AID: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:acousticsoftombak|acousticsoftombak]] and made on <tt>2012-02-17 01:53:09 UTC</tt>.

: This revision was by author [[User:acousticsoftombak|acousticsoftombak]] and made on <tt>2012-02-17 02:05:55 UTC</tt>.

: The original revision id was <tt>~~302669008~~</tt>.

: The original revision id was <tt>302670104</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

<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">~~For an intervallic system with~~ __**n**__ ~~divisions , __**AID**__ is considered as [[http~~://www.richland.edu/james/lecture/m116/sequences/arithmetic.html|arithmetic sequence]] with divisions of system as terms of sequence.

<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">For an intervallic system with __n__ divisions , [[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|__AID__]] is considered as [[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html|arithmetic sequence]] with divisions of system as terms of sequence.

If the first division is __**A1**__ and the last , __**An**__~~~~ , with common difference of __**d**__~~~~~~ , we~~ have :

If the first division is __A1__ and the last , __An__ , with common difference of __d__ , we have :

**A1 = A1 **

A1 = A1

** A2 = A1+d **

A2 = A1+d

** A3= A1+2d **

A3= A1+2d

** A4 = A1+3d **

A4 = A1+3d

** ……… **

………

**An = A1+(n-1)d**

An = A1+(n-1)d

**So sum of the divisions is** __**Sn**__ :

**Sn = __n[2A1+(n-1)d]__**

** 2**

As we can consider __**Sn**__ of system to be 1200 cent or anything else (octavic or non-octavic system ) then __**d**__

is most important to make an **__AID__** with **__n__** divisions with **__A1__**.important to make an __**AID**__ with **__n__** divisions with __**A1**__.

so, the common difference between divisions is :

**d = __2(Sn - nA1)__**

** n(n-1)**

By considering **Sn=1200** , **A1=70** , **n=12** , **__d__** will be 5.454545455 and our 12-tone scale is equal to:

==[[#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** ==

==Scales based on **AID** can be subsets of **EDO** if :==

== Scales based on **AID** can be subsets of **EDO** if : ==

1- we choose **d=0** so , **A1 = Sn/n**

Consider **__n__=8** and **__A1__=150** , then we have **8-EDO** .

2- for a constant __**n**__ 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__** , **__n__=8** and **__A1__=70** , then we have a subset of a **140-ED(1400.)** with Degrees as

**7 17 30 46 65 87 112 140 :**

~~ ~~

==[[#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** ==

~~==[[#TOC-And-now-for:]]And now for:==~~

~~==[[#TOC-Sn-1400-and-n-8-]]**Sn=1400 and** **n=8,**==~~

~~==[[#TOC-If-A1-175.0-then-we-have-8-AID-1400]]If **A1=175.0** then we have **8-AID(1400.)**==~~

==[[#TOC-If-A1-56-then-we-have-700-AID-1400.]]If **A1=56** then we have **700-AID(1400.)** with Degrees as **28 73 135 214 310 423 553 700**==

==[[#TOC-If-A1-87.5-then-we-have-112-AID-140]]If **A1=87.5** then we have **112-AID(1400.)** with Degrees as ** 7 16 27 40 55 72 91 112**==

==[[#TOC-1]] ==

~~==[[#TOC-2]] ==~~

~~==[[#TOC-3]]==~~

**__AID__** sytem shows different ascending , descending or linear trend of ~~change in~~ divisions ~~sizes due to relation between **n** and **A1** in __**AID**__&~~lt;span style="color: black~~; font-family: Arial~~; font-size: ~~10pt~~;"> ~~and **__EDO__** with equal degree:</span~~>

So sum of the divisions is __Sn__ :

~~*** If choosing**~~ **__A1</span~~>__ ~~greater than division size in equal degree~~ </span~~>__~~EDO~~__ ,~~ __d__ ~~is negative and __~~~~AID~~__~~****~~ ~~is descending.

