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Ursulismic chords: Difference between revisions

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'''Ursulismmic chords''' are [[essentially tempered chord]]s tempered by the ursulisma, [[375/374]].  
'''Ursulismic chords''' are [[dyadic chord|essentially tempered chords]] tempered by the ursulisma, [[375/374]].
 
There are 8 triads, 28 tetrads, 26 pentads and 7 hexads as 2.3.5.11.17 subgroup [[17-odd-limit]] essentially tempered chords.


The ursulismic triads include the following inversely related pairs:  
The ursulismic triads include the following inversely related pairs:  
* 1-5/4-17/10 with steps 5/4-15/11-20/17, and its inverse
* 1-5/4-22/15 with steps of 5/4-20/17-15/11, and its inverse  
* 1-5/4-22/15 with steps 5/4-20/17-15/11;  
* 1-20/17-22/15 with steps of 20/17-5/4-15/11;
* 1-17/15-5/4 with steps 17/15-11/10-8/5, and its inverse
* 1-6/5-15/11 with steps of 6/5-17/15-22/15, and its inverse
* 1-11/10-5/4 with steps 11/10-17/15-8/5;
* 1-17/15-15/11 with steps of 17/15-6/5-22/15;
* 1-17/10-15/8 with steps 17/10-11/10-16/15, and its inverse
* 1-17/15-5/4 with steps of 17/15-11/10-8/5, and its inverse  
* 1-11/10-15/8 with steps 11/10-17/10-16/15;  
* 1-11/10-5/4 with steps of 11/10-17/15-8/5;
* 1-22/15-5/3 with steps 22/15-17/15-6/5, and its inverse
* 1-11/10-20/17 with steps of 11/10-16/15-17/10, and its inverse
* 1-17/15-5/3 with steps 17/15-22/15-6/5.  
* 1-16/15-20/17 with steps of 16/15-11/10-17/10.
 
For tetrads, there are eight palindromic chords and ten pairs of chords in inverse relationship. The palindromic chords are
* 1-6/5-15/11-18/11 with steps of 6/5-17/15-6/5-11/9;
* 1-20/17-22/15-8/5 with steps of 20/17-5/4-12/11-5/4;
* 1-20/17-5/4-22/15 with steps of 20/17-17/16-20/17-15/11;
* 1-17/15-5/4-17/12 with steps of 17/15-11/10-17/15-24/17;
* 1-11/10-5/4-11/8 with steps of 11/10-17/15-11/10-16/11;
* 1-17/15-6/5-15/11 with steps of 17/15-18/17-17/15-22/15;
* 1-11/10-20/17-22/17 with steps of 11/10-16/15-11/10-17/11;
* 1-17/15-15/11-17/11 with steps of 17/15-6/5-17/15-22/17.


They can be extended to the following palindromic tetrads:
The inversely related pairs of chords are
* 1-11/10-17/10-15/8 with steps 11/10-17/11-11/10-16/15;  
* 1-5/4-3/2-17/10 with steps of 5/4-6/5-17/15-20/17, and its inverse
* 1-17/15-22/15-5/3 with steps 17/15-22/17-17/15-6/5.  
* 1-6/5-3/2-30/17 with steps of 6/5-5/4-20/17-17/15;
* 1-20/17-24/17-8/5 with steps of 20/17-6/5-17/15-5/4, and its inverse
* 1-17/15-15/11-8/5 with steps of 17/15-6/5-20/17-5/4;
* 1-6/5-15/11-3/2 with steps of 6/5-17/15-11/10-4/3, and its inverse
* 1-11/10-5/4-3/2 with steps of 11/10-17/15-6/5-4/3;
* 1-15/11-3/2-8/5 with steps of 15/11-11/10-16/15-5/4, and its inverse
* 1-11/10-3/2-15/8 with steps of 11/10-15/11-5/4-16/15;
* 1-5/4-11/8-22/15 with steps of 5/4-11/10-16/15-15/11, and its inverse
* 1-16/15-20/17-22/15 with steps of 16/15-11/10-5/4-15/11;
* 1-20/17-22/17-22/15 with steps of 20/17-11/10-17/15-15/11, and its inverse
* 1-17/15-5/4-22/15 with steps of 17/15-11/10-20/17-15/11;
* 1-15/11-3/2-17/10 with steps of 15/11-11/10-17/15-20/17, and its inverse
* 1-11/10-3/2-30/17 with steps of 11/10-15/11-20/17-17/15;
* 1-17/15-5/4-15/11 with steps of 17/15-11/10-12/11-22/15, and its inverse
* 1-12/11-6/5-15/11 with steps of 12/11-11/10-17/15-22/15;
* 1-17/15-5/4-4/3 with steps of 17/15-11/10-16/15-3/2, and its inverse
* 1-16/15-20/17-4/3 with steps of 16/15-11/10-17/15-3/2;
* 1-11/10-20/17-5/4 with steps of 11/10-16/15-17/16-8/5, and its inverse
* 1-17/16-17/15-5/4 with steps of 17/16-16/15-11/10-8/5.


