AID: Difference between revisions

Latest revision as of 05:29, 20 October 2024

Todo: cleanup

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

Arithmetic irrational divisions

For an intervallic system with n divisions, AID is considered as arithmetic sequence with divisions of system as terms of sequence.

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

A1 = A1

A2 = A1+d

A3 = A1+2d

A4 = A1+3d

...

An = A1+(n-1)d

So sum of the divisions is Sn :

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

As we can consider Sn of system to be 1200 cent or anything else (octavic or non-octavic system ) then d is most important to make an AID with n divisions with A1. So, the common difference between divisions is :

d =( 2(Sn - nA1))/(n(n-1))

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

0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0

Scales based on AID can be subsets of EDO if:

we choose d=0 so, A1 = Sn/n .. Consider n=8 and A1=150, then we have 8-EDO .
for a constant n and different A1, if d and (Sn/A1) are integers, we have a subset of EDO or EDI (Equal divisions of Interval).

Consider Sn = 1400 , n=8 and A1=70, then we have a subset of a 140-ED (1400.) with Degrees as 7 17 30 46 65 87 112 140 :

0.0 70.0 170.0 300.0 460.0 650.0 870.0 1120.0 1400.0

And now for Sn=1400 and n=8,

If A1=175.0 then we have 8-AID(1400.)
If A1=56 then we have 700-AID(1400.) with Degrees as 28 73 135 214 310 423 553 700
If A1=87.5 then we have 112-AID(1400.) with Degrees as 7 16 27 40 55 72 91 112

AID 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:

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

171.4285714 is point of intersection in these 3 trends:

