Biyatismic chords: Difference between revisions

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'''Biyatismic chords''' are [[Dyadic chord|essentially tempered dyadic chords]] tempered by the biyatisma, [[121/120]].

~~The [[biyatismic]] temperament contains many possibilities for essentially tempered dyadic chords including 2~~ triads and ~~6 tetrads in the [[11-odd-limit]] of the~~ 2.3.5.11 [[subgroup~~]], but all of these become essentially just in the~~ [[15-odd-limit]].

There are two triads, nine tetrads, seven pentads and one hexad as 2.3.5.11 subgroup [[11-odd-limit]] essentially tempered chords.

The two biyatismic triads are in inverse relationship:

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* 1-6/5-16/11 chord with steps of 6/5-11/9-11/8.

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

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

* 1-9/8-11/8-18/11 chord with steps of 9/8-11/9-6/5-11/9;

* 1-5/4-11/8-11/6 chord with steps of 5/4-11/10-4/3-11/10.

* 1-5/4-11/8-11/6 chord with steps of 5/4-11/10-4/3-11/10;

* 1-3/2-18/11-11/6 chord with steps of 3/2-11/10-10/9-11/10.

The inversely related pairs of chords are

* 1-11/8-5/3-11/6 ~~chord~~ with steps of 11/8-11/9-11/10-11/10 and 1-11/10-6/5-16/11 ~~chord~~ with steps of 11/10-11/10-11/9-11/8;

* 1-11/8-5/3-11/6 with steps of 11/8-11/9-11/10-11/10 and 1-11/10-6/5-16/11 with steps of 11/10-11/10-11/9-11/8;

* 1-5/4-3/2-11/6 ~~chord~~ with steps of 5/4-6/5-11/9-11/10 and 1-11/10-4/3-8/5 ~~chord~~ with steps of 11/10-11/9-6/5-5/4.

* 1-5/4-3/2-11/6 with steps of 5/4-6/5-11/9-11/10 and 1-11/10-4/3-8/5 with steps of 11/10-11/9-6/5-5/4;

* 1-11/8-18/11-11/6 with steps of 11/8-6/5-10/9-11/10 and 1-11/10-11/9-16/11 with steps of 11/10-10/9-6/5-11/8.

Equal temperaments with biyatismic chords include {{~~EDOs~~| 7, 9, 15, 16, 22, 24, 29, 31, 38, 46 and 53 }}.

For pentads, there are one palindromic chord and three pairs of chords in inverse relationship. The palindromic chord is

* 1-9/8-11/8-3/2-18/11 chord with steps of 9/8-11/9-11/10-11/10-11/9.

The inversely related pairs of chords are

* 1-11/8-3/2-5/3-11/6 with steps of 11/8-11/10-10/9-11/10-11/10 and 1-11/10-6/5-4/3-16/11 with steps of 11/10-11/10-10/9-11/10-11/8;

* 1-5/4-3/2-5/3-11/6 with steps of 5/4-6/5-10/9-11/10-11/10 and 1-11/10-6/5-4/3-8/5 with steps of 11/10-11/10-10/9-6/5-5/4;

* 1-5/4-11/8-3/2-11/6 with steps of 5/4-11/10-11/10-11/9-11/10 and 1-11/10-4/3-16/11-8/5 with steps of 11/10-11/9-11/10-11/10-5/4.

Finally there is a palindromic hexad,

* 1-5/4-11/8-3/2-5/3-11/6 with steps of 5/4-11/10-11/10-10/9-11/10-11/10.

Equal temperaments with biyatismic chords include {{Optimal ET sequence| 7, 9, 15, 16, 22, 24, 29, 31, 38, 46 and 53 }}.

