Graph-theoretic properties of scales: Difference between revisions

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>2013-02-17 21:29:13 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-02-17 21:33:09 UTC</tt>.<br>

: The original revision id was <tt>~~407729838~~</tt>.<br>

: The original revision id was <tt>407730416</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 87:

[[Zeus7tri]] is the tempering in zeus of 11/10, 5/4, 11/8, 3/2, 12/7, 15/8, 2. In 99et it has steps 0, 13, 32, 45, 58, 77, 90, 99, with an 11-limit consonance set {13, 13, 15, 17, 19, 22, 26, 28, 32, 35, 36, 41, 45, 48, 51, 54, 58, 63, 64, 67, 71, 73, 77, 80, 82, 84, 86, 86}. The scale contains two maximal hexads, consisting of the pentad of the notes 1 through 5 plus either 0 or 6. The automorphism group is of order 240, consisting of the direct product of the group of order two exchanging 0 and 6, and the symmetric group on the five other notes 1, 2, 3, 4, 5. It has vertex, edge and algebraic connectivities all 5.

The scale is interesting in that it has the trivalent property, so that there are exactly three specific intervals for each generic interval save the octave multiples. ~~Class~~ 2 ~~contains~~ the three thirds, 7/6, 6/5 and 5/4.

The scale is interesting in that it has the trivalent property, so that there are exactly three specific intervals for each generic interval save the octave multiples. It is a consonant class scale, with class 2 containing the three thirds, 7/6, 6/5 and 5/4.

[[image:graph of zeus7tri.png width="445" height="629"]]

Line 144:

=Ten note scales=

==Magic[10]==

~~Maic~~[10], the 10-note MOS of [[Magic family#Magic-11-limit|magic temperament]], can in [[104edo|104et]] be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2. It has 27 maximal cliques, 18 triads and 9 tetrads.

Magic[10], the 10-note MOS of [[Magic family#Magic-11-limit|magic temperament]], can in [[104edo|104et]] be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2. It has 27 maximal cliques, 18 triads and 9 tetrads.

Abstractly, the rather large group of automorphisms of order 288 is the direct product of the Klein four-group and the transitive group 6T13 of degree 6, which is the wreath product S3 ≀ S2. The four-group part acts on the notes from 1 to 4, and is generated by the involutions (1,4) and (2,3), and the 6T13 group, of order 72, acts on notes 5 through 10--or 5 through 9 and 0, if you prefer. It is generated by (5,10)(6,8)(7,9) together with (5,6), (6,7) and (8,9).

Line 305:

<a class="wiki_link" href="/Zeus7tri">Zeus7tri</a> is the tempering in zeus of 11/10, 5/4, 11/8, 3/2, 12/7, 15/8, 2. In 99et it has steps 0, 13, 32, 45, 58, 77, 90, 99, with an 11-limit consonance set {13, 13, 15, 17, 19, 22, 26, 28, 32, 35, 36, 41, 45, 48, 51, 54, 58, 63, 64, 67, 71, 73, 77, 80, 82, 84, 86, 86}. The scale contains two maximal hexads, consisting of the pentad of the notes 1 through 5 plus either 0 or 6. The automorphism group is of order 240, consisting of the direct product of the group of order two exchanging 0 and 6, and the symmetric group on the five other notes 1, 2, 3, 4, 5. It has vertex, edge and algebraic connectivities all 5.<br />

<br />

The scale is interesting in that it has the trivalent property, so that there are exactly three specific intervals for each generic interval save the octave multiples. ~~Class~~ 2 ~~contains~~ the three thirds, 7/6, 6/5 and 5/4.<br />

The scale is interesting in that it has the trivalent property, so that there are exactly three specific intervals for each generic interval save the octave multiples. It is a consonant class scale, with class 2 containing the three thirds, 7/6, 6/5 and 5/4.<br />

