Pentacircle chords: Difference between revisions

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**Imported revision 285352166 - 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>2011-12-13 02:15:38 UTC</tt>.<br>
: The original revision id was <tt>285352166</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 11-limit [[Dyadic chord#Essentially tempered dyadic chords|essentially tempered dyadic chord]] 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.


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


</pre></div>
And the inversely related pairs
<h4>Original HTML content:</h4>
* 1–11/8–11/7–16/9 with steps 11/8, 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 11-limit &lt;a class="wiki_link" href="/Dyadic%20chord#Essentially tempered dyadic chords"&gt;essentially tempered dyadic chord&lt;/a&gt; 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;
* 1–11/8–14/9–7/4 with steps 11/8, 9/8, 9/8, 8/7;  
&lt;br /&gt;
* 1–11/9–11/7–16/9 with steps 11/9, 9/7, 9/8, 9/8, and its inverse
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>
* 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]]

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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