{{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.]]
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<h4>Original Wikitext content:</h4>
<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.
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.
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.
There is peer-reviewed research which shows that convex scales are common, e.g. [[http://dare.uva.nl/en/record/190378]]. However, one problem with that study is that they used the [[Scala]] archive, which contains many scales explicitly constructed to be convex...
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.
==Formal definition==
==Examples==
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]].
===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
(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.)
===Convex set===
A convex set is a set that includes all convex combinations of its elements.
==Examples==
* Every [[MOSScales|MOS]] is convex.
* Every [[MOSScales|MOS]] is convex.
* In fact, every [[distributionally even]] scale is convex.
*In fact, every [[distributionally even]] scale is convex.
* Every [[Fokker blocks|Fokker block]] is convex.
* Every [[Fokker block]] is convex.
* Every untempered [[Tonality diamond|tonality diamond]] is convex.
* Every untempered [[tonality diamond]] is convex.
* [[Gallery of Z-polygon transversals]]</pre></div>
<h4>Original HTML content:</h4>
[[Category:Scale]]
<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 />
[[Category:Math]]
<br />
[[Category:Todo:clarify]]
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 />
<br />
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 />
<br />
There is peer-reviewed research which shows that convex scales are common, e.g. <a class="wiki_link_ext" href="http://dare.uva.nl/en/record/190378" rel="nofollow">http://dare.uva.nl/en/record/190378</a>. However, one problem with that study is that they used the <a class="wiki_link" href="/Scala">Scala</a> archive, which contains many scales explicitly constructed to be convex...<br />
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 />
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 />
(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.)<br />
<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>
Latest revision as of 20:39, 25 June 2025
Todo: expand
explain musical application --hkm (talk) 20:39, 25 June 2025 (UTC)
A convex set of 12 tones from the marvel lattice.
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 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. 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.
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.
