AID
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<span style="font-family: Times New Roman;"><span style="color: black; font-size: 11pt;"><span style="font-family: arial,sans-serif;">For an intervallic system with </span></span></span><span style="font-family: arial,sans-serif;">__<span style="color: black; font-size: 11pt;">**n**</span>__<span style="color: black; font-size: 11pt;"> divisions , </span>__<span style="color: black; font-size: 11pt;">**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; text-align: left;"><span style="font-family: Times New Roman;"><span style="font-family: arial,sans-serif;"><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><span style="color: black; font-size: 11pt;"> , with common difference of </span>__<span style="color: black; font-size: 11pt;">**d**</span>__</span><span style="color: black; font-size: 11pt;"><span style="font-family: arial,sans-serif;"> , we have :</span> </span></span> <span style="direction: ltr; text-align: left;"><span style="font-family: Times New Roman;"> </span><span style="font-family: arial,sans-serif;">**<span style="color: black; font-size: 11pt;">A1 = A1 </span>**</span></span> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;">**<span style="font-family: arial,sans-serif;"> A2 = A1+d </span>**</span></span> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;">**<span style="font-family: arial,sans-serif;"> A3= A1+2d </span>**</span></span> <span style="direction: ltr; text-align: left;">**<span style="font-family: arial,sans-serif;"><span style="color: black; font-size: 11pt;"> A4 = A1+3d </span></span>**</span> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;">**<span style="font-family: arial,sans-serif;"> ……… </span>**</span></span> <span style="direction: ltr; text-align: left;"><span style="font-family: Times New Roman;"><span style="color: black; font-size: 11pt;"><span style="font-family: arial,sans-serif;"> </span>**<span style="font-family: arial,sans-serif;">An = A1+(n-1)d</span>** </span></span></span> <span style="direction: ltr; text-align: left;"><span style="font-family: Times New Roman;"><span style="color: black; font-size: 11pt;"> <span style="font-family: arial,sans-serif;">**So sum of the divisions is** </span></span>__<span style="color: black; font-size: 11pt;"><span style="font-family: arial,sans-serif;">**Sn**</span></span>__<span style="color: black; font-size: 11pt;"> :</span></span></span> <span style="direction: ltr; text-align: center;"><span style="font-family: Times New Roman;"><span style="color: black; font-size: 11pt;"> **<span style="font-family: arial,sans-serif;"><span style="color: black; font-size: 11pt;">Sn = __n[2A1+(n-1)d]__</span></span>**</span></span></span> <span style="direction: ltr; text-align: center;"><span style="color: black; font-size: 11pt;">**<span style="font-family: arial,sans-serif;"> 2</span>**</span></span> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;"><span style="font-family: Times New Roman;"> </span></span><span style="font-family: Times New Roman;"><span style="color: black; font-size: 11pt;"> <span style="font-family: arial,sans-serif;">As we can consider </span></span></span><span style="font-family: arial,sans-serif;">__<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></span> <span style="direction: ltr; text-align: left;"><span style="font-family: arial,sans-serif;"><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> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;"><span style="font-family: Times New Roman;"><span style="font-family: arial,sans-serif;"> so, the common difference between divisions is :</span> </span></span></span> <span style="direction: ltr; text-align: center;"><span style="color: black; font-size: 11pt;"><span style="font-family: Times New Roman;"> <span style="font-family: arial,sans-serif;"> **<span style="color: black; font-size: 11pt;">d = __2(Sn - nA1)__</span>**</span></span></span></span> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;">**<span style="font-family: arial,sans-serif;"> n(n-1)</span>**</span></span> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;"><span style="font-family: Arial;"> By considering **Sn=1200** , **A1=70** , **n=12** , **__d__** will be 5.454545455 and our 12-tone <span style="font-family: arial,sans-serif;">scale is equal to</span>:</span></span></span> ==<span style="direction: ltr; text-align: center;">[[#TOC-0.070.0145.455226.364312.727404.545]]<span style="font-family: arial,sans-serif;"> **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: Times New Roman;"><span style="color: