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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | In a [[Regular_Temperaments|regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' in the interval lattice of the temperament. The "regular temperament" is often [[Just_intonation|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-19 15:48:45 UTC</tt>.<br>
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| : The original revision id was <tt>266530940</tt>.<br>
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| : The revision comment was: <tt>Reverted to Oct 19, 2011 12:27 pm</tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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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">In a [[Regular Temperaments|regular temperament]], a **convex scale** is a set of pitches that form a **convex set** in the interval lattice of the temperament. The "regular temperament" is often [[Just intonation|JI]], in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.
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| A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [[http://en.wikipedia.org/wiki/Convex_set|convex region]] of continuous space. See below for a more formal definition. | | A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [http://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. See below for a more formal definition. |
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| The **convex hull** or **convex closure** of a scale is the smallest convex scale that contains it. See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches. | | The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. See [[Gallery_of_Z-polygon_transversals|Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches. |
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| ==Formal definition== | | ==Formal definition== |
| The following definitions make sense in the context of any Z-[[http://en.wikipedia.org/wiki/Module_%28mathematics%29|module]], which is the same concept as an [[http://en.wikipedia.org/wiki/Abelian_group|abelian group]]. | | The following definitions make sense in the context of any Z-[http://en.wikipedia.org/wiki/Module_%28mathematics%29 module], which is the same concept as an [http://en.wikipedia.org/wiki/Abelian_group abelian group]. |
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| ===Convex combination=== | | ===Convex combination=== |
| A **convex combination** of a set of vectors is, intuitively, a weighted average of the vectors where all the weights are non-negative. Formally, b is a convex combination of a1, a2... whenever there exist non-negative integers c1, c2... such that | | A '''convex combination''' of a set of vectors is, intuitively, a weighted average of the vectors where all the weights are non-negative. Formally, b is a convex combination of a1, a2... whenever there exist non-negative integers c1, c2... such that |
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| $(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$ | | <math>$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$</math> |
| [[math]] | | |
| | Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent definition can be given in terms of the [http://en.wikipedia.org/wiki/Injective_hull injective hull] of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by |
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| | <math>$c = c_1 + c_2 + \dots + c_k$</math> |
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| Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent definition can be given in terms of the [[http://en.wikipedia.org/wiki/Injective_hull|injective hull]] of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by
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| $c = c_1 + c_2 + \dots + c_k$
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| we obtain | | we obtain |
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| $b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$ | | <math>$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$</math> |
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| where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers. | | where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers. |
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| ===Convex set=== | | ===Convex set=== |
| A convex set (sometimes called a Z-polytope) in a lattice is a set that includes all convex combinations of its elements. The convex closure of a set of lattice elements is the smallest convex set containing the set. | | A convex set (sometimes called a Z-polytope) in a lattice is a set that includes all convex combinations of its elements. The convex closure of a set of lattice elements is the smallest convex set containing the set. |
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| ==Examples== | | ==Examples== |
| * Every [[MOSScales|MOS]] is convex.
| | <ul><li>Every [[MOSScales|MOS]] is convex.</li><li>In fact, every [[distributionally_even|distributionally even]] scale is convex.</li><li>Every [[Fokker_blocks|Fokker block]] is convex.</li><li>Every untempered [[Tonality_diamond|tonality diamond]] is convex.</li><li>[[Gallery_of_Z-polygon_transversals|Gallery of Z-polygon transversals]]</li></ul> [[Category:math]] |
| * In fact, every [[distributionally even]] scale is convex.
| | [[Category:scale]] |
| * Every [[Fokker blocks|Fokker block]] is convex.
| | [[Category:theory]] |
| * Every untempered [[Tonality diamond|tonality diamond]] is convex.
