Graph-theoretic properties of scales: Difference between revisions

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=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 &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.

=Examples=  

The Zarlino scale, or "just diatonic" 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.

[[image:zarlino.png]]

The diatonic scale in 31edo consists of the notes 0, 5, 10, 13, 18, 23, 28, 31. In the 7-limit, it has the consonance set {6, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 25}, leading to a graph with the high degree of symmetry afforded by the dihedral group of order 14. This graph is in fact isomorphic to the graph of (5 or 7 limit) 7edo. The maximal cliques of the graph are the seven dyadic chord triads, one on each degree of the scale; three major, three minor, and one diminished. Because of the graph symmetry, the 7-limit diatonic scale can transpose to any key, mapping all dyadic triads to other dyadic triads.

[[image:diatonic7.png]]

[[image:star.png]]

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.

[[image:oktony.png]]

Orwell[9], the 9-note orwell MOS, has notes 0, 5, 12, 17, 24, 29, 36, 41, 48, 53 in [[53edo|53 equal]]. With the 11-limit consonance set {7, 8, 9, 10, 12, 14, 15, 17, 19, 22, 24, 26, 27, 29, 31, 34, 36, 38, 39, 41, 43, 44, 45, 46} it defines a graph of 31 edges with an automorphism group of order 96. The group consists of an [[http://mathworld.wolfram.com/OctahedralGroup.html|octahedronal group]] part which is transitive on the six notes from 2 to 7, that is {12, 17, 24, 29, 36, 41} of 53edo, generated by (2,3), (4,5),(6,7), (2,4)(3,5) and (4,6)(5,7), and an involution exchanging notes 1 and 8, that is, 5 and 48 of 53edo. The center of the MOS, note 0 in this numbering, stays where it is, and has degree six whereas all the other notes are degree seven.

[[image:orwell9.png]]

The standard 2)5 dekany is a [[Combination product sets|combination product set]], Cps([2,3,5,7,11], 2). It consists of ten notes associated to two-element subset of the set of the first five primes, {2,3,5,7,11}, and in one mode is 1-12/11-5/4-14/11-15/11-3/2-35/22-7/4-20/11-21/11, which we will take as its notes from note 0 to note 9. It has 30 edges, with connectivities 5 ≤ 6 ≤ 6, and the largest element of the Laplace spectrum is 8, so that the complementary graph is also connected. Its radius and diameter are both 2. The graph is known as the [[http://en.wikipedia.org/wiki/Johnson_graph|Johnson graph]] J(5,2). The 3)5 dekany, which is the inverse of the standard dekany, has the Johnson graph J(5,3) as its graph, which is graph-isomorphic to J(5,2).

[[image:dekany.png]]

The Ellis duodene is the 12-note 5-limit scale 1-16/15-9/8-6/5-5/4-4/3-45/32-3/2-8/5-5/3-9/5-15/8-2, and marvel tempering it leads to [[The Marveldene|marveldene]]. An excellent tuning for marvel is [[166edo|166et]], and in that the scale becomes 0, 16, 28, 44, 53, 69, 81, 97, 113, 122, 141, 150, 166. If we use the 15-limit consonance set, {16, 18, 19, 21, 23, 25, 28, 32, 34, 37, 40, 44, 48, 50, 53, 58, 60, 63, 69, 74, 76, 78, 81, 85, 88, 90, 92, 97, 103, 106, 108, 113, 116, 118, 122, 126, 129, 132, 134, 138, 141, 143, 145, 147, 148, 150}, we obtain a graph with 55 edges and an automorphism group of order 32. Abstractly, the automorphism group is the direct product of the group of the square with the Klein 4-group; as a permutation group it is the square (dihedral group D8 of order 8) part, generated by {(1,2)(5,6)(8,11)(9,10), (1,5), (2,6)}, and the two involutions flipping two notes, (3,4) flipping Eb and E, and (7,12) flipping C and G.

