Pentacircle chords: Difference between revisions

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It's important to single ou the palindromic triad cuz it's the most fundamental form of these chords. The rest can be viewed as variations on it
 
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A '''pentacircle chord''' is an [[essentially tempered dyadic chord]] in the 2.9.7.11 [[Just intonation subgroup|subgroup]] in the [[11-odd-limit]], [[tempering out]] the [[pentacircle comma]], [[896/891]].  
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|triads]] 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
The three pentacircle triads include a palindrome since it identifies [[14/11]] by a stack of two [[9/8]]'s:
* 1-9/7-16/11 with steps 9/7-9/8-11/8
* 1–9/8–14/11 with steps 9/8, 9/8, 11/7.  
* 1-11/7-16/9 with steps 11/7-9/8-9/8.  


There are six pentacircle [[Tetrad|tetrads]]:  
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 [[palindromic]] 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7
The tetrads include the palindromic
* the palindromic 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7
* 1–9/8–14/9–7/4 with steps 9/8, 11/8, 9/8, 8/7;
* 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
* 1–9/8–11/8–14/9 with steps 9/8, 11/9, 9/8, 9/7.
* 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 [[Pentad|pentads]], inversely related:
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.


* 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8
Finally, there are two pentacircle pentads, inversely related:
* 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8
* 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.
[[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]].


[[EDO|Equal divisions of the octave]] with pentacircle chords include [[22edo|22]], [[27edo|27]], [[41edo|41]], [[46edo|46]], [[58edo|58]], [[68edo|68]], [[80edo|80]], [[87edo|87]], [[121edo|121]], [[145edo|145]], [[167edo|167]], [[208edo|208]], [[266edo|266e]] and [[433edo|433bce]].
[[Category:11-odd-limit chords]]
 
[[Category:11-odd-limit]]
[[Category:Chords]]
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Pentacircle]]
[[Category:Pentacircle]]
[[Category:Tetrad]]

Latest revision as of 13:55, 11 October 2024

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.

Pentacircle chords are of pattern 1b in the 2.9.7.11 subgroup 11-odd-limit, meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.

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.

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
  • 1–11/8–14/9–7/4 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 temperaments with pentacircle chords include 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208, with 208edo giving the optimal patent val.

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