AID: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
Newer edit →
VisualWikitext
Wikispaces>acousticsoftombak
**Imported revision 302669008 - Original comment: **
 
Wikispaces>acousticsoftombak
**Imported revision 302670104 - Original comment: **
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:acousticsoftombak|acousticsoftombak]] and made on <tt>2012-02-17 01:53:09 UTC</tt>.<br>
: This revision was by author [[User:acousticsoftombak|acousticsoftombak]] and made on <tt>2012-02-17 02:05:55 UTC</tt>.<br>
: The original revision id was <tt>302669008</tt>.<br>
: The original revision id was <tt>302670104</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;For an intervallic system with &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style="font-family: arial,sans-serif;"&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**n**&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; divisions , &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**AID**&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; is considered as [[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html|arithmetic sequence]] with divisions of system as terms of sequence. &lt;/span&gt;&lt;/span&gt;
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;For an intervallic system with &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; divisions , &lt;/span&gt;&lt;span style="color: #be00ff; font-size: 11pt;"&gt;[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|__AID__]]&lt;/span&gt;&lt;span style="color: black; font-size: 11pt;"&gt; is considered as [[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html|arithmetic sequence]] with divisions of system as terms of sequence. &lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; If the first division is &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**A1**&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; and the last , &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**An**&lt;/span&gt;__&lt;/span&gt;&lt;span style="color: black; font-size: 11pt;"&gt; , with common difference of &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**d**&lt;/span&gt;__&lt;/span&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; , we have :&lt;/span&gt; &lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; If the first division is &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; and the last , &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;An&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; , with common difference of &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; , we have : &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt; &lt;/span&gt;&lt;span style="font-family: arial,sans-serif;"&gt;**&lt;span style="color: black; font-size: 11pt;"&gt;A1 = A1 &lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A1 = A1 &lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt; A2 = A1+d &lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A2 = A1+d &lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt; A3= A1+2d &lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A3= A1+2d &lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; A4 = A1+3d &lt;/span&gt;&lt;/span&gt;**&lt;/span&gt;
&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A4 = A1+3d&lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt; ……… &lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;……… &lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; &lt;/span&gt;**&lt;span style="font-family: arial,sans-serif;"&gt;An = A1+(n-1)d&lt;/span&gt;** &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;An = A1+(n-1)d&lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; &lt;span style="font-family: arial,sans-serif;"&gt;**So sum of the divisions is** &lt;/span&gt;&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;**Sn**&lt;/span&gt;&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; :&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; text-align: center;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; **&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;Sn = __n[2A1+(n-1)d]__&lt;/span&gt;&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; text-align: center;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt; 2&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: Times New Roman;"&gt; &lt;/span&gt;&lt;/span&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; &lt;span style="font-family: arial,sans-serif;"&gt;As we can consider &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style="font-family: arial,sans-serif;"&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**Sn**&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; of system to be 1200 cent or anything else (octavic or non-octavic system ) then &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**d**&lt;/span&gt;__ &lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; is most important to make an **__AID__** with **__n__** divisions with **__A1__**.important to make an &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**AID**&lt;/span&gt;__ with **__n__**&lt;span style="color: black; font-size: 11pt;"&gt; divisions with &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;**A1**&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;. &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; so, the common difference between divisions is :&lt;/span&gt; &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; text-align: center;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: Times New Roman;"&gt; &lt;span style="font-family: arial,sans-serif;"&gt;  **&lt;span style="color: black; font-size: 11pt;"&gt;d = __2(Sn - nA1)__&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt; n(n-1)&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: Arial;"&gt; By considering **Sn=1200** , **A1=70** , **n=12** , **__d__** will be 5.454545455 and our 12-tone &lt;span style="font-family: arial,sans-serif;"&gt;scale is equal to&lt;/span&gt;:&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
==&lt;span style="direction: ltr; text-align: center;"&gt;[[#TOC-0.070.0145.455226.364312.727404.545]]&lt;span style="font-family: arial,sans-serif;"&gt; **0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0**&lt;/span&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: Arial;"&gt; &lt;/span&gt;&lt;/span&gt; &lt;/span&gt;&lt;span style="font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;/span&gt;==
==&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;Scales based on **AID** can be subsets of **EDO** if :&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;==
==&lt;span style="direction: ltr; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; Scales based on **AID** can be subsets of **EDO** if :&lt;/span&gt; &lt;/span&gt;&lt;/span&gt;==
&lt;span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;1- we choose **d=0** so ,&lt;span style="color: black; font-size: 11pt;"&gt; **&lt;span style="color: black; font-size: 11pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;A1&lt;/span&gt; = Sn/n&lt;/span&gt;** &lt;/span&gt;  &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;Consider &lt;/span&gt;**__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;=8&lt;/span&gt;**&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; and &lt;/span&gt;**__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;=150&lt;/span&gt;**&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , then we have **8-EDO** .&lt;/span&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;2-&lt;span style="font-family: arial,sans-serif;"&gt; for a constant &lt;/span&gt;&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;**&lt;span style="font-family: arial,sans-serif;"&gt;n&lt;/span&gt;**&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; and different **__A1__**, if &lt;/span&gt;**__d__** and (**Sn/A1)** are **Integer number**&lt;/span&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt; , we have &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;a susbet of **EDO or EDI( Equal divisions of Interval)** .Consider **__Sn = 1400__** , &lt;/span&gt;**__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;=8&lt;/span&gt;**&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; and &lt;/span&gt;**__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;=70&lt;/span&gt;**&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , then we have a subset of a &lt;/span&gt;&lt;/span&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;**140-ED(1400.)** with Degrees as &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; **7 17 30 46 65 87 112 140 :**&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="color: red; font-family: Arial;"&gt; &lt;/span&gt;
==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: center;"&gt;&lt;span style="color: #000000;"&gt;[[#TOC-0.00070.000170.000300.000460.000650]] **0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000  1400.000** &lt;/span&gt;&lt;/span&gt;==
==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;&lt;span style="color: #000000;"&gt;[[#TOC-And-now-for:]]And now for:&lt;/span&gt;&lt;/span&gt;==
==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;[[#TOC-Sn-1400-and-n-8-]]**Sn=1400 and** **n=8,**&lt;/span&gt;==
==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;[[#TOC-If-A1-175.0-then-we-have-8-AID-1400]]If **A1=175.0** then we have **8-AID(1400.)**&lt;/span&gt;==
==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;[[#TOC-If-A1-56-then-we-have-700-AID-1400.]]If **A1=56** then we have **700-AID(1400.)** with Degrees as &lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;  **28 73 135 214 310 423 553 700**&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;==
==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;[[#TOC-If-A1-87.5-then-we-have-112-AID-140]]&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;If  **A1=87.5** then we have **112-AID(1400.)** with Degrees as **&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; 7 16 27 40 55 72 91 112&lt;/span&gt;&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;==
==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;[[#TOC-1]]&lt;span style="font-family: arial,sans-serif;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;==
==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: center;"&gt;[[#TOC-2]] &lt;/span&gt;==
==&lt;span style="direction: ltr; margin: 0cm 0cm 0pt 9pt; text-align: left;"&gt;[[#TOC-3]]&lt;/span&gt;==


