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 [[ | 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. | |||
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: | |||
There are fifteen swetismic [[ | |||
* 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; | ||
* | * 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 [[ | 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]] | ||