Graph-theoretic properties of scales: Difference between revisions
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<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>2014- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-18 13:44:02 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>509653300</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> | ||
<h4>Original Wikitext content:</h4> | <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">[[toc]] | <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">[[toc]] | ||
[[image:mathhazard.jpg align="center"]] | |||
=Graph of a scale= | =Graph of a scale= | ||
Given a [[periodic scale]], meaning a scale whose steps repeat, and assuming some multiple of the period is an interval of equivalence (usually this means the octave, ie interval of 2, which from now on we will assume is the interval of equivalence) then we may reduce the scale to a finite set S of pitch classes. This relates to the usual way of defining a scale, as used for instance by [[Scala]]. If we say 1-9/8-5/4-4/3-3/2-5/3-15/8-2 is a scale, we mean that each step of it represents a class of octave-equivalent pitches, so that "5/4" represents {...5/8, 5/4, 5/2, 5, 10 ...} and both "1" and "2" mean {...1/4, 1/2, 1, 2, 4...}. Suppose we have a finite set of pitches C strictly within the octave, so that s∊C entails 1 < s < 2, and suppose if s∊C then also 2/s∊C. The elements of C represent consonant pitch classes exclusive of the unison-octave class. | Given a [[periodic scale]], meaning a scale whose steps repeat, and assuming some multiple of the period is an interval of equivalence (usually this means the octave, ie interval of 2, which from now on we will assume is the interval of equivalence) then we may reduce the scale to a finite set S of pitch classes. This relates to the usual way of defining a scale, as used for instance by [[Scala]]. If we say 1-9/8-5/4-4/3-3/2-5/3-15/8-2 is a scale, we mean that each step of it represents a class of octave-equivalent pitches, so that "5/4" represents {...5/8, 5/4, 5/2, 5, 10 ...} and both "1" and "2" mean {...1/4, 1/2, 1, 2, 4...}. Suppose we have a finite set of pitches C strictly within the octave, so that s∊C entails 1 < s < 2, and suppose if s∊C then also 2/s∊C. The elements of C represent consonant pitch classes exclusive of the unison-octave class. | ||
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<!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 2em;"><a href="#Thirteen note scales-Orwell[13]">Orwell[13]</a></div> | <!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 2em;"><a href="#Thirteen note scales-Orwell[13]">Orwell[13]</a></div> | ||
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<!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Graph of a scale"></a><!-- ws:end:WikiTextHeadingRule:0 -->Graph of a scale</h1> | <!-- ws:end:WikiTextTocRule:100 --><br /> | ||
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Given a <a class="wiki_link" href="/periodic%20scale">periodic scale</a>, meaning a scale whose steps repeat, and assuming some multiple of the period is an interval of equivalence (usually this means the octave, ie interval of 2, which from now on we will assume is the interval of equivalence) then we may reduce the scale to a finite set S of pitch classes. This relates to the usual way of defining a scale, as used for instance by <a class="wiki_link" href="/Scala">Scala</a>. If we say 1-9/8-5/4-4/3-3/2-5/3-15/8-2 is a scale, we mean that each step of it represents a class of octave-equivalent pitches, so that &quot;5/4&quot; represents {...5/8, 5/4, 5/2, 5, 10 ...} and both &quot;1&quot; and &quot;2&quot; mean {...1/4, 1/2, 1, 2, 4...}. Suppose we have a finite set of pitches C strictly within the octave, so that s∊C entails 1 &lt; s &lt; 2, and suppose if s∊C then also 2/s∊C. The elements of C represent consonant pitch classes exclusive of the unison-octave class.