The connectivites of the marveldene go 6.222 ≤ 7 ≤ 8, and it has a radius of 1 and a diameter of 2, with center {0, 7}, ie C and G. The largest element of the Laplace spectrum is 12, so the complementary graph is not connected. It has eight maximum cliques, the septads [0, 1, 3, 5, 7, 8, 11], [0, 1, 3, 5, 7, 9, 11], [0, 1, 4, 5, 7, 8, 11], [0, 1, 4, 5, 7, 9, 11], [0, 2, 3, 6, 7, 8, 10], [0, 2, 3, 6, 7, 8, 11], [0, 2, 4, 6, 7, 8, 10] and [0, 2, 4, 6, 7, 8, 11], and two maximal pentads, [0, 3, 7, 9, 10] and [0, 4, 7, 9, 10].

Magic[13] is the 13-note MOS of [[Magic family#Magic-11-limit|magic temperament]], which in [[104edo|104et]] has notes 0, 5, 10, 28, 33, 38, 43, 61, 66, 71, 76, 94, 99, 104. Using 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} as the consonance set, we obtain a graph on 13 notes with 61 edges representing dyads. It has algebraic, vertex and edge connectivities all 8, and a disconnected graph complement. The radius and diameter are both 2.

Orwell[13], the 13-note MOS of orwell, has four more notes but the same set of 11-limit consonances as Orwell[9]. Now we have 0, 5, 7, 12, 17, 19, 24, 29, 34, 36, 41, 46, 48, 53, and while the graph of the scale still has a lot of symmetry, it is of an entirely different kind. This time the symmetry group is analogous to that of the diatonic scale, being the dihedral group D26 on 13 points; as with the 7-limit diatonic scale, the symmetries of Orwell[13] appear as inversion and the ability to transpose to any key while retaining the dyadic character of all dyadic chords, which however are much more numerous in kind and number. In particular, Orwell[13] has 26 maximal pentads and 13 maximal hexads on which the automorphism group faithfully acts.

The graph of Orwell[13] is 10-regular, has 65 edges, with connectivities 9.058 ≤ 10 ≤ 10, and radius and diameter both 2.</pre></div>

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  Given a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/Scala"&gt;Scala&lt;/a&gt;. 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 &amp;quot;5/4&amp;quot; represents {...5/8, 5/4, 5/2, 5, 10 ...} and both &amp;quot;1&amp;quot; and &amp;quot;2&amp;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 &amp;lt; s &amp;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.&lt;br /&gt;

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A &lt;a class="wiki_link" href="/dyadic%20chord"&gt;dyadic chord&lt;/a&gt; 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:&lt;br /&gt;

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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.&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Examples&lt;/h1&gt;

  &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Examples-The Zarlino scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;The Zarlino scale&lt;/h2&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Examples-The diatonic scale (Meantone[7])"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;The diatonic scale (Meantone[7])&lt;/h2&gt;

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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.&lt;br /&gt;

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&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;

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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.&lt;br /&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Examples-Orwell[9]"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Orwell[9]&lt;/h2&gt;

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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.&lt;br /&gt;

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The automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5.&lt;br /&gt;

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.&lt;br /&gt;

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The connectivites of the marveldene go 6.222 ≤ 7 ≤ 8, and it has a radius of 1 and a diameter of 2, with center {0, 7}, ie C and G. The largest element of the Laplace spectrum is 12, so the complementary graph is not connected. It has eight maximum cliques, the septads [0, 1, 3, 5, 7, 8, 11], [0, 1, 3, 5, 7, 9, 11], [0, 1, 4, 5, 7, 8, 11], [0, 1, 4, 5, 7, 9, 11], [0, 2, 3, 6, 7, 8, 10], [0, 2, 3, 6, 7, 8, 11], [0, 2, 4, 6, 7, 8, 10] and [0, 2, 4, 6, 7, 8, 11], and two maximal pentads, [0, 3, 7, 9, 10] and [0, 4, 7, 9, 10].&lt;br /&gt;

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The graph of Orwell[13] is 10-regular, has 65 edges, with connectivities 9.058 ≤ 10 ≤ 10, and radius and diameter both 2.&lt;/body&gt;&lt;/html&gt;</pre></div>