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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<span style="color: #000000; font-family: arial,sans-serif; font-size: 140%;">'''Arithmetic rational''' '''divisions of octave''' </span>
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>2012-02-24 00:21:56 UTC</tt>.<br>
: The original revision id was <tt>304675644</tt>.<br>
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<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: 140%;"&gt;**Arithmetic rational** **divisions of octave** &lt;/span&gt;


&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;**ARDO** (which is simplified as **[[http://sites.google.com/site/240edo/arithmeticrationaldivisionsofoctave|ADO]])** is an intervallic system &lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;considered as &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
<span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;">'''ARDO''' (which is simplified as '''[http://sites.google.com/site/240edo/arithmeticrationaldivisionsofoctave ADO])''' is an intervallic system <span style="color: black; font-family: arial,sans-serif; font-size: 15px;">considered as </span></span></span>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;[[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html|arithmetic sequence]] with divisions of system as &lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;terms of sequence. &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;If the first division is __**R1**__ (wich is ratio of C/C) and the last , __**Rn**__ &lt;/span&gt;&lt;span style="color: black; font-size: 15px;"&gt;(wich is ratio of 2C/C), with common difference of &lt;/span&gt;__&lt;span style="color: black; font-size: 15px;"&gt;**d**&lt;/span&gt;__&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: center;"&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;(which is **1/C**), we have : &lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;**R2 = R1+d** &lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;**R3= R1+2d** &lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;**&lt;span style="color: black; font-size: 15px;"&gt;R4 = R1+3d &lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;**………**&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt;**&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;Rn = R1+(n-1)d&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;


&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;Each consequent divisions like **R4** and **R3** have a difference of **d** with each other.The concept of division here is a bit different from **EDO** and other systems (which is the difference of cents of two consequent degree). In **ADO**, a division is frequency-related and is the ratio of each degree due to the first degree.For example ratio of 1.5 is the size of 3/2 in 12-ADO system.&lt;/span&gt;&lt;/span&gt;
<span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html arithmetic sequence] with divisions of system as <span style="color: black; font-family: arial,sans-serif; font-size: 15px;">terms of sequence. </span></span></span>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;For any **C-ADO** system with [[http://www.tonalsoft.com/enc/c/cardinality.aspx|**cardinality**]] of **C**, we have ratios related to different degrees of **m** as : &lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: center;"&gt;(C+m/C)&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;For example , in **12-ADO** the ratio related to the first degree is 13/12 .&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;**12-ADO** can be shown as series like: &lt;/span&gt;**&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;12:13&lt;/span&gt;****&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;:14:15:16:17:18:19:20:21:22:23:24&lt;/span&gt;**&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt; or &lt;/span&gt;**&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;12 13 &lt;/span&gt;**&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;14 15 16 17 18 19 20 21 22 23 24&lt;/span&gt; **&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;.&lt;/span&gt;**&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;For an **ADO** intervallic system with **n** divisions we have &lt;span style="font-family: arial,sans-serif;"&gt;unequal divisions of length &lt;/span&gt;by dividing string length to**&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;n&lt;/span&gt;** unequal divisions based on each degree ratios.If the first division has ratio of **R1** and length of **&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;L1&lt;/span&gt;** and the last, **Rn** and **&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;Ln&lt;/span&gt;** , we have: **Ln = 1/Rn** and if **Rn &gt;........&gt; R3 &gt; R2 &gt; R1** so : &lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;**&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;L1 &gt; L2 &gt; L3 &gt; …… &gt; Ln&lt;/span&gt;**&lt;/span&gt;
[[image:http://sites.google.com/site/240edo/ADO-4-custom-size-350-238.jpg align="center" link="http://sites.google.com/site/240edo/ADO-4.jpg"]]


