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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | The '''step pattern product''' is an operation on two abstract scale [[step pattern]]s of the same length that produces a new abstract step pattern. The operation does not concern the information of the relative sizes of the scale steps (so that for example, [[5L 2s|diatonic]] and [[antidiatonic]] are represented by the same pattern AABAAAB). These step patterns are called factors, and their product is taken by doing the following: |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:guest|guest]] and made on <tt>2013-01-23 03:16:23 UTC</tt>.<br>
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| : The original revision id was <tt>400665958</tt>.<br>
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| : The revision comment was: <tt></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>
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| <h4>Original Wikitext content:</h4>
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| <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">A **word** is a sequence of letters from some finite alphabet. Words can be used to represent scales; for example "aabaaab" represents the ([[meantone]]) major scale if 'a' represents a whole step and 'b' a half step.
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| [[http://www.shahdna.tk/news_world/|الاخبار ]]- [[http://www.shahdna.tk/sports/|اخبار الرياضة]]- [[http://www.shahdna.tk/woman/|عالم المراة]] | |
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| [[http://www.shahdna.tk/movie/|شاهدنا افلام اون لاين ريال مدريد برشلونة كورة ]]
| | * Take two step patterns. Conventionally, these are written with different sets of letters, for example AABAAAB and CCDCCDC. |
| [[http://www.shahdna.tk|www.shahdna.tk]]
| | * Pair up the entries in the step patterns: (A,C)(A,C)(B,D)(A,C)(A,C)(A,D)(B,C) |
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| | * Assign each ordered pair its own new symbol: EEFEEGH, again, conventionally written with a new set of letters. |
| Given two words of the same length ("factors"), their **product word** is the word whose alphabet consists of ordered pairs of the alphabets of its factor words, and whose //n//th letter is the ordered pair of the //n//th letters of its factors.
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| For example, the product word of "aabaaab" and "xxyxyxy" is "(a,x)(a,x)(b,y)(a,x)(a,y)(a,x)(b,y)". For brevity, we can substitute each ordered pair of letters by a new single letter and say this is equivalent to the word "rrsrtrs". This construction has an obvious generalization to the product of three or more words.
| | This construction has an obvious generalization to the product of three or more scales. |
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| The importance of product words in music theory is that every [[Fokker blocks|Fokker block]] can be expressed as the product word of two or more [[distributional evenness|distributionally even]] scales in a unique way. Fokker blocks are therefore equivalent to product words of DE scales of the same size. If one or both of the DE scales are rotated (into different [[mode|modes]]), then the product Fokker block scale is not always a mode, but is often a [[dome]] instead.</pre></div>
| | Once the resulting step pattern has been acquired, one may arbitrarily assign the different types of steps to different relative sizes. Any permutation is possible, such as LLm<sub>1</sub>LLm<sub>2</sub>s, ssLssm<sub>1</sub>m<sub>2</sub>, etc; these scales are all [[sister]]s. |
| <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>Product word</title></head><body>A <strong>word</strong> is a sequence of letters from some finite alphabet. Words can be used to represent scales; for example &quot;aabaaab&quot; represents the (<a class="wiki_link" href="/meantone">meantone</a>) major scale if 'a' represents a whole step and 'b' a half step.<br /> | | == Related properties == |
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| | * Every [[distributionally even]] ternary scale is the step pattern product of two MOS words. However, the step pattern product of two MOS scales (even if ternary) need not be distributionally even. |
| <a class="wiki_link_ext" href="http://www.shahdna.tk/news_world/" rel="nofollow">الاخبار </a>- <a class="wiki_link_ext" href="http://www.shahdna.tk/sports/" rel="nofollow">اخبار الرياضة</a>- <a class="wiki_link_ext" href="http://www.shahdna.tk/woman/" rel="nofollow">عالم المراة</a><br />
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| | == Utilization == |
| <a class="wiki_link_ext" href="http://www.shahdna.tk/movie/" rel="nofollow">شاهدنا افلام اون لاين ريال مدريد برشلونة كورة </a><br />
| | The step pattern product can be used to construct complex scales such as the [[Zarlino]] scale from MOS scales. The Zarlino scale (and its sisters) may be represented by taking the product of ABABABA and AABAAAB, resulting in ABCBABC. |
| <a class="wiki_link_ext" href="http://www.shahdna.tk" rel="nofollow">www.shahdna.tk</a><br />
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| </span><br />
| | In general, every [[Fokker block]] can be expressed as the product of two or more [[MOS scale]]s in a unique way. Fokker blocks are therefore equivalent to products of MOS scales of the same size. If one or both of the MOS scales are rotated into different [[mode]]s relative to the original inputs, then the product Fokker block scale is not always a mode, but is often a [[dome]] of the original Fokker block instead. |
| Given two words of the same length (&quot;factors&quot;), their <strong>product word</strong> is the word whose alphabet consists of ordered pairs of the alphabets of its factor words, and whose <em>n</em>th letter is the ordered pair of the <em>n</em>th letters of its factors.<br />
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| | == Mathematical context == |
| For example, the product word of &quot;aabaaab&quot; and &quot;xxyxyxy&quot; is &quot;(a,x)(a,x)(b,y)(a,x)(a,y)(a,x)(b,y)&quot;. For brevity, we can substitute each ordered pair of letters by a new single letter and say this is equivalent to the word &quot;rrsrtrs&quot;. This construction has an obvious generalization to the product of three or more words.<br />
| | In general mathematical theory (and more advanced pages on the wiki), what this page refers to as "step patterns" are called "words", and the "step pattern product" is a "product word". |
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| The importance of product words in music theory is that every <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> can be expressed as the product word of two or more <a class="wiki_link" href="/distributional%20evenness">distributionally even</a> scales in a unique way. Fokker blocks are therefore equivalent to product words of DE scales of the same size. If one or both of the DE scales are rotated (into different <a class="wiki_link" href="/mode">modes</a>), then the product Fokker block scale is not always a mode, but is often a <a class="wiki_link" href="/dome">dome</a> instead.</body></html></pre></div>
| | == Examples == |
| | ''See also: [[:Category:Product words]]'' |
| | * [[2a5b * 3a4b]] |
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| | [[Category:Math]] |
| | [[Category:Scale]] |
| | [[Category:Combinatorics on words]] |