MOS scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 143260827 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 143380977 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-20 00:40:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>143380977</tt>.<br> | ||
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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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If the period is assumed to be 2^(1/n) for some integer n, we can give instead the total number of large and small steps in the octave, instead of just the period, and this is commonly done. In this case, GCD(L, s) gives the number of periods in an octave. | If the period is assumed to be 2^(1/n) for some integer n, we can give instead the total number of large and small steps in the octave, instead of just the period, and this is commonly done. In this case, GCD(L, s) gives the number of periods in an octave. | ||
===Classification via the ? function=== | |||
Yet another way of classifying MOS is via [[http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function|Minkowski's ? function]]. Here ?(x) is a continuous increasing function from the real numbers to the real numbers which has some peculiar properties, one being that it sends rational numbers to [[http://en.wikipedia.org/wiki/Dyadic_rational|dyadic rationals]]. Hence if q is a rational number 0 < q < 1 in use in the mediant system of classifying MOS, r = ?(q) = A/2^n will be a dyadic rational number which can also be used. Note that the ? function is invertible, and it and its inverse function, the Box function, have code given for them in the algorithms section at the bottom of the article. | |||
The integer n in the denominator of r (with A assumed to be odd) is the order (or n+1 is, according to some sources) of q in the [[http://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree|Stern-Brocot tree]]. The two neighboring numbers of order n+1, which will be Box(r - 2^(-n-1) and Box(r + 2^(-n-1)) can also be expressed in terms of the ? and Box functions. If r represents a MOS, the range of possible values for a generator of the MOS will be Box(r) < g < Box(r + 2^(-n)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/11 < g < 1/10. | |||
===Names for MOS=== | |||
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]]. | ||
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If the period is assumed to be 2^(1/n) for some integer n, we can give instead the total number of large and small steps in the octave, instead of just the period, and this is commonly done. In this case, GCD(L, s) gives the number of periods in an octave.<br /> | If the period is assumed to be 2^(1/n) for some integer n, we can give instead the total number of large and small steps in the octave, instead of just the period, and this is commonly done. In this case, GCD(L, s) gives the number of periods in an octave.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="MOS scales-Classification of MOS-Classification via the ? function"></a><!-- ws:end:WikiTextHeadingRule:6 -->Classification via the ? function</h3> | |||
Yet another way of classifying MOS is via <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function" rel="nofollow">Minkowski's ? function</a>. Here ?(x) is a continuous increasing function from the real numbers to the real numbers which has some peculiar properties, one being that it sends rational numbers to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dyadic_rational" rel="nofollow">dyadic rationals</a>. Hence if q is a rational number 0 &lt; q &lt; 1 in use in the mediant system of classifying MOS, r = ?(q) = A/2^n will be a dyadic rational number which can also be used. Note that the ? function is invertible, and it and its inverse function, the Box function, have code given for them in the algorithms section at the bottom of the article.<br /> | |||
<br /> | |||
The integer n in the denominator of r (with A assumed to be odd) is the order (or n+1 is, according to some sources) of q in the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree" rel="nofollow">Stern-Brocot tree</a>. The two neighboring numbers of order n+1, which will be Box(r - 2^(-n-1) and Box(r + 2^(-n-1)) can also be expressed in terms of the ? and Box functions. If r represents a MOS, the range of possible values for a generator of the MOS will be Box(r) &lt; g &lt; Box(r + 2^(-n)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/11 &lt; g &lt; 1/10.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="MOS scales-Classification of MOS-Names for MOS"></a><!-- ws:end:WikiTextHeadingRule:8 -->Names for MOS</h3> | |||
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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="MOS scales-Classification of MOS-MOS in equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:10 -->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 /> | 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 /> | ||
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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 /> | 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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="MOS scales-Classification of MOS-Blackwood R constant"></a><!-- ws:end:WikiTextHeadingRule:12 -->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 /> | ||
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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&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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="MOS scales-Catalog of MOS"></a><!-- ws:end:WikiTextHeadingRule:14 -->Catalog of MOS</h2> | ||
Below is a list of MOS with number of elements from 5 to 10, plus some of the more significant larger MOS.<br /> | Below is a list of MOS with number of elements from 5 to 10, plus some of the more significant larger MOS.<br /> | ||
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Garibaldi[29] [12L 17s]<br /> | Garibaldi[29] [12L 17s]<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="MOS scales-MOS As Applied To Rhythms"></a><!-- ws:end:WikiTextHeadingRule:16 -->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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="MOS scales-Algorithms"></a><!-- ws:end:WikiTextHeadingRule:18 -->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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(u[1]+u[2]-numer(q)-denom(q))/(u[1]+u[2]) end:<br /> | (u[1]+u[2]-numer(q)-denom(q))/(u[1]+u[2]) end:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="MOS scales-MOSDiagrams"></a><!-- ws:end:WikiTextHeadingRule:20 --><a class="wiki_link" href="/MOSDiagrams">MOSDiagrams</a></h2> | ||
</body></html></pre></div> | </body></html></pre></div> | ||