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Charismic chords: Difference between revisions

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There are 2 triads, 13 tetrads, 17 pentads and 6 hexads as 2.3.5.17 subgroup [[17-odd-limit]] essentially tempered chords.
There are 2 triads, 13 tetrads, 17 pentads and 6 hexads as 2.3.5.17 subgroup [[17-odd-limit]] essentially tempered chords.


The two diasemismic triads are in inverse relationship:
The two diasemismic triads are in inverse relationship:  
* 1-5/4-24/17 with steps of 5/4-9/8-17/12;
* 1-5/4-24/17 with steps of 5/4-9/8-17/12;
* 1-9/8-24/17 with steps of 9/8-5/4-17/12.
* 1-9/8-24/17 with steps of 9/8-5/4-17/12.
Line 14: Line 14:
* 1-16/15-9/8-6/5 with steps of 16/15-18/17-16/15-5/3.
* 1-16/15-9/8-6/5 with steps of 16/15-18/17-16/15-5/3.


The inversely related pairs of chords are
The inversely related pairs of chords are  
* 1-5/4-24/17-15/8 with steps of 5/4-9/8-4/3-16/15 and its inverse 1-4/3-3/2-15/8 with steps of 4/3-9/8-5/4-16/15;
* 1-5/4-24/17-15/8 with steps of 5/4-9/8-4/3-16/15, and its inverse  
* 1-5/4-24/17-3/2 with steps of 5/4-9/8-16/15-4/3 and its inverse 1-16/15-6/5-3/2 with steps of 16/15-9/8-5/4-4/3;
* 1-4/3-3/2-15/8 with steps of 4/3-9/8-5/4-16/15;
* 1-5/4-4/3-24/17 with steps of 5/4-16/15-18/17-17/12 and its inverse 1-18/17-9/8-24/17 with steps of 18/17-16/15-5/4-17/12;
* 1-5/4-24/17-3/2 with steps of 5/4-9/8-16/15-4/3, and its inverse  
* 1-20/17-5/4-24/17 with steps 20/17-16/15-9/8-17/12 and its inverse 1-9/8-6/5-24/17 with steps of 9/8-16/15-20/17-17/12.
* 1-16/15-6/5-3/2 with steps of 16/15-9/8-5/4-4/3;
* 1-5/4-4/3-24/17 with steps of 5/4-16/15-18/17-17/12, and its inverse  
* 1-18/17-9/8-24/17 with steps of 18/17-16/15-5/4-17/12;
* 1-20/17-5/4-24/17 with steps 20/17-16/15-9/8-17/12, and its inverse  
* 1-9/8-6/5-24/17 with steps of 9/8-16/15-20/17-17/12.


For pentads, there are one palindromic chord and eight pairs of chords in inverse relationship. The palindromic chord is
For pentads, there are one palindromic chord and eight pairs of chords in inverse relationship. The palindromic chord is  
* 1-5/4-24/17-30/17-15/8 with steps of 5/4-9/8-5/4-16/15-16/15.
* 1-5/4-24/17-30/17-15/8 with steps of 5/4-9/8-5/4-16/15-16/15.