<br /> | Given a <a class="wiki_link" href="/periodic%20scale">periodic scale</a>, meaning a scale whose steps repeat, and assuming some multiple of the period is an interval of equivalence (usually this means the octave, ie interval of 2, which from now on we will assume is the interval of equivalence) then we may reduce the scale to a finite set S of pitch classes. This relates to the usual way of defining a scale, as used for instance by <a class="wiki_link" href="/Scala">Scala</a>. If we say 1-9/8-5/4-4/3-3/2-5/3-15/8-2 is a scale, we mean that each step of it represents a class of octave-equivalent pitches, so that &quot;5/4&quot; represents {...5/8, 5/4, 5/2, 5, 10 ...} and both &quot;1&quot; and &quot;2&quot; mean {...1/4, 1/2, 1, 2, 4...}. Suppose we have a finite set of pitches C strictly within the octave, so that s∊C entails 1 &lt; s &lt; 2, and suppose if s∊C then also 2/s∊C. The elements of C represent consonant pitch classes exclusive of the unison-octave class.<br /> | ||
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A <a class="wiki_link" href="/dyadic%20chord">dyadic chord</a> pentad is of genus 1, and any scale containing dyadic pentads is therefore not planar. The embedding of a pentad's graph on a torus is illustrated below:<br /> | A <a class="wiki_link" href="/dyadic%20chord">dyadic chord</a> pentad is of genus 1, and any scale containing dyadic pentads is therefore not planar. The embedding of a pentad's graph on a torus is illustrated below:<br /> | ||
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The edges leading from the four outer vertices wrap around to the opposite side, creating the torus embedding. On the other hand, a tetrad is of genus 0, since it can be drawn on a sphere as the verticies of a tetrahedron.<br /> | The edges leading from the four outer vertices wrap around to the opposite side, creating the torus embedding. On the other hand, a tetrad is of genus 0, since it can be drawn on a sphere as the verticies of a tetrahedron.<br /> | ||
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The <a class="wiki_link" href="/zarlino">Zarlino scale</a>, or &quot;just diatonic&quot; as it's often called, is the scale 1-9/8-5/4-4/3-3/2-5/3-15/8-2, with three major and two minor triads. It has a characteristic polynomial x(x+1)(x^2-3)(x^3-x^2-7*x-3) = x^7 - 11x^5 - 10x^4 + 21x^3 + 30x^2 + 9x. From the coefficients of this we can read off that it has 11 dyads and 5 triads, and from the roots, we find that its automorphism group is an elementary 2-group--in fact, it is of order 2, with the nontrivial automorphism being inversion. The three connectivities, algebraic, vertex, and edge, are 0.914 ≤ 2 ≤ 2. The scales radius is 2 and its diameter is 3. Its genus, of course, is 0, but less obviously its maximal genus is 2.<br /> | The <a class="wiki_link" href="/zarlino">Zarlino scale</a>, or &quot;just diatonic&quot; as it's often called, is the scale 1-9/8-5/4-4/3-3/2-5/3-15/8-2, with three major and two minor triads. It has a characteristic polynomial x(x+1)(x^2-3)(x^3-x^2-7*x-3) = x^7 - 11x^5 - 10x^4 + 21x^3 + 30x^2 + 9x. From the coefficients of this we can read off that it has 11 dyads and 5 triads, and from the roots, we find that its automorphism group is an elementary 2-group--in fact, it is of order 2, with the nontrivial automorphism being inversion. The three connectivities, algebraic, vertex, and edge, are 0.914 ≤ 2 ≤ 2. The scales radius is 2 and its diameter is 3. Its genus, of course, is 0, but less obviously its maximal genus is 2.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Seven note scales-The diatonic scale (Meantone[7])"></a><!-- ws:end:WikiTextHeadingRule:18 -->The diatonic scale (Meantone[7])</h2> | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Seven note scales-The diatonic scale (Meantone[7])"></a><!-- ws:end:WikiTextHeadingRule:18 -->The diatonic scale (Meantone[7])</h2> | ||
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The genus of the 7-limit diatonic scale is 1, with maximal genus of 4. The connectivities go 3.198 ≤ 4 ≤ 4, and the radius and diameter are both 2.<br /> | The genus of the 7-limit diatonic scale is 1, with maximal genus of 4. The connectivities go 3.198 ≤ 4 ≤ 4, and the radius and diameter are both 2.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Seven note scales-Gypsy"></a><!-- ws:end:WikiTextHeadingRule:20 -->Gypsy</h2> | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Seven note scales-Gypsy"></a><!-- ws:end:WikiTextHeadingRule:20 -->Gypsy</h2> | ||
