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Wikispaces>acousticsoftombak **Imported revision 302670104 - Original comment: ** |
Wikispaces>acousticsoftombak **Imported revision 302670396 - 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:acousticsoftombak|acousticsoftombak]] and made on <tt>2012-02-17 02: | : This revision was by author [[User:acousticsoftombak|acousticsoftombak]] and made on <tt>2012-02-17 02:09:10 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>302670396</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> | ||
<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"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">For an intervallic system with </span>__<span style="color: black; font-size: 11pt;">n</span>__<span style="color: black; font-size: 11pt;"> divisions , </span><span style="color: #be00ff; font-size: 11pt;">[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|__AID__]]</span><span style="color: black; font-size: 11pt;"> is considered as [[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html|arithmetic sequence]] with divisions of system as terms of sequence. </span></span> | <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"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><span style="color: #000000; font-family: arial,sans-serif; font-size: 18px;">**Arithmetic irrational divisions**</span><span style="color: #000000;"><span style="font-family: arial,sans-serif;"> </span></span></span> | ||
<span style="direction: ltr; display: block; | |||
<span style="direction: ltr; display: block | <span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">For an intervallic system with </span>__<span style="color: black; font-size: 11pt;">n</span>__<span style="color: black; font-size: 11pt;"> divisions , </span><span style="color: #be00ff; font-size: 11pt;">[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|__AID__]]</span><span style="color: black; font-size: 11pt;"> is considered as [[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html|arithmetic sequence]] with divisions of system as terms of sequence. </span></span> | ||
<span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;"><span style="color: black; font-size: 11pt;"> If the first division is </span>__<span style="color: black; font-size: 11pt;">A1</span>__<span style="color: black; font-size: 11pt;"> and the last , </span>__<span style="color: black; font-size: 11pt;">An</span>__<span style="color: black; font-size: 11pt;"> , with common difference of </span>__<span style="color: black; font-size: 11pt;">d</span>__<span style="color: black; font-size: 11pt;"> , we have : </span></span> | |||
<span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;">A1 = A1 </span> | |||
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A2 = A1+d </span> | <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A2 = A1+d </span> | ||
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A3= A1+2d </span> | <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A3= A1+2d </span> | ||
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<span style="direction: ltr; font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">So sum of the divisions is </span>__<span style="color: black; font-size: 11pt;">Sn</span>__<span style="color: black; font-size: 11pt;"> :</span></span> | <span style="direction: ltr; font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">So sum of the divisions is </span>__<span style="color: black; font-size: 11pt;">Sn</span>__<span style="color: black; font-size: 11pt;"> :</span></span> | ||
<span style="direction: ltr; display: block | <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"> **<span style="color: black; font-size: 11pt;">Sn =( __n[2A1+(n-1)d])/2__</span>**</span> | ||
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> | <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> | ||
</span> | |||
<span style="direction: ltr; display: block; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;"> As we can consider </span>__<span style="color: black; font-size: 11pt;">Sn</span>__<span style="color: black; font-size: 11pt;"> of system to be 1200 cent or anything else (octavic or non-octavic system ) then </span>__<span style="color: black; font-size: 11pt;">d</span>__<span style="color: black; font-size: 11pt;"> is most important to make an __AID__ with __n__ divisions with __A1__.important to make an </span>__<span style="color: black; font-size: 11pt;">AID</span>__ with __n__<span style="color: black; font-size: 11pt;"> divisions with </span>__<span style="color: black; font-size: 11pt;">A1</span>__<span style="color: black; font-size: 11pt;">.</span></span><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> so, the common difference between divisions is : </span></span> | <span style="direction: ltr; display: block; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;"> As we can consider </span>__<span style="color: black; font-size: 11pt;">Sn</span>__<span style="color: black; font-size: 11pt;"> of system to be 1200 cent or anything else (octavic or non-octavic system ) then </span>__<span style="color: black; font-size: 11pt;">d</span>__<span style="color: black; font-size: 11pt;"> is most important to make an __AID__ with __n__ divisions with __A1__.important to make an </span>__<span style="color: black; font-size: 11pt;">AID</span>__ with __n__<span style="color: black; font-size: 11pt;"> divisions with </span>__<span style="color: black; font-size: 11pt;">A1</span>__<span style="color: black; font-size: 11pt;">.</span></span><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> so, the common difference between divisions is : </span></span> | ||
<span style="direction: ltr; display: block | <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"> **<span style="color: black; font-size: 11pt;">d =( __2(Sn - nA1))/((n(n-1))__</span>**</span> | ||
<span style="direction: ltr; display: block | <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"> By considering Sn=1200 , A1=70 , n=12 , __d__ will be 5.454545455 and our 12-tone scale is equal to:</span> | ||
