Pentacircle chords: Difference between revisions

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A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 [[subgroup]] in the [[11-odd-limit]], [[tempering out]] the pentacircle comma, [[896/891]].

~~The pentacircle~~ [[~~triad~~]]~~s are three~~ in ~~number:~~

Pentacircle chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 1b]] in the 2.9.7.11 [[subgroup]] [[11-odd-limit]], meaning that there are 3 [[triad]]s, 6 [[tetrad]]s and 2 [[pentad]]s, for a total of 11 distinct chord structures.

* 1-9/7-16/9 ~~with steps 9/~~7-11~~/8-9/8;~~

* 1-9/7-16/11 ~~with steps 9/7~~-~~9/8~~-11~~/8;~~

* 1-11/7-16/9 with steps 11/7-9/8-9/8.

~~There are six~~ pentacircle [[~~tetrad~~]]s~~, including the palindromic~~

The three pentacircle triads include a palindrome since it identifies [[14/11]] by a stack of two [[9/8]]'s:

* 1-9/8~~-14~~/9-7/4 with steps 9/8-11/8-9/8-8/7;

* 1–9/8–14/11 with steps 9/8, 9/8, 11/7.

* 1-9/8-11/8~~-14~~/9 with steps 9/8-11/9-9/8-9/7.

And an inversely related pair:

* 1–9/8–16/11 with steps 9/8, 9/7, 11/8;

* 1–9/8–14/9 with steps 9/8, 11/8, 9/7;

The tetrads include the palindromic

* 1–9/8–14/9–7/4 with steps 9/8, 11/8, 9/8, 8/7;

* 1–9/8–11/8–14/9 with steps 9/8, 11/9, 9/8, 9/7.

And the inversely related pairs

* ~~1-11~~/~~8-11~~/~~7-16~~/9 with steps 11/8-8/7-9/8-9/8, and its inverse

* 1–11/8–11/7–16/9 with steps 11/8, 8/7, 9/8, 9/8, and its inverse

* ~~1-11~~/~~8-14~~/~~9-7~~/4 with steps 11/8-9/8-9/8-8/7;

* 1–11/8–14/9–7/4 with steps 11/8, 9/8, 9/8, 8/7;

* ~~1-11~~/~~9-11~~/~~7-16~~/9 with steps 11/9-9/7-9/8-9/8, and its inverse

* 1–11/9–11/7–16/9 with steps 11/9, 9/7, 9/8, 9/8, and its inverse

* ~~1-11~~/~~9-11~~/~~8-14~~/9 with steps 11/9~~-9/8-~~9/8~~-9/7.~~

* 1–11/9–11/8–14/9 with steps 11/9, 9/8, 9/8, 9/7.

~~Finally~~, ~~there are two pentacircle [[pentad]]s, inversely related:~~

* 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8, ~~and its inverse~~

* 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9~~/8-8~~/7~~-9/8~~.

~~The count of chords is triads~~: 3, ~~tetrads: 6~~, ~~pentads: 2~~, ~~for a total of~~ 11.

Finally, there are two pentacircle pentads, inversely related:

* 1–11/9–11/8–11/7–16/9 with steps 11/9, 9/8, 8/7, 9/8, 9/8, and its inverse

* 1–11/9–11/8–14/9–16/9 with steps 11/9, 9/8, 9/8, 8/7, 9/8.

[[~~EDO|~~Equal ~~divisions of the octave~~]] with pentacircle chords include {{EDOs| 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208 }}, with 208edo giving the [[optimal patent val]].

[[Equal temperament]]s with pentacircle chords include {{EDOs| 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208 }}, with 208edo giving the [[optimal patent val]].

[[Category:11-odd-limit]]

[[Category:11-odd-limit chords]]

[[Category:Essentially tempered chords]]

[[Category:~~Triad~~]]

[[Category:Triads]]

[[Category:~~Tetrad~~]]

[[Category:Tetrads]]

[[Category:~~Pentad~~]]

[[Category:Pentads]]

[[Category:Pentacircle]]

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 A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 [[subgroup]] in the [[11-odd-limit]], [[tempering out]] the pentacircle comma, [[896/891]].
-The pentacircle [[triad]]s are three in number:
+Pentacircle chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 1b]] in the 2.9.7.11 [[subgroup]] [[11-odd-limit]], meaning that there are 3 [[triad]]s, 6 [[tetrad]]s and 2 [[pentad]]s, for a total of 11 distinct chord structures.
-* 1-9/7-16/9 with steps 9/7-11/8-9/8;
-* 1-9/7-16/11 with steps 9/7-9/8-11/8;
-* 1-11/7-16/9 with steps 11/7-9/8-9/8.
-There are six pentacircle [[tetrad]]s, including the palindromic
+The three pentacircle triads include a palindrome since it identifies [[14/11]] by a stack of two [[9/8]]'s:
-* 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7;
+* 1–9/8–14/11 with steps 9/8, 9/8, 11/7.
-* 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7.
+And an inversely related pair:
+* 1–9/8–16/11 with steps 9/8, 9/7, 11/8;
+* 1–9/8–14/9 with steps 9/8, 11/8, 9/7;
+The tetrads include the palindromic
+* 1–9/8–14/9–7/4 with steps 9/8, 11/8, 9/8, 8/7;
+* 1–9/8–11/8–14/9 with steps 9/8, 11/9, 9/8, 9/7.
 And the inversely related pairs
-* 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8, and its inverse
+* 1–11/8–11/7–16/9 with steps 11/8, 8/7, 9/8, 9/8, and its inverse
-* 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7;
+* 1–11/8–14/9–7/4 with steps 11/8, 9/8, 9/8, 8/7;
-* 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8, and its inverse
+* 1–11/9–11/7–16/9 with steps 11/9, 9/7, 9/8, 9/8, and its inverse
-* 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7.
+* 1–11/9–11/8–14/9 with steps 11/9, 9/8, 9/8, 9/7.
-Finally, there are two pentacircle [[pentad]]s, inversely related:
-* 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8, and its inverse
-* 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.
-The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.
+Finally, there are two pentacircle pentads, inversely related:
+* 1–11/9–11/8–11/7–16/9 with steps 11/9, 9/8, 8/7, 9/8, 9/8, and its inverse
+* 1–11/9–11/8–14/9–16/9 with steps 11/9, 9/8, 9/8, 8/7, 9/8.
-[[EDO|Equal divisions of the octave]] with pentacircle chords include {{EDOs| 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208 }}, with 208edo giving the [[optimal patent val]].
+[[Equal temperament]]s with pentacircle chords include {{EDOs| 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208 }}, with 208edo giving the [[optimal patent val]].
-[[Category:11-odd-limit]]
+[[Category:11-odd-limit chords]]
 [[Category:Essentially tempered chords]]
-[[Category:Triad]]
+[[Category:Triads]]
-[[Category:Tetrad]]
+[[Category:Tetrads]]
-[[Category:Pentad]]
+[[Category:Pentads]]
 [[Category:Pentacircle]]