Graph-theoretic properties of scales: Difference between revisions
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<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]] | ||
=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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We can attribute various properties to a scale, given a choice of consonance set, by the presence of a graph-theoretic property in the graph of the scale. One key graph-theoretic property is connectivity. A graph is said to be //connected// if for any two vertices a and b, there is a path of edges between a and b. A scale is therefore connected if you can go from any one note to any other note by means of consonant intervals only. The graph G has an edge-connectivity ε if it is possible to disconnect the graph by removing ε edges, but no smaller number of edges will do. Similarly, it has vertex-connectivity ν if it is possible to disconnect the graph by removing ν vertices (notes of the scale) but no smaller number will do. The graph is //k-edge-connected// if ε ≥ k, and //k-vertex-connected// if ν ≥ k; these definitions transfer immediately to scales, and a high degree of connectivity may often be desired in a scale. | We can attribute various properties to a scale, given a choice of consonance set, by the presence of a graph-theoretic property in the graph of the scale. One key graph-theoretic property is connectivity. A graph is said to be //connected// if for any two vertices a and b, there is a path of edges between a and b. A scale is therefore connected if you can go from any one note to any other note by means of consonant intervals only. The graph G has an edge-connectivity ε if it is possible to disconnect the graph by removing ε edges, but no smaller number of edges will do. Similarly, it has vertex-connectivity ν if it is possible to disconnect the graph by removing ν vertices (notes of the scale) but no smaller number will do. The graph is //k-edge-connected// if ε ≥ k, and //k-vertex-connected// if ν ≥ k; these definitions transfer immediately to scales, and a high degree of connectivity may often be desired in a scale. | ||
A [[http://en.wikipedia.org/wiki/Clique_%28graph_theory%29|clique]] in a graph is a subgraph such that there is an edge connecting every | A [[http://en.wikipedia.org/wiki/Clique_%28graph_theory%29|clique]] in a graph is a subgraph such that there is an edge connecting every pair of vertices; that is, the subgraph is a complete graph. In music terms, a clique is a [[dyadic chord]]. Hence, counting cliques of various degrees (number of notes) is a relevant consideration when analyzing a scale. The various problems of this nature are called, collectively, the [[http://en.wikipedia.org/wiki/Clique_problem|clique problem]], and this unfortunately is computationally hard. However, for scales of the size most people would care to consider a brute-force approach suffices. | ||
Among all cliques, the [[http://mathworld.wolfram.com/MaximalClique.html|maximal cliques]], those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A //maximum clique// is a clique of largest size; these are always maximal but the converse does not always hold. Given any set of sets, we may form a graph with these as the set of vertices, by drawing an edge between any two sets with a nonempty intersection. In this way we can create graphs of all k-cliques (k note dyadic chords), maximal k-cliques, all maximal cliques, and so forth. Such graphs can illuminate harmonic relationships. | Among all cliques, the [[http://mathworld.wolfram.com/MaximalClique.html|maximal cliques]], those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A //maximum clique// is a clique of largest size; these are always maximal but the converse does not always hold. Given any set of sets, we may form a graph with these as the set of vertices, by drawing an edge between any two sets with a nonempty intersection. In this way we can create graphs of all k-cliques (k note dyadic chords), maximal k-cliques, all maximal cliques, and so forth. Such graphs can illuminate harmonic relationships. | ||
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=Examples= | =Examples= | ||
==The Zarlino scale== | ==The Zarlino scale== | ||
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. | 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]] | [[image:zarlino.png]] | ||
==The diatonic scale (Meantone[7])== | ==The diatonic scale (Meantone[7])== | ||
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. | 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. | ||
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[[image:diatonic7.png]] | [[image:diatonic7.png]] | ||
==Star== | ==Star== | ||
[[Star and Nova|Star]] is a [[star|scale]] of [[Starling temperaments#Valentine | [[Star and Nova|Star]] is a [[star|scale]] of [[Starling temperaments#Valentine%20temperament-11-limit|11-limit valentine temperament]], which in [[77edo|77et]] 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[[http://en.wikipedia.org/wiki/16-cell|16-cell]], 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, [[Star and Nova|Nova]], 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. | ||
