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>2011-03-~~18 11~~:59:23 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-22 20:30:07 UTC</tt>.<br>

: The original revision id was <tt>~~211787514~~</tt>.<br>

: The original revision id was <tt>278337108</tt>.<br>

: 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]]

=Definitions=

A MOS scale is constructed by iterating a generator g inside of a period R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 ~~<=~~ i < R. We stop only at points when there are exactly two sizes of intervals in the scale, the large interval L and the small interval s. If the period R is 1/P octave for some integer P, we obtain an octave-repeating [[periodic scale]] by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.

A MOS scale is constructed by iterating a generator g inside of a period R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 ≤ i < R. We stop only at points when there are exactly two sizes of intervals in the scale, the large interval L and the small interval s. If the period R is 1/P octave for some integer P, we obtain an octave-repeating [[periodic scale]] by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.

If we continue by adding another interval, it will either fall short of R, or exceed it, but either way we will now have an interval c = L-s, which has been called the "chroma". A MODMOS is in essence, a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a MODMOS. We can restrict this definition by requiring that no more than a certain percentage of the notes may be adjusted, or requiring that the monotonic ascending ordering of the notes by size be retained, or relaxing the condition that notes are adjusted by a single chroma so that they can be adjusted by any number of them, but we will take this as the basic definition of a MODMOS.

The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ~~+-N~~, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size c2 = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N, since we have limited ourselves to note adjustments of one chroma.

The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ∓N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size c2 = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N, since we have limited ourselves to note adjustments of one chroma.

=Examples=

Consider the MOS of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone, 78.049 cents. This is the interval whose adjustment of a note up or down is represented by a sharp # or flat b symbol. The diatonic scale has steps LLsLLLs, which in the key of C can be written CDEFGABC'. From the definition of a MODMOS, if we add sharps and flats to this, and do not get another diatonic scale, then we have a MODMOS. For example LsLLLLs, which is CDEbFGABC', is the melodic minor scale, which is therefore a MODMOS. The harmonic minor scale is CDEbFGAbBC', and is therefore also a MODMOS. However, the natural minor, CDEbFGAbBbC' is a mode of the diatonic scale, and a MOS rather than a MODMOS.

Consider the MOS of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone, 78.049 cents. This is the interval whose adjustment of a note up or down is represented by a sharp # or flat symbol. The diatonic scale has steps LLsLLLs, which in the key of C can be written CDEFGABC'. From the definition of a MODMOS, if we add sharps and flats to this, and do not get another diatonic scale, then we have a MODMOS. For example LsLLLLs, which is CDEbFGABC', is the melodic minor scale, which is therefore a MODMOS. The harmonic minor scale is CDEbFGAbBC', and is therefore also a MODMOS. However, the natural minor, CDEbFGAbBbC' is a mode of the diatonic scale, and a MOS rather than a MODMOS.

If we take the 12 note MOS rather than the seven notes of the diatonic scale, then the chroma is a diesis, which in 1/4 comma meantone is exactly 128/125, or 41.059 cents. However, other meantone tunings will give it a different size, and in [[50edo|50et]], for example, it is exactly 48 cents in size. Subtracting this adjusts a note twelve fifths up on the chain of fifths, and adding it adjusts it twelve notes down. If we take a chain of eleven 50et fifths up from the unison to create a 12-note MOS, so that we have a generator chain from 0 to 11, we may adjust the 3 up to 15 and the 7 up to 19. This leads to the scale called "smithgw_modmos12a.scl" in the [[http://www.huygens-fokker.org/docs/scales.zip|Scala Scale Archive]]. Another MODMOS of Meantone[12] in the archive is wreckpop, "smithgw_wreckpop.scl". This takes a gamut of Meantone[12] from -4 to 7, which we may call from Ab to F#, and adjusts the 5 (B) up a 48 cent diesis, and so down to -7 fifths, or Cb, and the -2 (Bb) down a diesis, and so up the chain of fifths to 10 (A#.)