<span style="font-family: Tahoma,Geneva,sans-serif;">[[#TOC-0.070.0145.455226.364312.727404.545]] **0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0**</span> | <span style="font-family: Tahoma,Geneva,sans-serif;">[[#TOC-0.070.0145.455226.364312.727404.545]] **0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0**</span> | ||
<span style="direction: ltr; display: block | <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;">Scales based on AID can be subsets of EDO if : Scales based on AID can be subsets of EDO if : </span><span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"><span style="color: black; font-size: 10pt;">1- we choose d=0 so ,<span style="color: black; font-size: 11pt;"> A1 = Sn/n</span> .. Consider </span>__<span style="color: black; font-size: 10pt;">n</span>__<span style="color: black; font-size: 10pt;">=8 and </span>__<span style="color: black; font-size: 10pt;">A1</span>__<span style="color: black; font-size: 10pt;">=150 , then we have 8-EDO .</span></span> | ||
<span style="direction: ltr; display: block; margin: 0cm 0cm 0pt; text-align: left | <span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"><span style="color: black; font-size: 10pt;">2- for a constant </span>__<span style="color: black; font-size: 10pt;">n</span>__<span style="color: black; font-size: 10pt;"> and different __A1__, if __d__ and (Sn/A1) are Integer number , we have a susbet of EDO or EDI( Equal divisions of Interval) .Consider __Sn = 1400__ , </span>__<span style="color: black; font-size: 10pt;">n</span>__<span style="color: black; font-size: 10pt;">=8 and </span>__<span style="color: black; font-size: 10pt;">A1</span>__<span style="color: black; font-size: 10pt;">=70 , then we have a subset of a 140-ED(1400.) with Degrees as 7 17 30 46 65 87 112 140 :</span></span> | ||
**0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000** | **0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000** | ||
<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;">[[#TOC-And-now-for:]]And now for </span><span style="font-family: Tahoma,Geneva,sans-serif;">Sn=1400 and n=8, | <span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;">[[#TOC-And-now-for:]]And now for </span><span style="font-family: Tahoma,Geneva,sans-serif;">Sn=1400 and n=8,</span> | ||
[[#TOC-If-A1-175.0-then-we-have-8-AID-1400]]If A1=175.0 then we have 8-AID(1400.) | <span style="font-family: Tahoma,Geneva,sans-serif;"> [[#TOC-If-A1-175.0-then-we-have-8-AID-1400]]If A1=175.0 then we have 8-AID(1400.)</span> | ||
[[#TOC-If-A1-56-then-we-have-700-AID-1400.]]If A1=56 then we have 700-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 28 73 135 214 310 423 553 700</span> | <span style="font-family: Tahoma,Geneva,sans-serif;"> [[#TOC-If-A1-56-then-we-have-700-AID-1400.]]If A1=56 then we have 700-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 28 73 135 214 310 423 553 700</span></span> | ||
[[#TOC-If-A1-87.5-then-we-have-112-AID-140]]<span style="color: black; font-size: 10pt;">If A1=87.5 then we have 112-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 7 16 27 40 55 72 91 112</span></span></span> | <span style="font-family: Tahoma,Geneva,sans-serif;"> [[#TOC-If-A1-87.5-then-we-have-112-AID-140]]<span style="color: black; font-size: 10pt;">If A1=87.5 then we have 112-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 7 16 27 40 55 72 91 112</span></span></span> | ||
<span style="font-family: Tahoma,Geneva,sans-serif;">__AID__ sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in __AID__<span style="color: black; font-size: 10pt;"> and __EDO__ with equal degree:</span></span> | <span style="font-family: Tahoma,Geneva,sans-serif;">__AID__ sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in __AID__<span style="color: black; font-size: 10pt;"> and __EDO__ with equal degree:</span></span> | ||
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==<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;">171.4285714 is point of intersection in these 3 trends:</span>== | ==<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;">171.4285714 is point of intersection in these 3 trends:</span>== | ||
[[image:http://sites.google.com/site/240edo/AIDO2.jpg align="center"]] | [[image:http://sites.google.com/site/240edo/AIDO2.jpg align="center"]] | ||
we can have different kinds of AID: | |||
<span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;">__<span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDO = Arithmetic irrational divisions of octave</span></span>__</span></span> | |||
<span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;">__<span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDINO = Arithmetic irrational divisions of irrational non-octave</span></span>__</span></span> | |||
<span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;">__<span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDRNO = Arithmetic irrational divisions of rational non-octave</span></span>__</span></span> | |||
<span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;">__<span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDRI = Arithmetic irrational divisions of rational interval</span></span>__</span></span> | |||
<span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;">__<span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDII = Arithmetic irrational divisions of irrational interval</span></span>__</span></span> | |||
