Graph-theoretic properties of scales: Difference between revisions

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[[image:diatonic7.png]]

==Star==  

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;

  The Zarlino scale, or &amp;quot;just diatonic&amp;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.&lt;br /&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;

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;a class="wiki_link" href="/Star%20and%20Nova"&gt;Star&lt;/a&gt; is a &lt;a class="wiki_link" href="/star"&gt;scale&lt;/a&gt; of &lt;a class="wiki_link" href="/Starling%20temperaments#Valentine%20temperament-11-limit"&gt;11-limit valentine temperament&lt;/a&gt;, which in &lt;a class="wiki_link" href="/77edo"&gt;77et&lt;/a&gt; is 0, 5, 20, 25, 40, 45, 57, 65, 77, with the 11-limit consonance set {10, 12, 13, 15, 17, 20, 22, 25, 27, 28, 32, 35, 37, 40, 42, 45, 49, 50, 52, 55, 57, 60, 62, 64, 65, 67}. Its eight vertices are connected via 24 edges; in four dimensions this can be identified as the eight verticies and 24 edges of the&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/16-cell" rel="nofollow"&gt;16-cell&lt;/a&gt;, or hexadecachoron, a four-dimensional regular polytope. The scale exhibits a very high degree of symmetry, with an automorphism group of order 384, the 8T44 transitive group S2≀S4. Star has a 9-limit twin scale, &lt;a class="wiki_link" href="/Star%20and%20Nova"&gt;Nova&lt;/a&gt;, with an isomorphic graph, which means all the statements about Star in this section, except for the fact that it is 11-limit, are equally true of Nova.&lt;br /&gt;

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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.&lt;br /&gt;

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  By &lt;a class="wiki_link" href="/Oktone"&gt;Oktone&lt;/a&gt; is meant the tempering of the octone scale, 1-15/14-60/49-5/4-10/7-3/2-12/7-7/4-2, in &lt;a class="wiki_link" href="/Breed%20family#Jove,%20aka%20Wonder"&gt;jove temperament&lt;/a&gt;. The tempering can be accomplished via &lt;a class="wiki_link" href="/202edo"&gt;202et&lt;/a&gt;, leading to the scale 0, 20, 59, 65, 104, 118, 157, 163, 202 which we can use together with the 11-limit diamond as a consonance set, {25, 28, 31, 34, 39, 45, 53, 59, 65, 70, 73, 84, 93, 98, 104, 109, 118, 129, 132, 137, 143, 149, 157, 163, 168, 171, 174, 177}. This leads to a highly accurate scale with a good deal of symmetry. The automorphism group of the graph is the direct product of the group of the square with an involution, ie an element of order two. The involution simultaneously exchanges notes 2 and 7, and 3 and 6; that is, it is in cycle terms (2, 7)(3,6), or (59, 163)(65, 157) in terms of 202et, and the square part permutes the cycle (0, 4, 1, 5) by the group of the square.&lt;br /&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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  Orwell[9], the 9-note orwell MOS, has notes 0, 5, 12, 17, 24, 29, 36, 41, 48, 53 in &lt;a class="wiki_link" href="/53edo"&gt;53 equal&lt;/a&gt;. 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 &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/OctahedralGroup.html" rel="nofollow"&gt;octahedronal group&lt;/a&gt; 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.&lt;br /&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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  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;

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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).&lt;br /&gt;

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  The standard 2)5 dekany is a &lt;a class="wiki_link" href="/Combination%20product%20sets"&gt;combination product set&lt;/a&gt;, 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Johnson_graph" rel="nofollow"&gt;Johnson graph&lt;/a&gt; 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).&lt;br /&gt;

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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.&lt;br /&gt;

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  Myna[11], the 11-note MOS of &lt;a class="wiki_link" href="/Starling%20temperaments#x5-limit%20(Mynic)-11-limit"&gt;myna temperament&lt;/a&gt;, can be specified in the &lt;a class="wiki_link" href="/89edo"&gt;89et&lt;/a&gt; tuning as the notes 0, 3, 20, 23, 26, 43, 46, 63, 66, 69, 86, 89. Using the 11-limit diamond as a consonance set gives a graph with 40 edges corresponding to 40 dyads. Its connectivities are 6 ≤ 7 ≤ 7, It has eight vertices of degree 7, and three of degree 8, with a radius and diameter both 2.&lt;br /&gt;

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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.&lt;br /&gt;

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  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 &lt;a class="wiki_link" href="/The%20Marveldene"&gt;Marveldene&lt;/a&gt;. An excellent tuning for marvel is &lt;a class="wiki_link" href="/166edo"&gt;166et&lt;/a&gt;, 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 four-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.&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 5-limit Fokker block 1-27/25-10/9-6/5-5/4-4/3-36/25-3/2-8/5-5/3-9/5-48/25-2, which has been called the &lt;a class="wiki_link" href="/duodene_skew"&gt;skewed duodene&lt;/a&gt; or magsyn3, serves as a transversal for the &lt;a class="wiki_link" href="/novadene"&gt;Novadene&lt;/a&gt;, which in 185et is 0, 19, 29, 48, 60, 77, 96, 108, 125, 137, 156, 173, 185 with consonances the 9-limit diamond, 29, 31, 36, 41, 48, 60, 67, 77, 89, 96, 108, 118, 125, 137, 144, 149, 154, 156.&lt;br /&gt;

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It should be noted that the Novadene has a couple of 14/11 thirds which are four cents sharp, and it makes sense to count them among the consonances of Novadene; however, doing so makes the graph less symmetrical.&lt;br /&gt;

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  Magic[13] is the 13-note MOS of &lt;a class="wiki_link" href="/Magic%20family#Magic-11-limit"&gt;magic temperament&lt;/a&gt;, which in &lt;a class="wiki_link" href="/104edo"&gt;104et&lt;/a&gt; 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.&lt;br /&gt;

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It has a disparate collection of 36 maximal cliques ranging from triads to hexads, and an automorphism group which abstractly is the direct product of two square groups. However, as a permutation group the two cycles on which the automorphisms act as the square group, (3, 7, 6, 10) and (4, 8, 5, 9), don't appear in isolation, and the group actually moves everything but the central note of the MOS, note 0 in this labeling.&lt;br /&gt;

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  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.&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>