&lt;span style="font-family: arial,sans-serif;"&gt;**__AID__** sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between **n** and **A1** in __**AID**__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; and **__EDO__** with equal degree:&lt;/span&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;
&lt;span style="direction: ltr; font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;So sum of the divisions is &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;Sn&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; :&lt;/span&gt;&lt;/span&gt;
*** If choosing** &lt;/span&gt;**__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; greater than division size in equal degree &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;EDO&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is negative and &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;AID&lt;/span&gt;__****&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is descending.
 
* If choosing &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; smaller than division size in equal degree &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;EDO&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is positive and &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;AID&lt;/span&gt;__****&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is ascending.
&lt;span style="direction: ltr; display: block; text-align: center;"&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; **&lt;span style="color: black; font-size: 11pt;"&gt;Sn = __n[2A1+(n-1)d]__&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
* If choosing &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; equal to division size in equal degree &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;EDO&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , &lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is zero.&lt;/span&gt;**</pre></div>
&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; **2**&lt;/span&gt;
 
&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; As we can consider &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;Sn&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; of system to be 1200 cent or anything else (octavic or non-octavic system ) then &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; is most important to make an __AID__ with __n__ divisions with __A1__.important to make an &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;AID&lt;/span&gt;__ with __n__&lt;span style="color: black; font-size: 11pt;"&gt; divisions with &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;.&lt;/span&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; so, the common difference between divisions is : &lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; display: block; text-align: center;"&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; **&lt;span style="color: black; font-size: 11pt;"&gt;d = __2(Sn - nA1)__&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; **n(n-1)**&lt;/span&gt;
 