<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;">[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|**Example : Baran scale**]]</span></pre></div> | <span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;">[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|**Example : Baran scale**]]</span></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>AID</title></head><body><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">For an intervallic system with </span><u><span style="color: black; font-size: 11pt;">n</span></u><span style="color: black; font-size: 11pt;"> divisions , </span><span style="color: #be00ff; font-size: 11pt;"><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow">__AID__</a></span><span style="color: black; font-size: 11pt;"> is considered as <a class="wiki_link_ext" href="http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html" rel="nofollow">arithmetic sequence</a> with divisions of system as terms of sequence. </span></span><br /> | <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>AID</title></head><body><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><span style="color: #000000; font-family: arial,sans-serif; font-size: 18px;"><strong>Arithmetic irrational divisions</strong></span><span style="color: #000000;"><span style="font-family: arial,sans-serif;"> </span></span></span><br /> | ||
<span style="direction: ltr; display: block; | <br /> | ||
<span style="direction: ltr; display: block | <span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">For an intervallic system with </span><u><span style="color: black; font-size: 11pt;">n</span></u><span style="color: black; font-size: 11pt;"> divisions , </span><span style="color: #be00ff; font-size: 11pt;"><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow">__AID__</a></span><span style="color: black; font-size: 11pt;"> is considered as <a class="wiki_link_ext" href="http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html" rel="nofollow">arithmetic sequence</a> with divisions of system as terms of sequence. </span></span><br /> | ||
<span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;"><span style="color: black; font-size: 11pt;"> If the first division is </span><u><span style="color: black; font-size: 11pt;">A1</span></u><span style="color: black; font-size: 11pt;"> and the last , </span><u><span style="color: black; font-size: 11pt;">An</span></u><span style="color: black; font-size: 11pt;"> , with common difference of </span><u><span style="color: black; font-size: 11pt;">d</span></u><span style="color: black; font-size: 11pt;"> , we have : </span></span><br /> | |||
<span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;">A1 = A1 </span><br /> | |||
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A2 = A1+d </span><br /> | <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A2 = A1+d </span><br /> | ||
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A3= A1+2d </span><br /> | <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A3= A1+2d </span><br /> | ||
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<span style="direction: ltr; font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">So sum of the divisions is </span><u><span style="color: black; font-size: 11pt;">Sn</span></u><span style="color: black; font-size: 11pt;"> :</span></span><br /> | <span style="direction: ltr; font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">So sum of the divisions is </span><u><span style="color: black; font-size: 11pt;">Sn</span></u><span style="color: black; font-size: 11pt;"> :</span></span><br /> | ||
<br /> | <br /> | ||
<span style="direction: ltr; display: block | <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"> <strong><span style="color: black; font-size: 11pt;">Sn =( <u>n[2A1+(n-1)d])/2</u></span></strong></span><br /> | ||
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> < | <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"><br /> | ||
</span><br /> | |||
<br /> | <br /> | ||
<span style="direction: ltr; display: block; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;"> As we can consider </span><u><span style="color: black; font-size: 11pt;">Sn</span></u><span style="color: black; font-size: 11pt;"> of system to be 1200 cent or anything else (octavic or non-octavic system ) then </span><u><span style="color: black; font-size: 11pt;">d</span></u><span style="color: black; font-size: 11pt;"> is most important to make an <u>AID</u> with <u>n</u> divisions with <u>A1</u>.important to make an </span><u><span style="color: black; font-size: 11pt;">AID</span></u> with <u>n</u><span style="color: black; font-size: 11pt;"> divisions with </span><u><span style="color: black; font-size: 11pt;">A1</span></u><span style="color: black; font-size: 11pt;">.</span></span><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> so, the common difference between divisions is : </span></span><br /> | <span style="direction: ltr; display: block; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;"> As we can consider </span><u><span style="color: black; font-size: 11pt;">Sn</span></u><span style="color: black; font-size: 11pt;"> of system to be 1200 cent or anything else (octavic or non-octavic system ) then </span><u><span style="color: black; font-size: 11pt;">d</span></u><span style="color: black; font-size: 11pt;"> is most important to make an <u>AID</u> with <u>n</u> divisions with <u>A1</u>.important to make an </span><u><span style="color: black; font-size: 11pt;">AID</span></u> with <u>n</u><span style="color: black; font-size: 11pt;"> divisions with </span><u><span style="color: black; font-size: 11pt;">A1</span></u><span style="color: black; font-size: 11pt;">.</span></span><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> so, the common difference between divisions is : </span></span><br /> | ||
<span style="direction: ltr; display: block | <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"> <strong><span style="color: black; font-size: 11pt;">d =( <u>2(Sn - nA1))/((n(n-1))</u></span></strong></span><br /> | ||
<br /> | <br /> | ||
<span style="direction: ltr; display: block | <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"> By considering Sn=1200 , A1=70 , n=12 , <u>d</u> will be 5.454545455 and our 12-tone scale is equal to:</span><br /> | ||