<br />

<img src="/file/view/graph%20of%20zeus7tri.png/402208970/445x629/graph%20of%20zeus7tri.png" alt="graph of zeus7tri.png" title="graph of zeus7tri.png" style="height: 629px; width: 445px;" /><br />

Line 362:

<h1 id="toc19"><a name="Ten note scales"></a>Ten note scales</h1>

<h2 id="toc20"><a name="Ten note scales-Magic[10]"></a>Magic[10]</h2>

~~Maic~~[10], the 10-note MOS of <a class="wiki_link" href="/Magic%20family#Magic-11-limit">magic temperament</a>, can in <a class="wiki_link" href="/104edo">104et</a> be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2. It has 27 maximal cliques, 18 triads and 9 tetrads.<br />

Magic[10], the 10-note MOS of <a class="wiki_link" href="/Magic%20family#Magic-11-limit">magic temperament</a>, can in <a class="wiki_link" href="/104edo">104et</a> be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2. It has 27 maximal cliques, 18 triads and 9 tetrads.<br />

<br />

Abstractly, the rather large group of automorphisms of order 288 is the direct product of the Klein four-group and the transitive group 6T13 of degree 6, which is the wreath product S3 ≀ S2. The four-group part acts on the notes from 1 to 4, and is generated by the involutions (1,4) and (2,3), and the 6T13 group, of order 72, acts on notes 5 through 10--or 5 through 9 and 0, if you prefer. It is generated by (5,10)(6,8)(7,9) together with (5,6), (6,7) and (8,9).<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>2013-02-17 21:29:13 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-02-17 21:33:09 UTC</tt>.<br>
-: The original revision id was <tt>407729838</tt>.<br>
+: The original revision id was <tt>407730416</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 87: / Line 87: @@
 [[Zeus7tri]] is the tempering in zeus of 11/10, 5/4, 11/8, 3/2, 12/7, 15/8, 2. In 99et it has steps 0, 13, 32, 45, 58, 77, 90, 99, with an 11-limit consonance set {13, 13, 15, 17, 19, 22, 26, 28, 32, 35, 36, 41, 45, 48, 51, 54, 58, 63, 64, 67, 71, 73, 77, 80, 82, 84, 86, 86}. The scale contains two maximal hexads, consisting of the pentad of the notes 1 through 5 plus either 0 or 6. The automorphism group is of order 240, consisting of the direct product of the group of order two exchanging 0 and 6, and the symmetric group on the five other notes 1, 2, 3, 4, 5. It has vertex, edge and algebraic connectivities all 5.
-The scale is interesting in that it has the trivalent property, so that there are exactly three specific intervals for each generic interval save the octave multiples. Class 2 contains the three thirds, 7/6, 6/5 and 5/4.
+The scale is interesting in that it has the trivalent property, so that there are exactly three specific intervals for each generic interval save the octave multiples. It is a consonant class scale, with class 2 containing the three thirds, 7/6, 6/5 and 5/4.
 [[image:graph of zeus7tri.png width="445" height="629"]]
@@ Line 144: / Line 144: @@
 =Ten note scales=
 ==Magic[10]==
-Maic[10], the 10-note MOS of [[Magic family#Magic-11-limit|magic temperament]], can in [[104edo|104et]] be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2. It has 27 maximal cliques, 18 triads and 9 tetrads.
+Magic[10], the 10-note MOS of [[Magic family#Magic-11-limit|magic temperament]], can in [[104edo|104et]] be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2. It has 27 maximal cliques, 18 triads and 9 tetrads.
 Abstractly, the rather large group of automorphisms of order 288 is the direct product of the Klein four-group and the transitive group 6T13 of degree 6, which is the wreath product S3 ≀ S2. The four-group part acts on the notes from 1 to 4, and is generated by the involutions (1,4) and (2,3), and the 6T13 group, of order 72, acts on notes 5 through 10--or 5 through 9 and 0, if you prefer. It is generated by (5,10)(6,8)(7,9) together with (5,6), (6,7) and (8,9).
