Periodic scale: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 401115018 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 401116536 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2013-01-24 | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2013-01-24 12:00:56 UTC</tt>.<br> | ||
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The set {s[i] | i∈ℤ} generates a group G, the **group of the scale**; this is a free, finitely generated subgroup of the reals ℝ. The **rank of the scale** is the rank of G. | The set {s[i] | i∈ℤ} generates a group G, the **group of the scale**; this is a free, finitely generated subgroup of the reals ℝ. The **rank of the scale** is the rank of G. | ||
**Epimorphic**: If there exists a homomorphism h: G <span style="line-height: 1.5;">→ ℤ so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by | **Epimorphic**: If there exists a homomorphism h: G <span style="line-height: 1.5;">→ ℤ so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by [[Yves Hellegouarch]]. The name comes from the fact that h is an epimorphism onto ℤ.</span> | ||
**[[http://en.wikipedia.org/wiki/Myhill%27s_property|Myhill's property]]** : A monotone scale in which every class but classes nP have exactly two elements has Myhill's property. If every such class has exactly three elements, it has the **trivalence property**. | **[[http://en.wikipedia.org/wiki/Myhill%27s_property|Myhill's property]]**: A monotone scale in which every class but classes nP have exactly two elements has Myhill's property. <span style="line-height: 1.5;">If every such class has exactly three elements, it has the </span>**<span style="line-height: 1.5;">trivalence property</span>**<span style="line-height: 1.5;">. Myhill's property is synonymous with </span>**<span style="line-height: 1.5;">strict </span>****<span style="line-height: 1.5;">[[xenharmonic/MOSScales|MOS]]</span>**<span style="line-height: 1.5;">, though some authors prefer to identify MOS itself with Myhill's property.</span> | ||
**Distributional evenness**: A monotone scale in which every class comes in exactly n elements is n-distributionally even, or **n-DE**. If n=2, then we can simply say that it is distributionally even. <span style="line-height: 1.5;">Distributional evenness is also synonymous with </span>**<span style="line-height: 1.5;">[[MOSScales|MOS]]</span>**<span style="line-height: 1.5;">, though some authors prefer a stricter definition of MOS identifying it with Myhill's property.</span> | |||
**Convexity**: The scale is [[Convex scale|convex]] if every convex combination of notes, meaning every ℕ-linear combination of scale notes, is a scale note. If the quasiperiod **P** is normalized so as to be positive and minimal, this is equivalent to the condition that the equivalence classes of the notes modulo the repetition interval **O** is a [[http://en.wikipedia.org/wiki/Convex_lattice_polytope|ℤ-polytope]] in the lattice defined by a basis for G mod **O**.</pre></div> | **Convexity**: The scale is [[Convex scale|convex]] if every convex combination of notes, meaning every ℕ-linear combination of scale notes, is a scale note. If the quasiperiod **P** is normalized so as to be positive and minimal, this is equivalent to the condition that the equivalence classes of the notes modulo the repetition interval **O** is a [[http://en.wikipedia.org/wiki/Convex_lattice_polytope|ℤ-polytope]] in the lattice defined by a basis for G mod **O**.</pre></div> | ||
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The set {s[i] | i∈ℤ} generates a group G, the <strong>group of the scale</strong>; this is a free, finitely generated subgroup of the reals ℝ. The <strong>rank of the scale</strong> is the rank of G.<br /> | The set {s[i] | i∈ℤ} generates a group G, the <strong>group of the scale</strong>; this is a free, finitely generated subgroup of the reals ℝ. The <strong>rank of the scale</strong> is the rank of G.<br /> | ||
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<strong>Epimorphic</strong>: If there exists a homomorphism h: G <span style="line-height: 1.5;">→ ℤ so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by </span><span style="line-height: 1.5;"><a class="wiki_link" href="/ | <strong>Epimorphic</strong>: If there exists a homomorphism h: G <span style="line-height: 1.5;">→ ℤ so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by <a class="wiki_link" href="/Yves%20Hellegouarch">Yves Hellegouarch</a>. The name comes from the fact that h is an epimorphism onto ℤ.</span><br /> | ||
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<strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Myhill%27s_property" rel="nofollow">Myhill's property</a></strong>: A monotone scale in which every class but classes nP have exactly two elements has Myhill's property. <span style="line-height: 1.5;">If every such class has exactly three elements, it has the </span><strong><span style="line-height: 1.5;">trivalence property</span></strong><span style="line-height: 1.5;">. Myhill's property is synonymous with </span><strong><span style="line-height: 1.5;">strict </span></strong><strong><span style="line-height: 1.5;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOSScales">MOS</a></span></strong><span style="line-height: 1.5;">, though some authors prefer to identify MOS itself with Myhill's property.</span><br /> | |||
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<strong>< | <strong>Distributional evenness</strong>: A monotone scale in which every class comes in exactly n elements is n-distributionally even, or <strong>n-DE</strong>. If n=2, then we can simply say that it is distributionally even. <span style="line-height: 1.5;">Distributional evenness is also synonymous with </span><strong><span style="line-height: 1.5;"><a class="wiki_link" href="/MOSScales">MOS</a></span></strong><span style="line-height: 1.5;">, though some authors prefer a stricter definition of MOS identifying it with Myhill's property.</span><br /> | ||
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<strong>Convexity</strong>: The scale is <a class="wiki_link" href="/Convex%20scale">convex</a> if every convex combination of notes, meaning every ℕ-linear combination of scale notes, is a scale note. If the quasiperiod <strong>P</strong> is normalized so as to be positive and minimal, this is equivalent to the condition that the equivalence classes of the notes modulo the repetition interval <strong>O</strong> is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_lattice_polytope" rel="nofollow">ℤ-polytope</a> in the lattice defined by a basis for G mod <strong>O</strong>.</body></html></pre></div> | <strong>Convexity</strong>: The scale is <a class="wiki_link" href="/Convex%20scale">convex</a> if every convex combination of notes, meaning every ℕ-linear combination of scale notes, is a scale note. If the quasiperiod <strong>P</strong> is normalized so as to be positive and minimal, this is equivalent to the condition that the equivalence classes of the notes modulo the repetition interval <strong>O</strong> is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_lattice_polytope" rel="nofollow">ℤ-polytope</a> in the lattice defined by a basis for G mod <strong>O</strong>.</body></html></pre></div> | ||