&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; By considering Sn=1200 , A1=70 , n=12 , __d__ will be 5.454545455 and our 12-tone scale is equal to:&lt;/span&gt;&lt;/span&gt;
&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;[[#TOC-0.070.0145.455226.364312.727404.545]] **0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0**&lt;/span&gt;
&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;Scales based on AID can be subsets of EDO if :&lt;/span&gt;&lt;/span&gt;&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; Scales based on AID can be subsets of EDO if : &lt;/span&gt;&lt;/span&gt;&lt;span style="direction: ltr; display: block; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 10pt;"&gt;1- we choose d=0 so ,&lt;span style="color: black; font-size: 11pt;"&gt; A1 = Sn/n&lt;/span&gt; .. Consider &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;=8 and &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;=150 , then we have 8-EDO .&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="direction: ltr; display: block; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 10pt;"&gt;2- for a constant &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; and different __A1__, if __d__ and (Sn/A1) are Integer number , we have a susbet of EDO or EDI( Equal divisions of Interval) .Consider __Sn = 1400__ , &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;=8 and &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;=70 , then we have a subset of a 140-ED(1400.) with Degrees as 7 17 30 46 65 87 112 140 :&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
**0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000**
&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;"&gt;[[#TOC-And-now-for:]]And now for &lt;/span&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;Sn=1400 and n=8,
[[#TOC-If-A1-175.0-then-we-have-8-AID-1400]]If A1=175.0 then we have 8-AID(1400.)
[[#TOC-If-A1-56-then-we-have-700-AID-1400.]]If A1=56 then we have 700-AID(1400.) with Degrees as &lt;span style="color: black; font-size: 10pt;"&gt; 28 73 135 214 310 423 553 700&lt;/span&gt;
[[#TOC-If-A1-87.5-then-we-have-112-AID-140]]&lt;span style="color: black; font-size: 10pt;"&gt;If A1=87.5 then we have 112-AID(1400.) with Degrees as &lt;span style="color: black; font-size: 10pt;"&gt; 7 16 27 40 55 72 91 112&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
 
&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;__AID__ sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in __AID__&lt;span style="color: black; font-size: 10pt;"&gt; and __EDO__ with equal degree:&lt;/span&gt;&lt;/span&gt;
* &lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;If choosing __&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; greater than division size in equal degree &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;EDO&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; , &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; is negative and &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;AID&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; is descending.&lt;/span&gt;&lt;/span&gt;
* &lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;If choosing __&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; smaller than division size in equal degree &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;EDO&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; , &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; is positive and &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;AID&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; is ascending.&lt;/span&gt;&lt;/span&gt;
* &lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;If choosing __&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; equal to division size in equal degree &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;EDO&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; , &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; is zero.&lt;/span&gt;**&lt;/span&gt;
 
[[image:http://sites.google.com/site/240edo/AIDO-custom-size-298-402.jpg align="center"]]
 
==&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;"&gt;171.4285714 is point of intersection in these 3 trends:&lt;/span&gt;==
 
&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;"&gt; &lt;/span&gt;
[[image:http://sites.google.com/site/240edo/AIDO2.jpg align="center"]]
 