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: Baran scale

@@ 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:x31eq|x31eq]] and made on <tt>2013-03-31 11:32:20 UTC</tt>.<br>
-: The original revision id was <tt>419047982</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="color: #000000; font-family: arial,sans-serif; font-size: 18px;"&gt;**Arithmetic irrational divisions**&lt;/span&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;For an intervallic system with &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; divisions , &lt;/span&gt;&lt;span style="color: #be00ff; font-size: 11pt;"&gt;[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|AID_]]&lt;/span&gt;&lt;span style="color: black; font-size: 11pt;"&gt; is considered as [[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html|arithmetic sequence]] with divisions of system as terms of sequence. &lt;/span&gt;&lt;/span&gt;
+==Arithmetic irrational divisions==
-&lt;span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; If the first division is &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; and the last , &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;An&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; , with common difference of &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; , we have : &lt;/span&gt;&lt;/span&gt;
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"&gt;A1 = A1&lt;/span&gt;
-&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A2 = A1+d &lt;/span&gt;
-&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A3= A1+2d &lt;/span&gt;
-&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A4 = A1+3d&lt;/span&gt;
-&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;……… &lt;/span&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;An = A1+(n-1)d&lt;/span&gt;
-&lt;span style="direction: ltr; font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;So sum of the divisions is &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;Sn&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; :&lt;/span&gt;&lt;/span&gt;
+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.
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"&gt;**&lt;span style="color: black; font-size: 11pt;"&gt;Sn =( __n[2A1+(n-1)d])/2__&lt;/span&gt;**&lt;/span&gt;
+If the first division is <u>A1</u> and the last, <u>An</u> , with common difference of <u>d</u> , we have :
+:: A1 = A1
+:: A2 = A1+d
+:: A3 = A1+2d
+:: A4 = A1+3d
+:: ...
+:: An = A1+(n-1)d
+So sum of the divisions is <u>Sn</u> :
+:: '''Sn =( <u>n[2A1+(n-1)d])/2</u>'''
-&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; As we can consider &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;Sn&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; of system to be 1200 cent or anything else (octavic or non-octavic system ) then &lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt;d&lt;/span&gt;__&lt;span style="color: black; font-size: 11pt;"&gt; is most important to make an AID with n divisions with A1.&lt;/span&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; So, the common difference between divisions is : &lt;/span&gt;&lt;/span&gt;
+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="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"&gt;**&lt;span style="color: black; font-size: 11pt;"&gt;d =( __2(Sn - nA1))/((n(n-1))__&lt;/span&gt;**&lt;/span&gt;
+:: '''d =( <u>2(Sn - nA1))/(n(n-1))</u>'''
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"&gt;By considering Sn=1200 , A1=70 , n=12 , d will be 5.454545455 and our 12-tone scale is equal to:&lt;/span&gt;
+By considering Sn=1200, A1=70, n=12, d will be 5.454545455 and our 12-tone scale is equal to:
-&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;
+:: '''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="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"&gt;Scales based on AID can be subsets of EDO if :&lt;/span&gt;&lt;span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="color: black; font-size: 10pt;"&gt;1- we choose d=0 so ,&lt;span style="color: black; font-size: 11pt;"&gt; A1 = Sn/n&lt;/span&gt; .. Consider &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;=8 and &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;=150 , then we have 8-EDO .&lt;/span&gt;&lt;/span&gt;
-&lt;span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="color: black; font-size: 10pt;"&gt;2- for a constant &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt; and different __A1__, if __d__ and (Sn/A1) are Integer number , we have a susbet of EDO or EDI( Equal divisions of Interval) .Consider __Sn = 1400__ , &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;=8 and &lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;__&lt;span style="color: black; font-size: 10pt;"&gt;=70 , then we have a subset of a 140-ED(1400.) with Degrees as [[tel/7 17 30 46 65 87 112|7 17 30 46 65 87 112]] 140 :&lt;/span&gt;&lt;/span&gt;
-**0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000**
-&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;"&gt;[[#TOC-And-now-for:]]And now for &lt;/span&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;Sn=1400 and n=8,&lt;/span&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt; [[#TOC-If-A1-175.0-then-we-have-8-AID-1400]]If A1=175.0 then we have 8-AID(1400.)&lt;/span&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt; [[#TOC-If-A1-56-then-we-have-700-AID-1400.]]If A1=56 then we have 700-AID(1400.) with Degrees as &lt;span style="color: black; font-size: 10pt;"&gt; [[tel/28 73 135 214 310|28 73 135 214 310]] [[tel/423 553 700|423 553 700]]&lt;/span&gt;&lt;/span&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt; [[#TOC-If-A1-87.5-then-we-have-112-AID-140]]&lt;span style="color: black; font-size: 10pt;"&gt;If A1=87.5 then we have 112-AID(1400.) with Degrees as &lt;span style="color: black; font-size: 10pt;"&gt; [[tel/7 16 27 40 55 72 91|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;
+Scales based on AID can be subsets of [[EDO]] if:
-* &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;
+# 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="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;
+# 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="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"]]
+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 :
+:: 0.0 70.0 170.0 300.0 460.0 650.0 870.0 1120.0 1400.0
-==&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;==
+And now for Sn=1400 and n=8,
+* If A1=175.0 then we have 8-AID(1400.)
+* If A1=56 then we have 700-AID(1400.) with Degrees as  28 73 135 214 310 423 553 700
+* If A1=87.5 then we have 112-AID(1400.) with Degrees as  7 16 27 40 55 72 91 112
+AID 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:
+* If choosing <u>A1</u> greater than division size in equal degree EDO, <u>d</u> is negative and <u>AID</u> is descending.
+* If choosing <u>A1</u> smaller than division size in equal degree EDO, <u>d</u> is positive and <u>AID</u> is ascending.
+* If choosing <u>A1</u> equal to division size in equal degree EDO, <u>d</u> is zero.