&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;This lengths are related to reverse of ratios in system.The above picture shows the differences between divisions of length in 12-ADO system . On the contrary , we have equal divisios of length in [[http://sites.google.com/site/240edo/equaldivisionsoflength(edl)|**EDL system**]]:&lt;/span&gt;&lt;/span&gt;
<span style="display: block; text-align: left;"><span style="font-family: Times New Roman;"><span style="font-family: arial,sans-serif;">If the first division is <u>'''R1'''</u> (wich is ratio of C/C) and the last , <u>'''Rn'''</u> </span><span style="color: black; font-size: 15px;">(wich is ratio of 2C/C), with common difference of </span><u><span style="color: black; font-size: 15px;">'''d'''</span></u></span></span>
[[image:http://sites.google.com/site/240edo/ADO-5-custom-size-346-235.jpg align="center" link="http://sites.google.com/site/240edo/ADO-5.jpg"]]


[[image:http://sites.google.com/site/240edo/ADO-3-custom-size-604-289.jpg align="center" link="http://sites.google.com/site/240edo/ADO-3.jpg"]]
<span style="display: block; text-align: center;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">(which is '''1/C'''), we have : </span></span>


&lt;span style="display: block; text-align: center;"&gt;
<span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">'''R2 = R1+d''' </span></span>
&lt;/span&gt;
&lt;span style="display: block; text-align: center;"&gt;**&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;__Relation between harmonics and ADO system__&lt;/span&gt;**&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;**ADO** (like **EDL)** is based on [[http://en.wikipedia.org/wiki/Superparticular_number|**Superparticular ratios**]] and [[http://en.wikipedia.org/wiki/Harmonic_series_%28music%29|**harmonic series**]]. Have a look at 12-ADO in this picture:&lt;/span&gt;&lt;/span&gt;
[[image:http://sites.google.com/site/240edo/ADO-2-custom-size-378-270.jpg align="center" link="http://sites.google.com/site/240edo/ADO-2.jpg"]]


<span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">'''R3= R1+2d''' </span></span>


&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;The above picture shows that **ADO** system is classified as :&lt;/span&gt;
<span style="display: block; text-align: left;"><span style="font-family: arial,sans-serif;">'''<span style="color: black; font-size: 15px;">R4 = R1+3d </span>'''</span></span>


&lt;span style="display: block; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;- System with unequal &lt;/span&gt;&lt;span style="color: blue; font-family: arial; font-size: 13px;"&gt;[[@http://tonalsoft.com/enc/e/epimorios.aspx|**epimorios**]]&lt;/span&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt; **(**[[http://en.wikipedia.org/wiki/Superparticular_number|**Superparticular**]]**)** divisions.&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
<span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">'''………'''</span></span>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;- System based on ascending series of superparticular ratios with descending sizes.&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;- System which covers superparticular ratios between harmonic of number C (in this example 12)to harmonic of Number 2C(in this example 24).&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;**- &lt;span style="font-family: arial,sans-serif;"&gt;[[http://sites.google.com/site/240edo/ADO-EDL.XLS|An spreadsheet showing relation between harmonics , superparticular ratios and ADO system]]&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial; font-size: 15px;"&gt;**-** &lt;span style="font-family: arial,sans-serif;"&gt;[[http://www.music.sc.edu/fs/bain/software/BainTheOvertoneSeries.pdf|The Overtone Series]]&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: center;"&gt;&lt;span style="color: black; font-family: arial; font-size: 15px;"&gt;**&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;__Relation between Otonality and ADO system__&lt;/span&gt;**&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;We can consider &lt;/span&gt;**&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;ADO&lt;/span&gt;**&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt; system as &lt;/span&gt;&lt;span style="color: blue; font-family: Arial; font-size: 13px;"&gt;[[@http://en.wikipedia.org/wiki/Otonal|**Otonal system**]]&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt; .**Otonality** is a term introduced by &lt;/span&gt;&lt;span style="color: blue; font-family: Arial; font-size: 13px;"&gt;[[@http://en.wikipedia.org/wiki/Harry_Partch|**Harry Partch**]]&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt; to describe chords whose notes are the overtones (multiples) of a given fixed tone.Considering ADO , an Otonality is a collection of pitches which can be expressed in ratios that have equal denominators. For example, 1/1, 5/4, and 3/2 form an Otonality because they can be written as 4/4, 5/4, 6/4. Every Otonality is therefore part of the [[http://en.wikipedia.org/wiki/Harmonic_series_%28music%29|**harmonic series**]]. nominator here is called "&lt;/span&gt;&lt;span style="color: blue; font-family: Arial; font-size: 13px;"&gt;[[@http://tonalsoft.com/enc/n/nexus.aspx|**Numerary nexus**]]&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;".An Otonality corresponds to an [[http://en.wikipedia.org/wiki/Arithmetic_series|**arithmetic series**]] of frequencies or a [[http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29|**harmonic series**]] of wavelengths or distances on a [[http://en.wikipedia.org/wiki/String_instrument|**string instrument**]].&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;**&lt;span style="color: black; font-family: 'Times New Roman'; font-size: 13px;"&gt;- &lt;/span&gt;__&lt;span style="color: windowtext; font-family: 'times new roman'; font-size: 16px;"&gt;[[http://240edo.googlepages.com/ADO-EDL.XLS|Fret position calculator (excel sheet ) based on EDL system and string length]]&lt;/span&gt;__**&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;&lt;span style="color: #0000ff; font-family: arial,sans-serif; font-size: 16px;"&gt;[[http://sites.google.com/site/240edo/ADOandEDO.xls|- How to approximate EDand ADO systems with each other?Download this file]]&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span style="display: block; text-align: center;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;**__&lt;span style="color: windowtext; font-family: arial,sans-serif; font-size: 16px;"&gt;Related to ADO&lt;/span&gt;__**&lt;/span&gt;&lt;/span&gt;