* If choosing __A1__ ~~smaller than division size in equal degree~~ __~~EDO~~__ , __d__ ~~is positive and~~ __~~AID~~__**** is ~~ascending.~~

**Sn = __n[2A1+(n-1)d]__**

* If choosing __A1</span~~>__ ~~equal to division size in equal degree~~ </span~~>__~~EDO~~__ ,~~ __d__&~~lt;span style="color: black; font-family: Arial; font-size: 10pt;"> is zero.**</pre></div>

**2**

As we can consider __Sn__ of system to be 1200 cent or anything else (octavic or non-octavic system ) then __d__ is most important to make an __AID__ with __n__ divisions with __A1__.important to make an __AID__ with __n__ divisions with __A1__. so, the common difference between divisions is :

**d = __2(Sn - nA1)__**

**n(n-1)**

By considering Sn=1200 , A1=70 , n=12 , __d__ will be 5.454545455 and our 12-tone scale is equal to:

[[#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**

Scales based on AID can be subsets of EDO if : Scales based on AID can be subsets of EDO if : 1- we choose d=0 so , A1 = Sn/n .. Consider __n__=8 and __A1__=150 , then we have 8-EDO .

2- for a constant __n__ 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__ , __n__=8 and __A1__=70 , then we have a subset of a 140-ED(1400.) with Degrees as 7 17 30 46 65 87 112 140 :

**0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000**

[[#TOC-And-now-for:]]And now for Sn=1400 and n=8,

[[#TOC-If-A1-175.0-then-we-have-8-AID-1400]]If A1=175.0 then we have 8-AID(1400.)

[[#TOC-If-A1-56-then-we-have-700-AID-1400.]]If A1=56 then we have 700-AID(1400.) with Degrees as 28 73 135 214 310 423 553 700

[[#TOC-If-A1-87.5-then-we-have-112-AID-140]]If A1=87.5 then we have 112-AID(1400.) with Degrees as 7 16 27 40 55 72 91 112

__AID__ sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in __AID__ and __EDO__ with equal degree:

* If choosing __A1__ greater than division size in equal degree __EDO__ , __d__ is negative and __AID__ is descending.

* If choosing __A1__ smaller than division size in equal degree __EDO__ , __d__ is positive and __AID__ is ascending.

* If choosing __A1__ equal to division size in equal degree __EDO__ , __d__ is zero.**

[[image:http://sites.google.com/site/240edo/AIDO-custom-size-298-402.jpg align="center"]]

==171.4285714 is point of intersection in these 3 trends:==

[[image:http://sites.google.com/site/240edo/AIDO2.jpg align="center"]]

[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|**Example : Baran scale**]]</pre></div>

<h4>Original HTML content:</h4>

~~<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0;~~

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #

AID sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in AID and EDO with equal degree:</span~~></span~~>

<~~ul><ul><ul><li>If choosing A1 greater than division size in equal degree </span~~><~~u><span~~ style=~~"color: black~~; ~~font-family: Arial~~; ~~font~~-~~size~~: ~~10pt~~;~~"&gt~~;~~EDO~~<~~/span&~~gt;&~~lt;/u&gt~~;<~~span style~~=~~"color: black~~; ~~font-family: Arial~~; ~~font-size~~: ~~10pt;"> , <~~/~~span>d<~~/~~span><~~/~~u> is negative and <~~/~~span~~>~~&lt~~;~~u>~~AID&lt~~;~~/span~~>~~&lt~~;~~/u> ~~is descending.&lt~~;~~/li~~>~~&lt~~;/ul>~~</ul&~~gt;&~~lt;li&gt~~;~~If choosing &~~lt;/~~span~~&~~gt;&lt~~;u>~~A1</u~~><~~span~~ style="~~color: black; font~~-~~family~~: ~~Arial; font-size: 10pt;~~"> ~~smaller than division size in equal degree~~ <~~/span>EDO<~~/~~span><~~/~~u> , d<~~/~~span~~><~~span style="color: black; font~~-~~family~~: ~~Arial; font-size~~: ~~10pt;"> is positive and ~~AID~~</~~span~~> is ascending.</li><li>If choosing A1<~~/span> equal to division size in equal degree EDO<~~/~~span><~~/~~u> , <~~/~~span>d<~~/~~span><~~/~~u> ~~is zero.~~**</li></ul></body></html></pre></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>