black; font-size: 11pt;"><span style="font-family: Arial;"> </span></span> </span><span style="font-family: arial,sans-serif;"> </span></span>== ==<span style="direction: ltr; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-size: 11pt;">Scales based on **AID** can be subsets of **EDO** if :</span></span></span>== ==<span style="direction: ltr; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-size: 11pt;"> Scales based on **AID** can be subsets of **EDO** if :</span> </span></span>== <span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;">1- we choose **d=0** so ,<span style="color: black; font-size: 11pt;"> **<span style="color: black; font-size: 11pt;"><span style="font-family: arial,sans-serif;">A1</span> = Sn/n</span>** </span> </span></span></span> <span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;">Consider </span>**__<span style="color: black; font-family: Arial; font-size: 10pt;">n</span>__<span style="color: black; font-family: Arial; font-size: 10pt;">=8</span>**<span style="color: black; font-family: Arial; font-size: 10pt;"> and </span>**__<span style="color: black; font-family: Arial; font-size: 10pt;">A1</span>__<span style="color: black; font-family: Arial; font-size: 10pt;">=150</span>**<span style="color: black; font-family: Arial; font-size: 10pt;"> , then we have **8-EDO** .</span></span><span style="color: black; font-family: Arial; font-size: 10pt;"><span style="font-family: arial,sans-serif;"> </span></span></span> <span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;">2-<span style="font-family: arial,sans-serif;"> for a constant </span></span>__<span style="color: black; font-family: Arial; font-size: 10pt;">**<span style="font-family: arial,sans-serif;">n</span>**</span>__<span style="color: black; font-family: Arial; font-size: 10pt;"><span style="font-family: arial,sans-serif;"> and different **__A1__**, if </span>**__d__** and (**Sn/A1)** are **Integer number**</span></span><span style="color: black; font-family: Arial; font-size: 10pt;"><span style="font-family: arial,sans-serif;"> , we have </span></span></span><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;">a susbet of **EDO or EDI( Equal divisions of Interval)** .Consider **__Sn = 1400__** , </span>**__<span style="color: black; font-family: Arial; font-size: 10pt;">n</span>__<span style="color: black; font-family: Arial; font-size: 10pt;">=8</span>**<span style="color: black; font-family: Arial; font-size: 10pt;"> and </span>**__<span style="color: black; font-family: Arial; font-size: 10pt;">A1</span>__<span style="color: black; font-family: Arial; font-size: 10pt;">=70</span>**<span style="color: black; font-family: Arial; font-size: 10pt;"> , then we have a subset of a </span></span><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;">**140-ED(1400.)** with Degrees as </span></span></span> <span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;"> **7 17 30 46 65 87 112 140 :**</span></span></span> <span style="color: red; font-family: Arial;"> </span> ==<span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: center;"><span style="color: #000000;">[[#TOC-0.00070.000170.000300.000460.000650]] **0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000** </span></span>== ==<span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"><span style="color: #000000;">[[#TOC-And-now-for:]]And now for:</span></span>== ==<span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;">[[#TOC-Sn-1400-and-n-8-]]**Sn=1400 and** **n=8,**</span>== ==<span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;">[[#TOC-If-A1-175.0-then-we-have-8-AID-1400]]If **A1=175.0** then we have **8-AID(1400.)**</span>== ==<span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;">[[#TOC-If-A1-56-then-we-have-700-AID-1400.]]If **A1=56** then we have **700-AID(1400.)** with Degrees as <span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;"> **28 73 135 214 310 423 553 700**</span></span></span>== ==<span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;">[[#TOC-If-A1-87.5-then-we-have-112-AID-140]]<span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;">If **A1=87.5** then we have **112-AID(1400.)