<span style="font-family: Tahoma,Geneva,sans-serif;"><!-- ws:start:WikiTextAnchorRule:2:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-0.070.0145.455226.364312.727404.545&quot; title=&quot;Anchor: TOC-0.070.0145.455226.364312.727404.545&quot;/&gt; --><a name="TOC-0.070.0145.455226.364312.727404.545"></a><!-- ws:end:WikiTextAnchorRule:2 --> <strong>0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0</strong></span><br /> | <span style="font-family: Tahoma,Geneva,sans-serif;"><!-- ws:start:WikiTextAnchorRule:2:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-0.070.0145.455226.364312.727404.545&quot; title=&quot;Anchor: TOC-0.070.0145.455226.364312.727404.545&quot;/&gt; --><a name="TOC-0.070.0145.455226.364312.727404.545"></a><!-- ws:end:WikiTextAnchorRule:2 --> <strong>0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0</strong></span><br /> | ||
<span style="direction: ltr; display: block | <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;">Scales based on AID can be subsets of EDO if : Scales based on AID can be subsets of EDO if : </span><span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"><span style="color: black; font-size: 10pt;">1- we choose d=0 so ,<span style="color: black; font-size: 11pt;"> A1 = Sn/n</span> .. Consider </span><u><span style="color: black; font-size: 10pt;">n</span></u><span style="color: black; font-size: 10pt;">=8 and </span><u><span style="color: black; font-size: 10pt;">A1</span></u><span style="color: black; font-size: 10pt;">=150 , then we have 8-EDO .</span></span><br /> | ||
<span style="direction: ltr; display: block; margin: 0cm 0cm 0pt; text-align: left | <span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"><span style="color: black; font-size: 10pt;">2- for a constant </span><u><span style="color: black; font-size: 10pt;">n</span></u><span style="color: black; font-size: 10pt;"> and different <u>A1</u>, if <u>d</u> and (Sn/A1) are Integer number , we have a susbet of EDO or EDI( Equal divisions of Interval) .Consider <u>Sn = 1400</u> , </span><u><span style="color: black; font-size: 10pt;">n</span></u><span style="color: black; font-size: 10pt;">=8 and </span><u><span style="color: black; font-size: 10pt;">A1</span></u><span style="color: black; font-size: 10pt;">=70 , then we have a subset of a 140-ED(1400.) with Degrees as 7 17 30 46 65 87 112 140 :</span></span><br /> | ||
<strong>0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000</strong><br /> | <strong>0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000</strong><br /> | ||
<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;"><!-- ws:start:WikiTextAnchorRule:3:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-And-now-for:&quot; title=&quot;Anchor: TOC-And-now-for:&quot;/&gt; --><a name="TOC-And-now-for:"></a><!-- ws:end:WikiTextAnchorRule:3 -->And now for </span><span style="font-family: Tahoma,Geneva,sans-serif;">Sn=1400 and n=8,<br /> | <span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;"><!-- ws:start:WikiTextAnchorRule:3:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-And-now-for:&quot; title=&quot;Anchor: TOC-And-now-for:&quot;/&gt; --><a name="TOC-And-now-for:"></a><!-- ws:end:WikiTextAnchorRule:3 -->And now for </span><span style="font-family: Tahoma,Geneva,sans-serif;">Sn=1400 and n=8,</span><br /> | ||
<!-- ws:start:WikiTextAnchorRule:4:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-If-A1-175.0-then-we-have-8-AID-1400&quot; title=&quot;Anchor: TOC-If-A1-175.0-then-we-have-8-AID-1400&quot;/&gt; --><a name="TOC-If-A1-175.0-then-we-have-8-AID-1400"></a><!-- ws:end:WikiTextAnchorRule:4 -->If A1=175.0 then we have 8-AID(1400.)<br /> | <span style="font-family: Tahoma,Geneva,sans-serif;"> <!-- ws:start:WikiTextAnchorRule:4:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-If-A1-175.0-then-we-have-8-AID-1400&quot; title=&quot;Anchor: TOC-If-A1-175.0-then-we-have-8-AID-1400&quot;/&gt; --><a name="TOC-If-A1-175.0-then-we-have-8-AID-1400"></a><!-- ws:end:WikiTextAnchorRule:4 -->If A1=175.0 then we have 8-AID(1400.)</span><br /> | ||
<!-- ws:start:WikiTextAnchorRule:5:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-If-A1-56-then-we-have-700-AID-1400.&quot; title=&quot;Anchor: TOC-If-A1-56-then-we-have-700-AID-1400.&quot;/&gt; --><a name="TOC-If-A1-56-then-we-have-700-AID-1400."></a><!-- ws:end:WikiTextAnchorRule:5 -->If A1=56 then we have 700-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 28 73 135 214 310 423 553 700</span><br /> | <span style="font-family: Tahoma,Geneva,sans-serif;"> <!-- ws:start:WikiTextAnchorRule:5:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-If-A1-56-then-we-have-700-AID-1400.&quot; title=&quot;Anchor: TOC-If-A1-56-then-we-have-700-AID-1400.&quot;/&gt; --><a name="TOC-If-A1-56-then-we-have-700-AID-1400."></a><!-- ws:end:WikiTextAnchorRule:5 -->If A1=56 then we have 700-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 28 73 135 214 310 423 553 700</span></span><br /> | ||
<!-- ws:start:WikiTextAnchorRule:6:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-If-A1-87.5-then-we-have-112-AID-140&quot; title=&quot;Anchor: TOC-If-A1-87.5-then-we-have-112-AID-140&quot;/&gt; --><a name="TOC-If-A1-87.5-then-we-have-112-AID-140"></a><!-- ws:end:WikiTextAnchorRule:6 --><span style="color: black; font-size: 10pt;">If A1=87.5 then we have 112-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 7 16 27 40 55 72 91 112</span></span></span><br /> | <span style="font-family: Tahoma,Geneva,sans-serif;"> <!-- ws:start:WikiTextAnchorRule:6:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-If-A1-87.5-then-we-have-112-AID-140&quot; title=&quot;Anchor: TOC-If-A1-87.5-then-we-have-112-AID-140&quot;/&gt; --><a name="TOC-If-A1-87.5-then-we-have-112-AID-140"></a><!-- ws:end:WikiTextAnchorRule:6 --><span style="color: black; font-size: 10pt;">If A1=87.5 then we have 112-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 7 16 27 40 55 72 91 112</span></span></span><br /> | ||
<br /> | <br /> | ||