<a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow">**Example : Baran scale**</a></body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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 01:53:09 UTC</tt>.<br>
+: This revision was by author [[User:acousticsoftombak|acousticsoftombak]] and made on <tt>2012-02-17 02:05:55 UTC</tt>.<br>
-: The original revision id was <tt>302669008</tt>.<br>
+: The original revision id was <tt>302670104</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>
 <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">&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;For an intervallic system with &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style="font-family: arial,sans-serif;"&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**n**&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; divisions , &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**AID**&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; is considered as [[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html|arithmetic sequence]] with divisions of system as terms of sequence. &lt;/span&gt;&lt;/span&gt;
+<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">&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;For an intervallic system with &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; divisions , &lt;/span&gt;&lt;span style="color: #be00ff; font-size: 11pt;"&gt;[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|__AID__]]&lt;/span&gt;&lt;span style="color: black; font-size: 11pt;"&gt; is considered as [[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html|arithmetic sequence]] with divisions of system as terms of sequence. &lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; If the first division is &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**A1**&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; and the last , &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**An**&lt;/span&gt;__&lt;/span&gt;&lt;span style="color: black; font-size: 11pt;"&gt; , with common difference of &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**d**&lt;/span&gt;__&lt;/span&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; , we have :&lt;/span&gt; &lt;/span&gt;&lt;/span&gt;
+&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; If the first division is &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; and the last , &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;An&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; , with common difference of &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; , we have : &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt; &lt;/span&gt;&lt;span style="font-family: arial,sans-serif;"&gt;**&lt;span style="color: black; font-size: 11pt;"&gt;A1 = A1 &lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
+&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A1 = A1 &lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt; A2 = A1+d &lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
+&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A2 = A1+d &lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt; A3= A1+2d &lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
+&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A3= A1+2d &lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; A4 = A1+3d &lt;/span&gt;&lt;/span&gt;**&lt;/span&gt;
+&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A4 = A1+3d&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt; ……… &lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
+&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;……… &lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; &lt;/span&gt;**&lt;span style="font-family: arial,sans-serif;"&gt;An = A1+(n-1)d&lt;/span&gt;** &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
+&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;An = A1+(n-1)d&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; &lt;span style="font-family: arial,sans-serif;"&gt;**So sum of the divisions is** &lt;/span&gt;&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;**Sn**&lt;/span&gt;&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; :&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: center;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; **&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;Sn = __n[2A1+(n-1)d]__&lt;/span&gt;&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: center;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt; 2&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: Times New Roman;"&gt; &lt;/span&gt;&lt;/span&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; &lt;span style="font-family: arial,sans-serif;"&gt;As we can consider &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style="font-family: arial,sans-serif;"&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**Sn**&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; of system to be 1200 cent or anything else (octavic or non-octavic system ) then &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**d**&lt;/span&gt;__ &lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; is most important to make an **__AID__** with **__n__** divisions with **__A1__**.important to make an &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**AID**&lt;/span&gt;__ with **__n__**&lt;span style="color: black; font-size: 11pt;"&gt; divisions with &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**A1**&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;. &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; so, the common difference between divisions is :&lt;/span&gt; &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: center;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: Times New Roman;"&gt; &lt;span style="font-family: arial,sans-serif;"&gt;  **&lt;span style="color: black; font-size: 11pt;"&gt;d = __2(Sn - nA1)__&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt; n(n-1)&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: Arial;"&gt; By considering **Sn=1200** , **A1=70** , **n=12** , **__d__** will be 5.454545455 and our 12-tone &lt;span style="font-family: arial,sans-serif;"&gt;scale is equal to&lt;/span&gt;:&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
-==&lt;span style="direction: ltr; text-align: center;"&gt;[[#TOC-0.070.0145.455226.364312.727404.545]]&lt;span style="font-family: arial,sans-serif;"&gt; **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**&lt;/span&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: Arial;"&gt; &lt;/span&gt;&lt;/span&gt; &lt;/span&gt;&lt;span style="font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;/span&gt;==
-==&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;Scales based on **AID** can be subsets of **EDO** if :&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;==