** with Degrees as **<span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;"> 7 16 27 40 55 72 91 112</span></span>**</span></span></span>== ==<span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;">[[#TOC-1]]<span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;"> </span></span></span>== ==<span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: center;">[[#TOC-2]] </span>== ==<span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;">[[#TOC-3]]</span>== <span style="font-family: arial,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-family: Arial; font-size: 10pt;"> and **__EDO__** with equal degree:</span></span><span style="color: black; font-family: Arial; font-size: 10pt;"> *** If choosing** </span>**__<span style="color: black; font-family: Arial; font-size: 10pt;">A1</span>__<span style="color: black; font-family: Arial; font-size: 10pt;"> greater than division size in equal degree </span>__<span style="color: black; font-family: Arial; font-size: 10pt;">EDO</span>__<span style="color: black; font-family: Arial; font-size: 10pt;"> , </span>__<span style="color: black; font-family: Arial; font-size: 10pt;">d</span>__<span style="color: black; font-family: Arial; font-size: 10pt;"> is negative and </span>__<span style="color: black; font-family: Arial; font-size: 10pt;">AID</span>__****<span style="color: black; font-family: Arial; font-size: 10pt;"> is descending. * If choosing </span>__<span style="color: black; font-family: Arial; font-size: 10pt;">A1</span>__<span style="color: black; font-family: Arial; font-size: 10pt;"> smaller than division size in equal degree </span>__<span style="color: black; font-family: Arial; font-size: 10pt;">EDO</span>__<span style="color: black; font-family: Arial; font-size: 10pt;"> , </span>__<span style="color: black; font-family: Arial; font-size: 10pt;">d</span>__<span style="color: black; font-family: Arial; font-size: 10pt;"> is positive and </span>__<span style="color: black; font-family: Arial; font-size: 10pt;">AID</span>__****<span style="color: black; font-family: Arial; font-size: 10pt;"> is ascending. * If choosing </span>__<span style="color: black; font-family: Arial; font-size: 10pt;">A1</span>__<span style="color: black; font-family: Arial; font-size: 10pt;"> equal to division size in equal degree </span>__<span style="color: black; font-family: Arial; font-size: 10pt;">EDO</span>__<span style="color: black; font-family: Arial; font-size: 10pt;"> , </span>__<span style="color: black; font-family: Arial; font-size: 10pt;">d</span>__<span style="color: black; font-family: Arial; font-size: 10pt;"> is zero.</span>**
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<html><head><title>AID</title></head><body><span style="font-family: Times New Roman;"><span style="color: black; font-size: 11pt;"><span style="font-family: arial,sans-serif;">For an intervallic system with </span></span></span><span style="font-family: arial,sans-serif;"><u><span style="color: black; font-size: 11pt;"><strong>n</strong></span></u><span style="color: black; font-size: 11pt;"> divisions , </span><u><span style="color: black; font-size: 11pt;"><strong>AID</strong></span></u><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; text-align: left;"><span style="font-family: Times New Roman;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-size: 11pt;"> If the first division is </span><u><span style="color: black; font-size: 11pt;"><strong>A1</strong></span></u><span style="color: black; font-size: 11pt;"> and the last , </span><u><span style="color: black; font-size: 11pt;"><strong>An</strong></span></u></span><span style="color: black; font-size: 11pt;"> , with common difference of </span><u><span style="color: black; font-size: 11pt;"><strong>d</strong></span></u></span><span style="color: black; font-size: 11pt;"><span style="font-family: arial,sans-serif;"> , we have :</span> </span></span><br /> <span style="direction: ltr; text-align: left;"><span style="font-family: Times New Roman;"> </span><span style="font-family: arial,sans-serif;"><strong><span style="color: black; font-size: 11pt;">A1 = A1 </span></strong></span></span><br /> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;"><strong><span style="font-family: arial,sans-serif;"> A2 = A1+d </span></strong></span></span><br /> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;"><strong><span style="font-family: arial,sans-serif;"> A3= A1+2d </span></strong></span></span><br /> <span style="direction: ltr; text-align: left;"><strong><span style="font-family: arial,sans-serif;"><span style="color: black; font-size: 11pt;"> A4 = A1+3d </span></span></strong></span><br /> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;"><strong><span style="font-family: arial,sans-serif;"> ……… </span></strong></span></span><br /> <span style="direction: ltr; text-align: left;"><span style="font-family: Times New Roman;"><span style="color: black; font-size: 11pt;"><span style="font-family: arial,sans-serif;"> </span><strong><span style="font-family: arial,sans-serif;">An = A1+(n-1)d</span></strong> </span></span></span><br /> <span style="direction: ltr; text-align: left;"><span style="font-family: Times New Roman;"><span style="color: black; font-size: 11pt;"> <span style="font-family: arial,sans-serif;"><strong>So sum of the divisions is</strong> </span></span><u><span style="color: black; font-size: 11pt;"><span style="font-family: arial,sans-serif;"><strong>Sn</strong></span></span></u><span style="color: black; font-size: 11pt;"> :</span></span></span><br /> <span style="direction: ltr; text-align: center;"><span style="font-family: Times New Roman;"><span style="color: black; font-size: 11pt;"> <strong><span style="font-family: arial,sans-serif;"><span style="color: black; font-size: 11pt;">Sn = <u>n[2A1+(n-1)d]</u></span></span></strong></span></span></span><br /> <span style="direction: ltr; text-align: center;"><span style="color: black; font-size: 11pt;"><strong><span style="font-family: arial,sans-serif;"> 2</span></strong></span></span><br /> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;"><span style="font-family: Times New Roman;"> </span></span><span style="font-family: Times New Roman;"><span style="color: black; font-size: 11pt;"> <span style="font-family: arial,sans-serif;">As we can consider </span></span></span><span style="font-family: arial,sans-serif;"><u><span style="color: black; font-size: 11pt;"><strong>Sn</strong></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;"><strong>d</strong></span></u> </span></span><br /> <span style="direction: ltr; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-size: 11pt;"> is most important to make an <strong><u>AID</u></strong> with <strong><u>n</u></strong> divisions with <strong><u>A1</u></strong>.important to make an </span><u><span style="color: black; font-size: 11pt;"><strong>AID</strong></span></u> with <strong><u>n</u></strong><span style="color: black; font-size: 11pt;"> divisions with </span><u><span style="color: black; font-size: 11pt;"><strong>A1</strong></span></u><span style="color: black; font-size: 11pt;">. </span></span></span><br /> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;"><span style="font-family: Times New Roman;"><span style="font-family: arial,sans-serif;"> so, the common difference between divisions is :</span> </span></span></span><br /> <span style="direction: ltr; text-align: center;"><span style="color: black; font-size: 11pt;"><span style="font-family: Times New Roman;"> <span style="font-family: arial,sans-serif;"> <strong><span style="color: black; font-size: 11pt;">d = <u>2(Sn - nA1)</u></span></strong></span></span></span></span><br /> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;"><strong><span style="font-family: arial,sans-serif;"> n(n-1)</span></strong></span></span><br /> <span style="direction: ltr; text-align: left;"><span style="color: black; font-size: 11pt;"><span style="font-family: Arial;"> By considering <strong>Sn=1200</strong> , <strong>A1=70</strong> , <strong>n=12</strong> , <strong><u>d</u></strong> will be 5.454545455 and our 12-tone <span style="font-family: arial,sans-serif;">scale is equal to</span>:</span></span></span><br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-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"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="direction: ltr; text-align: center;"><!-- ws:start:WikiTextAnchorRule:24:<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:24 --><span style="font-family: arial,sans-serif;"> <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><span style="font-family: Times New Roman;"><span style="color: black; font-size: 11pt;"><span style="font-family: Arial;"> </span></span> </span><span style="font-family: arial,sans-serif;"> </span></span></h2> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-Scales based on AID** can be subsets of **EDO if :"></a><!-- ws:end:WikiTextHeadingRule:2 --><span style="direction: ltr; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-size: 11pt;">Scales based on <strong>AID</strong> can be subsets of <strong>EDO</strong> if :</span></span></span></h2> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x-Scales based on AID** can be subsets of **EDO if :"></a><!