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| * [[Gallery of Z-polygon transversals]]</pre></div>
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| <h4>Original HTML content:</h4>
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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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Convex scale</title></head><body>In a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>, a <strong>convex scale</strong> is a set of pitches that form a <strong>convex set</strong> in the interval lattice of the temperament. The &quot;regular temperament&quot; is often <a class="wiki_link" href="/Just%20intonation">JI</a>, in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.<br />
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| A simple, easy-to-understand definition of a &quot;convex set&quot; in a lattice is the intersection of the lattice with any <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_set" rel="nofollow">convex region</a> of continuous space. See below for a more formal definition.<br />
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| The <strong>convex hull</strong> or <strong>convex closure</strong> of a scale is the smallest convex scale that contains it. See <a class="wiki_link" href="/Gallery%20of%20Z-polygon%20transversals">Gallery of Z-polygon transversals</a> for many scales that are the convex closures of interesting sets of pitches.<br />
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| <!-- ws:start:WikiTextHeadingRule:3:&lt;h2&gt; --><h2 id="toc0"><a name="x-Formal definition"></a><!-- ws:end:WikiTextHeadingRule:3 -->Formal definition</h2>
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| The following definitions make sense in the context of any Z-<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Module_%28mathematics%29" rel="nofollow">module</a>, which is the same concept as an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">abelian group</a>.<br />
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| <!-- ws:start:WikiTextHeadingRule:5:&lt;h3&gt; --><h3 id="toc1"><a name="x-Formal definition-Convex combination"></a><!-- ws:end:WikiTextHeadingRule:5 -->Convex combination</h3>
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| A <strong>convex combination</strong> of a set of vectors is, intuitively, a weighted average of the vectors where all the weights are non-negative. Formally, b is a convex combination of a1, a2... whenever there exist non-negative integers c1, c2... such that<br />
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| $(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$&lt;br/&gt;[[math]]
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| --><script type="math/tex">$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$</script><!-- ws:end:WikiTextMathRule:0 --><br />
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| Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent definition can be given in terms of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow">injective hull</a> of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by <br />
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| $c = c_1 + c_2 + \dots + c_k$&lt;br/&gt;[[math]]
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| --><script type="math/tex">$c = c_1 + c_2 + \dots + c_k$</script><!-- ws:end:WikiTextMathRule:1 --><br />
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| we obtain<br />
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| [[math]]&lt;br/&gt;
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| $b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$&lt;br/&gt;[[math]]
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| --><script type="math/tex">$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$</script><!-- ws:end:WikiTextMathRule:2 --><br />
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| where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:7:&lt;h3&gt; --><h3 id="toc2"><a name="x-Formal definition-Convex set"></a><!-- ws:end:WikiTextHeadingRule:7 -->Convex set</h3>
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| A convex set (sometimes called a Z-polytope) in a lattice is a set that includes all convex combinations of its elements. The convex closure of a set of lattice elements is the smallest convex set containing the set.<br />
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| <!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc3"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:9 -->Examples</h2>
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| <ul><li>Every <a class="wiki_link" href="/MOSScales">MOS</a> is convex.</li><li>In fact, every <a class="wiki_link" href="/distributionally%20even">distributionally even</a> scale is convex.</li><li>Every <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> is convex.</li><li>Every untempered <a class="wiki_link" href="/Tonality%20diamond">tonality diamond</a> is convex.</li><li><a class="wiki_link" href="/Gallery%20of%20Z-polygon%20transversals">Gallery of Z-polygon transversals</a></li></ul></body></html></pre></div>
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In a regular temperament, a convex scale is a set of pitches that form a convex set in the interval lattice of the temperament. The "regular temperament" is often JI, in which case the lattice is the familiar JI lattice, but convex scales exist for any regular temperament.
A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any convex region of continuous space. See below for a more formal definition.
The convex hull or convex closure of a scale is the smallest convex scale that contains it. See Gallery of Z-polygon transversals for many scales that are the convex closures of interesting sets of pitches.
Formal definition
The following definitions make sense in the context of any Z-module, which is the same concept as an abelian group.
Convex combination
A convex combination of a set of vectors is, intuitively, a weighted average of the vectors where all the weights are non-negative. Formally, b is a convex combination of a1, a2... whenever there exist non-negative integers c1, c2... such that
[math]\displaystyle{ $(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$ }[/math]
Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent definition can be given in terms of the injective hull of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by
[math]\displaystyle{ $c = c_1 + c_2 + \dots + c_k$ }[/math]
we obtain
[math]\displaystyle{ $b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$ }[/math]
where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers.
Convex set
A convex set (sometimes called a Z-polytope) in a lattice is a set that includes all convex combinations of its elements. The convex closure of a set of lattice elements is the smallest convex set containing the set.
Examples