&lt;span style="display: block; text-align: center;"&gt;&lt;span style="color: black; font-family: arial; font-size: 24px;"&gt;[[http://www.soundtransformations.btinternet.co.uk/Danerhudyarthemarmonicseries.htm|**Magic of Tone and the Art of Music by the late Dane Rhudyar**]]&lt;/span&gt;&lt;/span&gt;</pre></div>
<span style="display: block; text-align: left;"><span style="font-family: Times New Roman;">'''<span style="color: black; font-family: arial,sans-serif; font-size: 15px;">Rn = R1+(n-1)d</span>'''</span></span>
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;ADO&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="color: #000000; font-family: arial,sans-serif; font-size: 140%;"&gt;&lt;strong&gt;Arithmetic rational&lt;/strong&gt; &lt;strong&gt;divisions of octave&lt;/strong&gt; &lt;/span&gt;&lt;br /&gt;
<span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">Each consequent divisions like '''R4''' and '''R3''' have a difference of '''d''' with each other.The concept of division here is a bit different from '''EDO''' and other systems (which is the difference of cents of two consequent degree). In '''ADO''', a division is frequency-related and is the ratio of each degree due to the first degree.For example ratio of 1.5 is the size of 3/2 in 12-ADO system.</span></span>
&lt;br /&gt;
 
