MOS scale: Difference between revisions
Wikispaces>Andrew_Heathwaite **Imported revision 293735516 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 314368274 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-03-25 12:56:35 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>314368274</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 10: | Line 10: | ||
=Definition= | =Definition= | ||
An **MOS** or **Moment Of Symmetry** is a scale in which every interval except for the period comes in two sizes. The term "MOS," and the method of scale construction it entails, were invented by Erv Wilson in 1975. His original paper can be found at [[http://anaphoria.com/mos.PDF]]. There is also an introduction at [[http://anaphoria.com/wilsonintroMOS.html]]. | |||
Sometimes, scales are defined with respect to a period and an additional "equivalence interval," considered to be the interval at which pitch classes repeat. MOS's in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called **Multi-MOS**'s. MOS's in which the equivalence interval is equal the period are sometimes called **Strict MOS**'s. MOS's in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label. | |||
With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional Evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as //well-formed scales//, the term used in the 1989 paper by Norman Carey and David Clampitt. A great deal of interesting work has been done on scales in academic circles extending these ideas. | |||
=Mathematical Definition= | |||
An MOS specifically consists of: | |||
1. A period "P" (of any size but most commonly the octave or a 1/N fraction of an octave) | 1. A period "P" (of any size but most commonly the octave or a 1/N fraction of an octave) | ||
2. A generator "g" (of any size, for example 700 cents in 12 equal temperament) which is added repeatedly to make a chain of scale steps, starting from the unison or 0 cents scale step, and then reducing to within the period | 2. A generator "g" (of any size, for example 700 cents in 12 equal temperament) which is added repeatedly to make a chain of scale steps, starting from the unison or 0 cents scale step, and then reducing to within the period | ||
3. Exactly | 3. Exactly two sizes of scale steps (Large and small, often written "L" and "s") | ||
4. Where //each// number of scale steps, or generic interval, within the scale occurs in no more than two different sizes, and in exactly two if the interval is not a multiple of the period | 4. Where //each// number of scale steps, or generic interval, within the scale occurs in no more than two different sizes, and in exactly two if the interval is not a multiple of the period | ||
5. The unison or starting point of the scale is then allowed to be transferred to any scale degree--all the modes of an MOS are legal. | 5. The unison or starting point of the scale is then allowed to be transferred to any scale degree--all the modes of an MOS are legal. | ||
| Line 22: | Line 29: | ||
These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and the pseudo-Myhill property, meaning that not all non-unison classes have only two specific intervals. | These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and the pseudo-Myhill property, meaning that not all non-unison classes have only two specific intervals. | ||
=Mathematical Properties of MOS= | =Mathematical Properties of MOS= | ||
Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a [[http://en.wikipedia.org/wiki/Farey_sequence#Farey_neighbours|Farey pair]], meaning that a/b < c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 ≤ t ≤ 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the [[http://en.wikipedia.org/wiki/Mediant_%28mathematics%29|mediant]] (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b < g < c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones. | Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a [[http://en.wikipedia.org/wiki/Farey_sequence#Farey_neighbours|Farey pair]], meaning that a/b < c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 ≤ t ≤ 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the [[http://en.wikipedia.org/wiki/Mediant_%28mathematics%29|mediant]] (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b < g < c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones. | ||
| Line 337: | Line 340: | ||
==[[MOSDiagrams]]== </pre></div> | ==[[MOSDiagrams]]== </pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>MOSScales</title></head><body><!-- ws:start:WikiTextTocRule:28:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="# | <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>MOSScales</title></head><body><!-- ws:start:WikiTextTocRule:28:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Mathematical Definition">Mathematical Definition</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Mathematical Properties of MOS">Mathematical Properties of MOS</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#Visualizing MOS: Generator Chains, Pitch Space, and Hierarchies">Visualizing MOS: Generator Chains, Pitch Space, and Hierarchies</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Classification of MOS">Classification of MOS</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --> | <a href="#Catalog of MOS">Catalog of MOS</a><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --> | <a href="#Variations on MOS Scales">Variations on MOS Scales</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | <a href="#MOS As Applied To Rhythms">MOS As Applied To Rhythms</a><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --> | <a href="#Algorithms">Algorithms</a><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --> | ||
