Cuthbert chords: Difference between revisions

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'''Cuthbert chords''' are [[essentially tempered dyadic chord]]s tempered by the cuthbert comma, [[847/845]].

~~The most typical cuthbert triad is a palindrome~~ in the 2.5.7.11.13 [[subgroup]] [[13-odd-limit]], consisting of two [[13/11]]'s making up [[7/5]], which implies tempering by cuthbert, the 847/845 comma. It is, in other words, the 847/845-tempered version of

Cuthbert chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 1a]] in the 2.5.7.11.13 [[subgroup]] [[13-odd-limit]], meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.

* ~~1-13~~/~~11-7~~/5 with steps 13/11-13/11-10/7.

The most basic cuthbert triad is a palindrome, consisting of two [[13/11]]'s making up [[7/5]], which implies tempering by cuthbert, the 847/845 comma. It is, in other words, the 847/845-tempered version of

* 1–13/11–7/5 chord with steps of 13/11, 13/11, 10/7.

There is an inversely related pair which is more squeezed and fit for a sort of secundal harmony:

* 1-11/10-13/11 with steps 11/10-14/13~~-22~~/13, and its inverse

* 1–11/10–13/11 with steps of 11/10, 14/13, 22/13, and its inverse

* 1-14/13-13/11 with steps 14/13-11/10~~-22~~/13.

* 1–14/13–13/11 with steps of 14/13, 11/10, 22/13.

They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic tetrads are

* 1–11/10–13/11–13/10 chord with steps of 11/10, 14/13, 11/10, 20/13;

* 1–14/13–13/11–14/11 chord with steps of 14/13, 11/10, 14/13, 11/7.

The inversely related pairs of tetrads are

* 1–13/11–14/11–7/5 with steps of 13/11, 14/13, 11/10, 10/7, and its inverse

* 1–11/10–13/11–7/5 with steps of 11/10, 14/13, 13/11, 10/7;

* 1–13/11–13/10–7/5 with steps of 13/11, 11/10, 14/13, 10/7, and its inverse

* 1–14/13–13/11–7/5 with steps of 14/13, 11/10, 13/11, 10/7.

~~They can be extended to the following~~ inversely related ~~tetrads~~:

Then there is an inversely related pair of pentads:

* ~~1-13~~/~~11-14~~/~~11-7~~/5 with steps ~~13/11-14/13-~~11/10~~-10/7~~, ~~and its inverse~~

* 1–11/10–13/11–13/10–7/5 with steps of 11/10, 14/13, 11/10, 14/13, 10/7, and its inverse

* 1-11/10-13/11-7/5 with steps 11/10-14/13~~-13/11-10/7;~~

* 1–14/13–13/11–14/11–7/5 with steps of 14/13, 11/10, 14/13, 11/10, 10/7.

* 1-13/11~~-13~~/10~~-7/5 with steps 13/11-11/10-~~14/13-10/7, and its inverse

* ~~1-14~~/~~13-13~~/~~11-7~~/5 with steps 14/13-11/10-13/~~11-~~10/7.

~~Then there are two inversely related pentads:~~

Equal temperaments with cuthbert triads include {{Optimal ET sequence| 29, 33, 37, 41, 46, 50, 53, 58, 70, 87, 94, 99, 103, 111, 128, 140, 149, 177, 190, 198, 205, 227, 264, 284 and 388 }}.

* 1-11/10-13/11-13/10-7/5 with ~~steps 11/10-14/13-11/10-14/13-10/7~~, and ~~its inverse~~

* 1-14/13-13/11-14/11-7/5 with steps 14/13-11/10-14/13-11/10-10/7.

== Garibert tetrad ==

The first cuthbert triad can be extended to the '''garibert tetrad''', which is the {[[275/273]], 847/845} garibert tempering of a tetrad,

* ~~1-13~~/~~11-7~~/5-[[5/3]] with steps of size 13/11-13/11-13/11-[[6/5]].

* 1–13/11–7/5–[[5/3]] with steps of size 13/11, 13/11, 13/11, [[6/5]].

Equal temperaments with ~~cuthbert triads~~ include {{~~EDOs~~| 29~~, 33~~, 37, 41~~, 46, 50~~, 53~~, 58, 70, 87,~~ 94~~, 99, 103, 111, 128, 140, 149, 177, 190, 198, 205, 227, 264, 284 and 388~~ }}~~. Equal temperaments with garibert tetrads include 41, 53, and 94~~; and it is a characteristic chord of [[13-limit]] [[garibaldi temperament]].

Equal temperaments with the garibert tetrad include {{Optimal ET sequence| 16, 29, 37, 41, 53 and 94 }}; and it is a characteristic chord of [[13-limit]] [[garibaldi temperament]].