 
&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;"&gt;[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|**Example : Baran scale**]]&lt;/span&gt;</pre></div>
<h4>Original HTML content:</h4>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0;
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #
  &lt;br /&gt;
  &lt;br /&gt;
&lt;span style="font-family: arial,sans-serif;"&gt;&lt;strong&gt;&lt;u&gt;AID&lt;/u&gt;&lt;/strong&gt; sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between &lt;strong&gt;n&lt;/strong&gt; and &lt;strong&gt;A1&lt;/strong&gt; in &lt;u&gt;&lt;strong&gt;AID&lt;/strong&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; and &lt;strong&gt;&lt;u&gt;EDO&lt;/u&gt;&lt;/strong&gt; with equal degree:&lt;/span&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;&lt;br /&gt;
&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;"&gt; &lt;/span&gt;&lt;br /&gt;
&lt;ul&gt;&lt;ul&gt;&lt;ul&gt;&lt;li&gt;If choosing&lt;strong&gt; &lt;/span&gt;&lt;/strong&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; greater than division size in equal degree &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;EDO&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is negative and &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;AID&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is descending.&lt;/li&gt;&lt;/ul&gt;&lt;/ul&gt;&lt;li&gt;If choosing &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; smaller than division size in equal degree &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;EDO&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is positive and &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;AID&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is ascending.&lt;/li&gt;&lt;li&gt;If choosing &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; equal to division size in equal degree &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;EDO&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-family: Arial; font-size: 10pt;"&gt; is zero.&lt;/span&gt;**&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
&lt;!-- ws:start:WikiTextRemoteImageRule:16:&amp;lt;div style=&amp;quot;text-align: center&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://sites.google.com/site/240edo/AIDO2.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/div&amp;gt; --&gt;&lt;div style="text-align: center"&gt;&lt;img src="http://sites.google.com/site/240edo/AIDO2.jpg" alt="external image AIDO2.jpg" title="external image AIDO2.jpg" /&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:16 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;"&gt;&lt;a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow"&gt;**Example : Baran scale**&lt;/a&gt;&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 02:05, 17 February 2012

IMPORTED REVISION FROM WIKISPACES

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

This revision was by author acousticsoftombak and made on 2012-02-17 02:05:55 UTC.
The original revision id was 302670104.
The revision comment was:

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

Original Wikitext content:

<span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">For an intervallic system with </span>__<span style="color: black; font-size: 11pt;">n</span>__<span style="color: black; font-size: 11pt;"> divisions , </span><span style="color: #be00ff; font-size: 11pt;">[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|__AID__]]</span><span style="color: black; font-size: 11pt;"> is considered as [[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html|arithmetic sequence]] with divisions of system as terms of sequence. </span></span>
<span style="direction: ltr; display: block; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;"> If the first division is </span>__<span style="color: black; font-size: 11pt;">A1</span>__<span style="color: black; font-size: 11pt;"> and the last , </span>__<span style="color: black; font-size: 11pt;">An</span>__<span style="color: black; font-size: 11pt;"> , with common difference of </span>__<span style="color: black; font-size: 11pt;">d</span>__<span style="color: black; font-size: 11pt;"> , we have : </span></span></span>
<span style="direction: ltr; display: block; text-align: left;"><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A1 = A1 </span></span>
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A2 = A1+d </span>
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A3= A1+2d </span>
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A4 = A1+3d</span>
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">……… </span>
<span style="font-family: Tahoma,Geneva,sans-serif;">An = A1+(n-1)d</span>

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

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

<span style="direction: ltr; display: block; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;"> As we can consider </span>__<span style="color: black; font-size: 11pt;">Sn</span>__<span style="color: black; font-size: 11pt;"> of system to be 1200 cent or anything else (octavic or non-octavic system ) then </span>__<span style="color: black; font-size: 11pt;">d</span>__<span style="color: black; font-size: 11pt;"> is most important to make an __AID__ with __n__ divisions with __A1__.important to make an </span>__<span style="color: black; font-size: 11pt;">AID</span>__ with __n__<span style="color: black; font-size: 11pt;"> divisions with </span>__<span style="color: black; font-size: 11pt;">A1</span>__<span style="color: black; font-size: 11pt;">.</span></span><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> so, the common difference between divisions is : </span></span>
<span style="direction: ltr; display: block; text-align: center;"><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> **<span style="color: black; font-size: 11pt;">d = __2(Sn - nA1)__</span>**</span></span>
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> **n(n-1)**</span>