The graph of Star is 6-regular, with every vertex connected to six others (on the 16-cell only opposite verticies fail to connect), and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected, it consists of the four edges connecting opposite verticies of the 16-cell. It has 16 maximal cliques, all of which are tetrads, corresponding to the 16 cells of the 16-cell, and 32 triads extendable to these tetrads, corresponding to the 32 triangular faces of the 16-cell. The graph of the maximal cliques is 14-regular, with each tetrad linking to all others but its relative complement in the whole scale, that is the set of notes from note 0 to note 7, the "opposite vertex". The relative complement of each dyadic tetrad is also a dyadic tetrad. The graph is known to be of genus 1, so it can be drawn on a square with opposite edges identified. | The graph of Star is 6-regular, with every vertex connected to six others (on the 16-cell only opposite verticies fail to connect), and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected, it consists of the four edges connecting opposite verticies of the 16-cell. It has 16 maximal cliques, all of which are tetrads, corresponding to the 16 cells of the 16-cell, and 32 triads extendable to these tetrads, corresponding to the 32 triangular faces of the 16-cell. The graph of the maximal cliques is 14-regular, with each tetrad linking to all others but its relative complement in the whole scale, that is the set of notes from note 0 to note 7, the "opposite vertex". The relative complement of each dyadic tetrad is also a dyadic tetrad. The graph is known to be of genus 1, so it can be drawn on a square with opposite edges identified. | ||
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[[image:http://upload.wikimedia.org/wikipedia/commons/a/a0/16-cell.gif]] | [[image:http://upload.wikimedia.org/wikipedia/commons/a/a0/16-cell.gif]] | ||
==Oktone== | ==Oktone== | ||
By [[Oktone]] 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 [[Breed family#Jove, | By [[Oktone]] 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 [[Breed family#Jove,%20aka%20Wonder|jove temperament]]. The tempering can be accomplished via [[202edo|202et]], 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. | ||
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. | 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. | ||
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[[image:oktony.png]] | [[image:oktony.png]] | ||
==Orwell[9]== | ==Orwell[9]== | ||
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. | 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. | ||
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[[image:orwell9.png]] | [[image:orwell9.png]] | ||
==Magic[10]== | ==Magic[10]== | ||
Maic[10], the 10-note MOS of [[Magic family#Magic-11-limit|magic temperament]], can in [[104edo|104et]] 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. | Maic[10], the 10-note MOS of [[Magic family#Magic-11-limit|magic temperament]], can in [[104edo|104et]] 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. | ||
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[[image:magic10.png]] | [[image:magic10.png]] | ||
==The dekany== | ==The dekany== | ||
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). | 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). | ||
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[[image:dekany.png]] | [[image:dekany.png]] | ||
==Myna[11]== | ==Myna[11]== | ||
Myna[11], the 11-note MOS of [[Starling temperaments#x5-limit (Mynic)-11-limit|myna temperament]], can be specified in the [[89edo|89et]] 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. | Myna[11], the 11-note MOS of [[Starling temperaments#x5-limit%20(Mynic)-11-limit|myna temperament]], can be specified in the [[89edo|89et]] 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. | ||
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. | 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. | ||
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[[image:myna11.png]] | [[image:myna11.png]] | ||
==The Marveldene== | ==The Marveldene== | ||
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 [[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 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. | 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 [[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 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. | ||
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]. | 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]. | ||
==The Novadene== | ==The Novadene== | ||
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 [[duodene_skew|skewed duodene]] or magsyn3, serves as a transversal for the [[novadene|Novadene]], 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. | 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 [[duodene_skew|skewed duodene]] or magsyn3, serves as a transversal for the [[novadene|Novadene]], 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. | ||
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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. | 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. | ||
==Magic[13]== | ==Magic[13]== | ||
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. | 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. | ||
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. | 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. | ||
==Orwell[13]== | ==Orwell[13]== | ||
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. | 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. | ||
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<!-- ws:end:WikiTextTocRule:58 --> | <!-- ws:end:WikiTextTocRule:58 --><!-- 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: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> | |||