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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 /> | 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 /> | ||
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<em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3" rel="nofollow">Benny</a></em><br /> | <em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3" rel="nofollow">Benny</a></em><br /> | ||
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Elfjove7 has an edge and vertex connectivity of 3, a radius of 1 with a diameter of 2, and is of genus 1.<br /> | Elfjove7 has an edge and vertex connectivity of 3, a radius of 1 with a diameter of 2, and is of genus 1.<br /> | ||
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<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/jupiter's%20rations.mp3" rel="nofollow">Jupiter's Rations</a> by Andrew Heathwaite<br /> | <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/jupiter's%20rations.mp3" rel="nofollow">Jupiter's Rations</a> by Andrew Heathwaite<br /> | ||
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The first note of any of the 16 tetrads of Star can be either 0 or 1, the second 2 or 3, the third 4 or 5, and the fourth 6 or 7. Any of the resulting 16 possible combinations will not have the forbidden intervals (0, 1), (2, 3), (4, 5), or (6, 7) which are too small to give 11-limit consonances.<br /> | The first note of any of the 16 tetrads of Star can be either 0 or 1, the second 2 or 3, the third 4 or 5, and the fourth 6 or 7. Any of the resulting 16 possible combinations will not have the forbidden intervals (0, 1), (2, 3), (4, 5), or (6, 7) which are too small to give 11-limit consonances.<br /> | ||
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<a class="wiki_link_ext" href="http://micro.soonlabel.com/star/20120830-77et-star.mp3" rel="nofollow">77et Star</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br /> | <a class="wiki_link_ext" href="http://micro.soonlabel.com/star/20120830-77et-star.mp3" rel="nofollow">77et Star</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br /> | ||
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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 /> | 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 /> | ||
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<a class="wiki_link_ext" href="http://archive.org/download/HighOktoneElgar/oktelg.mp3" rel="nofollow">High Oktone Elgar</a><br /> | <a class="wiki_link_ext" href="http://archive.org/download/HighOktoneElgar/oktelg.mp3" rel="nofollow">High Oktone Elgar</a><br /> | ||
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Zeus8tri has three maximal cliques, the hexads [0, 1, 2, 4, 5, 6], [1, 2, 3, 4, 5, 6], [1, 2, 3, 5, 6, 7], each of which consists of the four notes in the center plus {0, 4}, {3, 4}, or {3, 7}. Instead of having a single circle of thirds like zeus7tri, it has two, both of which are the 15-limit valinorsmic tetrads with steps 6/5, 11/9, 6/5, 8/7 which we have not counted among the (11-limit) chords of zeus8tri. Adding 126/125 to the commas of zeus leads to valentine temperament, which does not damage the tuning accuracy of zeus very much. In valentine, zeus8tri becomes a scale of Graham complexity 13, with generator steps -7, -5, -3, -1, 0, 2, 4, 6.<br /> | Zeus8tri has three maximal cliques, the hexads [0, 1, 2, 4, 5, 6], [1, 2, 3, 4, 5, 6], [1, 2, 3, 5, 6, 7], each of which consists of the four notes in the center plus {0, 4}, {3, 4}, or {3, 7}. Instead of having a single circle of thirds like zeus7tri, it has two, both of which are the 15-limit valinorsmic tetrads with steps 6/5, 11/9, 6/5, 8/7 which we have not counted among the (11-limit) chords of zeus8tri. Adding 126/125 to the commas of zeus leads to valentine temperament, which does not damage the tuning accuracy of zeus very much. In valentine, zeus8tri becomes a scale of Graham complexity 13, with generator steps -7, -5, -3, -1, 0, 2, 4, 6.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="Nine note scales"></a><!-- ws:end:WikiTextHeadingRule:38 -->Nine note scales</h1> | <!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="Nine note scales"></a><!-- ws:end:WikiTextHeadingRule:38 -->Nine note scales</h1> | ||