<span style="direction: ltr; display: block; text-align: left;"><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> By considering Sn=1200 , A1=70 , n=12 , __d__ will be 5.454545455 and our 12-tone scale is equal to:</span></span>
<span style="font-family: Tahoma,Geneva,sans-serif;">[[#TOC-0.070.0145.455226.364312.727404.545]] **0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0**</span>
<span style="direction: ltr; display: block; text-align: left;"><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">Scales based on AID can be subsets of EDO if :</span></span><span style="direction: ltr; display: block; text-align: left;"><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> Scales based on AID can be subsets of EDO if : </span></span><span style="direction: ltr; display: block; margin: 0cm 0cm 0pt; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 10pt;">1- we choose d=0 so ,<span style="color: black; font-size: 11pt;"> A1 = Sn/n</span> .. Consider </span>__<span style="color: black; font-size: 10pt;">n</span>__<span style="color: black; font-size: 10pt;">=8 and </span>__<span style="color: black; font-size: 10pt;">A1</span>__<span style="color: black; font-size: 10pt;">=150 , then we have 8-EDO .</span></span></span>
<span style="direction: ltr; display: block; margin: 0cm 0cm 0pt; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 10pt;">2- for a constant </span>__<span style="color: black; font-size: 10pt;">n</span>__<span style="color: black; font-size: 10pt;"> and different __A1__, if __d__ and (Sn/A1) are Integer number , we have a susbet of EDO or EDI( Equal divisions of Interval) .Consider __Sn = 1400__ , </span>__<span style="color: black; font-size: 10pt;">n</span>__<span style="color: black; font-size: 10pt;">=8 and </span>__<span style="color: black; font-size: 10pt;">A1</span>__<span style="color: black; font-size: 10pt;">=70 , then we have a subset of a 140-ED(1400.) with Degrees as 7 17 30 46 65 87 112 140 :</span></span></span>
**0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000**
<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;">[[#TOC-And-now-for:]]And now for </span><span style="font-family: Tahoma,Geneva,sans-serif;">Sn=1400 and n=8,
[[#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 <span style="color: black; font-size: 10pt;"> 28 73 135 214 310 423 553 700</span>
[[#TOC-If-A1-87.5-then-we-have-112-AID-140]]<span style="color: black; font-size: 10pt;">If A1=87.5 then we have 112-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 7 16 27 40 55 72 91 112</span></span></span>

<span style="font-family: Tahoma,Geneva,sans-serif;">__AID__ sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in __AID__<span style="color: black; font-size: 10pt;"> and __EDO__ with equal degree:</span></span>
* <span style="font-family: Tahoma,Geneva,sans-serif;">If choosing __<span style="color: black; font-size: 10pt;">A1</span>__<span style="color: black; font-size: 10pt;"> greater than division size in equal degree </span>__<span style="color: black; font-size: 10pt;">EDO</span>__<span style="color: black; font-size: 10pt;"> , </span>__<span style="color: black; font-size: 10pt;">d</span>__<span style="color: black; font-size: 10pt;"> is negative and </span>__<span style="color: black; font-size: 10pt;">AID</span>__<span style="color: black; font-size: 10pt;"> is descending.</span></span>
* <span style="font-family: Tahoma,Geneva,sans-serif;">If choosing __<span style="color: black; font-size: 10pt;">A1</span>__<span style="color: black; font-size: 10pt;"> smaller than division size in equal degree </span>__<span style="color: black; font-size: 10pt;">EDO</span>__<span style="color: black; font-size: 10pt;"> , </span>__<span style="color: black; font-size: 10pt;">d</span>__<span style="color: black; font-size: 10pt;"> is positive and </span>__<span style="color: black; font-size: 10pt;">AID</span>__<span style="color: black; font-size: 10pt;"> is ascending.</span></span>
* <span style="font-family: Tahoma,Geneva,sans-serif;">If choosing __<span style="color: black; font-size: 10pt;">A1</span>__<span style="color: black; font-size: 10pt;"> equal to division size in equal degree </span>__<span style="color: black; font-size: 10pt;">EDO</span>__<span style="color: black; font-size: 10pt;"> , </span>__<span style="color: black; font-size: 10pt;">d</span>__<span style="color: black; font-size: 10pt;"> is zero.</span>**</span>

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

==<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;">171.4285714 is point of intersection in these 3 trends:</span>== 

<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;"> </span>
[[image:http://sites.google.com/site/240edo/AIDO2.jpg align="center"]]


<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;">[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|**Example : Baran scale**]]</span>

Original HTML content:

<html><head><title>AID</title></head><body><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">For an intervallic system with </span><u><span style="color: black; font-size: 11pt;">n</span></u><span style="color: black; font-size: 11pt;"> divisions , </span><span style="color: #be00ff; font-size: 11pt;"><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow">__AID__</a></span><span style="color: black; font-size: 11pt;"> is considered as <a class="wiki_link_ext" href="http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html" rel="nofollow">arithmetic sequence</a> with divisions of system as terms of sequence. </span></span><br />
<span style="direction: ltr; display: block; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><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></span><br />
<span style="direction: ltr; display: block; text-align: left;"><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A1 = A1 </span></span><br />
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A2 = A1+d </span><br />
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A3= A1+2d </span><br />
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">A4 = A1+3d</span><br />
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">……… </span><br />
<span style="font-family: Tahoma,Geneva,sans-serif;">An = A1+(n-1)d</span><br />
<br />
<span style="direction: ltr; font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;">So sum of the divisions is </span><u><span style="color: black; font-size: 11pt;">Sn</span></u><span style="color: black; font-size: 11pt;"> :</span></span><br />
<br />
<span style="direction: ltr; display: block; text-align: center;"><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> <strong><span style="color: black; font-size: 11pt;">Sn = <u>n[2A1+(n-1)d]</u></span></strong></span></span><br />
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> <strong>2</strong></span><br />
<br />
<span style="direction: ltr; display: block; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><span style="color: black; font-size: 11pt;"> As we can consider </span><u><span style="color: black; font-size: 11pt;">Sn</span></u><span style="color: black; font-size: 11pt;"> of system to be 1200 cent or anything else (octavic or non-octavic system ) then </span><u><span style="color: black; font-size: 11pt;">d</span></u><span style="color: black; font-size: 11pt;"> is most important to make an <u>AID</u> with <u>n</u> divisions with <u>A1</u>.important to make an </span><u><span style="color: black; font-size: 11pt;">AID</span></u> with <u>n</u><span style="color: black; font-size: 11pt;"> divisions with </span><u><span style="color: black; font-size: 11pt;">A1</span></u><span style="color: black; font-size: 11pt;">.</span></span><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> so, the common difference between divisions is : </span></span><br />
<span style="direction: ltr; display: block; text-align: center;"><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> <strong><span style="color: black; font-size: 11pt;">d = <u>2(Sn - nA1)</u></span></strong></span></span><br />
<span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> <strong>n(n-1)</strong></span><br />
<br />
<span style="direction: ltr; display: block; text-align: left;"><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> By considering Sn=1200 , A1=70 , n=12 , <u>d</u> will be 5.454545455 and our 12-tone scale is equal to:</span></span><br />
<span style="font-family: Tahoma,Geneva,sans-serif;"><!-- ws:start:WikiTextAnchorRule:2:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-0.070.0145.455226.364312.727404.545&quot; title=&quot;Anchor: TOC-0.070.0145.455226.364312.727404.545&quot;/&gt; --><a name="TOC-0.070.0145.455226.364312.727404.545"></a><!-- ws:end:WikiTextAnchorRule:2 --> <strong>0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0</strong></span><br />
<span style="direction: ltr; display: block; text-align: left;"><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;">Scales based on AID can be subsets of EDO if :</span></span><span style="direction: ltr; display: block; text-align: left;"><span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"> Scales based on AID can be subsets of EDO if : </span></span><span style="direction: ltr; display: block; margin: 0cm 0cm 0pt; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><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><br />
<span style="direction: ltr; display: block; margin: 0cm 0cm 0pt; text-align: left;"><span style="font-family: Tahoma,Geneva,sans-serif;"><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></span><br />
<strong>0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000</strong><br />