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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="#Definitions">Definitions</a> | <a href="#Examples">Examples</a> | <a href="#MODMOS in Jazz">MODMOS in Jazz</a> | <a href="#Proper MODMOS">Proper MODMOS</a> | <a href="#Outline for General Algorithm">Outline for General Algorithm</a>

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

A MOS scale is constructed by iterating a generator g inside of a period R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 ~~&lt;=~~ i &lt; R. We stop only at points when there are exactly two sizes of intervals in the scale, the large interval L and the small interval s. If the period R is 1/P octave for some integer P, we obtain an octave-repeating <a class="wiki_link" href="/periodic%20scale">periodic scale</a> by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.<br />

A MOS scale is constructed by iterating a generator g inside of a period R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 ≤ i &lt; R. We stop only at points when there are exactly two sizes of intervals in the scale, the large interval L and the small interval s. If the period R is 1/P octave for some integer P, we obtain an octave-repeating <a class="wiki_link" href="/periodic%20scale">periodic scale</a> by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.<br />

<br />

If we continue by adding another interval, it will either fall short of R, or exceed it, but either way we will now have an interval c = L-s, which has been called the &quot;chroma&quot;. A MODMOS is in essence, a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a MODMOS. We can restrict this definition by requiring that no more than a certain percentage of the notes may be adjusted, or requiring that the monotonic ascending ordering of the notes by size be retained, or relaxing the condition that notes are adjusted by a single chroma so that they can be adjusted by any number of them, but we will take this as the basic definition of a MODMOS. <br />

<br />

The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ~~+-N~~, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size c2 = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N, since we have limited ourselves to note adjustments of one chroma.<br />

The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ∓N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size c2 = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N, since we have limited ourselves to note adjustments of one chroma.<br />

<br />

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

Consider the MOS of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone, 78.049 cents. This is the interval whose adjustment of a note up or down is represented by a sharp # or flat b symbol. The diatonic scale has steps LLsLLLs, which in the key of C can be written CDEFGABC'. From the definition of a MODMOS, if we add sharps and flats to this, and do not get another diatonic scale, then we have a MODMOS. For example LsLLLLs, which is CDEbFGABC', is the melodic minor scale, which is therefore a MODMOS. The harmonic minor scale is CDEbFGAbBC', and is therefore also a MODMOS. However, the natural minor, CDEbFGAbBbC' is a mode of the diatonic scale, and a MOS rather than a MODMOS.<br />

Consider the MOS of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone, 78.049 cents. This is the interval whose adjustment of a note up or down is represented by a sharp # or flat symbol. The diatonic scale has steps LLsLLLs, which in the key of C can be written CDEFGABC'. From the definition of a MODMOS, if we add sharps and flats to this, and do not get another diatonic scale, then we have a MODMOS. For example LsLLLLs, which is CDEbFGABC', is the melodic minor scale, which is therefore a MODMOS. The harmonic minor scale is CDEbFGAbBC', and is therefore also a MODMOS. However, the natural minor, CDEbFGAbBbC' is a mode of the diatonic scale, and a MOS rather than a MODMOS.<br />

<br />

If we take the 12 note MOS rather than the seven notes of the diatonic scale, then the chroma is a diesis, which in 1/4 comma meantone is exactly 128/125, or 41.059 cents. However, other meantone tunings will give it a different size, and in <a class="wiki_link" href="/50edo">50et</a>, for example, it is exactly 48 cents in size. Subtracting this adjusts a note twelve fifths up on the chain of fifths, and adding it adjusts it twelve notes down. If we take a chain of eleven 50et fifths up from the unison to create a 12-note MOS, so that we have a generator chain from 0 to 11, we may adjust the 3 up to 15 and the 7 up to 19. This leads to the scale called &quot;smithgw_modmos12a.scl&quot; in the <a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/scales.zip" rel="nofollow">Scala Scale Archive</a>. Another MODMOS of Meantone[12] in the archive is wreckpop, &quot;smithgw_wreckpop.scl&quot;. This takes a gamut of Meantone[12] from -4 to 7, which we may call from Ab to F#, and adjusts the 5 (B) up a 48 cent diesis, and so down to -7 fifths, or Cb, and the -2 (Bb) down a diesis, and so up the chain of fifths to 10 (A#.)<br />

