindromic tetrads are:


* 1-7/6-9/7-3/2 with steps 7/6-11/10-7/6-4/3;
* 1-7/6-9/7-3/2 with steps 7/6-11/10-7/6-4/3;
* 1-6/5-7/5-12/7 with steps 6/5-7/6-11/9-7/6;
* 1-6/5-7/5-12/7 with steps 6/5-7/6-11/9-7/6;
* and 1-9/7-7/5-9/5 with steps 9/7-12/11-9/7-10/9.
* 1-9/7-7/5-9/5 with steps 9/7-12/11-9/7-10/9.


The six pairs are:
The six pairs are:


* 1-9/7-3/2-11/6 with steps 9/7-7/6-11/9-12/11 and 1-7/6-3/2-18/11 with steps 7/6-9/7-12/11-11/9;
* 1-9/7-3/2-11/6 with steps 9/7-7/6-11/9-12/11, and its inverse
* 1-7/6-9/7-10/7 with steps 7/6-11/10-10/9-7/5 and 1-7/5-14/9-12/7 with steps 7/5-10/9-11/10-7/6;
* 1-7/6-3/2-18/11 with steps 7/6-9/7-12/11-11/9;
* 1-7/6-9/7-18/11 with steps 7/6-11/10-14/11-11/9 and 1-14/11-7/5-18/11 with steps 14/11-11/10-7/6-11/9;
* 1-7/6-9/7-10/7 with steps 7/6-11/10-10/9-7/5, and its inverse
* 1-7/6-10/7-11/6 with steps 7/6-11/9-9/7-12/11 and 1-9/7-11/7-11/6 with steps 9/7-11/9-7/6-12/11;
* 1-7/5-14/9-12/7 with steps 7/5-10/9-11/10-7/6;
* 1-7/6-9/7-11/6 with steps 7/6-11/10-10/7-12/11 and 1-10/7-11/7-11/6 with steps 10/7-11/10-7/6-12/11;
* 1-7/6-9/7-18/11 with steps 7/6-11/10-14/11-11/9, and its inverse
* and 1-10/7-14/9-12/7 with steps 10/7-12/11-11/10-7/6 and 1-7/6-9/7-7/5 with steps 7/6-11/10-12/11-10/7.
* 1-14/11-7/5-18/11 with steps 14/11-11/10-7/6-11/9;
* 1-7/6-10/7-11/6 with steps 7/6-11/9-9/7-12/11, and its inverse
* 1-9/7-11/7-11/6 with steps 9/7-11/9-7/6-12/11;
* 1-7/6-9/7-11/6 with steps 7/6-11/10-10/7-12/11, and its inverse
* 1-10/7-11/7-11/6 with steps 10/7-11/10-7/6-12/11;
* and 1-10/7-14/9-12/7 with steps 10/7-12/11-11/10-7/6, and its inverse
* 1-7/6-9/7-7/5 with steps 7/6-11/10-12/11-10/7.


Finally, there are six swetismic [[Pentad|pentads]] coming in three pairs:
Finally, there are six swetismic [[pentad]]s coming in three pairs:


* 1-7/6-9/7-3/2-11/6 with steps 7/6-11/10-7/6-11/9-12/11 and 1-7/6-9/7-3/2-18/11 with steps 7/6-11/10-7/6-12/11-11/9;
* 1-7/6-9/7-3/2-11/6 with steps 7/6-11/10-7/6-11/9-12/11, and its inverse
* 1-7/6-10/7-5/3-11/6 with steps 7/6-11/9-7/6-11/10-12/11 and 1-7/6-10/7-5/3-20/11 with steps 7/6-11/9-7/6-12/11-11/10;
* 1-7/6-9/7-3/2-18/11 with steps 7/6-11/10-7/6-12/11-11/9;
* and 1-7/6-9/7-10/7-11/6 with steps 7/6-11/10-10/9-9/7-12/11 and 1-9/7-10/7-11/7-11/6 with steps 9/7-10/9-11/10-7/6-12/11.
* 1-7/6-10/7-5/3-11/6 with steps 7/6-11/9-7/6-11/10-12/11, and its inverse
* 1-7/6-10/7-5/3-20/11 with steps 7/6-11/9-7/6-12/11-11/10;
* and 1-7/6-9/7-10/7-11/6 with steps 7/6-11/10-10/9-9/7-12/11, and its inverse
* 1-9/7-10/7-11/7-11/6 with steps 9/7-10/9-11/10-7/6-12/11.


The 11 limit swetismic chords number triads: 6, tetrads: 15, pentads: 6 for a total of 27.
The 11-odd-limit swetismic chords number triads: 6, tetrads: 15, pentads: 6 for a total of 27.