<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;"><!-- ws:start:WikiTextAnchorRule:3:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-And-now-for:&quot; title=&quot;Anchor: TOC-And-now-for:&quot;/&gt; --><a name="TOC-And-now-for:"></a><!-- ws:end:WikiTextAnchorRule:3 -->And now for </span><span style="font-family: Tahoma,Geneva,sans-serif;">Sn=1400 and n=8,<br />
<!-- ws:start:WikiTextAnchorRule:4:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-If-A1-175.0-then-we-have-8-AID-1400&quot; title=&quot;Anchor: TOC-If-A1-175.0-then-we-have-8-AID-1400&quot;/&gt; --><a name="TOC-If-A1-175.0-then-we-have-8-AID-1400"></a><!-- ws:end:WikiTextAnchorRule:4 -->If A1=175.0 then we have 8-AID(1400.)<br />
<!-- ws:start:WikiTextAnchorRule:5:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-If-A1-56-then-we-have-700-AID-1400.&quot; title=&quot;Anchor: TOC-If-A1-56-then-we-have-700-AID-1400.&quot;/&gt; --><a name="TOC-If-A1-56-then-we-have-700-AID-1400."></a><!-- ws:end:WikiTextAnchorRule:5 -->If A1=56 then we have 700-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 28 73 135 214 310 423 553 700</span><br />
<!-- ws:start:WikiTextAnchorRule:6:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@TOC-If-A1-87.5-then-we-have-112-AID-140&quot; title=&quot;Anchor: TOC-If-A1-87.5-then-we-have-112-AID-140&quot;/&gt; --><a name="TOC-If-A1-87.5-then-we-have-112-AID-140"></a><!-- ws:end:WikiTextAnchorRule:6 --><span style="color: black; font-size: 10pt;">If A1=87.5 then we have 112-AID(1400.) with Degrees as <span style="color: black; font-size: 10pt;"> 7 16 27 40 55 72 91 112</span></span></span><br />
<br />
<span style="font-family: Tahoma,Geneva,sans-serif;"><u>AID</u> sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in <u>AID</u><span style="color: black; font-size: 10pt;"> and <u>EDO</u> with equal degree:</span></span><br />
<ul><li><span style="font-family: Tahoma,Geneva,sans-serif;">If choosing <u><span style="color: black; font-size: 10pt;">A1</span></u><span style="color: black; font-size: 10pt;"> greater than division size in equal degree </span><u><span style="color: black; font-size: 10pt;">EDO</span></u><span style="color: black; font-size: 10pt;"> , </span><u><span style="color: black; font-size: 10pt;">d</span></u><span style="color: black; font-size: 10pt;"> is negative and </span><u><span style="color: black; font-size: 10pt;">AID</span></u><span style="color: black; font-size: 10pt;"> is descending.</span></span></li><li><span style="font-family: Tahoma,Geneva,sans-serif;">If choosing <u><span style="color: black; font-size: 10pt;">A1</span></u><span style="color: black; font-size: 10pt;"> smaller than division size in equal degree </span><u><span style="color: black; font-size: 10pt;">EDO</span></u><span style="color: black; font-size: 10pt;"> , </span><u><span style="color: black; font-size: 10pt;">d</span></u><span style="color: black; font-size: 10pt;"> is positive and </span><u><span style="color: black; font-size: 10pt;">AID</span></u><span style="color: black; font-size: 10pt;"> is ascending.</span></span></li><li><span style="font-family: Tahoma,Geneva,sans-serif;">If choosing <u><span style="color: black; font-size: 10pt;">A1</span></u><span style="color: black; font-size: 10pt;"> equal to division size in equal degree </span><u><span style="color: black; font-size: 10pt;">EDO</span></u><span style="color: black; font-size: 10pt;"> , </span><u><span style="color: black; font-size: 10pt;">d</span></u><span style="color: black; font-size: 10pt;"> is zero.</span>**</span></li></ul><br />
<!-- ws:start:WikiTextRemoteImageRule:15:&lt;div style=&quot;text-align: center&quot;&gt;&lt;img src=&quot;http://sites.google.com/site/240edo/AIDO-custom-size-298-402.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/div&gt; --><div style="text-align: center"><img src="http://sites.google.com/site/240edo/AIDO-custom-size-298-402.jpg" alt="external image AIDO-custom-size-298-402.jpg" title="external image AIDO-custom-size-298-402.jpg" /></div><!-- ws:end:WikiTextRemoteImageRule:15 --><br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-171.4285714 is point of intersection in these 3 trends:"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;">171.4285714 is point of intersection in these 3 trends:</span></h2>
 <br />
<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;"> </span><br />
<!-- ws:start:WikiTextRemoteImageRule:16:&lt;div style=&quot;text-align: center&quot;&gt;&lt;img src=&quot;http://sites.google.com/site/240edo/AIDO2.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/div&gt; --><div style="text-align: center"><img src="http://sites.google.com/site/240edo/AIDO2.jpg" alt="external image AIDO2.jpg" title="external image AIDO2.jpg" /></div><!-- ws:end:WikiTextRemoteImageRule:16 --><br />
<br />
<span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;"><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow">**Example : Baran scale**</a></span></body></html>
Retrieved from "https://en.xen.wiki/index.php?title=AID&oldid=11131"