[[Category:13-odd-limit]]

[[Category:13-odd-limit chords]]

[[Category:Essentially tempered chords]]

[[Category:~~Triad~~]]

[[Category:Triads]]

[[Category:~~Tetrad~~]]

[[Category:Tetrads]]

[[Category:~~Pentad~~]]

[[Category:Pentads]]

[[Category:Cuthbert]]

[[Category:Garibaldi]]

[[Category:Garibert]]

[[Category:Gassormic]]

@@ Line 1: / Line 1: @@
 '''Cuthbert chords''' are [[essentially tempered dyadic chord]]s tempered by the cuthbert comma, [[847/845]].
-The most typical cuthbert triad is a palindrome in the 2.5.7.11.13 [[subgroup]] [[13-odd-limit]], consisting of two [[13/11]]'s making up [[7/5]], which implies tempering by cuthbert, the 847/845 comma. It is, in other words, the 847/845-tempered version of
+Cuthbert chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 1a]] in the 2.5.7.11.13 [[subgroup]] [[13-odd-limit]], meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.
-* 1-13/11-7/5 with steps 13/11-13/11-10/7.
+The most basic cuthbert triad is a palindrome, consisting of two [[13/11]]'s making up [[7/5]], which implies tempering by cuthbert, the 847/845 comma. It is, in other words, the 847/845-tempered version of
+* 1–13/11–7/5 chord with steps of 13/11, 13/11, 10/7.
 There is an inversely related pair which is more squeezed and fit for a sort of secundal harmony:
-* 1-11/10-13/11 with steps 11/10-14/13-22/13, and its inverse
+* 1–11/10–13/11 with steps of 11/10, 14/13, 22/13, and its inverse
-* 1-14/13-13/11 with steps 14/13-11/10-22/13.
+* 1–14/13–13/11 with steps of 14/13, 11/10, 22/13.
+They can be extended to the following tetrads, with two palindromic chords and two pairs of chords in inverse relationship. The palindromic tetrads are
+* 1–11/10–13/11–13/10 chord with steps of 11/10, 14/13, 11/10, 20/13;
+* 1–14/13–13/11–14/11 chord with steps of 14/13, 11/10, 14/13, 11/7.
+The inversely related pairs of tetrads are
+* 1–13/11–14/11–7/5 with steps of 13/11, 14/13, 11/10, 10/7, and its inverse
+* 1–11/10–13/11–7/5 with steps of 11/10, 14/13, 13/11, 10/7;
+* 1–13/11–13/10–7/5 with steps of 13/11, 11/10, 14/13, 10/7, and its inverse
+* 1–14/13–13/11–7/5 with steps of 14/13, 11/10, 13/11, 10/7.
-They can be extended to the following inversely related tetrads:
+Then there is an inversely related pair of pentads:
-* 1-13/11-14/11-7/5 with steps 13/11-14/13-11/10-10/7, and its inverse
+* 1–11/10–13/11–13/10–7/5 with steps of 11/10, 14/13, 11/10, 14/13, 10/7, and its inverse
-* 1-11/10-13/11-7/5 with steps 11/10-14/13-13/11-10/7;
+* 1–14/13–13/11–14/11–7/5 with steps of 14/13, 11/10, 14/13, 11/10, 10/7.
-* 1-13/11-13/10-7/5 with steps 13/11-11/10-14/13-10/7, and its inverse
-* 1-14/13-13/11-7/5 with steps 14/13-11/10-13/11-10/7.
-Then there are two inversely related pentads:
+Equal temperaments with cuthbert triads include {{Optimal ET sequence| 29, 33, 37, 41, 46, 50, 53, 58, 70, 87, 94, 99, 103, 111, 128, 140, 149, 177, 190, 198, 205, 227, 264, 284 and 388 }}.
-* 1-11/10-13/11-13/10-7/5 with steps 11/10-14/13-11/10-14/13-10/7, and its inverse
-* 1-14/13-13/11-14/11-7/5 with steps 14/13-11/10-14/13-11/10-10/7.
 == Garibert tetrad ==
 The first cuthbert triad can be extended to the '''garibert tetrad''', which is the {[[275/273]], 847/845} garibert tempering of a tetrad,
-* 1-13/11-7/5-[[5/3]] with steps of size 13/11-13/11-13/11-[[6/5]].
+* 1–13/11–7/5–[[5/3]] with steps of size 13/11, 13/11, 13/11, [[6/5]].
-Equal temperaments with cuthbert triads include {{EDOs| 29, 33, 37, 41, 46, 50, 53, 58, 70, 87, 94, 99, 103, 111, 128, 140, 149, 177, 190, 198, 205, 227, 264, 284 and 388 }}. Equal temperaments with garibert tetrads include 41, 53, and 94; and it is a characteristic chord of [[13-limit]] [[garibaldi temperament]].
+Equal temperaments with the garibert tetrad include {{Optimal ET sequence| 16, 29, 37, 41, 53 and 94 }}; and it is a characteristic chord of [[13-limit]] [[garibaldi temperament]].
-[[Category:13-odd-limit]]
+[[Category:13-odd-limit chords]]
 [[Category:Essentially tempered chords]]
-[[Category:Triad]]
+[[Category:Triads]]
-[[Category:Tetrad]]
+[[Category:Tetrads]]
-[[Category:Pentad]]
+[[Category:Pentads]]
 [[Category:Cuthbert]]
 [[Category:Garibaldi]]
 [[Category:Garibert]]
 [[Category:Gassormic]]