AID: Difference between revisions

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<~~h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

'''Arithmetic irrational divisions'''

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

~~: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-12-29 11:41:53 UTC</tt>. ~~

~~: The original revision id was <tt>602894478</tt>. ~~

~~: The revision comment was: <tt>removed tel links</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">**Arithmetic irrational divisions**</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. ~~~~

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</span~~>__ , we have : <~~/span~~>~~

~~A1 = A1~~

~~A2 = A1+d ~~

~~A3= A1+2d ~~

~~A4 = A1+3d~~

~~……… ~~

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

~~<~~span style="direction: ltr; font-family: Tahoma,Geneva,sans-serif;"~~><~~span style="color: black; font-size: 11pt;"~~>So sum of~~ the ~~divisions~~ is ~~<~~/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~~><~~/span~~>~~

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

~~<~~span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: ~~center~~;"~~>**Sn =( __n[2A1+(n-1)d])/2__**<~~/span~~>~~

A1 = A1

A2 = A1+d

A3= A1+2d

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. So, the common difference between divisions is : 

A4 = A1+3d

~~**d =( __2(Sn - nA1))/((n(n-1))__**</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:

………

[[#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 :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

An = A1+(n-1)d

__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:

So sum of the divisions is Sn :

* If choosing __<~~span style="color: black; font-size: ~~10pt~~;">A1__ greater than division size in equal degree __EDO__ , __d__ is ~~negative and __AID<~~/span~~>__<~~span style="color: black; font-size: ~~10pt~~;"~~> is descending.<~~/span~~><~~/~~span>

* If choosing __A1__ smaller than division size in equal degree <~~/span~~>__EDO</span>__ , __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~~"]]

'''Sn =( n[2A1+(n-1)d])/2'''

==~~&lt~~;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~~>==~~

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. 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:

'''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 :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'''

And now for Sn=1400 and n=8,

If A1=175.0 then we have 8-AID(1400.)

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

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:

<ul><li>If choosing A1 greater than division size in equal degree EDO , d is negative and AID is descending.</li><li>If choosing A1 smaller than division size in equal degree EDO , d is positive and AID is ascending.</li><li>If choosing A1 equal to division size in equal degree EDO , d is zero.**</li></ul>

External image: http://sites.google.com/site/240edo/AIDO-custom-size-298-402.jpg

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</div>

[[Category:IMPORTDEBUG - Change External Images]]

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

External image: http://sites.google.com/site/240edo/AIDO2.jpg

: WARNING: MediaWiki doesn't have very good support for external images.

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</div>

[[Category:IMPORTDEBUG - Change External Images]]

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

we can have different kinds of AID:

__AIDO = Arithmetic irrational divisions of octave__

__AIDINO = Arithmetic irrational divisions of irrational non-octave__

__AIDRNO = Arithmetic irrational divisions of rational non-octave__

__AIDRI = Arithmetic irrational divisions of rational interval__

__AIDII = Arithmetic irrational divisions of irrational interval__

~~<~~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></pre></div>~~

AIDO = Arithmetic irrational divisions of octave

~~<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="~~color~~: ~~#000000~~; font-family: ~~arial~~,sans-serif; font-size: ~~18px~~;"~~>Arithmetic irrational divisions ~~

AIDINO = Arithmetic irrational divisions of irrational non-octave

~~ ~~

~~<~~span style="font-family: ~~Tahoma,Geneva~~,sans-serif;"~~><~~span style="~~color~~: ~~black~~; font-~~size~~: ~~11pt;"&gt~~;~~For an intervallic system with &lt~~;~~/span>&lt~~;u~~><~~span style="color: ~~black~~; font-~~size~~: ~~11pt~~;"~~>n<~~/span~~><~~/u~~><~~span style="~~color~~: ~~black~~; font-~~size~~: ~~11pt;"> divisions~~ , ~~&lt~~;~~/span&gt~~;~~&lt~~;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~~;~~"&gt~~; ~~is considered as &lt~~;~~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~~> ~~