== See also ==

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[[Category:Triads]]

[[Category:Tetrads]]

[[Category:Pentads]]

[[Category:Hexads]]

[[Category:Biyatismic]]

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 '''Biyatismic chords''' are [[Dyadic chord|essentially tempered dyadic chords]] tempered by the biyatisma, [[121/120]].
-The [[biyatismic]] temperament contains many possibilities for essentially tempered dyadic chords including 2 triads and 6 tetrads in the [[11-odd-limit]] of the 2.3.5.11 [[subgroup]], but all of these become essentially just in the [[15-odd-limit]].
+There are two triads, nine tetrads, seven pentads and one hexad as 2.3.5.11 subgroup [[11-odd-limit]] essentially tempered chords.
 The two biyatismic triads are in inverse relationship:
@@ Line 7: / Line 7: @@
 * 1-6/5-16/11 chord with steps of 6/5-11/9-11/8.
-For tetrads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are
+For tetrads, there are three palindromic chords and three pairs of chords in inverse relationship. The palindromic chords are
 * 1-9/8-11/8-18/11 chord with steps of 9/8-11/9-6/5-11/9;
-* 1-5/4-11/8-11/6 chord with steps of 5/4-11/10-4/3-11/10.
+* 1-5/4-11/8-11/6 chord with steps of 5/4-11/10-4/3-11/10;
+* 1-3/2-18/11-11/6 chord with steps of 3/2-11/10-10/9-11/10.
 The inversely related pairs of chords are
-* 1-11/8-5/3-11/6 chord with steps of 11/8-11/9-11/10-11/10 and 1-11/10-6/5-16/11 chord with steps of 11/10-11/10-11/9-11/8;
+* 1-11/8-5/3-11/6 with steps of 11/8-11/9-11/10-11/10 and 1-11/10-6/5-16/11 with steps of 11/10-11/10-11/9-11/8;
-* 1-5/4-3/2-11/6 chord with steps of 5/4-6/5-11/9-11/10 and 1-11/10-4/3-8/5 chord with steps of 11/10-11/9-6/5-5/4.
+* 1-5/4-3/2-11/6 with steps of 5/4-6/5-11/9-11/10 and 1-11/10-4/3-8/5 with steps of 11/10-11/9-6/5-5/4;
+* 1-11/8-18/11-11/6 with steps of 11/8-6/5-10/9-11/10 and 1-11/10-11/9-16/11 with steps of 11/10-10/9-6/5-11/8.
-Equal temperaments with biyatismic chords include {{EDOs| 7, 9, 15, 16, 22, 24, 29, 31, 38, 46 and 53 }}.
+For pentads, there are one palindromic chord and three pairs of chords in inverse relationship. The palindromic chord is
+* 1-9/8-11/8-3/2-18/11 chord with steps of 9/8-11/9-11/10-11/10-11/9.
+The inversely related pairs of chords are
+* 1-11/8-3/2-5/3-11/6 with steps of 11/8-11/10-10/9-11/10-11/10 and 1-11/10-6/5-4/3-16/11 with steps of 11/10-11/10-10/9-11/10-11/8;
+* 1-5/4-3/2-5/3-11/6 with steps of 5/4-6/5-10/9-11/10-11/10 and 1-11/10-6/5-4/3-8/5 with steps of 11/10-11/10-10/9-6/5-5/4;
+* 1-5/4-11/8-3/2-11/6 with steps of 5/4-11/10-11/10-11/9-11/10 and 1-11/10-4/3-16/11-8/5 with steps of 11/10-11/9-11/10-11/10-5/4.
+Finally there is a palindromic hexad,
+* 1-5/4-11/8-3/2-5/3-11/6 with steps of 5/4-11/10-11/10-10/9-11/10-11/10.
+Equal temperaments with biyatismic chords include {{Optimal ET sequence| 7, 9, 15, 16, 22, 24, 29, 31, 38, 46 and 53 }}.
 == See also ==
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 [[Category:Triads]]
 [[Category:Tetrads]]
+[[Category:Pentads]]
+[[Category:Hexads]]
 [[Category:Biyatismic]]