Graph-theoretic properties of scales: Difference between revisions

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However arrived at, the scale in [[197edo|197et]] is 0, 19, 63, 82, 115, 134, 178, 197. It has two graphs of interest, since the graphs of 7-limit relations and of 9-limit relations are not isomorphic, but the automorphism groups (of order 16) of these graphs are. The 7-limit consonance set is {38, 44, 52, 63, 82, 96, 101, 115, 134, 145, 153, 159, 197} and the 9-limit set is {30, 33, 38, 44, 52, 63, 71, 82, 96, 101, 115, 126, 134, 145, 153, 159, 164, 167, 197}. The difference is {30, 33, 71, 126, 164, 167}. In Gypsy, the 9-limit intervals occur between 5/4 and 8/5, tempered to 9/7, and between 4/3 and 3/2.

If we call the automorphism group of the 7-limit graph G7 and that of the 9-limit group G9, then G7 is guaranteed to send 7-limit intervals to 7-limit intervals, but will not necessarily send 9-limit intervals to a consonance. G9 must send 9-limit intervals to 9-limit intervals, but may send a 7-limit interval to the 9-limit. The intersection G7∩G9 is a group of order eight which sends 7-limit intervals to 7-limit intervals, and strictly 9-limit intervals, those of [[Kees height]] 9, to strictly 9-limit intervals; it can't send such intervals to the 7-limit without the inverse sending a 7-limit interval to the 9-limit. G7∩G9 = {0123456, 0132546, 0145236, 0154326, 0623451, 0632541, 0645231, 0654321}, G7\G9 = {0125436, 0134526, 0143256, 0152346, 0625431, 0634521, 0643251, 0652341}, and G9\G7 = {0123546, 0132456, 0145326, 0154236, 0623541, 0632451, 0645321, 0654231}. G7 and G9 are intransitive; one orbit consists of the fixed center interval 0, and an involution exchanges the extreme intervals 0 and 6. The other four points are permuted by the group of the square in two different representations for G7 and G9.

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However arrived at, the scale in <a class="wiki_link" href="/197edo">197et</a> is 0, 19, 63, 82, 115, 134, 178, 197. It has two graphs of interest, since the graphs of 7-limit relations and of 9-limit relations are not isomorphic, but the automorphism groups (of order 16) of these graphs are. The 7-limit consonance set is {38, 44, 52, 63, 82, 96, 101, 115, 134, 145, 153, 159, 197} and the 9-limit set is {30, 33, 38, 44, 52, 63, 71, 82, 96, 101, 115, 126, 134, 145, 153, 159, 164, 167, 197}. The difference is {30, 33, 71, 126, 164, 167}. In Gypsy, the 9-limit intervals occur between 5/4 and 8/5, tempered to 9/7, and between 4/3 and 3/2.<br />

<br />

If we call the automorphism group of the 7-limit graph G7 and that of the 9-limit group G9, then G7 is guaranteed to send 7-limit intervals to 7-limit intervals, but will not necessarily send 9-limit intervals to a consonance. G9 must send 9-limit intervals to 9-limit intervals, but may send a 7-limit interval to the 9-limit. The intersection G7∩G9 is a group of order eight which sends 7-limit intervals to 7-limit intervals, and strictly 9-limit intervals, those of <a class="wiki_link" href="/Kees%20height">Kees height</a> 9, to strictly 9-limit intervals; it can't send such intervals to the 7-limit without the inverse sending a 7-limit interval to the 9-limit. G7∩G9 = {0123456, 0132546, 0145236, 0154326, 0623451, 0632541, 0645231, 0654321}, G7\G9 = {0125436, 0134526, 0143256, 0152346, 0625431, 0634521, 0643251, 0652341}, and G9\G7 = {0123546, 0132456, 0145326, 0154236, 0623541, 0632451, 0645321, 0654231}. G7 and G9 are intransitive; one orbit consists of the fixed center interval 0, and an involution exchanges the extreme intervals 0 and 6. The other four points are permuted by the group of the square in two different representations for G7 and G9.<br />