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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We can attribute various properties to a scale, given a choice of consonance set, by the presence of a graph-theoretic property in the graph of the scale. One key graph-theoretic property is connectivity. A graph is said to be <em>connected</em> if for any two vertices a and b, there is a path of edges between a and b. A scale is therefore connected if you can go from any one note to any other note by means of consonant intervals only. The graph G has an edge-connectivity ε if it is possible to disconnect the graph by removing ε edges, but no smaller number of edges will do. Similarly, it has vertex-connectivity ν if it is possible to disconnect the graph by removing ν vertices (notes of the scale) but no smaller number will do. The graph is <em>k-edge-connected</em> if ε ≥ k, and <em>k-vertex-connected</em> if ν ≥ k; these definitions transfer immediately to scales, and a high degree of connectivity may often be desired in a scale.<br /> | We can attribute various properties to a scale, given a choice of consonance set, by the presence of a graph-theoretic property in the graph of the scale. One key graph-theoretic property is connectivity. A graph is said to be <em>connected</em> if for any two vertices a and b, there is a path of edges between a and b. A scale is therefore connected if you can go from any one note to any other note by means of consonant intervals only. The graph G has an edge-connectivity ε if it is possible to disconnect the graph by removing ε edges, but no smaller number of edges will do. Similarly, it has vertex-connectivity ν if it is possible to disconnect the graph by removing ν vertices (notes of the scale) but no smaller number will do. The graph is <em>k-edge-connected</em> if ε ≥ k, and <em>k-vertex-connected</em> if ν ≥ k; these definitions transfer immediately to scales, and a high degree of connectivity may often be desired in a scale.<br /> | ||
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A <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Clique_%28graph_theory%29" rel="nofollow">clique</a> in a graph is a subgraph such that there is an edge connecting every | A <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Clique_%28graph_theory%29" rel="nofollow">clique</a> in a graph is a subgraph such that there is an edge connecting every pair of vertices; that is, the subgraph is a complete graph. In music terms, a clique is a <a class="wiki_link" href="/dyadic%20chord">dyadic chord</a>. Hence, counting cliques of various degrees (number of notes) is a relevant consideration when analyzing a scale. The various problems of this nature are called, collectively, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Clique_problem" rel="nofollow">clique problem</a>, and this unfortunately is computationally hard. However, for scales of the size most people would care to consider a brute-force approach suffices.<br /> | ||
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Among all cliques, the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/MaximalClique.html" rel="nofollow">maximal cliques</a>, those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A <em>maximum clique</em> is a clique of largest size; these are always maximal but the converse does not always hold. Given any set of sets, we may form a graph with these as the set of vertices, by drawing an edge between any two sets with a nonempty intersection. In this way we can create graphs of all k-cliques (k note dyadic chords), maximal k-cliques, all maximal cliques, and so forth. Such graphs can illuminate harmonic relationships.<br /> | Among all cliques, the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/MaximalClique.html" rel="nofollow">maximal cliques</a>, those not contained in larger cliques, are of special interest; they might be regarded as the basic chords of the scale. A <em>maximum clique</em> is a clique of largest size; these are always maximal but the converse does not always hold. Given any set of sets, we may form a graph with these as the set of vertices, by drawing an edge between any two sets with a nonempty intersection. In this way we can create graphs of all k-cliques (k note dyadic chords), maximal k-cliques, all maximal cliques, and so forth. Such graphs can illuminate harmonic relationships.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:12 -->Examples</h1> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:12 -->Examples</h1> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Examples-The Zarlino scale"></a><!-- ws:end:WikiTextHeadingRule:14 -->The Zarlino scale</h2> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Examples-The Zarlino scale"></a><!-- ws:end:WikiTextHeadingRule:14 -->The Zarlino scale</h2> | ||
The Zarlino scale, 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 Zarlino scale, 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:16:&lt;h2&gt; --><h2 id="toc8"><a name="Examples-The diatonic scale (Meantone[7])"></a><!-- ws:end:WikiTextHeadingRule:16 -->The diatonic scale (Meantone[7])</h2> | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Examples-The diatonic scale (Meantone[7])"></a><!-- ws:end:WikiTextHeadingRule:16 -->The diatonic scale (Meantone[7])</h2> | ||
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.<br /> | 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.<br /> | ||
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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:18:&lt;h2&gt; --><h2 id="toc9"><a name="Examples-Star"></a><!-- ws:end:WikiTextHeadingRule:18 -->Star</h2> | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Examples-Star"></a><!-- ws:end:WikiTextHeadingRule:18 -->Star</h2> | ||
<a class="wiki_link" href="/Star%20and%20Nova">Star</a> is a <a class="wiki_link" href="/star">scale</a> of <a class="wiki_link" href="/Starling%20temperaments#Valentine | <a class="wiki_link" href="/Star%20and%20Nova">Star</a> is a <a class="wiki_link" href="/star">scale</a> of <a class="wiki_link" href="/Starling%20temperaments#Valentine%20temperament-11-limit">11-limit valentine temperament</a>, which in <a class="wiki_link" href="/77edo">77et</a> 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<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/16-cell" rel="nofollow">16-cell</a>, 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, <a class="wiki_link" href="/Star%20and%20Nova">Nova</a>, 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.<br /> | ||