<span style="font-family: Tahoma,Geneva,sans-serif;"><u>AID</u> sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in <u>AID</u><span style="color: black; font-size: 10pt;"> and <u>EDO</u> with equal degree:</span></span><br /> | <span style="font-family: Tahoma,Geneva,sans-serif;"><u>AID</u> sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in <u>AID</u><span style="color: black; font-size: 10pt;"> and <u>EDO</u> with equal degree:</span></span><br /> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-171.4285714 is point of intersection in these 3 trends:"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;">171.4285714 is point of intersection in these 3 trends:</span></h2> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-171.4285714 is point of intersection in these 3 trends:"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;">171.4285714 is point of intersection in these 3 trends:</span></h2> | ||
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<br /> | |||
<!-- ws:start:WikiTextRemoteImageRule:16:&lt;div style=&quot;text-align: center&quot;&gt;&lt;img src=&quot;http://sites.google.com/site/240edo/AIDO2.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/div&gt; --><div style="text-align: center"><img src="http://sites.google.com/site/240edo/AIDO2.jpg" alt="external image AIDO2.jpg" title="external image AIDO2.jpg" /></div><!-- ws:end:WikiTextRemoteImageRule:16 --><br /> | <!-- ws:start:WikiTextRemoteImageRule:16:&lt;div style=&quot;text-align: center&quot;&gt;&lt;img src=&quot;http://sites.google.com/site/240edo/AIDO2.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/div&gt; --><div style="text-align: center"><img src="http://sites.google.com/site/240edo/AIDO2.jpg" alt="external image AIDO2.jpg" title="external image AIDO2.jpg" /></div><!-- ws:end:WikiTextRemoteImageRule:16 -->we can have different kinds of AID:<br /> | ||
<span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><u><span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDO = Arithmetic irrational divisions of octave</span></span></u></span></span><br /> | |||
<span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><u><span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDINO = Arithmetic irrational divisions of irrational non-octave</span></span></u></span></span><br /> | |||
<span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><u><span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDRNO = Arithmetic irrational divisions of rational non-octave</span></span></u></span></span><br /> | |||
<span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><u><span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDRI = Arithmetic irrational divisions of rational interval</span></span></u></span></span><br /> | |||
<span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><u><span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDII = Arithmetic irrational divisions of irrational interval</span></span></u></span></span><br /> | |||
<br /> | <br /> | ||
<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;"><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow">**Example : Baran scale**</a></span></body></html></pre></div> | <span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;"><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow">**Example : Baran scale**</a></span></body></html></pre></div> | ||
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<span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><span style="color: #000000; font-family: arial,sans-serif; font-size: 18px;">**Arithmetic irrational divisions**</span><span style="color: #000000;"><span style="font-family: arial,sans-serif;"> </span></span></span> <span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">For an intervallic system with </span>__<span style="color: black; font-size: 11pt;">n</span>__<span style="color: black; font-size: 11pt;"> divisions , </span><span style="color: #be00ff; font-size: 11pt;">[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|__AID__]]</span><span style="color: black; font-size: 11pt;"> is considered as [[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html|arithmetic sequence]] with divisions of system as terms of sequence. </span></span> <span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;"><span style="color: black; font-size: 11pt;"> If the first division is </span>__<span style="color: black; font-size: 11pt;">A1</span>__<span style="color: black; font-size: 11pt;"> and the last , </span>__<span style="color: black; font-size: 11pt;">An</span>__<span style="color: black; font-size: 11pt;"> , with common difference of </span>__<span style="color: black; font-size: 11pt;">d</span>__<span style="color: black; font-size: 11pt;"> , we have : </span></span> <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;">A1 = A1 </span> <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A2 = A1+d </span> <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A3= A1+2d </span> <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A4 = A1+3d</span> <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">……… </span> <span style="font-family: Tahoma,Geneva,sans-serif;">An = A1+(n-1)d</span> <span style="direction: ltr; font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">So sum of the divisions is </span>__<span style="color: black; font-size: 11pt;">Sn</span>__<span style="color: black; font-size: 11pt;"> :</span></span> <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"> **<span style="color: black; font-size: 11pt;">Sn =( __n[2A1+(n-1)d])/2__</span>**</span> <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> </span> <span style="direction: ltr; display: block; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;"> As we can consider </span>__<span style="color: black; font-size: 11pt;">Sn</span>__<span style="color: black; font-size: 11pt;"> of system to be 1200 cent or anything else (octavic or non-octavic system ) then </span>__<span style="color: black; font-size: 11pt;">d</span>__<span style="color: black; font-size: 11pt;"> is most important to make an __AID__ with __n__ divisions with __A1__.important to make an </span>__<span style="color: black; font-size: 11pt;">AID</span>__ with __n__<span style="color: black; font-size: 11pt;"> divisions with </span>__<span style="color: black; font-size: 11pt;">A1</span>__<span style="color: black; font-size: 11pt;">.