The inversely related pairs of chords are
The inversely related pairs of chords are  
* 1-9/8-5/4-24/17-3/2 with steps of 9/8-10/9-9/8-16/15-4/3 and its inverse 1-9/8-5/4-24/17-15/8 with steps of 9/8-10/9-9/8-4/3-16/15;
* 1-9/8-5/4-24/17-3/2 with steps of 9/8-10/9-9/8-16/15-4/3, and its inverse  
* 1-5/4-24/17-3/2-15/8 with steps of 5/4-9/8-16/15-5/4-16/15 and its inverse 1-5/4-4/3-5/3-16/9 with steps of 5/4-16/15-5/4-16/15-9/8;
* 1-9/8-5/4-24/17-15/8 with steps of 9/8-10/9-9/8-4/3-16/15;
* 1-5/4-4/3-24/17-15/8 with steps of 5/4-16/15-18/17-4/3-16/15 and its inverse 1-5/4-4/3-16/9-15/8 with steps of 5/4-16/15-4/3-18/17-16/15;
* 1-5/4-24/17-3/2-15/8 with steps of 5/4-9/8-16/15-5/4-16/15, and its inverse  
* 1-5/4-4/3-24/17-3/2 with steps of 5/4-16/15-18/17-16/15-4/3 and its inverse 1-16/15-9/8-6/5-3/2 with steps of 16/15-18/17-16/15-5/4-4/3;
* 1-5/4-4/3-5/3-16/9 with steps of 5/4-16/15-5/4-16/15-9/8;
* 1-20/17-5/4-4/3-15/8 with steps of 20/17-16/15-16/15-24/17-16/15 and its inverse 1-20/17-5/4-30/17-15/8 with steps of 20/17-16/15-24/17-16/15-16/15;
* 1-5/4-4/3-24/17-15/8 with steps of 5/4-16/15-18/17-4/3-16/15, and its inverse  
* 1-16/15-5/4-4/3-3/2 with steps of 16/15-20/17-16/15-9/8-4/3 and its inverse 1-9/8-6/5-24/17-3/2 with steps of 9/8-16/15-20/17-16/15-4/3;
* 1-5/4-4/3-16/9-15/8 with steps of 5/4-16/15-4/3-18/17-16/15;
* 1-20/17-5/4-4/3-24/17 with steps of 20/17-16/15-16/15-18/17-17/12 and its inverse 1-18/17-9/8-6/5-24/17 with steps of 18/17-16/15-16/15-20/17-17/12;
* 1-5/4-4/3-24/17-3/2 with steps of 5/4-16/15-18/17-16/15-4/3, and its inverse  
* 1-5/4-24/17-3/2-30/17 with steps of 5/4-9/8-16/15-20/17-17/15 and its inverse 1-16/15-6/5-3/2-17/10 with steps of 16/15-9/8-5/4-17/15-20/17.
* 1-16/15-9/8-6/5-3/2 with steps of 16/15-18/17-16/15-5/4-4/3;
* 1-20/17-5/4-4/3-15/8 with steps of 20/17-16/15-16/15-24/17-16/15, and its inverse  
* 1-20/17-5/4-30/17-15/8 with steps of 20/17-16/15-24/17-16/15-16/15;
* 1-16/15-5/4-4/3-3/2 with steps of 16/15-20/17-16/15-9/8-4/3, and its inverse  
* 1-9/8-6/5-24/17-3/2 with steps of 9/8-16/15-20/17-16/15-4/3;
* 1-20/17-5/4-4/3-24/17 with steps of 20/17-16/15-16/15-18/17-17/12, and its inverse  
* 1-18/17-9/8-6/5-24/17 with steps of 18/17-16/15-16/15-20/17-17/12;
* 1-5/4-24/17-3/2-30/17 with steps of 5/4-9/8-16/15-20/17-17/15, and its inverse  
* 1-16/15-6/5-3/2-17/10 with steps of 16/15-9/8-5/4-17/15-20/17.


For hexads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are
For hexads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are  
* 1-9/8-5/4-24/17-3/2-15/8 with steps of 9/8-10/9-9/8-16/15-5/4-16/15;
* 1-9/8-5/4-24/17-3/2-15/8 with steps of 9/8-10/9-9/8-16/15-5/4-16/15;
* 1-5/4-4/3-24/17-3/2-15/8 with steps of 5/4-16/15-18/17-16/15-5/4-16/15.
* 1-5/4-4/3-24/17-3/2-15/8 with steps of 5/4-16/15-18/17-16/15-5/4-16/15.


The inversely related pairs of chords are
The inversely related pairs of chords are  
* 1-16/15-5/4-4/3-17/12-3/2 with steps of 16/15-20/17-16/15-16/15-18/17-4/3 and its inverse 1-18/17-9/8-6/5-24/17-3/2 with steps of 18/17-16/15-16/15-20/17-16/15-4/3;
* 1-16/15-5/4-4/3-17/12-3/2 with steps of 16/15-20/17-16/15-16/15-18/17-4/3, and its inverse  
* 1-5/4-24/17-3/2-30/17-15/8 with steps of 5/4-9/8-16/15-20/17-16/15-16/15 and its inverse 1-16/15-5/4-4/3-3/2-15/8 with steps of 16/15-20/17-16/15-9/8-5/4-16/15.
* 1-18/17-9/8-6/5-24/17-3/2 with steps of 18/17-16/15-16/15-20/17-16/15-4/3;
* 1-5/4-24/17-3/2-30/17-15/8 with steps of 5/4-9/8-16/15-20/17-16/15-16/15, and its inverse  
* 1-16/15-5/4-4/3-3/2-15/8 with steps of 16/15-20/17-16/15-9/8-5/4-16/15.


Equal temperaments with diasemismic chords include {{Optimal ET sequence|7, 9, 10, 12, 15, 19, 22, 31, 34, 46, 53, 65, 77, 87, 99 and 111}}.
Equal temperaments with diasemismic chords include {{Optimal ET sequence|9, 10, 12, 15, 19, 22, 31, 34, 46, 53, 65, 77, 87, 99 and 111}}.


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