MOS scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 143005075 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 143015127 - Original comment: ** |
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<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:genewardsmith|genewardsmith]] and made on <tt>2010-05-18 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-18 20:15:36 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>143015127</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> | ||
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Since numbers tend to be dry, Graham Breed has proposed a [[MOSNamingScheme|naming scheme for MOS scales]]. | Since numbers tend to be dry, Graham Breed has proposed a [[MOSNamingScheme|naming scheme for MOS scales]]. | ||
===MOS in equal temperaments=== | |||
In an equal temperament, all intervals are integer multiples of a smallest unit. If the equal temperament is N-EDO and the period is an octave, the sizes of the large and small steps will be p/N and q/N, with p > q. We then have L(p/N) + s(q/N) = 1, which on multiplying through by N gives us | |||
Lp + sq = N. | |||
which is a linear diophantine equation. Solving this by standard methods, and requiring L and s to be positive, gives us the [L, s] pair for the MOS. If some other quantity of equal steps gives the period, we may make the appropriate adjustment. | |||
===Blackwood R constant=== | ===Blackwood R constant=== | ||
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When R is less than 1, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing g with 1 - g if necessary, we can reduce always to the case where R>1 (or R<1 if we prefer.) | When R is less than 1, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing g with 1 - g if necessary, we can reduce always to the case where R>1 (or R<1 if we prefer.) | ||
Below is a list of MOS with number of elements from 5 to 10 | ==Catalog of MOS== | ||
Below is a list of MOS with number of elements from 5 to 10. | |||
Not all mathematical possibilities are listed - solutions of the equation that would yield too "exotic" scale steps (too small/too big diffference between s and L) are excluded. (The concrete - sort of arbitrary - restrictions applied were: a solution appears if 7/6 < L/s < 5.) | Not all mathematical possibilities are listed - solutions of the equation that would yield too "exotic" scale steps (too small/too big diffference between s and L) are excluded. (The concrete - sort of arbitrary - restrictions applied were: a solution appears if 7/6 < L/s < 5.) | ||
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Since numbers tend to be dry, Graham Breed has proposed a <a class="wiki_link" href="/MOSNamingScheme">naming scheme for MOS scales</a>.<br /> | Since numbers tend to be dry, Graham Breed has proposed a <a class="wiki_link" href="/MOSNamingScheme">naming scheme for MOS scales</a>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="MOS scales-Classification of MOS-Blackwood R constant"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="MOS scales-Classification of MOS-MOS in equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:6 -->MOS in equal temperaments</h3> | ||
In an equal temperament, all intervals are integer multiples of a smallest unit. If the equal temperament is N-EDO and the period is an octave, the sizes of the large and small steps will be p/N and q/N, with p &gt; q. We then have L(p/N) + s(q/N) = 1, which on multiplying through by N gives us<br /> | |||
<br /> | |||
Lp + sq = N.<br /> | |||
<br /> | |||
which is a linear diophantine equation. Solving this by standard methods, and requiring L and s to be positive, gives us the [L, s] pair for the MOS. If some other quantity of equal steps gives the period, we may make the appropriate adjustment.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="MOS scales-Classification of MOS-Blackwood R constant"></a><!-- ws:end:WikiTextHeadingRule:8 -->Blackwood R constant</h3> | |||
In the context of the &quot;recognizable diatonic&quot; scales deriving from the Farey pair (1/2, 3/5) <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr." rel="nofollow">Easley Blackwood Jr.</a> defined a characterizing constant R which we may generalize to any MOS as follows. If a/b &lt; g &lt; c/d is a generator with the given Farey pair, take the ratio of relative errors R = (bg - a)/(c - dg). Since this is a ratio of positive numbers, it is positive. As g tends towards a/b it tends to zero, and as g goes to c/d R goes to infinity. When g equals (a + c)/(b + d) it takes the value 1, and the range of propriety is 1/2 &lt;= R &lt;= 2.<br /> | In the context of the &quot;recognizable diatonic&quot; scales deriving from the Farey pair (1/2, 3/5) <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr." rel="nofollow">Easley Blackwood Jr.</a> defined a characterizing constant R which we may generalize to any MOS as follows. If a/b &lt; g &lt; c/d is a generator with the given Farey pair, take the ratio of relative errors R = (bg - a)/(c - dg). Since this is a ratio of positive numbers, it is positive. As g tends towards a/b it tends to zero, and as g goes to c/d R goes to infinity. When g equals (a + c)/(b + d) it takes the value 1, and the range of propriety is 1/2 &lt;= R &lt;= 2.<br /> | ||
