Pentacircle chords: Difference between revisions

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**Imported revision 287093074 - Original comment: **
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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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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]].  
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-17 17:15:07 UTC</tt>.<br>
: The original revision id was <tt>287093074</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A //pentacircle chord// is an [[Dyadic chord#Essentially tempered dyadic chords|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.


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.  


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.


</pre></div>
Finally, there are two pentacircle pentads, inversely related:  
<h4>Original HTML content:</h4>
* 1–11/9–11/8–11/7–16/9 with steps 11/9, 9/8, 8/7, 9/8, 9/8, and its inverse
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;pentacircle chords&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;pentacircle chord&lt;/em&gt; is an &lt;a class="wiki_link" href="/Dyadic%20chord#Essentially tempered dyadic chords"&gt;essentially tempered dyadic chord&lt;/a&gt; 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.&lt;br /&gt;
* 1–11/9–11/8–14/9–16/9 with steps 11/9, 9/8, 9/8, 8/7, 9/8.  
&lt;br /&gt;
 
The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.&lt;br /&gt;
[[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]].
&lt;br /&gt;
 
Equal divisions with pentacircle chords include 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, 208, 266e and 433bce.&lt;/body&gt;&lt;/html&gt;</pre></div>
[[Category:11-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Pentacircle]]

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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