@@ Line 305: / Line 305: @@
   &lt;a class="wiki_link" href="/Zeus7tri"&gt;Zeus7tri&lt;/a&gt; is the tempering in zeus of 11/10, 5/4, 11/8, 3/2, 12/7, 15/8, 2. In 99et it has steps 0, 13, 32, 45, 58, 77, 90, 99, with an 11-limit consonance set {13, 13, 15, 17, 19, 22, 26, 28, 32, 35, 36, 41, 45, 48, 51, 54, 58, 63, 64, 67, 71, 73, 77, 80, 82, 84, 86, 86}. The scale contains two maximal hexads, consisting of the pentad of the notes 1 through 5 plus either 0 or 6. The automorphism group is of order 240, consisting of the direct product of the group of order two exchanging 0 and 6, and the symmetric group on the five other notes 1, 2, 3, 4, 5. It has vertex, edge and algebraic connectivities all 5.&lt;br /&gt;
 &lt;br /&gt;
-The scale is interesting in that it has the trivalent property, so that there are exactly three specific intervals for each generic interval save the octave multiples. Class 2 contains the three thirds, 7/6, 6/5 and 5/4.&lt;br /&gt;
+The scale is interesting in that it has the trivalent property, so that there are exactly three specific intervals for each generic interval save the octave multiples. It is a consonant class scale, with class 2 containing the three thirds, 7/6, 6/5 and 5/4.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextLocalImageRule:95:&amp;lt;img src=&amp;quot;/file/view/graph%20of%20zeus7tri.png/402208970/445x629/graph%20of%20zeus7tri.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 629px; width: 445px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/graph%20of%20zeus7tri.png/402208970/445x629/graph%20of%20zeus7tri.png" alt="graph of zeus7tri.png" title="graph of zeus7tri.png" style="height: 629px; width: 445px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:95 --&gt;&lt;br /&gt;
@@ Line 362: / Line 362: @@
 &lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc19"&gt;&lt;a name="Ten note scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Ten note scales&lt;/h1&gt;
   &lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Ten note scales-Magic[10]"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;Magic[10]&lt;/h2&gt;
-  Maic[10], the 10-note MOS of &lt;a class="wiki_link" href="/Magic%20family#Magic-11-limit"&gt;magic temperament&lt;/a&gt;, can in &lt;a class="wiki_link" href="/104edo"&gt;104et&lt;/a&gt; be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2. It has 27 maximal cliques, 18 triads and 9 tetrads.&lt;br /&gt;
+  Magic[10], the 10-note MOS of &lt;a class="wiki_link" href="/Magic%20family#Magic-11-limit"&gt;magic temperament&lt;/a&gt;, can in &lt;a class="wiki_link" href="/104edo"&gt;104et&lt;/a&gt; be expressed as the scale 0, 23, 28, 33, 38, 61, 66, 71, 94, 99, 104. If we use the 11-limit diamond, {13, 15, 15, 18, 20, 23, 28, 30, 33, 36, 38, 43, 48, 51, 53, 56, 61, 66, 68, 71, 74, 76, 81, 84, 86, 89, 89, 91}, for consonances we get a graph with ten verticies and 34 edges, with algebraic, vertex, and edge connectivity all 6. Its radius and diameter are both 2. It has 27 maximal cliques, 18 triads and 9 tetrads.&lt;br /&gt;
 &lt;br /&gt;
 Abstractly, the rather large group of automorphisms of order 288 is the direct product of the Klein four-group and the transitive group 6T13 of degree 6, which is the wreath product S3 ≀ S2. The four-group part acts on the notes from 1 to 4, and is generated by the involutions (1,4) and (2,3), and the 6T13 group, of order 72, acts on notes 5 through 10--or 5 through 9 and 0, if you prefer. It is generated by (5,10)(6,8)(7,9) together with (5,6), (6,7) and (8,9).&lt;br /&gt;