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The graph has 16 maximal cliques, eight tetrads and eight pentads. All of the tetrads contain note 0, and all of the pentads notes 1 and 8. All three connectivites equal 6, the radius and diameter are both 2, and the graph complement is disconnected.<br /> | The graph has 16 maximal cliques, eight tetrads and eight pentads. All of the tetrads contain note 0, and all of the pentads notes 1 and 8. All three connectivites equal 6, the radius and diameter are both 2, and the graph complement is disconnected.<br /> | ||
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<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Grood/Mountain_Village_TarkanGrood.mp3" rel="nofollow">Mountain Village</a> by <a class="wiki_link" href="/Tarkan%20Grood">Tarkan Grood</a><br /> | <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Grood/Mountain_Village_TarkanGrood.mp3" rel="nofollow">Mountain Village</a> by <a class="wiki_link" href="/Tarkan%20Grood">Tarkan Grood</a><br /> | ||
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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 /> | 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 /> | ||
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<!-- ws:start:WikiTextHeadingRule:48:&lt;h2&gt; --><h2 id="toc24"><a name="Ten note scales-The dekany"></a><!-- ws:end:WikiTextHeadingRule:48 -->The dekany</h2> | <!-- ws:start:WikiTextHeadingRule:48:&lt;h2&gt; --><h2 id="toc24"><a name="Ten note scales-The dekany"></a><!-- ws:end:WikiTextHeadingRule:48 -->The dekany</h2> | ||
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Though it has only ten notes, an attempt to compute the genus of the dekany using SAGE caused it to wander off into the weeds and never return, or at least not when it was allowed to run overnight. An inquiry of someone who has published on the Johnson graphs revealed he had no idea what the genus of J(5,2) was, and it may very well not be known. However, the inequalities above show the genus must be at least 1.<br /> | Though it has only ten notes, an attempt to compute the genus of the dekany using SAGE caused it to wander off into the weeds and never return, or at least not when it was allowed to run overnight. An inquiry of someone who has published on the Johnson graphs revealed he had no idea what the genus of J(5,2) was, and it may very well not be known. However, the inequalities above show the genus must be at least 1.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:50:&lt;h1&gt; --><h1 id="toc25"><a name="Eleven note scales"></a><!-- ws:end:WikiTextHeadingRule:50 -->Eleven note scales</h1> | <!-- ws:start:WikiTextHeadingRule:50:&lt;h1&gt; --><h1 id="toc25"><a name="Eleven note scales"></a><!-- ws:end:WikiTextHeadingRule:50 -->Eleven note scales</h1> | ||
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The automorphism group of order 24 is the direct product of an involution and the group of the hexagon, which act on disjoint notes of the scale. The involution is (0,1)(5,6) and the hexagon group (dihedral group of order 12) permutes the cycle (2,7,4,9,3,8); this cycle together with the two involutions (2,3),(7,9) and (3,4),(7,8) generate the hexagon group.<br /> | The automorphism group of order 24 is the direct product of an involution and the group of the hexagon, which act on disjoint notes of the scale. The involution is (0,1)(5,6) and the hexagon group (dihedral group of order 12) permutes the cycle (2,7,4,9,3,8); this cycle together with the two involutions (2,3),(7,9) and (3,4),(7,8) generate the hexagon group.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:54:&lt;h1&gt; --><h1 id="toc27"><a name="Twelve note scales"></a><!-- ws:end:WikiTextHeadingRule:54 -->Twelve note scales</h1> | <!-- ws:start:WikiTextHeadingRule:54:&lt;h1&gt; --><h1 id="toc27"><a name="Twelve note scales"></a><!-- ws:end:WikiTextHeadingRule:54 -->Twelve note scales</h1> | ||