-- ws:end:WikiTextHeadingRule:4 --><span style="direction: ltr; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-size: 11pt;"> Scales based on <strong>AID</strong> can be subsets of <strong>EDO</strong> if :</span> </span></span></h2> <span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;">1- we choose <strong>d=0</strong> so ,<span style="color: black; font-size: 11pt;"> <strong><span style="color: black; font-size: 11pt;"><span style="font-family: arial,sans-serif;">A1</span> = Sn/n</span></strong> </span> </span></span></span><br /> <span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;">Consider </span><strong><u><span style="color: black; font-family: Arial; font-size: 10pt;">n</span></u><span style="color: black; font-family: Arial; font-size: 10pt;">=8</span></strong><span style="color: black; font-family: Arial; font-size: 10pt;"> and </span><strong><u><span style="color: black; font-family: Arial; font-size: 10pt;">A1</span></u><span style="color: black; font-family: Arial; font-size: 10pt;">=150</span></strong><span style="color: black; font-family: Arial; font-size: 10pt;"> , then we have <strong>8-EDO</strong> .</span></span><span style="color: black; font-family: Arial; font-size: 10pt;"><span style="font-family: arial,sans-serif;"> </span></span></span><br /> <span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;">2-<span style="font-family: arial,sans-serif;"> for a constant </span></span><u><span style="color: black; font-family: Arial; font-size: 10pt;"><strong><span style="font-family: arial,sans-serif;">n</span></strong></span></u><span style="color: black; font-family: Arial; font-size: 10pt;"><span style="font-family: arial,sans-serif;"> and different <strong><u>A1</u></strong>, if </span><strong><u>d</u></strong> and (<strong>Sn/A1)</strong> are <strong>Integer number</strong></span></span><span style="color: black; font-family: Arial; font-size: 10pt;"><span style="font-family: arial,sans-serif;"> , we have </span></span></span><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;">a susbet of <strong>EDO or EDI( Equal divisions of Interval)</strong> .Consider <strong><u>Sn = 1400</u></strong> , </span><strong><u><span style="color: black; font-family: Arial; font-size: 10pt;">n</span></u><span style="color: black; font-family: Arial; font-size: 10pt;">=8</span></strong><span style="color: black; font-family: Arial; font-size: 10pt;"> and </span><strong><u><span style="color: black; font-family: Arial; font-size: 10pt;">A1</span></u><span style="color: black; font-family: Arial; font-size: 10pt;">=70</span></strong><span style="color: black; font-family: Arial; font-size: 10pt;"> , then we have a subset of a </span></span><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;"><strong>140-ED(1400.)</strong> with Degrees as </span></span></span><br /> <span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;"> <strong>7 17 30 46 65 87 112 140 :</strong></span></span></span><br /> <span style="color: red; font-family: Arial;"> </span><br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x-0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000"></a><!-- ws:end:WikiTextHeadingRule:6 --><span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: center;"><span style="color: #000000;"><!-- ws:start:WikiTextAnchorRule:25:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@TOC-0.00070.000170.000300.000460.000650" title="Anchor: TOC-0.00070.000170.000300.000460.000650"/> --><a name="TOC-0.00070.000170.000300.000460.000650"></a><!-- ws:end:WikiTextAnchorRule:25 --> <strong>0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000</strong> </span></span></h2> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="x-And now for:"></a><!-- ws:end:WikiTextHeadingRule:8 --><span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"><span style="color: #000000;"><!-- ws:start:WikiTextAnchorRule:26:<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:26 -->And now for:</span></span></h2> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="x-Sn=1400 and** **n=8,"></a><!-- ws:end:WikiTextHeadingRule:10 --><span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"><!-- ws:start:WikiTextAnchorRule:27:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@TOC-Sn-1400-and-n-8-" title="Anchor: TOC-Sn-1400-and-n-8-"/> --><a name="TOC-Sn-1400-and-n-8-"></a><!-- ws:end:WikiTextAnchorRule:27 --><strong>Sn=1400 and</strong> <strong>n=8,</strong></span></h2> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="x-If A1=175.0** then we have **8-AID(1400.)"></a><!-- ws:end:WikiTextHeadingRule:12 --><span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"><!-- ws:start:WikiTextAnchorRule:28:<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:28 -->If <strong>A1=175.0</strong> then we have <strong>8-AID(1400.)</strong></span></h2> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="x-If A1=56** then we have **700-AID(1400.)** with Degrees as **28 73 135 214 310 423 553 700"></a><!