-==&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; Scales based on **AID** can be subsets of **EDO** if :&lt;/span&gt; &lt;/span&gt;&lt;/span&gt;==
-&lt;span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;1- we choose **d=0** so ,&lt;span style="color: black; font-size: 11pt;"&gt; **&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;A1&lt;/span&gt; = Sn/n&lt;/span&gt;** &lt;/span&gt;   &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;Consider &lt;/span&gt;**__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;=8&lt;/span&gt;**&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; and &lt;/span&gt;**__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;=150&lt;/span&gt;**&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , then we have **8-EDO** .&lt;/span&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;2-&lt;span style="font-family: arial,sans-serif;"&gt; for a constant &lt;/span&gt;&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt;n&lt;/span&gt;**&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; and different **__A1__**, if &lt;/span&gt;**__d__** and (**Sn/A1)** are **Integer number**&lt;/span&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; , we have &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;a susbet of **EDO or EDI( Equal divisions of Interval)** .Consider **__Sn = 1400__** , &lt;/span&gt;**__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;=8&lt;/span&gt;**&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; and &lt;/span&gt;**__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;=70&lt;/span&gt;**&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , then we have a subset of a &lt;/span&gt;&lt;/span&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;**140-ED(1400.)** with Degrees as &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; **7 17 30 46 65 87 112 140 :**&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
-&lt;span style="color: red; font-family: Arial;"&gt; &lt;/span&gt;
-==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: center;"&gt;&lt;span style="color: #000000;"&gt;[[#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** &lt;/span&gt;&lt;/span&gt;==
-==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;&lt;span style="color: #000000;"&gt;[[#TOC-And-now-for:]]And now for:&lt;/span&gt;&lt;/span&gt;==
-==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;[[#TOC-Sn-1400-and-n-8-]]**Sn=1400 and** **n=8,**&lt;/span&gt;==
-==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;[[#TOC-If-A1-175.0-then-we-have-8-AID-1400]]If **A1=175.0** then we have **8-AID(1400.)**&lt;/span&gt;==
-==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;[[#TOC-If-A1-56-then-we-have-700-AID-1400.]]If **A1=56** then we have **700-AID(1400.)** with Degrees as &lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;  **28 73 135 214 310 423 553 700**&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;==
-==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;[[#TOC-If-A1-87.5-then-we-have-112-AID-140]]&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;If  **A1=87.5** then we have **112-AID(1400.)** with Degrees as **&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; 7 16 27 40 55 72 91 112&lt;/span&gt;&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;==
-==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;[[#TOC-1]]&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;==
-==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: center;"&gt;[[#TOC-2]] &lt;/span&gt;==
-==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;[[#TOC-3]]&lt;/span&gt;==
-&lt;span style="font-family: arial,sans-serif;"&gt;**__AID__** sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between **n** and **A1** in __**AID**__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; and **__EDO__** with equal degree:&lt;/span&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;
+&lt;span style="direction: ltr; font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;So sum of the divisions is &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;Sn&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; :&lt;/span&gt;&lt;/span&gt;
-*** If choosing** &lt;/span&gt;**__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; greater than division size in equal degree &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;EDO&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is negative and &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;AID&lt;/span&gt;__****&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is descending.
-* If choosing &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; smaller than division size in equal degree &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;EDO&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is positive and &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;AID&lt;/span&gt;__****&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is ascending.
+&lt;span style="direction: ltr; display: block; text-align: center;"&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; **&lt;span style="color: black; font-size: 11pt;"&gt;Sn = __n[2A1+(n-1)d]__&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
-* If choosing &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; equal to division size in equal degree &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;EDO&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is zero.&lt;/span&gt;**</pre></div>
+&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; **2**&lt;/span&gt;
+&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; As we can consider &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;Sn&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; of system to be 1200 cent or anything else (octavic or non-octavic system ) then &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; is most important to make an __AID__ with __n__ divisions with __A1__.important to make an &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;AID&lt;/span&gt;__ with __n__&lt;span style="color: black; font-size: 11pt;"&gt; divisions with &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;.&lt;/span&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; so, the common difference between divisions is : &lt;/span&gt;&lt;/span&gt;
+&lt;span style="direction: ltr; display: block; text-align: center;"&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; **&lt;span style="color: black; font-size: 11pt;"&gt;d = __2(Sn - nA1)__&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
+&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; **n(n-1)**&lt;/span&gt;
+&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; By considering Sn=1200 , A1=70 , n=12 , __d__ will be 5.454545455 and our 12-tone scale is equal to:&lt;/span&gt;&lt;/span&gt;
