MODMOS scale: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:~~genewardsmith~~|~~genewardsmith~~]] and made on <tt>~~2014~~-06-~~20 19~~:19:19 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2016-01-30 23:54:21 UTC</tt>.<br>

: The original revision id was <tt>~~514557760~~</tt>.<br>

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: The revision comment was: <tt></tt><br>

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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|flat]]

=Introduction=

A scale is considered to be an **MOS** if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. **MODMOS** scales generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of chromatic alterations to an MOS. The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS (the fourths come in three sizes), they can be obtained by applying one chromatic alteration each to one of the modes of the diatonic MOS.

A scale is considered to be an **MOS** if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. **MODMOS** scales generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of "chromatic alterations" to an MOS. The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS (the fourths come in three sizes), they can be obtained by applying one chromatic alteration each to one of the modes of the diatonic MOS.

Numerous options exist for the choice of chromatic alteration, all of which can be obtained by combining and subtracting intervals from within the MOS. The most common is alteration by **chroma**, where the chroma is the difference between any pair of intervals sharing the same interval class.

This choice of chromatic alteration interval is so common that the term MODMOS has generally come to be associated only with those scales being altered by chroma. In the exposition below, we give a formal treatment of MODMOS's that looks only at chroma-altered scales. These scales are distinguished<span style="line-height: 1.5;"> by the sense that they are </span><span style="line-height: 1.5;">[[@Periodic scale|epimorphic]]</span><span style="line-height: 1.5;">, and hence of special musical interest. </span>However, alterations by other intervals may also be useful.

=Definitions=

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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>MODMOS Scales</title></head><body><a href="#Introduction">Introduction</a> | <a href="#Definitions">Definitions</a> | <a href="#Examples">Examples</a>

<h1 id="toc0"><a name="Introduction"></a>Introduction</h1>

A scale is considered to be an <strong>MOS</strong> if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. <strong>MODMOS</strong> scales generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of chromatic alterations to an MOS. The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS (the fourths come in three sizes), they can be obtained by applying one chromatic alteration each to one of the modes of the diatonic MOS.<br />

A scale is considered to be an <strong>MOS</strong> if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. <strong>MODMOS</strong> scales generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of &quot;chromatic alterations&quot; to an MOS. The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS (the fourths come in three sizes), they can be obtained by applying one chromatic alteration each to one of the modes of the diatonic MOS.<br />

<br />

Numerous options exist for the choice of chromatic alteration, all of which can be obtained by combining and subtracting intervals from within the MOS. The most common is alteration by <strong>chroma</strong>, where the chroma is the difference between any pair of intervals sharing the same interval class.<br />

<br />

This choice of chromatic alteration interval is so common that the term MODMOS has generally come to be associated only with those scales being altered by chroma. In the exposition below, we give a formal treatment of MODMOS's that looks only at chroma-altered scales. These scales are distinguished<span style="line-height: 1.5;"> by the sense that they are </span><span style="line-height: 1.5;"><a class="wiki_link" href="/Periodic%20scale" target="_blank">epimorphic</a></span><span style="line-height: 1.5;">, and hence of special musical interest. </span>However, alterations by other intervals may also be useful.<br />

<br />

<h1 id="toc1"><a name="Definitions"></a>Definitions</h1>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-06-20 19:19:19 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2016-01-30 23:54:21 UTC</tt>.<br>
-: The original revision id was <tt>514557760</tt>.<br>
+: The original revision id was <tt>573461251</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 8: / Line 8: @@
 <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|flat]]
 =Introduction=
-A scale is considered to be an **MOS** if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. **MODMOS** scales generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of chromatic alterations to an MOS. The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS (the fourths come in three sizes), they can be obtained by applying one chromatic alteration each to one of the modes of the diatonic MOS.
+A scale is considered to be an **MOS** if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. **MODMOS** scales generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of "chromatic alterations" to an MOS. The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS (the fourths come in three sizes), they can be obtained by applying one chromatic alteration each to one of the modes of the diatonic MOS.
 Numerous options exist for the choice of chromatic alteration, all of which can be obtained by combining and subtracting intervals from within the MOS. The most common is alteration by **chroma**, where the chroma is the difference between any pair of intervals sharing the same interval class.
+This choice of chromatic alteration interval is so common that the term MODMOS has generally come to be associated only with those scales being altered by chroma. In the exposition below, we give a formal treatment of MODMOS's that looks only at chroma-altered scales. These scales are distinguished&lt;span style="line-height: 1.5;"&gt; by the sense that they are &lt;/span&gt;&lt;span style="line-height: 1.5;"&gt;[[@Periodic scale|epimorphic]]&lt;/span&gt;&lt;span style="line-height: 1.5;"&gt;, and hence of special musical interest. &lt;/span&gt;However, alterations by other intervals may also be useful.
 =Definitions=
@@ Line 32: / Line 34: @@
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;MODMOS Scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Introduction"&gt;Introduction&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt; | &lt;a href="#Definitions"&gt;Definitions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt; | &lt;a href="#Examples"&gt;Examples&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Introduction&lt;/h1&gt;
-  A scale is considered to be an &lt;strong&gt;MOS&lt;/strong&gt; if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. &lt;strong&gt;MODMOS&lt;/strong&gt; scales generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of chromatic alterations to an MOS. The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS (the fourths come in three sizes), they can be obtained by applying one chromatic alteration each to one of the modes of the diatonic MOS.&lt;br /&gt;
+  A scale is considered to be an &lt;strong&gt;MOS&lt;/strong&gt; if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. &lt;strong&gt;MODMOS&lt;/strong&gt; scales generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of &amp;quot;chromatic alterations&amp;quot; to an MOS. The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS (the fourths come in three sizes), they can be obtained by applying one chromatic alteration each to one of the modes of the diatonic MOS.&lt;br /&gt;
 &lt;br /&gt;
 Numerous options exist for the choice of chromatic alteration, all of which can be obtained by combining and subtracting intervals from within the MOS. The most common is alteration by &lt;strong&gt;chroma&lt;/strong&gt;, where the chroma is the difference between any pair of intervals sharing the same interval class.&lt;br /&gt;
+&lt;br /&gt;
+This choice of chromatic alteration interval is so common that the term MODMOS has generally come to be associated only with those scales being altered by chroma. In the exposition below, we give a formal treatment of MODMOS's that looks only at chroma-altered scales. These scales are distinguished&lt;span style="line-height: 1.5;"&gt; by the sense that they are &lt;/span&gt;&lt;span style="line-height: 1.5;"&gt;&lt;a class="wiki_link" href="/Periodic%20scale" target="_blank"&gt;epimorphic&lt;/a&gt;&lt;/span&gt;&lt;span style="line-height: 1.5;"&gt;, and hence of special musical interest. &lt;/span&gt;However, alterations by other intervals may also be useful.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definitions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Definitions&lt;/h1&gt;