-- ws:end:WikiTextHeadingRule:14 --><span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"><!-- ws:start:WikiTextAnchorRule:29:<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:29 -->If <strong>A1=56</strong> then we have <strong>700-AID(1400.)</strong> with Degrees as <span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;"> <strong>28 73 135 214 310 423 553 700</strong></span></span></span></h2> <!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="x-If A1=87.5** then we have **112-AID(1400.)** with Degrees as ** 7 16 27 40 55 72 91 112"></a><!-- ws:end:WikiTextHeadingRule:16 --><span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"><!-- ws:start:WikiTextAnchorRule:30:<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:30 --><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;">If <strong>A1=87.5</strong> then we have <strong>112-AID(1400.)</strong> with Degrees as <strong><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;"> 7 16 27 40 55 72 91 112</span></span></strong></span></span></span></h2> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><!-- ws:end:WikiTextHeadingRule:18 --><span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"><!-- ws:start:WikiTextAnchorRule:31:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@TOC-1" title="Anchor: TOC-1"/> --><a name="TOC-1"></a><!-- ws:end:WikiTextAnchorRule:31 --><span style="font-family: arial,sans-serif;"><span style="color: black; font-family: Arial; font-size: 10pt;"> </span></span></span></h2> <!-- ws:start:WikiTextHeadingRule:20:<h2> --><h2 id="toc10"><!-- ws:end:WikiTextHeadingRule:20 --><span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: center;"><!-- ws:start:WikiTextAnchorRule:32:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@TOC-2" title="Anchor: TOC-2"/> --><a name="TOC-2"></a><!-- ws:end:WikiTextAnchorRule:32 --> </span></h2> <!-- ws:start:WikiTextHeadingRule:22:<h2> --><h2 id="toc11"><!-- ws:end:WikiTextHeadingRule:22 --><span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"><!-- ws:start:WikiTextAnchorRule:33:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@TOC-3" title="Anchor: TOC-3"/> --><a name="TOC-3"></a><!-- ws:end:WikiTextAnchorRule:33 --></span></h2> <br /> <span style="font-family: arial,sans-serif;"><strong><u>AID</u></strong> sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between <strong>n</strong> and <strong>A1</strong> in <u><strong>AID</strong></u><span style="color: black; font-family: Arial; font-size: 10pt;"> and <strong><u>EDO</u></strong> with equal degree:</span></span><span style="color: black; font-family: Arial; font-size: 10pt;"><br /> <ul><ul><ul><li>If choosing<strong> </span></strong><u><span style="color: black; font-family: Arial; font-size: 10pt;">A1</span></u><span style="color: black; font-family: Arial; font-size: 10pt;"> greater than division size in equal degree </span><u><span style="color: black; font-family: Arial; font-size: 10pt;">EDO</span></u><span style="color: black; font-family: Arial; font-size: 10pt;"> , </span><u><span style="color: black; font-family: Arial; font-size: 10pt;">d</span></u><span style="color: black; font-family: Arial; font-size: 10pt;"> is negative and </span><u><span style="color: black; font-family: Arial; font-size: 10pt;">AID</span></u><span style="color: black; font-family: Arial; font-size: 10pt;"> is descending.</li></ul></ul><li>If choosing </span><u><span style="color: black; font-family: Arial; font-size: 10pt;">A1</span></u><span style="color: black; font-family: Arial; font-size: 10pt;"> smaller than division size in equal degree </span><u><span style="color: black; font-family: Arial; font-size: 10pt;">EDO</span></u><span style="color: black; font-family: Arial; font-size: 10pt;"> , </span><u><span style="color: black; font-family: Arial; font-size: 10pt;">d</span></u><span style="color: black; font-family: Arial; font-size: 10pt;"> is positive and </span><u><span style="color: black; font-family: Arial; font-size: 10pt;">AID</span></u><span style="color: black; font-family: Arial; font-size: 10pt;"> is ascending.</li><li>If choosing </span><u><span style="color: black; font-family: Arial; font-size: 10pt;">A1</span></u><span style="color: black; font-family: Arial; font-size: 10pt;"> equal to division size in equal degree </span><u><span style="color: black; font-family: Arial; font-size: 10pt;">EDO</span></u><span style="color: black; font-family: Arial; font-size: 10pt;"> , </span><u><span style="color: black; font-family: Arial; font-size: 10pt;">d</span></u><span style="color: black; font-family: Arial; font-size: 10pt;"> is zero.</span>**</li></ul></body></html>