</span></span><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> so, the common difference between divisions is : </span></span> <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"> **<span style="color: black; font-size: 11pt;">d =( __2(Sn - nA1))/((n(n-1))__</span>**</span> <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"> By considering Sn=1200 , A1=70 , n=12 , __d__ will be 5.454545455 and our 12-tone scale is equal to:</span> <span style="font-family: Tahoma,Geneva,sans-serif;">[[#TOC-0.070.0145.455226.364312.727404.545]] **0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0**</span> <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;">Scales based on AID can be subsets of EDO if : Scales based on AID can be subsets of EDO if : </span><span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"><span style="color: black; font-size: 10pt;">1- we choose d=0 so ,<span style="color: black; font-size: 11pt;"> A1 = Sn/n</span> .. Consider </span>__<span style="color: black; font-size: 10pt;">n</span>__<span style="color: black; font-size: 10pt;">=8 and </span>__<span style="color: black; font-size: 10pt;">A1</span>__<span style="color: black; font-size: 10pt;">=150 , then we have 8-EDO .</span></span> <span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"><span style="color: black; font-size: 10pt;">2- for a constant </span>__<span style="color: black; font-size: 10pt;">n</span>__<span style="color: black; font-size: 10pt;"> and different __A1__, if __d__ and (Sn/A1) are Integer number , we have a susbet of EDO or EDI( Equal divisions of Interval) .Consider __Sn = 1400__ , </span>__<span style="color: black; font-size: 10pt;">n</span>__<span style="color: black; font-size: 10pt;">=8 and </span>__<span style="color: black; font-size: 10pt;">A1</span>__<span style="color: black; font-size: 10pt;">=70 , then we have a subset of a 140-ED(1400.) with Degrees as 7 17 30 46 65 87 112 140 :</span></span> **0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000** <span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;">[[#TOC-And-now-for:]]And now for </span><span style="font-family: Tahoma,Geneva,sans-serif;">Sn=1400 and n=8,</span> <span style="font-family: Tahoma,Geneva,sans-serif;"> [[#TOC-If-A1-175.0-then-we-have-8-AID-1400]]If A1=175.0 then we have 8-AID(1400.)</span> <span style="font-family: Tahoma,Geneva,sans-serif;"> [[#TOC-If-A1-56-then-we-have-700-AID-1400.]]If A1=56 then we have 700-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 28 73 135 214 310 423 553 700</span></span> <span style="font-family: Tahoma,Geneva,sans-serif;"> [[#TOC-If-A1-87.5-then-we-have-112-AID-140]]<span style="color: black; font-size: 10pt;">If A1=87.5 then we have 112-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 7 16 27 40 55 72 91 112</span></span></span> <span style="font-family: Tahoma,Geneva,sans-serif;">__AID__ sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in __AID__<span style="color: black; font-size: 10pt;"> and __EDO__ with equal degree:</span></span> * <span style="font-family: Tahoma,Geneva,sans-serif;">If choosing __<span style="color: black; font-size: 10pt;">A1</span>__<span style="color: black; font-size: 10pt;"> greater than division size in equal degree </span>__<span style="color: black; font-size: 10pt;">EDO</span>__<span style="color: black; font-size: 10pt;"> , </span>__<span style="color: black; font-size: 10pt;">d</span>__<span style="color: black; font-size: 10pt;"> is negative and </span>__<span style="color: black; font-size: 10pt;">AID</span>__<span style="color: black; font-size: 10pt;"> is descending.</span></span> * <span style="font-family: Tahoma,Geneva,sans-serif;">If choosing __<span style="color: black; font-size: 10pt;">A1</span>__<span style="color: black; font-size: 10pt;"> smaller than division size in equal degree </span>__<span style="color: black; font-size: 10pt;">EDO</span>__<span style="color: black; font-size: 10pt;"> , </span>__<span style="color: black; font-size: 10pt;">d</span>__<span style="color: black; font-size: 10pt;"> is positive and </span>__<span style="color: black; font-size: 10pt;">AID</span>__<span style="color: black; font-size: 10pt;"> is ascending.</span></span> * <span style="font-family: Tahoma,Geneva,sans-serif;">If choosing __<span style="color: black; font-size: 10pt;">A1</span>__<span style="color: black; font-size: 10pt;"> equal to division size in equal degree </span>__<span style="color: black; font-size: 10pt;">EDO</span>__<span style="color: black; font-size: 10pt;"> , </span>__<span style="color: black; font-size: 10pt;">d</span>__<span style="color: black; font-size: 10pt;"> is zero.</span>**</span> [[image:http://sites.google.com/site/240edo/AIDO-custom-size-298-402.jpg align="center"]] ==<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;">171.4285714 is point of intersection in these 3 trends:</span>== [[image:http://sites.google.com/site/240edo/AIDO2.jpg align="center"]] we can have different kinds of AID: <span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;">__<span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDO = Arithmetic irrational divisions of octave</span></span>__</span></span> <span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;">__<span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDINO = Arithmetic irrational divisions of irrational non-octave</span></span>__</span></span> <span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;">__<span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDRNO = Arithmetic irrational divisions of rational non-octave</span></span>__</span></span> <span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;">__<span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDRI = Arithmetic irrational divisions of rational interval</span></span>__</span></span> <span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;">__<span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDII = Arithmetic irrational divisions of irrational interval</span></span>__</span></span> <span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;">[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|**Example : Baran scale**]]</span>