-[[image:http://sites.google.com/site/240edo/AIDO2.jpg align="center"]]
+[[file:AIDO-custom-size-298-402.jpg]]
-we can have different kinds of AID:
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;__&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDO = Arithmetic irrational divisions of octave&lt;/span&gt;__&lt;/span&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;__&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDINO = Arithmetic irrational divisions of irrational non-octave&lt;/span&gt;__&lt;/span&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;__&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDRNO = Arithmetic irrational divisions of rational non-octave&lt;/span&gt;__&lt;/span&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;__&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDRI = Arithmetic irrational divisions of rational interval&lt;/span&gt;__&lt;/span&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;__&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDII = Arithmetic irrational divisions of irrational interval&lt;/span&gt;__&lt;/span&gt;
-&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;"&gt;[[http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29|**Example : Baran scale**]]&lt;/span&gt;</pre></div>
-<h4>Original HTML content:</h4>
+.4285714 is point of intersection in these 3 trends:
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;AID&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="color: #000000; font-family: arial,sans-serif; font-size: 18px;"&gt;&lt;strong&gt;Arithmetic irrational divisions&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;br /&gt;
+[[file:AIDO2.jpg]]
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;For an intervallic system with &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;n&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; divisions , &lt;/span&gt;&lt;span style="color: #be00ff; font-size: 11pt;"&gt;&lt;a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow"&gt;AID_&lt;/a&gt;&lt;/span&gt;&lt;span style="color: black; font-size: 11pt;"&gt; is considered as &lt;a class="wiki_link_ext" href="http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html" rel="nofollow"&gt;arithmetic sequence&lt;/a&gt; with divisions of system as terms of sequence. &lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; If the first division is &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; and the last , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;An&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; , with common difference of &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; , we have : &lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"&gt;A1 = A1&lt;/span&gt;&lt;br /&gt;
+We can have different kinds of AID:
-&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A2 = A1+d &lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A3= A1+2d &lt;/span&gt;&lt;br /&gt;
+* AIDO = Arithmetic irrational divisions of octave
-&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;A4 = A1+3d&lt;/span&gt;&lt;br /&gt;
+* AIDINO = Arithmetic irrational divisions of irrational non-octave
-&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt;……… &lt;/span&gt;&lt;br /&gt;
+* AIDRNO = Arithmetic irrational divisions of rational non-octave
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;An = A1+(n-1)d&lt;/span&gt;&lt;br /&gt;
+* AIDRI = Arithmetic irrational divisions of rational interval
-&lt;br /&gt;
+* AIDII = Arithmetic irrational divisions of irrational interval
-&lt;span style="direction: ltr; font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt;So sum of the divisions is &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;Sn&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; :&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;br /&gt;
+Example: [http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29 Baran scale]
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"&gt;&lt;strong&gt;&lt;span style="color: black; font-size: 11pt;"&gt;Sn =( &lt;u&gt;n[2A1+(n-1)d])/2&lt;/u&gt;&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;br /&gt;
+[[Category:AID]]
-&lt;br /&gt;
+[[Category:Shaahin Mohajeri]]
-&lt;br /&gt;
-&lt;span style="direction: ltr; display: block; text-align: left;"&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;span style="color: black; font-size: 11pt;"&gt; As we can consider &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;Sn&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; of system to be 1200 cent or anything else (octavic or non-octavic system ) then &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 11pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 11pt;"&gt; is most important to make an AID with n divisions with A1.&lt;/span&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt;"&gt; So, the common difference between divisions is : &lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: center;"&gt;&lt;strong&gt;&lt;span style="color: black; font-size: 11pt;"&gt;d =( &lt;u&gt;2(Sn - nA1))/((n(n-1))&lt;/u&gt;&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"&gt;By considering Sn=1200 , A1=70 , n=12 , d will be 5.454545455 and our 12-tone scale is equal to:&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:2:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@TOC-0.070.0145.455226.364312.727404.545&amp;quot; title=&amp;quot;Anchor: TOC-0.070.0145.455226.364312.727404.545&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-0.070.0145.455226.364312.727404.545"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:2 --&gt; &lt;strong&gt;0.0 70.0 145.455 226.364 312.727 404.545 501.818 604.545 712.727 826.364 945.455 1070.0 1200.0&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: black; direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 11pt; text-align: left;"&gt;Scales based on AID can be subsets of EDO if :&lt;/span&gt;&lt;span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="color: black; font-size: 10pt;"&gt;1- we choose d=0 so ,&lt;span style="color: black; font-size: 11pt;"&gt; A1 = Sn/n&lt;/span&gt; .. Consider &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;=8 and &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;=150 , then we have 8-EDO .&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="direction: ltr; display: block; font-family: Tahoma,Geneva,sans-serif; margin: 0cm 0cm 0pt; text-align: left;"&gt;&lt;span style="color: black; font-size: 10pt;"&gt;2- for a constant &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; and different &lt;u&gt;A1&lt;/u&gt;, if &lt;u&gt;d&lt;/u&gt; and (Sn/A1) are Integer number , we have a susbet of EDO or EDI( Equal divisions of Interval) .Consider &lt;u&gt;Sn = 1400&lt;/u&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;n&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;=8 and &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;=70 , then we have a subset of a 140-ED(1400.) with Degrees as &lt;a class="wiki_link" href="http://tel.wikispaces.com/7%2017%2030%2046%2065%2087%20112"&gt;7 17 30 46 65 87 112&lt;/a&gt; 140 :&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;strong&gt;0.000 70.000 170.000 300.000 460.000 650.000 870.000 1120.000 1400.000&lt;/strong&gt;&lt;br /&gt;