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The graph of Star is 6-regular, with every vertex connected to six others (on the 16-cell only opposite verticies fail to connect), and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected, it consists of the four edges connecting opposite verticies of the 16-cell. It has 16 maximal cliques, all of which are tetrads, corresponding to the 16 cells of the 16-cell, and 32 triads extendable to these tetrads, corresponding to the 32 triangular faces of the 16-cell. The graph of the maximal cliques is 14-regular, with each tetrad linking to all others but its relative complement in the whole scale, that is the set of notes from note 0 to note 7, the &quot;opposite vertex&quot;. The relative complement of each dyadic tetrad is also a dyadic tetrad. The graph is known to be of genus 1, so it can be drawn on a square with opposite edges identified.<br /> | The graph of Star is 6-regular, with every vertex connected to six others (on the 16-cell only opposite verticies fail to connect), and its algebraic, vertex and edge connectivities are all 6. The largest element of the Laplace spectrum is 8, so its complementary graph is not connected, it consists of the four edges connecting opposite verticies of the 16-cell. It has 16 maximal cliques, all of which are tetrads, corresponding to the 16 cells of the 16-cell, and 32 triads extendable to these tetrads, corresponding to the 32 triangular faces of the 16-cell. The graph of the maximal cliques is 14-regular, with each tetrad linking to all others but its relative complement in the whole scale, that is the set of notes from note 0 to note 7, the &quot;opposite vertex&quot;. The relative complement of each dyadic tetrad is also a dyadic tetrad. The graph is known to be of genus 1, so it can be drawn on a square with opposite edges identified.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Examples-Oktone"></a><!-- ws:end:WikiTextHeadingRule:20 -->Oktone</h2> | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Examples-Oktone"></a><!-- ws:end:WikiTextHeadingRule:20 -->Oktone</h2> | ||
By <a class="wiki_link" href="/Oktone">Oktone</a> 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 <a class="wiki_link" href="/Breed%20family#Jove, | By <a class="wiki_link" href="/Oktone">Oktone</a> 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 <a class="wiki_link" href="/Breed%20family#Jove,%20aka%20Wonder">jove temperament</a>. The tempering can be accomplished via <a class="wiki_link" href="/202edo">202et</a>, 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.<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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<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Examples-Orwell[9]"></a><!-- ws:end:WikiTextHeadingRule:22 -->Orwell[9]</h2> | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Examples-Orwell[9]"></a><!-- ws:end:WikiTextHeadingRule:22 -->Orwell[9]</h2> | ||
Orwell[9], the 9-note orwell MOS, has notes 0, 5, 12, 17, 24, 29, 36, 41, 48, 53 in <a class="wiki_link" href="/53edo">53 equal</a>. 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 <a class="wiki_link_ext" href="http://mathworld.wolfram.com/OctahedralGroup.html" rel="nofollow">octahedronal group</a> 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.<br /> | Orwell[9], the 9-note orwell MOS, has notes 0, 5, 12, 17, 24, 29, 36, 41, 48, 53 in <a class="wiki_link" href="/53edo">53 equal</a>. 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 <a class="wiki_link_ext" href="http://mathworld.wolfram.com/OctahedralGroup.html" rel="nofollow">octahedronal group</a> 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.<br /> | ||
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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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<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Examples-Magic[10]"></a><!-- ws:end:WikiTextHeadingRule:24 -->Magic[10]</h2> | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Examples-Magic[10]"></a><!-- ws:end:WikiTextHeadingRule:24 -->Magic[10]</h2> | ||
Maic[10], the 10-note MOS of <a class="wiki_link" href="/Magic%20family#Magic-11-limit">magic temperament</a>, can in <a class="wiki_link" href="/104edo">104et</a> 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.<br /> | Maic[10], the 10-note MOS of <a class="wiki_link" href="/Magic%20family#Magic-11-limit">magic temperament</a>, can in <a class="wiki_link" href="/104edo">104et</a> 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.<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 wr 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 wr 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:26:&lt;h2&gt; --><h2 id="toc13"><a name="Examples-The dekany"></a><!-- ws:end:WikiTextHeadingRule:26 -->The dekany</h2> | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Examples-The dekany"></a><!-- ws:end:WikiTextHeadingRule:26 -->The dekany</h2> | ||
The standard 2)5 dekany is a <a class="wiki_link" href="/Combination%20product%20sets">combination product set</a>, 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Johnson_graph" rel="nofollow">Johnson graph</a> 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).<br /> | The standard 2)5 dekany is a <a class="wiki_link" href="/Combination%20product%20sets">combination product set</a>, 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Johnson_graph" rel="nofollow">Johnson graph</a> 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).<br /> | ||