Original HTML content:
<html><head><title>AID</title></head><body><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><span style="color: #000000; font-family: arial,sans-serif; font-size: 18px;"><strong>Arithmetic irrational divisions</strong></span><span style="color: #000000;"><span style="font-family: arial,sans-serif;"> </span></span></span><br /> <br /> <span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">For an intervallic system with </span><u><span style="color: black; font-size: 11pt;">n</span></u><span style="color: black; font-size: 11pt;"> divisions , </span><span style="color: #be00ff; font-size: 11pt;"><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow">__AID__</a></span><span style="color: black; font-size: 11pt;"> is considered as <a class="wiki_link_ext" href="http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html" rel="nofollow">arithmetic sequence</a> with divisions of system as terms of sequence. </span></span><br /> <span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;"><span style="color: black; font-size: 11pt;"> If the first division is </span><u><span style="color: black; font-size: 11pt;">A1</span></u><span style="color: black; font-size: 11pt;"> and the last , </span><u><span style="color: black; font-size: 11pt;">An</span></u><span style="color: black; font-size: 11pt;"> , with common difference of </span><u><span style="color: black; font-size: 11pt;">d</span></u><span style="color: black; font-size: 11pt;"> , we have : </span></span><br /> <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;">A1 = A1 </span><br /> <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A2 = A1+d </span><br /> <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A3= A1+2d </span><br /> <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A4 = A1+3d</span><br /> <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">……… </span><br /> <span style="font-family: Tahoma,Geneva,sans-serif;">An = A1+(n-1)d</span><br /> <br /> <span style="direction: ltr; font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">So sum of the divisions is </span><u><span style="color: black; font-size: 11pt;">Sn</span></u><span style="color: black; font-size: 11pt;"> :</span></span><br /> <br /> <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"> <strong><span style="color: black; font-size: 11pt;">Sn =( <u>n[2A1+(n-1)d])/2</u></span></strong></span><br /> <span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"><br /> </span><br /> <br /> <span style="direction: ltr; display: block; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;"> As we can consider </span><u><span style="color: black; font-size: 11pt;">Sn</span></u><span style="color: black; font-size: 11pt;"> of system to be 1200 cent or anything else (octavic or non-octavic system ) then </span><u><span style="color: black; font-size: 11pt;">d</span></u><span style="color: black; font-size: 11pt;"> is most important to make an <u>AID</u> with <u>n</u> divisions with <u>A1</u>.important to make an </span><u><span style="color: black; font-size: 11pt;">AID</span></u> with <u>n</u><span style="color: black; font-size: 11pt;"> divisions with </span><u><span style="color: black; font-size: 11pt;">A1</span></u><span style="color: black; font-size: 11pt;">.</span></span><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> so, the common difference between divisions is : </span></span><br /> <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"> <strong><span style="color: black; font-size: 11pt;">d =( <u>2(Sn - nA1))/((n(n-1))</u></span></strong></span><br /> <br /> <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"> By considering Sn=1200 , A1=70 , n=12 , <u>d</u> will be 5.454545455 and our 12-tone scale is equal to:</span><br /> <span style="font-family: Tahoma,Geneva,sans-serif;"><!-- ws:start:WikiTextAnchorRule:2:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@TOC-0.070.0145.455226.364312.727404.545" title="Anchor: TOC-0.070.0145.455226.364312.727404.545"/> --><a name="TOC-0.070.0145.455226.364312.727404.545"></a><!-- ws:end:WikiTextAnchorRule:2 --> <strong>0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0</strong></span><br /> <span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;">Scales based on AID can be subsets of EDO if : Scales based on AID can be subsets of EDO if : </span><span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"><span style="color: black; font-size: 10pt;">1- we choose d=0 so ,<span style="color: black; font-size: 11pt;"> A1 = Sn/n</span> .. Consider </span><u><span style="color: black; font-size: 10pt;">n</span></u><span style="color: black; font-size: 10pt;">=8 and </span><u><span style="color: black; font-size: 10pt;">A1</span></u><span style="color: black; font-size: 10pt;">=150 , then we have 8-EDO .</span></span><br /> <span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"><span style="color: black; font-size: 10pt;">2- for a constant </span><u><span style="color: black; font-size: 10pt;">n</span></u><span style="color: black; font-size: 10pt;"> and different <u>A1</u>, if <u>d</u> and (Sn/A1) are Integer number , we have a susbet of EDO or EDI( Equal divisions of Interval) .Consider <u>Sn = 1400</u> , </span><u><span style="color: black; font-size: 10pt;">n</span></u><span style="color: black; font-size: 10pt;">=8 and </span><u><span style="color: black; font-size: 10pt;">A1</span></u><span style="color: black; font-size: 10pt;">=70 , then we have a subset of a 140-ED(1400.) with Degrees as 7 17 30 46 65 87 112 140 :</span></span><br /> <strong>0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000</strong><br /> <span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;"><!