Charismic chords: Difference between revisions

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'''Diasemismic chords''' are [[Dyadic chord|essentially tempered dyadic chords]] tempered by the diasemisma, [[256/255]].

~~The [[diasemismic]] temperament contains many possibilities for essentially tempered dyadic chords including two~~ triads and ~~twelve tetrads in the~~ [[17-odd-limit]] ~~of the 2.3.5.17 subgroup~~.

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:

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* 1-9/8-24/17 with steps of 9/8-5/4-17/12.

For tetrads, there are ~~four~~ palindromic chords and four pairs of chords in inverse relationship. The palindromic chords are

For tetrads, there are five palindromic chords and four pairs of chords in inverse relationship. The palindromic chords are

* 1-5/4-24/17-30/17 with steps of 5/4-9/8-5/4-17/15;

* 1-9/8-5/4-24/17 with steps of 9/8-10/9-9/8-17/12;

* 1-16/15-4/3-17/12 with steps of 16/15-5/4-16/15-24/17;

* 1-16/15-5/4-4/3 with steps of 16/15-20/17-16/15-3/2;

* 1-16/15-9/8-6/5 with steps of 16/15-18/17-16/15-5/3.

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* 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.

~~Remarkable~~ hexads ~~include following~~ two palindromic chords:

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.

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;

* 1-5/4-24/17-3/2-15/8 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;

* 1-5/4-4/3-24/17-15/8 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;

* 1-5/4-4/3-24/17-3/2 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;

* 1-20/17-5/4-4/3-15/8 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

* 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.

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;

* 1-5/4-24/17-3/2-30/17-15/8 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}}.

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[[Category:Triads]]

[[Category:Tetrads]]

[[Category:Pentads]]

[[Category:Hexads]]

[[Category:Diasemismic]]

@@ Line 1: / Line 1: @@
 '''Diasemismic chords''' are [[Dyadic chord|essentially tempered dyadic chords]] tempered by the diasemisma, [[256/255]].
-The [[diasemismic]] temperament contains many possibilities for essentially tempered dyadic chords including two triads and twelve tetrads in the [[17-odd-limit]] of the 2.3.5.17 subgroup.
+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:
@@ Line 7: / Line 7: @@
 * 1-9/8-24/17 with steps of 9/8-5/4-17/12.
-For tetrads, there are four palindromic chords and four pairs of chords in inverse relationship. The palindromic chords are
+For tetrads, there are five palindromic chords and four pairs of chords in inverse relationship. The palindromic chords are
 * 1-5/4-24/17-30/17 with steps of 5/4-9/8-5/4-17/15;
 * 1-9/8-5/4-24/17 with steps of 9/8-10/9-9/8-17/12;
 * 1-16/15-4/3-17/12 with steps of 16/15-5/4-16/15-24/17;
+* 1-16/15-5/4-4/3 with steps of 16/15-20/17-16/15-3/2;
 * 1-16/15-9/8-6/5 with steps of 16/15-18/17-16/15-5/3.
@@ Line 19: / Line 20: @@
 * 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.
-Remarkable hexads include following two palindromic chords:
+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.
+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;
+* 1-5/4-24/17-3/2-15/8 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;
+* 1-5/4-4/3-24/17-15/8 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;
+* 1-5/4-4/3-24/17-3/2 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;
+* 1-20/17-5/4-4/3-15/8 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
 * 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.
+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;
+* 1-5/4-24/17-3/2-30/17-15/8 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}}.
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 [[Category:Triads]]
 [[Category:Tetrads]]
+[[Category:Pentads]]
 [[Category:Hexads]]
 [[Category:Diasemismic]]