AIDRNO = Arithmetic irrational divisions of rational non-octave

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

~~A1 = A1 ~~

AIDRI = Arithmetic irrational divisions of rational interval

~~A2 = A1+d ~~

~~A3= A1+2d ~~

AIDII = Arithmetic irrational divisions of irrational interval

~~A4 = A1+3d ~~

~~……… ~~

[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29 **Example : Baran scale**] [[Category:AID]]

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

[[Category:Shaahin_Mohajeri]]

~~ ~~

[[Category:todo:cleanup]]

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. 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:

<a name="TOC-0.070.0145.455226.364312.727404.545"></a> 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 :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 ~~

<a name="TOC-And-now-for:"></a>And now for Sn=1400 and n=8,

<a name="TOC-If-A1-175.0-then-we-have-8-AID-1400"></a>If A1=175.0 then we have 8-AID(1400.)

<a name="TOC-If-A1-56-then-we-have-700-AID-1400."></a>If A1=56 then we have 700-AID(1400.) with Degrees as 28 73 135 214 310 423 553 700

<a name="TOC-If-A1-87.5-then-we-have-112-AID-140"></a>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:

<ul><li>If choosing A1 greater than division size in equal degree EDO , d is negative and AID is descending.</li><li>If choosing A1 smaller than division size in equal degree EDO , d is positive and AID is ascending.</li><li>If choosing A1 equal to division size in equal degree EDO , d is zero.**</li></ul>

<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>

<h2 id="toc0"><a name="x-171.4285714 is point of intersection in these 3 trends:"></a>171.4285714 is point of intersection in these 3 trends:</h2>

~~ ~~

<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>we can have different kinds of AID:

AIDO = Arithmetic irrational divisions of octave

AIDINO = Arithmetic irrational divisions of irrational non-octave

AIDRNO = Arithmetic irrational divisions of rational non-octave

AIDRI = Arithmetic irrational divisions of rational interval

AIDII = Arithmetic irrational divisions of irrational interval

~~ ~~

<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>
+<span style="color: #000000; font-family: arial,sans-serif; font-size: 18px;">'''Arithmetic irrational divisions'''</span>
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-12-29 11:41:53 UTC</tt>.<br>
-: The original revision id was <tt>602894478</tt>.<br>
-: The revision comment was: <tt>removed tel links</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="color: #000000; font-family: arial,sans-serif; font-size: 18px;"&gt;**Arithmetic irrational divisions**&lt;/span&gt;
-&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;
+<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;">[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>
-&lt;span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;"&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 style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"&gt;A1 = A1&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="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A3= A1+2d &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="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&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; 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;
+<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>
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"&gt;**&lt;span style="color: black; font-size: 11pt;"&gt;Sn =( __n[2A1+(n-1)d])/2__&lt;/span&gt;**&lt;/span&gt;
+<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>
-&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.&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;
+<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A4 = A1+3d</span>
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"&gt;**&lt;span style="color: black; font-size: 11pt;"&gt;d =( __2(Sn - nA1))/((n(n-1))__&lt;/span&gt;**&lt;/span&gt;
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"&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;
+<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">……… </span>
-&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="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"&gt;Scales based on AID can be subsets of EDO if :&lt;/span&gt;&lt;span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"&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 style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"&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;
-**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,&lt;/span&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&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="font-family: Tahoma,Geneva,sans-serif;"&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="color: black; font-size: 10pt;"&gt; 28 73 135 214 310 423 553 700&lt;/span&gt;&lt;/span&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&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;
+<span style="font-family: Tahoma,Geneva,sans-serif;">An = A1+(n-1)d</span>
-&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;
+<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>
-* &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"]]
+<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 =( <u>n[2A1+(n-1)d])/2</u></span>'''</span>
-==&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;==
+<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 AID with n divisions with A1.</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 =( <u>2(Sn - nA1))/((n(n-1))</u></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;"> '''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 :</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>
+<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>
+'''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;">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;"> If A1=175.0 then we have 8-AID(1400.)</span>
+<span style="font-family: Tahoma,Geneva,sans-serif;"> 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;"> <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;"><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>
+<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>
+<div class='external-image-warning' style='background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block'>
+External image: http://sites.google.com/site/240edo/AIDO-custom-size-298-402.jpg<br>
+: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
+: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
+</div>
+[[Category:IMPORTDEBUG - Change External Images]]
+==<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>==
+<div class='external-image-warning' style='background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block'>
+External image: http://sites.google.com/site/240edo/AIDO2.jpg<br>
+: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
+: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
+</div>
+[[Category:IMPORTDEBUG - Change External Images]]
-[[image:http://sites.google.com/site/240edo/AIDO2.jpg align="center"]]
 we can have different kinds of AID:
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;__&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDO = Arithmetic irrational divisions of octave&lt;/span&gt;__&lt;/span&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;__&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDINO = Arithmetic irrational divisions of irrational non-octave&lt;/span&gt;__&lt;/span&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;__&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDRNO = Arithmetic irrational divisions of rational non-octave&lt;/span&gt;__&lt;/span&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;__&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDRI = Arithmetic irrational divisions of rational interval&lt;/span&gt;__&lt;/span&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;__&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDII = Arithmetic irrational divisions of irrational interval&lt;/span&gt;__&lt;/span&gt;
-&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>
+<span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"><u><span style="color: #000000; font-family: arial,sans-serif;">AIDO = Arithmetic irrational divisions of octave</span></u></span>
-<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;AID&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="color: #000000; font-family: arial,sans-serif; font-size: 18px;"&gt;&lt;strong&gt;Arithmetic irrational divisions&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
+<span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"><u><span style="color: #000000; font-family: arial,sans-serif;">AIDINO = Arithmetic irrational divisions of irrational non-octave</span></u></span>
-&lt;br /&gt;
-&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;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;n&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; divisions , &lt;/span&gt;&lt;span style="color: #be00ff; font-size: 11pt;"&gt;&lt;a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow"&gt;AID_&lt;/a&gt;&lt;/span&gt;&lt;span style="color: black; font-size: 11pt;"&gt; is considered as &lt;a class="wiki_link_ext" href="http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html" rel="nofollow"&gt;arithmetic sequence&lt;/a&gt; with divisions of system as terms of sequence. &lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+<span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"><u><span style="color: #000000; font-family: arial,sans-serif;">AIDRNO = Arithmetic irrational divisions of rational non-octave</span></u></span>
-&lt;span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; If the first division is &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; and the last , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;An&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; , with common difference of &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; , we have : &lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"&gt;A1 = A1&lt;/span&gt;&lt;br /&gt;