@@ 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>2011-03-18 11:59:23 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-22 20:30:07 UTC</tt>.<br>
-: The original revision id was <tt>211787514</tt>.<br>
+: The original revision id was <tt>278337108</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]]
 =Definitions=
-A MOS scale is constructed by iterating a generator g inside of a period R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 &lt;= i &lt; R. We stop only at points when there are exactly two sizes of intervals in the scale, the large interval L and the small interval s. If the period R is 1/P octave for some integer P, we obtain an octave-repeating [[periodic scale]] by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.
+A MOS scale is constructed by iterating a generator g inside of a period R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 ≤ i &lt; R. We stop only at points when there are exactly two sizes of intervals in the scale, the large interval L and the small interval s. If the period R is 1/P octave for some integer P, we obtain an octave-repeating [[periodic scale]] by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.
 If we continue by adding another interval, it will either fall short of R, or exceed it, but either way we will now have an interval c = L-s, which has been called the "chroma". A MODMOS is in essence, a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a MODMOS. We can restrict this definition by requiring that no more than a certain percentage of the notes may be adjusted, or requiring that the monotonic ascending ordering of the notes by size be retained, or relaxing the condition that notes are adjusted by a single chroma so that they can be adjusted by any number of them, but we will take this as the basic definition of a MODMOS.
-The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by +-N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size c2 = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N, since we have limited ourselves to note adjustments of one chroma.
+The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ∓N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size c2 = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N, since we have limited ourselves to note adjustments of one chroma.
 =Examples=
-Consider the MOS of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone, 78.049 cents. This is the interval whose adjustment of a note up or down is represented by a sharp # or flat b symbol. The diatonic scale has steps LLsLLLs, which in the key of C can be written CDEFGABC'. From the definition of a MODMOS, if we add sharps and flats to this, and do not get another diatonic scale, then we have a MODMOS. For example LsLLLLs, which is CDEbFGABC', is the melodic minor scale, which is therefore a MODMOS. The harmonic minor scale is CDEbFGAbBC', and is therefore also a MODMOS. However, the natural minor, CDEbFGAbBbC' is a mode of the diatonic scale, and a MOS rather than a MODMOS.
+Consider the MOS of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone, 78.049 cents. This is the interval whose adjustment of a note up or down is represented by a sharp # or flat  symbol. The diatonic scale has steps LLsLLLs, which in the key of C can be written CDEFGABC'. From the definition of a MODMOS, if we add sharps and flats to this, and do not get another diatonic scale, then we have a MODMOS. For example LsLLLLs, which is CDEbFGABC', is the melodic minor scale, which is therefore a MODMOS. The harmonic minor scale is CDEbFGAbBC', and is therefore also a MODMOS. However, the natural minor, CDEbFGAbBbC' is a mode of the diatonic scale, and a MOS rather than a MODMOS.
 If we take the 12 note MOS rather than the seven notes of the diatonic scale, then the chroma is a diesis, which in 1/4 comma meantone is exactly 128/125, or 41.059 cents. However, other meantone tunings will give it a different size, and in [[50edo|50et]], for example, it is exactly 48 cents in size. Subtracting this adjusts a note twelve fifths up on the chain of fifths, and adding it adjusts it twelve notes down. If we take a chain of eleven 50et fifths up from the unison to create a 12-note MOS, so that we have a generator chain from 0 to 11, we may adjust the 3 up to 15 and the 7 up to 19. This leads to the scale called "smithgw_modmos12a.scl" in the [[http://www.huygens-fokker.org/docs/scales.zip|Scala Scale Archive]]. Another MODMOS of Meantone[12] in the archive is wreckpop, "smithgw_wreckpop.scl". This takes a gamut of Meantone[12] from -4 to 7, which we may call from Ab to F#, and adjusts the 5 (B) up a 48 cent diesis, and so down to -7 fifths, or Cb, and the -2 (Bb) down a diesis, and so up the chain of fifths to 10 (A#.)
@@ Line 75: / Line 75: @@
 <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:10:&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:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Definitions"&gt;Definitions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Examples"&gt;Examples&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt; | &lt;a href="#MODMOS in Jazz"&gt;MODMOS in Jazz&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Proper MODMOS"&gt;Proper MODMOS&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#Outline for General Algorithm"&gt;Outline for General Algorithm&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definitions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definitions&lt;/h1&gt;