And the following inversely related tetrads:  
For pentads, there are thirteen pairs of chords in inverse relationship:  
* 1-5/4-3/2-17/10 with steps 5/4-6/5-17/15-20/17, and its inverse
* 1-6/5-15/11-3/2-17/10 with steps of 6/5-17/15-11/10-17/15-20/17, and its inverse
* 1-6/5-3/2-30/17 with steps 6/5-5/4-20/17-17/15;
* 1-11/10-5/4-3/2-30/17 with steps of 11/10-17/15-6/5-20/17-17/15;
* 1-11/10-3/2-15/8 with steps 11/10-15/11-5/4-16/15, and its inverse
* 1-6/5-24/17-3/2-30/17 with steps of 6/5-20/17-17/16-20/17-17/15, and its inverse  
* 1-15/11-3/2-8/5 with steps 15/11-11/10-16/15-5/4;  
* 1-17/16-5/4-3/2-17/10 with steps of 17/16-20/17-6/5-17/15-20/17;
* 1-11/10-5/4-3/2 with steps 11/10-17/15-6/5-4/3, and its inverse
* 1-6/5-15/11-3/2-30/17 with steps of 6/5-17/15-11/10-20/17-17/15, and its inverse
* 1-6/5-15/11-3/2 with steps 6/5-17/15-11/10-4/3;  
* 1-11/10-5/4-3/2-17/10 with steps of 11/10-17/15-6/5-17/15-20/17;
* 1-3/2-17/10-15/8 with steps 3/2-17/15-11/10-16/15, and its inverse
* 1-6/5-15/11-3/2-18/11 with steps of 6/5-17/15-11/10-12/11-11/9, and its inverse
* 1-3/2-8/5-30/17 with steps 3/2-16/15-11/10-17/15;
* 1-11/10-5/4-3/2-11/6 with steps of 11/10-17/15-6/5-11/9-12/11;
* 1-5/4-3/2-17/10-15/8 with steps of 5/4-6/5-17/15-11/10-16/15, and its inverse  
* 1-6/5-3/2-8/5-30/17 with steps of 6/5-5/4-16/15-11/10-17/15;
* 1-5/4-17/12-3/2-17/10 with steps of 5/4-17/15-18/17-17/15-20/17, and its inverse
* 1-18/17-6/5-3/2-30/17 with steps of 18/17-17/15-5/4-20/17-17/15;
* 1-5/4-15/11-3/2-17/10 with steps of 5/4-12/11-11/10-17/15-20/17, and its inverse
* 1-11/10-6/5-3/2-30/17 with steps of 11/10-12/11-5/4-20/17-17/15;
* 1-11/10-11/8-3/2-15/8 with steps of 11/10-5/4-12/11-5/4-16/15, and its inverse  
* 1-12/11-15/11-3/2-8/5 with steps of 12/11-5/4-11/10-16/15-5/4;
* 1-15/11-3/2-8/5-17/10 with steps of 15/11-11/10-16/15-17/16-20/17, and its inverse
* 1-11/10-3/2-30/17-15/8 with steps of 11/10-15/11-20/17-17/16-16/15;
* 1-15/11-3/2-8/5-30/17 with steps of 15/11-11/10-16/15-11/10-17/15, and its inverse
* 1-11/10-3/2-17/10-15/8 with steps of 11/10-15/11-17/15-11/10-16/15;
* 1-15/11-3/2-17/10-15/8 with steps of 15/11-11/10-17/15-11/10-16/15, and its inverse  
* 1-11/10-3/2-8/5-30/17 with steps of 11/10-15/11-16/15-11/10-17/15;
* 1-24/17-3/2-8/5-30/17 with steps of 24/17-17/16-16/15-11/10-17/15, and its inverse
* 1-17/16-3/2-17/10-15/8 with steps of 17/16-24/17-17/15-11/10-16/15;
* 1-11/10-5/4-11/8-3/2 with steps of 11/10-17/15-11/10-12/11-4/3, and its inverse
* 1-12/11-6/5-15/11-3/2 with steps of 12/11-11/10-17/15-11/10-4/3.


* 1-17/15-5/4-22/15 with steps 17/15-11/10-20/17-15/11, and its inverse
For hexads, there are three palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are
* 1-11/10-5/4-17/10 with steps 11/10-17/15-15/11-20/17.  
* 1-17/16-5/4-17/12-3/2-17/10 with steps of 17/16-20/17-17/15-18/17-17/15-20/17;
* 1-11/10-5/4-3/2-17/10-15/8 with steps of 11/10-17/15-6/5-17/15-11/10-16/15;
* 1-11/10-5/4-11/8-3/2-11/6 with steps of 11/10-17/15-11/10-12/11-11/9-12/11.


For pentads, there are
The inversely related pairs of chords are  
* 1-11/10-5/4-3/2-15/8 with steps 11/10-17/15-6/5-5/4-16/15, and its inverse
* 1-6/5-24/17-3/2-8/5-30/17 with steps of 6/5-20/17-17/16-16/15-11/10-17/15, and its inverse
* 1-6/5-15/11-3/2-8/5 with steps 1-6/5-17/15-11/10-16/15-5/4;
* 1-17/16-5/4-3/2-17/10-15/8 with steps of 17/16-20/17-6/5-17/15-11/10-16/15;
* 1-11/10-5/4-3/2-17/10 with steps 11/10-17/15-6/5-17/15-20/17, and its inverse
* 1-5/4-15/11-3/2-17/10-15/8 with steps of 5/4-12/11-11/10-17/15-11/10-16/15, and its inverse  
* 1-6/5-15/11-3/2-30/17 with steps 6/5-17/15-11/10-20/17-17/15;  
* 1-11/10-6/5-3/2-8/5-30/17 with steps of 11/10-12/11-5/4-16/15-11/10-17/15.
* 1-11/10-3/2-17/10-15/8 with steps 11/10-15/11-17/15-11/10-16/15, and its inverse
* 1-15/11-3/2-8/5-30/17 with steps 15/11-11/10-16/15-11/10-17/15;


Finally, there is a palindromic hexad:
Equal temperaments with ursulismic chords include {{Optimal ET sequence|9, 12, 16, 22, 31, 34, 38, 65, 72, 87, 94, 137 and 159}}.
* 1-11/10-5/4-3/2-17/10-15/8 with steps 11/10-17/15-6/5-17/15-11/10-16/15.  


[[Category:17-odd-limit]]
[[Category:17-odd-limit]]
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