Pentacircle chords: Difference between revisions

(6 intermediate revisions by 3 users not shown)

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 triads are three in number, 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; and 1-11/7-16/9 with steps 11/7-9/8-9/8. There are six pentacircle tetrads: the palindromic 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7; the palindromic 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7; the inverse pair 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8 and 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7; and the inverse pair 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8 and 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7. 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 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.

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

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.

~~Equal divisions with~~ pentacircle ~~chords~~ include ~~22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167~~, ~~208~~, ~~266e and 433bce~~.

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

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

[[Category:11-odd-limit]]

And an inversely related pair:

~~[[Category:Chords~~]]

* 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

* 1–11/8–14/9–7/4 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

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

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.

[[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 chords]]

[[Category:Essentially tempered chords]]

[[Category:Triads]]

[[Category:Tetrads]]

[[Category:Pentads]]

[[Category:Pentacircle]]

~~[[Category:Tetrad]]~~

~~[[Category:Todo:add links]]~~

@@ 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 triads are three in number, 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; and 1-11/7-16/9 with steps 11/7-9/8-9/8. There are six pentacircle tetrads: the palindromic 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7; the palindromic 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7; the inverse pair 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8 and 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7; and the inverse pair 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8 and 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7. 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 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.
+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 count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.
+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.
-Equal divisions with pentacircle chords include 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, 208, 266e and 433bce.
+The three pentacircle triads include a palindrome since it identifies [[14/11]] by a stack of two [[9/8]]'s:
+* 1–9/8–14/11 with steps 9/8, 9/8, 11/7.
-[[Category:11-odd-limit]]
+And an inversely related pair:
-[[Category:Chords]]
+* 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
+* 1–11/8–14/9–7/4 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
+* 1–11/9–11/8–14/9 with steps 11/9, 9/8, 9/8, 9/7.
+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.
+[[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 chords]]
 [[Category:Essentially tempered chords]]
+[[Category:Triads]]
+[[Category:Tetrads]]
+[[Category:Pentads]]
 [[Category:Pentacircle]]
-[[Category:Tetrad]]
-[[Category:Todo:add links]]