Graph-theoretic properties of scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 394683768 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 394685102 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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 19: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-12-26 19:46:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>394685102</tt>.<br> | ||
: The revision comment was: <tt></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> | 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: | ||
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. | 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. | ||
The 9-limit graph has eight maximal cliques, four triads, each containing 0, and four tetrads, each containing both 1 and 6. The triads consist of 0 plus any of the dyads {2, 4}, {2, 5}, {3, 4} or {3, 5}, and the tetrads consist of {1, 6} plus any of the same set of dyads. | |||
==Star== | ==Star== | ||
| Line 232: | Line 234: | ||
<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 /> | 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 /> | ||
<br /> | |||
The 9-limit graph has eight maximal cliques, four triads, each containing 0, and four tetrads, each containing both 1 and 6. The triads consist of 0 plus any of the dyads {2, 4}, {2, 5}, {3, 4} or {3, 5}, and the tetrads consist of {1, 6} plus any of the same set of dyads.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Examples-Star"></a><!-- ws:end:WikiTextHeadingRule:20 -->Star</h2> | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Examples-Star"></a><!-- ws:end:WikiTextHeadingRule:20 -->Star</h2> | ||