Graph-theoretic properties of scales: Difference between revisions

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[[image:star.png]]

==~~The oktony~~==

==Oktone==

By ~~the oktony~~ is meant the tempering of the octone scale, 1-15/14-60/49-5/4-10/7-3/2-12/7-7/4-2, in [[Breed family#Jove, aka Wonder|jove temperament]]. The tempering can be accomplished via [[202edo|202et]], leading to the scale 0, 20, 59, 65, 104, 118, 157, 163, 202 which we can use together with the 11-limit diamond as a consonance set, {25, 28, 31, 34, 39, 45, 53, 59, 65, 70, 73, 84, 93, 98, 104, 109, 118, 129, 132, 137, 143, 149, 157, 163, 168, 171, 174, 177}. This leads to a highly accurate scale with a good deal of symmetry. The automorphism group of the graph is the direct product of the group of the square with an involution, ie an element of order two. The involution simultaneously exchanges notes 2 and 7, and 3 and 6; that is, it is in cycle terms (2, 7)(3,6), or (59, 163)(65, 157) in terms of 202et, and the square part permutes the cycle (0, 4, 1, 5) by the group of the square.

By [[oktone]] is meant the tempering of the octone scale, 1-15/14-60/49-5/4-10/7-3/2-12/7-7/4-2, in [[Breed family#Jove, aka Wonder|jove temperament]]. The tempering can be accomplished via [[202edo|202et]], leading to the scale 0, 20, 59, 65, 104, 118, 157, 163, 202 which we can use together with the 11-limit diamond as a consonance set, {25, 28, 31, 34, 39, 45, 53, 59, 65, 70, 73, 84, 93, 98, 104, 109, 118, 129, 132, 137, 143, 149, 157, 163, 168, 171, 174, 177}. This leads to a highly accurate scale with a good deal of symmetry. The automorphism group of the graph is the direct product of the group of the square with an involution, ie an element of order two. The involution simultaneously exchanges notes 2 and 7, and 3 and 6; that is, it is in cycle terms (2, 7)(3,6), or (59, 163)(65, 157) in terms of 202et, and the square part permutes the cycle (0, 4, 1, 5) by the group of the square.

The graph has twelve maximal cliques, all tetrads, of which four connect with all of the other tetrads, and eight connect with all but one. It has two vertices of degree five and six of degree six, with connectivities 4.586 ≤ 5 ≤ 5, and radius and diameter both 2.

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<img src="/file/view/star.png/359553295/star.png" alt="star.png" title="star.png" /><br />

<br />

<h2 id="toc10"><a name="Examples-~~The oktony~~"></a>~~The oktony~~</h2>

<h2 id="toc10"><a name="Examples-Oktone"></a>Oktone</h2>

By ~~the oktony~~ is meant the tempering of the octone scale, 1-15/14-60/49-5/4-10/7-3/2-12/7-7/4-2, in <a class="wiki_link" href="/Breed%20family#Jove, aka Wonder">jove temperament</a>. The tempering can be accomplished via <a class="wiki_link" href="/202edo">202et</a>, leading to the scale 0, 20, 59, 65, 104, 118, 157, 163, 202 which we can use together with the 11-limit diamond as a consonance set, {25, 28, 31, 34, 39, 45, 53, 59, 65, 70, 73, 84, 93, 98, 104, 109, 118, 129, 132, 137, 143, 149, 157, 163, 168, 171, 174, 177}. This leads to a highly accurate scale with a good deal of symmetry. The automorphism group of the graph is the direct product of the group of the square with an involution, ie an element of order two. The involution simultaneously exchanges notes 2 and 7, and 3 and 6; that is, it is in cycle terms (2, 7)(3,6), or (59, 163)(65, 157) in terms of 202et, and the square part permutes the cycle (0, 4, 1, 5) by the group of the square.<br />

By <a class="wiki_link" href="/oktone">oktone</a> is meant the tempering of the octone scale, 1-15/14-60/49-5/4-10/7-3/2-12/7-7/4-2, in <a class="wiki_link" href="/Breed%20family#Jove, aka Wonder">jove temperament</a>. The tempering can be accomplished via <a class="wiki_link" href="/202edo">202et</a>, leading to the scale 0, 20, 59, 65, 104, 118, 157, 163, 202 which we can use together with the 11-limit diamond as a consonance set, {25, 28, 31, 34, 39, 45, 53, 59, 65, 70, 73, 84, 93, 98, 104, 109, 118, 129, 132, 137, 143, 149, 157, 163, 168, 171, 174, 177}. This leads to a highly accurate scale with a good deal of symmetry. The automorphism group of the graph is the direct product of the group of the square with an involution, ie an element of order two. The involution simultaneously exchanges notes 2 and 7, and 3 and 6; that is, it is in cycle terms (2, 7)(3,6), or (59, 163)(65, 157) in terms of 202et, and the square part permutes the cycle (0, 4, 1, 5) by the group of the square.<br />