+&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;[[#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**&lt;/span&gt;
+&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;Scales based on AID can be subsets of EDO if :&lt;/span&gt;&lt;/span&gt;&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; Scales based on AID can be subsets of EDO if : &lt;/span&gt;&lt;/span&gt;&lt;span style="direction: ltr; display: block; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 10pt;"&gt;1- we choose d=0 so ,&lt;span style="color: black; font-size: 11pt;"&gt; A1 = Sn/n&lt;/span&gt; .. Consider &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;=8 and &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;=150 , then we have 8-EDO .&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
+&lt;span style="direction: ltr; display: block; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 10pt;"&gt;2- for a constant &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; 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__ , &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;=8 and &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;=70 , then we have a subset of a 140-ED(1400.) with Degrees as 7 17 30 46 65 87 112 140 :&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
+**0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000**
+&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;"&gt;[[#TOC-And-now-for:]]And now for &lt;/span&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;Sn=1400 and n=8,
+[[#TOC-If-A1-175.0-then-we-have-8-AID-1400]]If A1=175.0 then we have 8-AID(1400.)
+[[#TOC-If-A1-56-then-we-have-700-AID-1400.]]If A1=56 then we have 700-AID(1400.) with Degrees as &lt;span style="color: black; font-size: 10pt;"&gt; 28 73 135 214 310 423 553 700&lt;/span&gt;
+[[#TOC-If-A1-87.5-then-we-have-112-AID-140]]&lt;span style="color: black; font-size: 10pt;"&gt;If A1=87.5 then we have 112-AID(1400.) with Degrees as &lt;span style="color: black; font-size: 10pt;"&gt; 7 16 27 40 55 72 91 112&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
+&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;__AID__ sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in __AID__&lt;span style="color: black; font-size: 10pt;"&gt; and __EDO__ with equal degree:&lt;/span&gt;&lt;/span&gt;
+* &lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;If choosing __&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; greater than division size in equal degree &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;EDO&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; , &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; is negative and &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;AID&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; is descending.&lt;/span&gt;&lt;/span&gt;
+* &lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;If choosing __&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; smaller than division size in equal degree &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;EDO&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; , &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; is positive and &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;AID&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; is ascending.&lt;/span&gt;&lt;/span&gt;
+* &lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;If choosing __&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; equal to division size in equal degree &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;EDO&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; , &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; is zero.&lt;/span&gt;**&lt;/span&gt;
+[[image:http://sites.google.com/site/240edo/AIDO-custom-size-298-402.jpg align="center"]]
+==&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;"&gt;171.4285714 is point of intersection in these 3 trends:&lt;/span&gt;==
+&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;"&gt; &lt;/span&gt;
+[[image:http://sites.google.com/site/240edo/AIDO2.jpg align="center"]]
+&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;"&gt;[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|**Example : Baran scale**]]&lt;/span&gt;</pre></div>
 <h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #
   &lt;br /&gt;
-&lt;span style="font-family: arial,sans-serif;"&gt;&lt;strong&gt;&lt;u&gt;AID&lt;/u&gt;&lt;/strong&gt; sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between &lt;strong&gt;n&lt;/strong&gt; and &lt;strong&gt;A1&lt;/strong&gt; in &lt;u&gt;&lt;strong&gt;AID&lt;/strong&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; and &lt;strong&gt;&lt;u&gt;EDO&lt;/u&gt;&lt;/strong&gt; with equal degree:&lt;/span&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;&lt;br /&gt;
+&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;"&gt; &lt;/span&gt;&lt;br /&gt;
-&lt;ul&gt;&lt;ul&gt;&lt;ul&gt;&lt;li&gt;If choosing&lt;strong&gt; &lt;/span&gt;&lt;/strong&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; greater than division size in equal degree &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;EDO&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is negative and &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;AID&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is descending.&lt;/li&gt;&lt;/ul&gt;&lt;/ul&gt;&lt;li&gt;If choosing &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; smaller than division size in equal degree &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;EDO&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is positive and &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;AID&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is ascending.&lt;/li&gt;&lt;li&gt;If choosing &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; equal to division size in equal degree &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;EDO&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is zero.&lt;/span&gt;**&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;!-- ws:start:WikiTextRemoteImageRule:16:&amp;lt;div style=&amp;quot;text-align: center&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://sites.google.com/site/240edo/AIDO2.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/div&amp;gt; --&gt;&lt;div style="text-align: center"&gt;&lt;img src="http://sites.google.com/site/240edo/AIDO2.jpg" alt="external image AIDO2.jpg" title="external image AIDO2.jpg" /&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:16 --&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;"&gt;&lt;a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow"&gt;**Example : Baran scale**&lt;/a&gt;&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>