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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 an [[essentially tempered chord]] tempered by the swetisma, [[540/539]].


There are six swetismic [[Triad|triads]], consisting of three pairs of chords in inverse relationship:
Swetismic chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 2]] in the [[11-odd-limit]], meaning that there are 6 [[triad]]s, 15 [[tetrad]]s and 6 [[pentad]]s, for a total of 27 distinct chord structures.


* 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 six swetismic triads, consisting of three pairs of chords in inverse relationship:
* 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;
* 1–7/5–12/7 with steps 7/5, 11/9, 7/6, and its inverse
* 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.
* 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 [[Tetrad|tetrads]], consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. The palindromic tetrads are:
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;
* 1–7/6–9/7–3/2 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;  
* 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 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 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;  
* 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.
If we are willing to consider the [[15-odd-limit]], there are also 15-odd-limit swetismic tetrads, including something important for functional harmony. The ''swetismic dominant seventh chord'' is a tempering of  
* 1–9/7–3/2–7/4 with steps 9/7, 7/6, 7/6, 8/7.  


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.
Its inversion might be called the ''swetismic half-diminished chord'', a tempering of  
* 1–7/6–3/2–12/7 with steps 7/6, 9/7, 8/7, 7/6.  


[[Category:11-odd-limit]]
We also have
* 1–11/9–10/7–5/3 with steps 11/9, 7/6, 7/6, 6/5; and its inverse
* 1–7/6–15/11–5/3 with steps 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 chords]]
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Triads]]
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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}}.
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