<br /> | <br /> | ||
When R is less than 1, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing g with 1 - g if necessary, we can reduce always to the case where R&gt;1 (or R&lt;1 if we prefer.)<br /> | When R is less than 1, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing g with 1 - g if necessary, we can reduce always to the case where R&gt;1 (or R&lt;1 if we prefer.)<br /> | ||
<br /> | <br /> | ||
Below is a list of MOS with number of elements from 5 to 10 | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="MOS scales-Catalog of MOS"></a><!-- ws:end:WikiTextHeadingRule:10 -->Catalog of MOS</h2> | ||
Below is a list of MOS with number of elements from 5 to 10.<br /> | |||
Not all mathematical possibilities are listed - solutions of the equation that would yield too &quot;exotic&quot; scale steps (too small/too big diffference between s and L) are excluded. (The concrete - sort of arbitrary - restrictions applied were: a solution appears if 7/6 &lt; L/s &lt; 5.)<br /> | Not all mathematical possibilities are listed - solutions of the equation that would yield too &quot;exotic&quot; scale steps (too small/too big diffference between s and L) are excluded. (The concrete - sort of arbitrary - restrictions applied were: a solution appears if 7/6 &lt; L/s &lt; 5.)<br /> | ||
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<span style="color: #0000ee;"> </span><br /> | <span style="color: #0000ee;"> </span><br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="MOS scales-MOS As Applied To Rhythms"></a><!-- ws:end:WikiTextHeadingRule:12 -->MOS As Applied To Rhythms</h2> | ||
David Canright was the first to suggest Fibonacci Rhythms in 1/1. This lead to Kraig Grady to be the first to apply MOS patterns to rhythms. Two papers on the subject can be found here <a class="wiki_link_ext" href="http://anaphoria.com/hora.PDF" rel="nofollow">http://anaphoria.com/hora.PDF</a> and <a class="wiki_link_ext" href="http://anaphoria.com/horo2.PDF" rel="nofollow">http://anaphoria.com/horo2.PDF</a><br /> | David Canright was the first to suggest Fibonacci Rhythms in 1/1. This lead to Kraig Grady to be the first to apply MOS patterns to rhythms. Two papers on the subject can be found here <a class="wiki_link_ext" href="http://anaphoria.com/hora.PDF" rel="nofollow">http://anaphoria.com/hora.PDF</a> and <a class="wiki_link_ext" href="http://anaphoria.com/horo2.PDF" rel="nofollow">http://anaphoria.com/horo2.PDF</a><br /> | ||
MOS structures and thinking can be applied to the design of rhythms as well. See <a class="wiki_link" href="/MOS%20Rhythm%20Tutorial">MOS Rhythm Tutorial</a><br /> | MOS structures and thinking can be applied to the design of rhythms as well. See <a class="wiki_link" href="/MOS%20Rhythm%20Tutorial">MOS Rhythm Tutorial</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="MOS scales-Algorithms"></a><!-- ws:end:WikiTextHeadingRule:14 -->Algorithms</h2> | ||
Below is some Maple code for various mathematical routines having to do with MOS. If you have access to Maple, you can of course copy and run these programs. Even if you do not, since Maple code makes better pseudocode than most languages or computer algebra packages afford, it can be used as pseudocode. For that purpose, it will be helpful to know that &quot;modp(x, n)&quot; means reducing x mod the integer n to 0, 1, ..., n-1 not only when x is an integer, but also when it is a rational number with denominator prime to n. In that case, p/q mod n = r means p = qr mod n.<br /> | Below is some Maple code for various mathematical routines having to do with MOS. If you have access to Maple, you can of course copy and run these programs. Even if you do not, since Maple code makes better pseudocode than most languages or computer algebra packages afford, it can be used as pseudocode. For that purpose, it will be helpful to know that &quot;modp(x, n)&quot; means reducing x mod the integer n to 0, 1, ..., n-1 not only when x is an integer, but also when it is a rational number with denominator prime to n. In that case, p/q mod n = r means p = qr mod n.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="MOS scales-MOSDiagrams"></a><!-- ws:end:WikiTextHeadingRule:16 --><a class="wiki_link" href="/MOSDiagrams">MOSDiagrams</a></h2> | ||
</body></html></pre></div> | </body></html></pre></div> | ||