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

@@ 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>2012-12-26 12:19:42 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-12-26 19:25:18 UTC</tt>.<br>
-: The original revision id was <tt>394652244</tt>.<br>
+: The original revision id was <tt>394683768</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 72: / Line 72: @@
 However arrived at, the scale in [[197edo|197et]] is 0, 19, 63, 82, 115, 134, 178, 197. It has two graphs of interest, since the graphs of 7-limit relations and of 9-limit relations are not isomorphic, but the automorphism groups (of order 16) of these graphs are. The 7-limit consonance set is {38, 44, 52, 63, 82, 96, 101, 115, 134, 145, 153, 159, 197} and the 9-limit set is {30, 33, 38, 44, 52, 63, 71, 82, 96, 101, 115, 126, 134, 145, 153, 159, 164, 167, 197}. The difference is {30, 33, 71, 126, 164, 167}. In Gypsy, the 9-limit intervals occur between 5/4 and 8/5, tempered to 9/7, and between 4/3 and 3/2.
+If we call the automorphism group of the 7-limit graph G7 and that of the 9-limit group G9, then G7 is guaranteed to send 7-limit intervals to 7-limit intervals, but will not necessarily send 9-limit intervals to a consonance. G9 must send 9-limit intervals to 9-limit intervals, but may send a 7-limit interval to the 9-limit. The intersection G7∩G9 is a group of order eight which sends 7-limit intervals to 7-limit intervals, and strictly 9-limit intervals, those of [[Kees height]] 9, to strictly 9-limit intervals; it can't send such intervals to the 7-limit without the inverse sending a 7-limit interval to the 9-limit. G7∩G9 = {0123456, 0132546, 0145236, 0154326, 0623451, 0632541, 0645231, 0654321}, G7\G9 = {0125436, 0134526, 0143256, 0152346, 0625431, 0634521, 0643251, 0652341}, and G9\G7 = {0123546, 0132456, 0145326, 0154236, 0623541, 0632451, 0645321, 0654231}. G7 and G9 are intransitive; one orbit consists of the fixed center interval 0, and an involution exchanges the extreme intervals 0 and 6. The other four points are permuted by the group of the square in two different representations for G7 and G9.
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 However arrived at, the scale in &lt;a class="wiki_link" href="/197edo"&gt;197et&lt;/a&gt; is 0, 19, 63, 82, 115, 134, 178, 197. It has two graphs of interest, since the graphs of 7-limit relations and of 9-limit relations are not isomorphic, but the automorphism groups (of order 16) of these graphs are. The 7-limit consonance set is {38, 44, 52, 63, 82, 96, 101, 115, 134, 145, 153, 159, 197} and the 9-limit set is {30, 33, 38, 44, 52, 63, 71, 82, 96, 101, 115, 126, 134, 145, 153, 159, 164, 167, 197}. The difference is {30, 33, 71, 126, 164, 167}. In Gypsy, the 9-limit intervals occur between 5/4 and 8/5, tempered to 9/7, and between 4/3 and 3/2.&lt;br /&gt;
+&lt;br /&gt;
+If we call the automorphism group of the 7-limit graph G7 and that of the 9-limit group G9, then G7 is guaranteed to send 7-limit intervals to 7-limit intervals, but will not necessarily send 9-limit intervals to a consonance. G9 must send 9-limit intervals to 9-limit intervals, but may send a 7-limit interval to the 9-limit. The intersection G7∩G9 is a group of order eight which sends 7-limit intervals to 7-limit intervals, and strictly 9-limit intervals, those of &lt;a class="wiki_link" href="/Kees%20height"&gt;Kees height&lt;/a&gt; 9, to strictly 9-limit intervals; it can't send such intervals to the 7-limit without the inverse sending a 7-limit interval to the 9-limit. G7∩G9 = {0123456, 0132546, 0145236, 0154326, 0623451, 0632541, 0645231, 0654321}, G7\G9 = {0125436, 0134526, 0143256, 0152346, 0625431, 0634521, 0643251, 0652341}, and G9\G7 = {0123546, 0132456, 0145326, 0154236, 0623541, 0632451, 0645321, 0654231}. G7 and G9 are intransitive; one orbit consists of the fixed center interval 0, and an involution exchanges the extreme intervals 0 and 6. The other four points are permuted by the group of the square in two different representations for G7 and G9.&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-Star"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Star&lt;/h2&gt;