MOS scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 243034753 - Original comment: ** |
Wikispaces>igliashon **Imported revision 245924913 - 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:igliashon|igliashon]] and made on <tt>2011-08-14 19:50:17 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>245924913</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 14: | Line 14: | ||
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. | 3. Exactly two 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 | 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. | ||
Condition Four is [[http://en.wikipedia.org/wiki/Myhill%27s_property|Myhill's property]] where, as a [[periodic scale]], the scale has every generic interval aside from the initial unison interval and intervals some number of periods from it having exactly two specific intervals. Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a [[http://en.wikipedia.org/wiki/Continued_fraction|convergent or semiconvergent]] of the ratio g/P of the generator and the period. | Condition Four is [[http://en.wikipedia.org/wiki/Myhill%27s_property|Myhill's property]] where, as a [[periodic scale]], the scale has every generic interval aside from the initial unison interval and intervals some number of periods from it having exactly two specific intervals. Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a [[http://en.wikipedia.org/wiki/Continued_fraction|convergent or semiconvergent]] of the ratio g/P of the generator and the period. | ||
| Line 25: | Line 25: | ||
The term MOS and the scale construction method were invented by Erv Wilson in 1975. His original paper can be found here [[http://anaphoria.com/mos.PDF]]. There is also an introduction [[http://anaphoria.com/wilsonintroMOS.html]]. In academic music theory, MOS are known as //well-formed scales// and the introduction of the concept is attributed to a 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. | The term MOS and the scale construction method were invented by Erv Wilson in 1975. His original paper can be found here [[http://anaphoria.com/mos.PDF]]. There is also an introduction [[http://anaphoria.com/wilsonintroMOS.html]]. In academic music theory, MOS are known as //well-formed scales// and the introduction of the concept is attributed to a 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 Properties of MOS= | =Mathematical Properties of MOS= | ||
| Line 37: | Line 36: | ||
=Classification of MOS= | =Classification of MOS= | ||
Since MOS scales always consist of some number of large steps and some number of small steps, they can be classified simply by the number of large steps and the number of small steps, in the form #L#s--e.g., the diatonic scale can be described as [5L2s] (5 large steps and 2 small steps) or simply [5, 2]. It is typical to ignore the period when specifying MOS scales and instead use the number of large and small steps that make up the interval of equivalence (typically assumed to be the octave--a frequency ratio of 2/1--unless otherwise specified). For instance, the diminished scale in 12-TET is typically classified as 4L4s rather than 1L1s, since there are 4 large and 4 small steps that make up an octave. | |||
Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace g with 1-g and use the complementary pair if g is in the left hand side. This method is rarely used in discussions, however. | |||
The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the "Ls" routine) and for starting from an Ls pair and going to the mediant (the "medi" routine.) The Ls routine uses [[http://en.wikipedia.org/wiki/Modular_multiplicative_inverse|modular inverses]], whereas the medi routine uses continued fractions. | The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the "Ls" routine) and for starting from an Ls pair and going to the mediant (the "medi" routine.) The Ls routine uses [[http://en.wikipedia.org/wiki/Modular_multiplicative_inverse|modular inverses]], whereas the medi routine uses continued fractions. | ||
| Line 316: | Line 317: | ||
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. | 3. Exactly two 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 | 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 /> | ||
<br /> | <br /> | ||
Condition Four is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Myhill%27s_property" rel="nofollow">Myhill's property</a> where, as a <a class="wiki_link" href="/periodic%20scale">periodic scale</a>, the scale has every generic interval aside from the initial unison interval and intervals some number of periods from it having exactly two specific intervals. Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction" rel="nofollow">convergent or semiconvergent</a> of the ratio g/P of the generator and the period.<br /> | Condition Four is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Myhill%27s_property" rel="nofollow">Myhill's property</a> where, as a <a class="wiki_link" href="/periodic%20scale">periodic scale</a>, the scale has every generic interval aside from the initial unison interval and intervals some number of periods from it having exactly two specific intervals. Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction" rel="nofollow">convergent or semiconvergent</a> of the ratio g/P of the generator and the period.<br /> | ||
| Line 327: | Line 328: | ||
<br /> | <br /> | ||
The term MOS and the scale construction method were invented by Erv Wilson in 1975. His original paper can be found here <a class="wiki_link_ext" href="http://anaphoria.com/mos.PDF" rel="nofollow">http://anaphoria.com/mos.PDF</a>. There is also an introduction <a class="wiki_link_ext" href="http://anaphoria.com/wilsonintroMOS.html" rel="nofollow">http://anaphoria.com/wilsonintroMOS.html</a>. In academic music theory, MOS are known as <em>well-formed scales</em> and the introduction of the concept is attributed to a 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 /> | The term MOS and the scale construction method were invented by Erv Wilson in 1975. His original paper can be found here <a class="wiki_link_ext" href="http://anaphoria.com/mos.PDF" rel="nofollow">http://anaphoria.com/mos.PDF</a>. There is also an introduction <a class="wiki_link_ext" href="http://anaphoria.com/wilsonintroMOS.html" rel="nofollow">http://anaphoria.com/wilsonintroMOS.html</a>. In academic music theory, MOS are known as <em>well-formed scales</em> and the introduction of the concept is attributed to a 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 /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><!-- ws:end:WikiTextHeadingRule:2 --> </h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><!-- ws:end:WikiTextHeadingRule:2 --> </h1> | ||
<!-- 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> | <!-- 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> | ||
| Line 339: | Line 339: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Classification of MOS"></a><!-- ws:end:WikiTextHeadingRule:6 -->Classification of MOS</h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Classification of MOS"></a><!-- ws:end:WikiTextHeadingRule:6 -->Classification of MOS</h1> | ||
Since MOS scales always consist of some number of large steps and some number of small steps, they can be classified simply by the number of large steps and the number of small steps, in the form #L#s--e.g., the diatonic scale can be described as [5L2s] (5 large steps and 2 small steps) or simply [5, 2]. It is typical to ignore the period when specifying MOS scales and instead use the number of large and small steps that make up the interval of equivalence (typically assumed to be the octave--a frequency ratio of 2/1--unless otherwise specified). For instance, the diminished scale in 12-TET is typically classified as 4L4s rather than 1L1s, since there are 4 large and 4 small steps that make up an octave. <br /> | |||
<br /> | |||
Alternatively, we could give a mediant for a Farey pair associated to the MOS, where this mediant is less than any generator for the MOS. In other words, we use the right hand part of the Farey pair interval, which means we must replace g with 1-g and use the complementary pair if g is in the left hand side. This method is rarely used in discussions, however.<br /> | |||
<br /> | <br /> | ||
The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the &quot;Ls&quot; routine) and for starting from an Ls pair and going to the mediant (the &quot;medi&quot; routine.) The Ls routine uses <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Modular_multiplicative_inverse" rel="nofollow">modular inverses</a>, whereas the medi routine uses continued fractions.<br /> | The two systems are equivalent; in the Algorithms section you will find code for routines starting from the mediant and going to the Ls pair (the &quot;Ls&quot; routine) and for starting from an Ls pair and going to the mediant (the &quot;medi&quot; routine.) The Ls routine uses <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Modular_multiplicative_inverse" rel="nofollow">modular inverses</a>, whereas the medi routine uses continued fractions.<br /> | ||