-- ws:start:WikiTextAnchorRule:3:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@TOC-And-now-for:" title="Anchor: TOC-And-now-for:"/> --><a name="TOC-And-now-for:"></a><!-- ws:end:WikiTextAnchorRule:3 -->And now for </span><span style="font-family: Tahoma,Geneva,sans-serif;">Sn=1400 and n=8,</span><br /> <span style="font-family: Tahoma,Geneva,sans-serif;"> <!-- ws:start:WikiTextAnchorRule:4:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@TOC-If-A1-175.0-then-we-have-8-AID-1400" title="Anchor: TOC-If-A1-175.0-then-we-have-8-AID-1400"/> --><a name="TOC-If-A1-175.0-then-we-have-8-AID-1400"></a><!-- ws:end:WikiTextAnchorRule:4 -->If A1=175.0 then we have 8-AID(1400.)</span><br /> <span style="font-family: Tahoma,Geneva,sans-serif;"> <!-- ws:start:WikiTextAnchorRule:5:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@TOC-If-A1-56-then-we-have-700-AID-1400." title="Anchor: TOC-If-A1-56-then-we-have-700-AID-1400."/> --><a name="TOC-If-A1-56-then-we-have-700-AID-1400."></a><!-- ws:end:WikiTextAnchorRule:5 -->If A1=56 then we have 700-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 28 73 135 214 310 423 553 700</span></span><br /> <span style="font-family: Tahoma,Geneva,sans-serif;"> <!-- ws:start:WikiTextAnchorRule:6:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@TOC-If-A1-87.5-then-we-have-112-AID-140" title="Anchor: TOC-If-A1-87.5-then-we-have-112-AID-140"/> --><a name="TOC-If-A1-87.5-then-we-have-112-AID-140"></a><!-- ws:end:WikiTextAnchorRule:6 --><span style="color: black; font-size: 10pt;">If A1=87.5 then we have 112-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 7 16 27 40 55 72 91 112</span></span></span><br /> <br /> <span style="font-family: Tahoma,Geneva,sans-serif;"><u>AID</u> sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in <u>AID</u><span style="color: black; font-size: 10pt;"> and <u>EDO</u> with equal degree:</span></span><br /> <ul><li><span style="font-family: Tahoma,Geneva,sans-serif;">If choosing <u><span style="color: black; font-size: 10pt;">A1</span></u><span style="color: black; font-size: 10pt;"> greater than division size in equal degree </span><u><span style="color: black; font-size: 10pt;">EDO</span></u><span style="color: black; font-size: 10pt;"> , </span><u><span style="color: black; font-size: 10pt;">d</span></u><span style="color: black; font-size: 10pt;"> is negative and </span><u><span style="color: black; font-size: 10pt;">AID</span></u><span style="color: black; font-size: 10pt;"> is descending.</span></span></li><li><span style="font-family: Tahoma,Geneva,sans-serif;">If choosing <u><span style="color: black; font-size: 10pt;">A1</span></u><span style="color: black; font-size: 10pt;"> smaller than division size in equal degree </span><u><span style="color: black; font-size: 10pt;">EDO</span></u><span style="color: black; font-size: 10pt;"> , </span><u><span style="color: black; font-size: 10pt;">d</span></u><span style="color: black; font-size: 10pt;"> is positive and </span><u><span style="color: black; font-size: 10pt;">AID</span></u><span style="color: black; font-size: 10pt;"> is ascending.</span></span></li><li><span style="font-family: Tahoma,Geneva,sans-serif;">If choosing <u><span style="color: black; font-size: 10pt;">A1</span></u><span style="color: black; font-size: 10pt;"> equal to division size in equal degree </span><u><span style="color: black; font-size: 10pt;">EDO</span></u><span style="color: black; font-size: 10pt;"> , </span><u><span style="color: black; font-size: 10pt;">d</span></u><span style="color: black; font-size: 10pt;"> is zero.</span>**</span></li></ul><br /> <!-- ws:start:WikiTextRemoteImageRule:15:<div style="text-align: center"><img src="http://sites.google.com/site/240edo/AIDO-custom-size-298-402.jpg" alt="" title="" /></div> --><div style="text-align: center"><img src="http://sites.google.com/site/240edo/AIDO-custom-size-298-402.jpg" alt="external image AIDO-custom-size-298-402.jpg" title="external image AIDO-custom-size-298-402.jpg" /></div><!-- ws:end:WikiTextRemoteImageRule:15 --><br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-171.4285714 is point of intersection in these 3 trends:"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;">171.4285714 is point of intersection in these 3 trends:</span></h2> <br /> <br /> <!-- ws:start:WikiTextRemoteImageRule:16:<div style="text-align: center"><img src="http://sites.google.com/site/240edo/AIDO2.jpg" alt="" title="" /></div> --><div style="text-align: center"><img src="http://sites.google.com/site/240edo/AIDO2.jpg" alt="external image AIDO2.jpg" title="external image AIDO2.jpg" /></div><!-- ws:end:WikiTextRemoteImageRule:16 -->we can have different kinds of AID:<br /> <span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><u><span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDO = Arithmetic irrational divisions of octave</span></span></u></span></span><br /> <span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><u><span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDINO = Arithmetic irrational divisions of irrational non-octave</span></span></u></span></span><br /> <span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><u><span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDRNO = Arithmetic irrational divisions of rational non-octave</span></span></u></span></span><br /> <span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><u><span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDRI = Arithmetic irrational divisions of rational interval</span></span></u></span></span><br /> <span style="text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif; font-size: 14px;"><u><span style="color: #000000;"><span style="font-family: arial,sans-serif;">AIDII = Arithmetic irrational divisions of irrational interval</span></span></u></span></span><br /> <br /> <span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;"><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow">**Example : Baran scale**</a></span></body></html>