Xenharmonic Wiki

AID: Difference between revisions

VisualWikitext
Wikispaces>acousticsoftombak
**Imported revision 302669008 - Original comment: **
 
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
18,452 edits
mNo edit summary
 
(13 intermediate revisions by 8 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{todo|inline=1|cleanup|comment=If it is deprecated and cannot be updated, then just add the [[Template:Deprecated]] message box to the top of the page and delete this todo notice}}
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>
: The original revision id was <tt>302669008</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;
&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; 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; 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="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="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="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="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="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;
==Arithmetic irrational divisions==
*** 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.
For an intervallic system with <u>n</u> 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 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>
 
<h4>Original HTML content:</h4>
If the first division is <u>A1</u> and the last, <u>An</u> , with common difference of <u>d</u> , we have :  
<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="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;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;strong&gt;n&lt;/strong&gt;&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; divisions , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;strong&gt;AID&lt;/strong&gt;&lt;/span&gt;&lt;/u&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;
:: A1 = A1
&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;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;strong&gt;A1&lt;/strong&gt;&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;&lt;strong&gt;An&lt;/strong&gt;&lt;/span&gt;&lt;/u&gt;&lt;/span&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;&lt;strong&gt;d&lt;/strong&gt;&lt;/span&gt;&lt;/u&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;br /&gt;
:: A2 = A1+d
&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;strong&gt;&lt;span style="color: black; font-size: 11pt;"&gt;A1 = A1 &lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
:: A3 = A1+2d
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;strong&gt;&lt;span style="font-family: arial,sans-serif;"&gt; A2 = A1+d &lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
:: A4 = A1+3d
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;strong&gt;&lt;span style="font-family: arial,sans-serif;"&gt; A3= A1+2d &lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
:: ...
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;strong&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;/strong&gt;&lt;/span&gt;&lt;br /&gt;
:: An = A1+(n-1)d
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;strong&gt;&lt;span style="font-family: arial,sans-serif;"&gt; ……… &lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&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;strong&gt;&lt;span style="font-family: arial,sans-serif;"&gt;An = A1+(n-1)d&lt;/span&gt;&lt;/strong&gt; &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
So sum of the divisions is <u>Sn</u> :
&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;strong&gt;So sum of the divisions is&lt;/strong&gt; &lt;/span&gt;&lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;strong&gt;Sn&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; :&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
:: '''Sn =( <u>n[2A1+(n-1)d])/2</u>'''
&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;strong&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;Sn = &lt;u&gt;n[2A1+(n-1)d]&lt;/u&gt;&lt;/span&gt;&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 
&lt;span style="direction: ltr; text-align: center;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;strong&gt;&lt;span style="font-family: arial,sans-serif;"&gt; 2&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&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;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;strong&gt;Sn&lt;/strong&gt;&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;&lt;strong&gt;d&lt;/strong&gt;&lt;/span&gt;&lt;/u&gt; &lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
As we can consider <u>Sn</u> of system to be 1200 cent or anything else (octavic or non-octavic system ) then <u>d</u> is most important to make an AID with n divisions with A1. So, the common difference between divisions is :  
&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 &lt;strong&gt;&lt;u&gt;AID&lt;/u&gt;&lt;/strong&gt; with &lt;strong&gt;&lt;u&gt;n&lt;/u&gt;&lt;/strong&gt; divisions with &lt;strong&gt;&lt;u&gt;A1&lt;/u&gt;&lt;/strong&gt;.important to make an &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;strong&gt;AID&lt;/strong&gt;&lt;/span&gt;&lt;/u&gt; with &lt;strong&gt;&lt;u&gt;n&lt;/u&gt;&lt;/strong&gt;&lt;span style="color: black; font-size: 11pt;"&gt; divisions with &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;strong&gt;A1&lt;/strong&gt;&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;. &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
:: '''d =( <u>2(Sn - nA1))/(n(n-1))</u>'''
&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;br /&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;strong&gt;&lt;span style="color: black; font-size: 11pt;"&gt;d = &lt;u&gt;2(Sn - nA1)&lt;/u&gt;&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
By considering Sn=1200, A1=70, n=12, d will be 5.454545455 and our 12-tone scale is equal to:
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;strong&gt;&lt;span style="font-family: arial,sans-serif;"&gt; n(n-1)&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&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 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 &lt;strong&gt;Sn=1200&lt;/strong&gt; , &lt;strong&gt;A1=70&lt;/strong&gt; , &lt;strong&gt;n=12&lt;/strong&gt; , &lt;strong&gt;&lt;u&gt;d&lt;/u&gt;&lt;/strong&gt; 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;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="direction: ltr; text-align: center;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:24:&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:24 --&gt;&lt;span style="font-family: arial,sans-serif;"&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;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;/h2&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Scales based on AID** can be subsets of **EDO if :"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&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 &lt;strong&gt;AID&lt;/strong&gt; can be subsets of &lt;strong&gt;EDO&lt;/strong&gt; if :&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/h2&gt;
Scales based on AID can be subsets of [[EDO]] if:
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Scales based on AID** can be subsets of **EDO if :"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&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 &lt;strong&gt;AID&lt;/strong&gt; can be subsets of &lt;strong&gt;EDO&lt;/strong&gt; if :&lt;/span&gt; &lt;/span&gt;&lt;/span&gt;&lt;/h2&gt;