<!-- ws:end:WikiTextTocRule:43 --><hr /> | <!-- ws:end:WikiTextTocRule:43 --><hr /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | ||
An <strong>MOS</strong> or <strong>Moment Of Symmetry</strong> is a scale in which every interval except for the period comes in two sizes. The term &quot;MOS,&quot; and the method of scale construction it entails, were invented by Erv Wilson in 1975. His original paper can be found at <a class="wiki_link_ext" href="http://anaphoria.com/mos.PDF" rel="nofollow">http://anaphoria.com/mos.PDF</a>. There is also an introduction at <a class="wiki_link_ext" href="http://anaphoria.com/wilsonintroMOS.html" rel="nofollow">http://anaphoria.com/wilsonintroMOS.html</a>.<br /> | |||
<br /> | |||
Sometimes, scales are defined with respect to a period and an additional &quot;equivalence interval,&quot; considered to be the interval at which pitch classes repeat. MOS's in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called <strong>Multi-MOS</strong>'s. MOS's in which the equivalence interval is equal the period are sometimes called <strong>Strict MOS</strong>'s. MOS's in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.<br /> | |||
<br /> | |||
With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term <a class="wiki_link" href="/Distributional%20Evenness">distributionally even scale</a>, with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as <em>well-formed scales</em>, the term used in the 1989 paper by Norman Carey and David Clampitt. A great deal of interesting work has been done on scales in academic circles extending these ideas.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Mathematical Definition"></a><!-- ws:end:WikiTextHeadingRule:2 -->Mathematical Definition</h1> | |||
An MOS specifically consists of:<br /> | |||
<br /> | <br /> | ||
1. A period &quot;P&quot; (of any size but most commonly the octave or a 1/N fraction of an octave)<br /> | 1. A period &quot;P&quot; (of any size but most commonly the octave or a 1/N fraction of an octave)<br /> | ||
2. A generator &quot;g&quot; (of any size, for example 700 cents in 12 equal temperament) which is added repeatedly to make a chain of scale steps, starting from the unison or 0 cents scale step, and then reducing to within the period<br /> | 2. A generator &quot;g&quot; (of any size, for example 700 cents in 12 equal temperament) which is added repeatedly to make a chain of scale steps, starting from the unison or 0 cents scale step, and then reducing to within the period<br /> | ||
3. Exactly | 3. Exactly two sizes of scale steps (Large and small, often written &quot;L&quot; and &quot;s&quot;)<br /> | ||
4. Where <em>each</em> number of scale steps, or generic interval, within the scale occurs in no more than two different sizes, and in exactly two if the interval is not a multiple of the period<br /> | 4. Where <em>each</em> number of scale steps, or generic interval, within the scale occurs in no more than two different sizes, and in exactly two if the interval is not a multiple of the period<br /> | ||
5. The unison or starting point of the scale is then allowed to be transferred to any scale degree--all the modes of an MOS are legal.<br /> | 5. The unison or starting point of the scale is then allowed to be transferred to any scale degree--all the modes of an MOS are legal.<br /> | ||
| Line 353: | Line 363: | ||
These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and the pseudo-Myhill property, meaning that not all non-unison classes have only two specific intervals.<br /> | These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and the pseudo-Myhill property, meaning that not all non-unison classes have only two specific intervals.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Mathematical Properties of MOS"></a><!-- ws:end:WikiTextHeadingRule:4 -->Mathematical Properties of MOS</h1> | |||
Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Farey_sequence#Farey_neighbours" rel="nofollow">Farey pair</a>, meaning that a/b &lt; c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 ≤ t ≤ 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mediant_%28mathematics%29" rel="nofollow">mediant</a> (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b &lt; g &lt; c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones.<br /> | Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Farey_sequence#Farey_neighbours" rel="nofollow">Farey pair</a>, meaning that a/b &lt; c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 ≤ t ≤ 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mediant_%28mathematics%29" rel="nofollow">mediant</a> (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b &lt; g &lt; c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones.<br /> | ||
<br /> | <br /> | ||