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.

The pentacircle triads are

* 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 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.

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* 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:

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.

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

[[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]].

@@ Line 1: / Line 1: @@
 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.
+The pentacircle triads are
 * 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 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.
@@ Line 16: / Line 18: @@
 * 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:
+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.
-The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.
 [[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]].