# we choose d=0 so, A1 = Sn/n .. Consider <u>n</u>=8 and <u>A1</u>=150, then we have [[8edo|8-EDO]] .
&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 &lt;strong&gt;d=0&lt;/strong&gt; so ,&lt;span style="color: black; font-size: 11pt;"&gt; &lt;strong&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;/strong&gt; &lt;/span&gt;  &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
# for a constant <u>n</u> and different <u>A1</u>, if <u>d</u> and (Sn/A1) are integers, we have a subset of EDO or [[Equal-step tuning|EDI (Equal divisions of Interval)]].
&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;strong&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;n&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;=8&lt;/span&gt;&lt;/strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; and &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;=150&lt;/span&gt;&lt;/strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , then we have &lt;strong&gt;8-EDO&lt;/strong&gt; .&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;br /&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;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;&lt;strong&gt;&lt;span style="font-family: arial,sans-serif;"&gt;n&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; and different &lt;strong&gt;&lt;u&gt;A1&lt;/u&gt;&lt;/strong&gt;, if &lt;/span&gt;&lt;strong&gt;&lt;u&gt;d&lt;/u&gt;&lt;/strong&gt; and (&lt;strong&gt;Sn/A1)&lt;/strong&gt; are &lt;strong&gt;Integer number&lt;/strong&gt;&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 &lt;strong&gt;EDO or EDI( Equal divisions of Interval)&lt;/strong&gt; .Consider &lt;strong&gt;&lt;u&gt;Sn = 1400&lt;/u&gt;&lt;/strong&gt; , &lt;/span&gt;&lt;strong&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;n&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;=8&lt;/span&gt;&lt;/strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; and &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;=70&lt;/span&gt;&lt;/strong&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;&lt;strong&gt;140-ED(1400.)&lt;/strong&gt; with Degrees as &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
Consider <u>Sn = 1400</u> , <u>n</u>=8 and <u>A1</u>=70, then we have a subset of a 140-ED (1400.) with Degrees as 7 17 30 46 65 87 112 140 :
&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; &lt;strong&gt;7 17 30 46 65 87 112 140 :&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
:: 0.0 70.0 170.0 300.0 460.0 650.0 870.0 1120.0 1400.0
&lt;span style="color: red; font-family: Arial;"&gt; &lt;/span&gt;&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000  1400.000"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: center;"&gt;&lt;span style="color: #000000;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:25:&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-0.00070.000170.000300.000460.000650&amp;quot; title=&amp;quot;Anchor: TOC-0.00070.000170.000300.000460.000650&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-0.00070.000170.000300.000460.000650"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:25 --&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;/span&gt;&lt;/span&gt;&lt;/h2&gt;
And now for Sn=1400 and n=8,
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-And now for:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;&lt;span style="color: #000000;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:26:&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:26 --&gt;And now for:&lt;/span&gt;&lt;/span&gt;&lt;/h2&gt;
* If A1=175.0 then we have 8-AID(1400.)
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x-Sn=1400 and** **n=8,"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:27:&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-Sn-1400-and-n-8-&amp;quot; title=&amp;quot;Anchor: TOC-Sn-1400-and-n-8-&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-Sn-1400-and-n-8-"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:27 --&gt;&lt;strong&gt;Sn=1400 and&lt;/strong&gt; &lt;strong&gt;n=8,&lt;/strong&gt;&lt;/span&gt;&lt;/h2&gt;
* If A1=56 then we have 700-AID(1400.) with Degrees as  28 73 135 214 310 423 553 700
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x-If A1=175.0** then we have **8-AID(1400.)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:28:&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:28 --&gt;If &lt;strong&gt;A1=175.0&lt;/strong&gt; then we have &lt;strong&gt;8-AID(1400.)&lt;/strong&gt;&lt;/span&gt;&lt;/h2&gt;
* If A1=87.5 then we have 112-AID(1400.) with Degrees as  7 16 27 40 55 72 91 112
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="x-If A1=56** then we have **700-AID(1400.)** with Degrees as  **28 73 135 214 310 423 553 700"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:29:&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:29 --&gt;If &lt;strong&gt;A1=56&lt;/strong&gt; then we have &lt;strong&gt;700-AID(1400.)&lt;/strong&gt; with Degrees as &lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;  &lt;strong&gt;28 73 135 214 310 423 553 700&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/h2&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="x-If  A1=87.5** then we have **112-AID(1400.)** with Degrees as ** 7 16 27 40 55 72 91 112"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:30:&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:30 --&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;If  &lt;strong&gt;A1=87.5&lt;/strong&gt; then we have &lt;strong&gt;112-AID(1400.)&lt;/strong&gt; with Degrees as &lt;strong&gt;&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;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/h2&gt;
AID system 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:
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:31:&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-1&amp;quot; title=&amp;quot;Anchor: TOC-1&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-1"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:31 --&gt;&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;/h2&gt;
* If choosing <u>A1</u> greater than division size in equal degree EDO, <u>d</u> is negative and <u>AID</u> is descending.
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: center;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:32:&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-2&amp;quot; title=&amp;quot;Anchor: TOC-2&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-2"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:32 --&gt; &lt;/span&gt;&lt;/h2&gt;
* If choosing <u>A1</u> smaller than division size in equal degree EDO, <u>d</u> is positive and <u>AID</u> is ascending.
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:33:&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-3&amp;quot; title=&amp;quot;Anchor: TOC-3&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-3"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:33 --&gt;&lt;/span&gt;&lt;/h2&gt;
* If choosing <u>A1</u> equal to division size in equal degree EDO, <u>d</u> is zero.
&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;
[[file:AIDO-custom-size-298-402.jpg]]
&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>
 
 
171.4285714 is point of intersection in these 3 trends:
 
[[file:AIDO2.jpg]]
 
 
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
 
Example: [http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29 Baran scale]
 
[[Category:AID]]
[[Category:Shaahin Mohajeri]]
Retrieved from "https://en.xen.wiki/w/AID"