Mathematics of MOS: Difference between revisions
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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: | : This revision was by author [[User:kraiggrady|kraiggrady]] and made on <tt>2013-09-30 02:43:32 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>455293488</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | <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]] | ||
=Mathematical Definition of MOS= | =Mathematical Definition of MOS= | ||
An MOS specifically consists of: | An MOS specifically consists of: | ||
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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. No more than 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. | ||
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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== | ==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. | 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 define the range of propriety, can also be expressed in terms of the ? and Box functions as Box(r - 2^(-n-1) and Box(r + 2^(-n-1)). 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)), and the proper generators will be Box(r) < g < Box(r + 2^(-n-1)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/21 < g < 1/10, and will be proper if 2/21 < g < 3/31. Hence 7/72, a generator for miracle temperament, will define a MOS but it will not be proper since 7/72 > 3/31 = Box(3/2048 + 1/4096)). | 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 define the range of propriety, can also be expressed in terms of the ? and Box functions as Box(r - 2^(-n-1) and Box(r + 2^(-n-1)). 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)), and the proper generators will be Box(r) < g < Box(r + 2^(-n-1)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/21 < g < 1/10, and will be proper if 2/21 < g < 3/31. Hence 7/72, a generator for miracle temperament, will define a MOS but it will not be proper since 7/72 > 3/31 = Box(3/2048 + 1/4096)). | ||
==MOS in equal temperaments== | ==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 | 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 | ||
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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. | 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== | ||
In the context of the "recognizable diatonic" scales deriving from the Farey pair [1/2, 3/5] [[http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr.|Easley Blackwood Jr.]] defined a characterizing constant R which we may generalize to any MOS as follows. If a/b < g < 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 <= R <= 2. | In the context of the "recognizable diatonic" scales deriving from the Farey pair [1/2, 3/5] [[http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr.|Easley Blackwood Jr.]] defined a characterizing constant R which we may generalize to any MOS as follows. If a/b < g < 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 <= R <= 2. | ||
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w := n[nops(n)]; | w := n[nops(n)]; | ||
if type(x, rational) and modp(denom(x), 2)=0 then RETURN(w) fi; | if type(x, rational) and modp(denom(x), 2)=0 then RETURN(w) fi; | ||
evalf(w) end: | evalf(w) end:</pre></div> | ||
</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>Mathematics of MOS</title></head><body><!-- ws:start:WikiTextTocRule:16:&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:16 --><!-- ws:start:WikiTextTocRule:17: --><a href="#Mathematical Definition of MOS">Mathematical Definition of MOS</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Mathematical Properties of MOS">Mathematical Properties of MOS</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Visualizing MOS: Generator Chains, Pitch Space, and Hierarchies">Visualizing MOS: Generator Chains, Pitch Space, and Hierarchies</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Classification of MOS">Classification of MOS</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Algorithms">Algorithms</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <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>Mathematics of MOS</title></head><body><!-- ws:start:WikiTextTocRule:16:&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:16 --><!-- ws:start:WikiTextTocRule:17: --><a href="#Mathematical Definition of MOS">Mathematical Definition of MOS</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Mathematical Properties of MOS">Mathematical Properties of MOS</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Visualizing MOS: Generator Chains, Pitch Space, and Hierarchies">Visualizing MOS: Generator Chains, Pitch Space, and Hierarchies</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Classification of MOS">Classification of MOS</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Algorithms">Algorithms</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | ||
<!-- ws:end:WikiTextTocRule:25 --> | <!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Mathematical Definition of MOS"></a><!-- ws:end:WikiTextHeadingRule:0 -->Mathematical Definition of MOS</h1> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Mathematical Definition of MOS"></a><!-- ws:end:WikiTextHeadingRule:0 -->Mathematical Definition of MOS</h1> | |||
An MOS specifically consists of:<br /> | 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. | 3. No more than 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 /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Classification of MOS-Classification via the ? function"></a><!-- ws:end:WikiTextHeadingRule:8 -->Classification via the ? function</h2> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Classification of MOS-Classification via the ? function"></a><!-- ws:end:WikiTextHeadingRule:8 -->Classification via the ? function</h2> | ||
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 /> | 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 /> | <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 define the range of propriety, can also be expressed in terms of the ? and Box functions as Box(r - 2^(-n-1) and Box(r + 2^(-n-1)). 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)), and the proper generators will be Box(r) &lt; g &lt; Box(r + 2^(-n-1)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/21 &lt; g &lt; 1/10, and will be proper if 2/21 &lt; g &lt; 3/31. Hence 7/72, a generator for miracle temperament, will define a MOS but it will not be proper since 7/72 &gt; 3/31 = Box(3/2048 + 1/4096)).<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 define the range of propriety, can also be expressed in terms of the ? and Box functions as Box(r - 2^(-n-1) and Box(r + 2^(-n-1)). 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)), and the proper generators will be Box(r) &lt; g &lt; Box(r + 2^(-n-1)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/21 &lt; g &lt; 1/10, and will be proper if 2/21 &lt; g &lt; 3/31. Hence 7/72, a generator for miracle temperament, will define a MOS but it will not be proper since 7/72 &gt; 3/31 = Box(3/2048 + 1/4096)).<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Classification of MOS-MOS in equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:10 -->MOS in equal temperaments</h2> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Classification of MOS-MOS in equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:10 -->MOS in equal temperaments</h2> | ||
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 /> | ||
<br /> | <br /> | ||
Lp + sq = N.<br /> | Lp + sq = N.<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Classification of MOS-Blackwood R constant"></a><!-- ws:end:WikiTextHeadingRule:12 -->Blackwood R constant</h2> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Classification of MOS-Blackwood R constant"></a><!-- ws:end:WikiTextHeadingRule:12 -->Blackwood R constant</h2> | ||
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 /> | ||