If we are willing to consider the [[15-odd-limit]], there are also 15-odd-limit swetismic tetrads of 1-9/7-3/2-7/4, 1-7/6-3/2-12/7, 1-11/9-10/7-5/3 and 1-7/6-15/11-5/3 with steps 9/7-7/6-7/6-8/7, 7/6-9/7-8/7-7/6, 11/9-7/6-7/6-6/5 and 7/6-7/6-11/9-6/5.
If we are willing to consider the [[15-odd-limit]], there are also 15-odd-limit swetismic tetrads of 1-9/7-3/2-7/4, 1-7/6-3/2-12/7, 1-11/9-10/7-5/3 and 1-7/6-15/11-5/3 with steps 9/7-7/6-7/6-8/7, 7/6-9/7-8/7-7/6, 11/9-7/6-7/6-6/5 and 7/6-7/6-11/9-6/5.
[[Equal temperament]]s with swetismic chords include {{EDOs|19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 205, 224, 354, 537, 578, 761d, 1115de, 1339de, 1491de, 1715de and 1845de }}.


[[Category:11-odd-limit]]
[[Category:11-odd-limit]]
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[[Category:Pentads]]
[[Category:Pentads]]
[[Category:Swetismic]]
[[Category:Swetismic]]
[[EDO|Equal temperaments]] with swetismic chords include {{EDOs|19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 205, 224, 354, 537, 578, 761d, 1115de, 1339de, 1491de, 1715de and 1845de}}.

Revision as of 16:55, 8 August 2023

A swetismic chord is a swetismic (540/539) 11-odd-limit, essentially tempered chord.

There are six swetismic triads, consisting of three pairs of chords in inverse relationship:

  • 1-7/5-12/7 with steps 7/5-11/9-7/6, and its inverse
  • 1-7/6-10/7 with steps 7/6-11/9-7/5;
  • 1-7/6-9/7 with steps 7/6-11/10-14/9, and its inverse
  • 1-14/9-12/7 with steps 14/9-11/10-7/6;
  • 1-9/7-7/5 with steps 9/7-12/11-10/7, and its inverse
  • 1-10/7-14/9 with steps 10/7-12/11-9/7.

There are fifteen swetismic tetrads, consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. The palindromic tetrads are:

  • 1-7/6-9/7-3/2 with steps 7/6-11/10-7/6-4/3;
  • 1-6/5-7/5-12/7 with steps 6/5-7/6-11/9-7/6;
  • 1-9/7-7/5-9/5 with steps 9/7-12/11-9/7-10/9.

The six pairs are:

  • 1-9/7-3/2-11/6 with steps 9/7-7/6-11/9-12/11, and its inverse
  • 1-7/6-3/2-18/11 with steps 7/6-9/7-12/11-11/9;
  • 1-7/6-9/7-10/7 with steps 7/6-11/10-10/9-7/5, and its inverse
  • 1-7/5-14/9-12/7 with steps 7/5-10/9-11/10-7/6;
  • 1-7/6-9/7-18/11 with steps 7/6-11/10-14/11-11/9, and its inverse
  • 1-14/11-7/5-18/11 with steps 14/11-11/10-7/6-11/9;
  • 1-7/6-10/7-11/6 with steps 7/6-11/9-9/7-12/11, and its inverse
  • 1-9/7-11/7-11/6 with steps 9/7-11/9-7/6-12/11;
  • 1-7/6-9/7-11/6 with steps 7/6-11/10-10/7-12/11, and its inverse
  • 1-10/7-11/7-11/6 with steps 10/7-11/10-7/6-12/11;
  • and 1-10/7-14/9-12/7 with steps 10/7-12/11-11/10-7/6, and its inverse
  • 1-7/6-9/7-7/5 with steps 7/6-11/10-12/11-10/7.

Finally, there are six swetismic pentads coming in three pairs:

  • 1-7/6-9/7-3/2-11/6 with steps 7/6-11/10-7/6-11/9-12/11, and its inverse
  • 1-7/6-9/7-3/2-18/11 with steps 7/6-11/10-7/6-12/11-11/9;
  • 1-7/6-10/7-5/3-11/6 with steps 7/6-11/9-7/6-11/10-12/11, and its inverse
  • 1-7/6-10/7-5/3-20/11 with steps 7/6-11/9-7/6-12/11-11/10;
  • and 1-7/6-9/7-10/7-11/6 with steps 7/6-11/10-10/9-9/7-12/11, and its inverse
  • 1-9/7-10/7-11/7-11/6 with steps 9/7-10/9-11/10-7/6-12/11.

The 11-odd-limit swetismic chords number triads: 6, tetrads: 15, pentads: 6 for a total of 27.

If we are willing to consider the 15-odd-limit, there are also 15-odd-limit swetismic tetrads of 1-9/7-3/2-7/4, 1-7/6-3/2-12/7, 1-11/9-10/7-5/3 and 1-7/6-15/11-5/3 with steps 9/7-7/6-7/6-8/7, 7/6-9/7-8/7-7/6, 11/9-7/6-7/6-6/5 and 7/6-7/6-11/9-6/5.

Equal temperaments with swetismic chords include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 205, 224, 354, 537, 578, 761d, 1115de, 1339de, 1491de, 1715de and 1845de.

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