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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Todo|inline=1|expand|comment=explain musical application --[[User:Hkm|hkm]] ([[User talk:Hkm|talk]]) 20:39, 25 June 2025 (UTC)}}[[File:Lattice Marvel Convex12.png|400px|thumb|A convex set of 12 tones from the marvel lattice.]] |
| 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> | |
| : 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.
| | In a [[regular temperament]], a '''convex scale''' is a set of pitches that form a '''convex set''' (also called a Z-polytope) 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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| 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.
| | A simple, easy-to-understand definition of a "convex set" in a lattice is the intersection of the lattice with any [https://en.wikipedia.org/wiki/Convex_set convex region] of continuous space. Alternatively, a convex set in a lattice is a set where any weighted average of elements (where no element has negative weight) is within the set if it is on the lattice. |
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| ==Formal definition==
| | The '''convex hull''' or '''convex closure''' of a scale is the smallest convex scale that contains it. (Every scale has a unique convex hull.) See [[Gallery of Z-polygon transversals]] for many scales that are the convex closures of interesting sets of pitches. |
| 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=== | | ==Examples== |
| 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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| [[math]]
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| $(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$
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| [[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
| | * Every [[MOSScales|MOS]] is convex. |
| [[math]] | | *In fact, every [[distributionally even]] scale is convex. |
| $c = c_1 + c_2 + \dots + c_k$
| | * Every [[Fokker block]] is convex. |
| [[math]] | | * Every untempered [[tonality diamond]] is convex. |
| we obtain
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| [[math]] | |
| $b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$
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| [[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.
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| ===Convex set===
| | [[Category:Scale]] |
| 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.
| | [[Category:Math]] |
| | | [[Category:Todo:clarify]] |
| ==Examples==
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| * Every [[MOSScales|MOS]] is convex.
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| * In fact, every [[distributionally even]] scale is convex.
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| * Every [[Fokker blocks|Fokker block]] is convex.
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| * 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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| <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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| <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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| <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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| <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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| <!-- ws:start:WikiTextMathRule:0:
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| [[math]]&lt;br/&gt;
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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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| <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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| <!-- ws:start:WikiTextMathRule:1:
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| [[math]]&lt;br/&gt;
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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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| <!-- ws:start:WikiTextMathRule:2:
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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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| <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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