-&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;!-- ws:start:WikiTextAnchorRule:3:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@TOC-And-now-for:&amp;quot; title=&amp;quot;Anchor: TOC-And-now-for:&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-And-now-for:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:3 --&gt;And now for &lt;/span&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;Sn=1400 and n=8,&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt; &lt;!-- ws:start:WikiTextAnchorRule:4:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@TOC-If-A1-175.0-then-we-have-8-AID-1400&amp;quot; title=&amp;quot;Anchor: TOC-If-A1-175.0-then-we-have-8-AID-1400&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-If-A1-175.0-then-we-have-8-AID-1400"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:4 --&gt;If A1=175.0 then we have 8-AID(1400.)&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt; &lt;!-- ws:start:WikiTextAnchorRule:5:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@TOC-If-A1-56-then-we-have-700-AID-1400.&amp;quot; title=&amp;quot;Anchor: TOC-If-A1-56-then-we-have-700-AID-1400.&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-If-A1-56-then-we-have-700-AID-1400."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:5 --&gt;If A1=56 then we have 700-AID(1400.) with Degrees as &lt;span style="color: black; font-size: 10pt;"&gt; &lt;a class="wiki_link" href="http://tel.wikispaces.com/28%2073%20135%20214%20310"&gt;28 73 135 214 310&lt;/a&gt; &lt;a class="wiki_link" href="http://tel.wikispaces.com/423%20553%20700"&gt;423 553 700&lt;/a&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt; &lt;!-- ws:start:WikiTextAnchorRule:6:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@TOC-If-A1-87.5-then-we-have-112-AID-140&amp;quot; title=&amp;quot;Anchor: TOC-If-A1-87.5-then-we-have-112-AID-140&amp;quot;/&amp;gt; --&gt;&lt;a name="TOC-If-A1-87.5-then-we-have-112-AID-140"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:6 --&gt;&lt;span style="color: black; font-size: 10pt;"&gt;If A1=87.5 then we have 112-AID(1400.) with Degrees as &lt;span style="color: black; font-size: 10pt;"&gt; &lt;a class="wiki_link" href="http://tel.wikispaces.com/7%2016%2027%2040%2055%2072%2091"&gt;7 16 27 40 55 72 91&lt;/a&gt; 112&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;&lt;u&gt;AID&lt;/u&gt; sytem shows different ascending , descending or linear trend of change in divisions sizes due to relation between n and A1 in &lt;u&gt;AID&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; and &lt;u&gt;EDO&lt;/u&gt; with equal degree:&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;If choosing &lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; greater than division size in equal degree &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;EDO&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; is negative and &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;AID&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; is descending.&lt;/span&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;If choosing &lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; smaller than division size in equal degree &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;EDO&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; is positive and &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;AID&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; is ascending.&lt;/span&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Tahoma,Geneva,sans-serif;"&gt;If choosing &lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;A1&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; equal to division size in equal degree &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;EDO&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; , &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 10pt;"&gt;d&lt;/span&gt;&lt;/u&gt;&lt;span style="color: black; font-size: 10pt;"&gt; is zero.&lt;/span&gt;**&lt;/span&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:15:&amp;lt;div style=&amp;quot;text-align: center&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://sites.google.com/site/240edo/AIDO-custom-size-298-402.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/div&amp;gt; --&gt;&lt;div style="text-align: center"&gt;&lt;img src="http://sites.google.com/site/240edo/AIDO-custom-size-298-402.jpg" alt="external image AIDO-custom-size-298-402.jpg" title="external image AIDO-custom-size-298-402.jpg" /&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:15 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-171.4285714 is point of intersection in these 3 trends:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 13.3333px;"&gt;171.4285714 is point of intersection in these 3 trends:&lt;/span&gt;&lt;/h2&gt;
- &lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:16:&amp;lt;div style=&amp;quot;text-align: center&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://sites.google.com/site/240edo/AIDO2.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/div&amp;gt; --&gt;&lt;div style="text-align: center"&gt;&lt;img src="http://sites.google.com/site/240edo/AIDO2.jpg" alt="external image AIDO2.jpg" title="external image AIDO2.jpg" /&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:16 --&gt;we can have different kinds of AID:&lt;br /&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;&lt;u&gt;&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDO = Arithmetic irrational divisions of octave&lt;/span&gt;&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;&lt;u&gt;&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDINO = Arithmetic irrational divisions of irrational non-octave&lt;/span&gt;&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;&lt;u&gt;&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDRNO = Arithmetic irrational divisions of rational non-octave&lt;/span&gt;&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;&lt;u&gt;&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDRI = Arithmetic irrational divisions of rational interval&lt;/span&gt;&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 14px; text-align: left;"&gt;&lt;u&gt;&lt;span style="color: #000000; font-family: arial,sans-serif;"&gt;AIDII = Arithmetic irrational divisions of irrational interval&lt;/span&gt;&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;span style="color: #000000; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; text-decoration: none;"&gt;&lt;a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticirrationaldivisions%28aid%29" rel="nofollow"&gt;**Example : Baran scale**&lt;/a&gt;&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

AID: Difference between revisions

Latest revision as of 05:29, 20 October 2024

Arithmetic irrational divisions

Navigation menu

Search