<br />

The graph has twelve maximal cliques, all tetrads, of which four connect with all of the other tetrads, and eight connect with all but one. It has two vertices of degree five and six of degree six, with connectivities 4.586 ≤ 5 ≤ 5, and radius and diameter both 2.<br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2012-08-25 13:41:35 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-25 14:06:41 UTC</tt>.<br>
-: The original revision id was <tt>359876169</tt>.<br>
+: The original revision id was <tt>359878357</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>
@@ Line 74: / Line 74: @@
 [[image:star.png]]
-==The oktony==
+==Oktone==
-By the oktony is meant the tempering of the octone scale, 1-15/14-60/49-5/4-10/7-3/2-12/7-7/4-2, in [[Breed family#Jove, aka Wonder|jove temperament]]. The tempering can be accomplished via [[202edo|202et]], leading to the scale 0, 20, 59, 65, 104, 118, 157, 163, 202 which we can use together with the 11-limit diamond as a consonance set, {25, 28, 31, 34, 39, 45, 53, 59, 65, 70, 73, 84, 93, 98, 104, 109, 118, 129, 132, 137, 143, 149, 157, 163, 168, 171, 174, 177}. This leads to a highly accurate scale with a good deal of symmetry. The automorphism group of the graph is the direct product of the group of the square with an involution, ie an element of order two. The involution simultaneously exchanges notes 2 and 7, and 3 and 6; that is, it is in cycle terms (2, 7)(3,6), or (59, 163)(65, 157) in terms of 202et, and the square part permutes the cycle (0, 4, 1, 5) by the group of the square.
+By [[oktone]] is meant the tempering of the octone scale, 1-15/14-60/49-5/4-10/7-3/2-12/7-7/4-2, in [[Breed family#Jove, aka Wonder|jove temperament]]. The tempering can be accomplished via [[202edo|202et]], leading to the scale 0, 20, 59, 65, 104, 118, 157, 163, 202 which we can use together with the 11-limit diamond as a consonance set, {25, 28, 31, 34, 39, 45, 53, 59, 65, 70, 73, 84, 93, 98, 104, 109, 118, 129, 132, 137, 143, 149, 157, 163, 168, 171, 174, 177}. This leads to a highly accurate scale with a good deal of symmetry. The automorphism group of the graph is the direct product of the group of the square with an involution, ie an element of order two. The involution simultaneously exchanges notes 2 and 7, and 3 and 6; that is, it is in cycle terms (2, 7)(3,6), or (59, 163)(65, 157) in terms of 202et, and the square part permutes the cycle (0, 4, 1, 5) by the group of the square.
 The graph has twelve maximal cliques, all tetrads, of which four connect with all of the other tetrads, and eight connect with all but one. It has two vertices of degree five and six of degree six, with connectivities 4.586 ≤ 5 ≤ 5, and radius and diameter both 2.
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 &lt;!-- ws:start:WikiTextLocalImageRule:50:&amp;lt;img src=&amp;quot;/file/view/star.png/359553295/star.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/star.png/359553295/star.png" alt="star.png" title="star.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:50 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Examples-The oktony"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;The oktony&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Examples-Oktone"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Oktone&lt;/h2&gt;
-By the oktony is meant the tempering of the octone scale, 1-15/14-60/49-5/4-10/7-3/2-12/7-7/4-2, in &lt;a class="wiki_link" href="/Breed%20family#Jove, aka Wonder"&gt;jove temperament&lt;/a&gt;. The tempering can be accomplished via &lt;a class="wiki_link" href="/202edo"&gt;202et&lt;/a&gt;, leading to the scale 0, 20, 59, 65, 104, 118, 157, 163, 202 which we can use together with the 11-limit diamond as a consonance set, {25, 28, 31, 34, 39, 45, 53, 59, 65, 70, 73, 84, 93, 98, 104, 109, 118, 129, 132, 137, 143, 149, 157, 163, 168, 171, 174, 177}. This leads to a highly accurate scale with a good deal of symmetry. The automorphism group of the graph is the direct product of the group of the square with an involution, ie an element of order two. The involution simultaneously exchanges notes 2 and 7, and 3 and 6; that is, it is in cycle terms (2, 7)(3,6), or (59, 163)(65, 157) in terms of 202et, and the square part permutes the cycle (0, 4, 1, 5) by the group of the square.&lt;br /&gt;
+By &lt;a class="wiki_link" href="/oktone"&gt;oktone&lt;/a&gt; is meant the tempering of the octone scale, 1-15/14-60/49-5/4-10/7-3/2-12/7-7/4-2, in &lt;a class="wiki_link" href="/Breed%20family#Jove, aka Wonder"&gt;jove temperament&lt;/a&gt;. The tempering can be accomplished via &lt;a class="wiki_link" href="/202edo"&gt;202et&lt;/a&gt;, leading to the scale 0, 20, 59, 65, 104, 118, 157, 163, 202 which we can use together with the 11-limit diamond as a consonance set, {25, 28, 31, 34, 39, 45, 53, 59, 65, 70, 73, 84, 93, 98, 104, 109, 118, 129, 132, 137, 143, 149, 157, 163, 168, 171, 174, 177}. This leads to a highly accurate scale with a good deal of symmetry. The automorphism group of the graph is the direct product of the group of the square with an involution, ie an element of order two. The involution simultaneously exchanges notes 2 and 7, and 3 and 6; that is, it is in cycle terms (2, 7)(3,6), or (59, 163)(65, 157) in terms of 202et, and the square part permutes the cycle (0, 4, 1, 5) by the group of the square.&lt;br /&gt;
 &lt;br /&gt;
 The graph has twelve maximal cliques, all tetrads, of which four connect with all of the other tetrads, and eight connect with all but one. It has two vertices of degree five and six of degree six, with connectivities 4.586 ≤ 5 ≤ 5, and radius and diameter both 2.&lt;br /&gt;