-  A MOS scale is constructed by iterating a generator g inside of a period R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 &amp;lt;= i &amp;lt; R. We stop only at points when there are exactly two sizes of intervals in the scale, the large interval L and the small interval s. If the period R is 1/P octave for some integer P, we obtain an octave-repeating &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.&lt;br /&gt;
+  A MOS scale is constructed by iterating a generator g inside of a period R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 ≤ i &amp;lt; R. We stop only at points when there are exactly two sizes of intervals in the scale, the large interval L and the small interval s. If the period R is 1/P octave for some integer P, we obtain an octave-repeating &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.&lt;br /&gt;
 &lt;br /&gt;
 If we continue by adding another interval, it will either fall short of R, or exceed it, but either way we will now have an interval c = L-s, which has been called the &amp;quot;chroma&amp;quot;. A MODMOS is in essence, a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a MODMOS. We can restrict this definition by requiring that no more than a certain percentage of the notes may be adjusted, or requiring that the monotonic ascending ordering of the notes by size be retained, or relaxing the condition that notes are adjusted by a single chroma so that they can be adjusted by any number of them, but we will take this as the basic definition of a MODMOS. &lt;br /&gt;
 &lt;br /&gt;
-The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by +-N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size c2 = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N, since we have limited ourselves to note adjustments of one chroma.&lt;br /&gt;
+The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ∓N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size c2 = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N, since we have limited ourselves to note adjustments of one chroma.&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="Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Examples&lt;/h1&gt;
-  Consider the MOS of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone, 78.049 cents. This is the interval whose adjustment of a note up or down is represented by a sharp # or flat b symbol. The diatonic scale has steps LLsLLLs, which in the key of C can be written CDEFGABC'. From the definition of a MODMOS, if we add sharps and flats to this, and do not get another diatonic scale, then we have a MODMOS. For example LsLLLLs, which is CDEbFGABC', is the melodic minor scale, which is therefore a MODMOS. The harmonic minor scale is CDEbFGAbBC', and is therefore also a MODMOS. However, the natural minor, CDEbFGAbBbC' is a mode of the diatonic scale, and a MOS rather than a MODMOS.&lt;br /&gt;
+  Consider the MOS of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone, 78.049 cents. This is the interval whose adjustment of a note up or down is represented by a sharp # or flat  symbol. The diatonic scale has steps LLsLLLs, which in the key of C can be written CDEFGABC'. From the definition of a MODMOS, if we add sharps and flats to this, and do not get another diatonic scale, then we have a MODMOS. For example LsLLLLs, which is CDEbFGABC', is the melodic minor scale, which is therefore a MODMOS. The harmonic minor scale is CDEbFGAbBC', and is therefore also a MODMOS. However, the natural minor, CDEbFGAbBbC' is a mode of the diatonic scale, and a MOS rather than a MODMOS.&lt;br /&gt;
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 If we take the 12 note MOS rather than the seven notes of the diatonic scale, then the chroma is a diesis, which in 1/4 comma meantone is exactly 128/125, or 41.059 cents. However, other meantone tunings will give it a different size, and in &lt;a class="wiki_link" href="/50edo"&gt;50et&lt;/a&gt;, for example, it is exactly 48 cents in size. Subtracting this adjusts a note twelve fifths up on the chain of fifths, and adding it adjusts it twelve notes down. If we take a chain of eleven 50et fifths up from the unison to create a 12-note MOS, so that we have a generator chain from 0 to 11, we may adjust the 3 up to 15 and the 7 up to 19. This leads to the scale called &amp;quot;smithgw_modmos12a.scl&amp;quot; in the &lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/scales.zip" rel="nofollow"&gt;Scala Scale Archive&lt;/a&gt;. Another MODMOS of Meantone[12] in the archive is wreckpop, &amp;quot;smithgw_wreckpop.scl&amp;quot;. This takes a gamut of Meantone[12] from -4 to 7, which we may call from Ab to F#, and adjusts the 5 (B) up a 48 cent diesis, and so down to -7 fifths, or Cb, and the -2 (Bb) down a diesis, and so up the chain of fifths to 10 (A#.)&lt;br /&gt;