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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.<br /> | 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.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Examples-Myna[11]"></a><!-- ws:end:WikiTextHeadingRule:28 -->Myna[11]</h2> | <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Examples-Myna[11]"></a><!-- ws:end:WikiTextHeadingRule:28 -->Myna[11]</h2> | ||
Myna[11], the 11-note MOS of <a class="wiki_link" href="/Starling%20temperaments#x5-limit (Mynic)-11-limit">myna temperament</a>, can be specified in the <a class="wiki_link" href="/89edo">89et</a> 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.<br /> | Myna[11], the 11-note MOS of <a class="wiki_link" href="/Starling%20temperaments#x5-limit%20(Mynic)-11-limit">myna temperament</a>, can be specified in the <a class="wiki_link" href="/89edo">89et</a> 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.<br /> | ||
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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:30:&lt;h2&gt; --><h2 id="toc15"><a name="Examples-The Marveldene"></a><!-- ws:end:WikiTextHeadingRule:30 -->The Marveldene</h2> | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Examples-The Marveldene"></a><!-- ws:end:WikiTextHeadingRule:30 -->The Marveldene</h2> | ||
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 <a class="wiki_link" href="/The%20Marveldene">Marveldene</a>. An excellent tuning for marvel is <a class="wiki_link" href="/166edo">166et</a>, 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.<br /> | 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 <a class="wiki_link" href="/The%20Marveldene">Marveldene</a>. An excellent tuning for marvel is <a class="wiki_link" href="/166edo">166et</a>, 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.<br /> | ||
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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].<br /> | 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].<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Examples-The Novadene"></a><!-- ws:end:WikiTextHeadingRule:32 -->The Novadene</h2> | <!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Examples-The Novadene"></a><!-- ws:end:WikiTextHeadingRule:32 -->The Novadene</h2> | ||
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 <a class="wiki_link" href="/duodene_skew">skewed duodene</a> or magsyn3, serves as a transversal for the <a class="wiki_link" href="/novadene">Novadene</a>, 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.<br /> | 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 <a class="wiki_link" href="/duodene_skew">skewed duodene</a> or magsyn3, serves as a transversal for the <a class="wiki_link" href="/novadene">Novadene</a>, 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.<br /> | ||
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The graph of Novadene has 46 edges between its 12 vertices, and an automorphism group of order 48 isomorphic to the group of the octahedron. The automorphisms permute the four notes with note-number divisible by 3, namely 0, 3, 6 and 9, in all 24 possible ways and the other vertices in 48 ways, so there are two separate orbits. The degrees of 0, 3, 6 and 9 are 9, and of the other verticies 7. It has algebraic, vertex and edge connectivities of 5 ≤ 7 ≤ 7, and a radius and diameter both 2. It has four pentads which are its maximum cliques, and 23 other maximal cliques in the form of tetrads.<br /> | The graph of Novadene has 46 edges between its 12 vertices, and an automorphism group of order 48 isomorphic to the group of the octahedron. The automorphisms permute the four notes with note-number divisible by 3, namely 0, 3, 6 and 9, in all 24 possible ways and the other vertices in 48 ways, so there are two separate orbits. The degrees of 0, 3, 6 and 9 are 9, and of the other verticies 7. It has algebraic, vertex and edge connectivities of 5 ≤ 7 ≤ 7, and a radius and diameter both 2. It has four pentads which are its maximum cliques, and 23 other maximal cliques in the form of tetrads.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Examples-Magic[13]"></a><!-- ws:end:WikiTextHeadingRule:34 -->Magic[13]</h2> | <!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Examples-Magic[13]"></a><!-- ws:end:WikiTextHeadingRule:34 -->Magic[13]</h2> | ||
Magic[13] is the 13-note MOS of <a class="wiki_link" href="/Magic%20family#Magic-11-limit">magic temperament</a>, which in <a class="wiki_link" href="/104edo">104et</a> 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.<br /> | Magic[13] is the 13-note MOS of <a class="wiki_link" href="/Magic%20family#Magic-11-limit">magic temperament</a>, which in <a class="wiki_link" href="/104edo">104et</a> 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.<br /> | ||
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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.<br /> | 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.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Examples-Orwell[13]"></a><!-- ws:end:WikiTextHeadingRule:36 -->Orwell[13]</h2> | <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Examples-Orwell[13]"></a><!-- ws:end:WikiTextHeadingRule:36 -->Orwell[13]</h2> | ||
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.<br /> | 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.<br /> | ||
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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.</body></html></pre></div> | The graph of Orwell[13] is 10-regular, has 65 edges, with connectivities 9.058 ≤ 10 ≤ 10, and radius and diameter both 2.</body></html></pre></div> | ||