+<span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"><u><span style="color: #000000; font-family: arial,sans-serif;">AIDRI = Arithmetic irrational divisions of rational interval</span></u></span>
-&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A2 = A1+d &lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A3= A1+2d &lt;/span&gt;&lt;br /&gt;
+<span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"><u><span style="color: #000000; font-family: arial,sans-serif;">AIDII = Arithmetic irrational divisions of irrational interval</span></u></span>
-&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A4 = A1+3d&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;……… &lt;/span&gt;&lt;br /&gt;
+<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>      [[Category:AID]]
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;An = A1+(n-1)d&lt;/span&gt;&lt;br /&gt;
+[[Category:Shaahin_Mohajeri]]
-&lt;br /&gt;
+[[Category:todo:cleanup]]
-&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;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;Sn&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; :&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"&gt;&lt;strong&gt;&lt;span style="color: black; font-size: 11pt;"&gt;Sn =( &lt;u&gt;n[2A1+(n-1)d])/2&lt;/u&gt;&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&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;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;Sn&lt;/span&gt;&lt;/u&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;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; is most important to make an AID with n divisions with A1.&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;br /&gt;
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"&gt;&lt;strong&gt;&lt;span style="color: black; font-size: 11pt;"&gt;d =( &lt;u&gt;2(Sn - nA1))/((n(n-1))&lt;/u&gt;&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"&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;br /&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:2:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@TOC-0.070.0145.455226.364312.727404.545&amp;quot; title=&amp;quot;Anchor: TOC-0.070.0145.455226.364312.727404.545&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-0.070.0145.455226.364312.727404.545"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:2 --&gt; &lt;strong&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;/strong&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"&gt;Scales based on AID can be subsets of EDO if :&lt;/span&gt;&lt;span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"&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;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;=8 and &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;=150 , then we have 8-EDO .&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="color: black; font-size: 10pt;"&gt;2- for a constant &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; and different &lt;u&gt;A1&lt;/u&gt;, if &lt;u&gt;d&lt;/u&gt; and (Sn/A1) are Integer number , we have a susbet of EDO or EDI( Equal divisions of Interval) .Consider &lt;u&gt;Sn = 1400&lt;/u&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;=8 and &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&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;br /&gt;
-&lt;strong&gt;0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000&lt;/strong&gt;&lt;br /&gt;
-&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:3:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@TOC-And-now-for:&amp;quot; title=&amp;quot;Anchor: TOC-And-now-for:&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-And-now-for:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:3 --&gt;And now for &lt;/span&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;Sn=1400 and n=8,&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt; &lt;!-- ws:start:WikiTextAnchorRule:4:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@TOC-If-A1-175.0-then-we-have-8-AID-1400&amp;quot; title=&amp;quot;Anchor: TOC-If-A1-175.0-then-we-have-8-AID-1400&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-If-A1-175.0-then-we-have-8-AID-1400"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:4 --&gt;If A1=175.0 then we have 8-AID(1400.)&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt; &lt;!-- ws:start:WikiTextAnchorRule:5:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@TOC-If-A1-56-then-we-have-700-AID-1400.&amp;quot; title=&amp;quot;Anchor: TOC-If-A1-56-then-we-have-700-AID-1400.&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-If-A1-56-then-we-have-700-AID-1400."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:5 --&gt;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;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt; &lt;!-- ws:start:WikiTextAnchorRule:6:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@TOC-If-A1-87.5-then-we-have-112-AID-140&amp;quot; title=&amp;quot;Anchor: TOC-If-A1-87.5-then-we-have-112-AID-140&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-If-A1-87.5-then-we-have-112-AID-140"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:6 --&gt;&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;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;u&gt;AID&lt;/u&gt; sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in &lt;u&gt;AID&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; and &lt;u&gt;EDO&lt;/u&gt; with equal degree:&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;If choosing &lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; greater than division size in equal degree &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;EDO&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; is negative and &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;AID&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; is descending.&lt;/span&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;If choosing &lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; smaller than division size in equal degree &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;EDO&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; is positive and &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;AID&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; is ascending.&lt;/span&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;If choosing &lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; equal to division size in equal degree &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;EDO&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; is zero.&lt;/span&gt;**&lt;/span&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:15:&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/AIDO-custom-size-298-402.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/AIDO-custom-size-298-402.jpg" alt="external image AIDO-custom-size-298-402.jpg" title="external image AIDO-custom-size-298-402.jpg" /&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:15 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-171.4285714 is point of intersection in these 3 trends:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&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;/h2&gt;
- &lt;br /&gt;
-&lt;br /&gt;
-&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;we can have different kinds of AID:&lt;br /&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;&lt;u&gt;&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDO = Arithmetic irrational divisions of octave&lt;/span&gt;&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;&lt;u&gt;&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDINO = Arithmetic irrational divisions of irrational non-octave&lt;/span&gt;&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;&lt;u&gt;&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDRNO = Arithmetic irrational divisions of rational non-octave&lt;/span&gt;&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;&lt;u&gt;&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDRI = Arithmetic irrational divisions of rational interval&lt;/span&gt;&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;&lt;u&gt;&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDII = Arithmetic irrational divisions of irrational interval&lt;/span&gt;&lt;/u&gt;&lt;/span&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>