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;&lt;strong&gt;ARDO&lt;/strong&gt; (which is simplified as &lt;strong&gt;&lt;a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticrationaldivisionsofoctave" rel="nofollow"&gt;ADO&lt;/a&gt;)&lt;/strong&gt; is an intervallic system &lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;considered as &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
<span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">For any '''C-ADO''' system with [http://www.tonalsoft.com/enc/c/cardinality.aspx **cardinality**] of '''C''', we have ratios related to different degrees of '''m''' as : </span></span>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;&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 &lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;terms of sequence. &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;If the first division is &lt;u&gt;&lt;strong&gt;R1&lt;/strong&gt;&lt;/u&gt; (wich is ratio of C/C) and the last , &lt;u&gt;&lt;strong&gt;Rn&lt;/strong&gt;&lt;/u&gt; &lt;/span&gt;&lt;span style="color: black; font-size: 15px;"&gt;(wich is ratio of 2C/C), with common difference of &lt;/span&gt;&lt;u&gt;&lt;span style="color: black; font-size: 15px;"&gt;&lt;strong&gt;d&lt;/strong&gt;&lt;/span&gt;&lt;/u&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
<span style="display: block; text-align: center;">(C+m/C)</span>
&lt;span style="display: block; text-align: center;"&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;(which is &lt;strong&gt;1/C&lt;/strong&gt;), we have : &lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;&lt;strong&gt;R2 = R1+d&lt;/strong&gt; &lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
<span style="display: block; text-align: left;"><span style="font-family: arial,sans-serif;">For example , in '''12-ADO''' the ratio related to the first degree is 13/12 .</span></span>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;&lt;strong&gt;R3= R1+2d&lt;/strong&gt; &lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;strong&gt;&lt;span style="color: black; font-size: 15px;"&gt;R4 = R1+3d &lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
<span style="display: block; text-align: left;"><span style="font-family: arial,sans-serif;">'''12-ADO''' can be shown as series like: </span>'''<span style="color: black; font-family: Arial; font-size: 13px;">12:13</span>''''''<span style="color: black; font-family: Arial; font-size: 13px;">:14:15:16:17:18:19:20:21:22:23:24</span>'''<span style="color: black; font-family: arial; font-size: 13px;"> or </span>'''<span style="color: black; font-family: arial; font-size: 13px;">12 13 </span>'''<span style="color: black; font-family: Arial; font-size: 13px;">14 15 16 17 18 19 20 21 22 23 24</span> '''<span style="color: black; font-family: Arial; font-size: 13px;">.</span>'''</span>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;&lt;strong&gt;………&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;strong&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;Rn = R1+(n-1)d&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
<span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">For an '''ADO''' intervallic system with '''n''' divisions we have <span style="font-family: arial,sans-serif;">unequal divisions of length </span>by dividing string length to'''<span style="color: black; font-family: Arial; font-size: 13px;">n</span>''' unequal divisions based on each degree ratios.If the first division has ratio of '''R1''' and length of '''<span style="color: black; font-family: Arial; font-size: 13px;">L1</span>''' and the last, '''Rn''' and '''<span style="color: black; font-family: Arial; font-size: 13px;">Ln</span>''' , we have: '''Ln = 1/Rn''' and if '''Rn &gt;........&gt; R3 &gt; R2 &gt; R1''' so : </span></span>
&lt;br /&gt;
 
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;Each consequent divisions like &lt;strong&gt;R4&lt;/strong&gt; and &lt;strong&gt;R3&lt;/strong&gt; have a difference of &lt;strong&gt;d&lt;/strong&gt; with each other.The concept of division here is a bit different from &lt;strong&gt;EDO&lt;/strong&gt; and other systems (which is the difference of cents of two consequent degree). In &lt;strong&gt;ADO&lt;/strong&gt;, a division is frequency-related and is the ratio of each degree due to the first degree.For example ratio of 1.5 is the size of 3/2 in 12-ADO system.&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
<span style="display: block; text-align: left;">'''<span style="color: black; font-family: Arial; font-size: 13px;">L1 &gt; L2 &gt; L3 &gt; …… &gt; Ln</span>'''</span>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;For any &lt;strong&gt;C-ADO&lt;/strong&gt; system with &lt;a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/c/cardinality.aspx" rel="nofollow"&gt;**cardinality**&lt;/a&gt; of &lt;strong&gt;C&lt;/strong&gt;, we have ratios related to different degrees of &lt;strong&gt;m&lt;/strong&gt; as : &lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 
&lt;span style="display: block; text-align: center;"&gt;(C+m/C)&lt;/span&gt;&lt;br /&gt;
[http://sites.google.com/site/240edo/ADO-4.jpg
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;For example , in &lt;strong&gt;12-ADO&lt;/strong&gt; the ratio related to the first degree is 13/12 .&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="font-family: arial,sans-serif;"&gt;&lt;strong&gt;12-ADO&lt;/strong&gt; can be shown as series like: &lt;/span&gt;&lt;strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;12:13&lt;/span&gt;&lt;/strong&gt;&lt;strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;:14:15:16:17:18:19:20:21:22:23:24&lt;/span&gt;&lt;/strong&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt; or &lt;/span&gt;&lt;strong&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;12 13 &lt;/span&gt;&lt;/strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;14 15 16 17 18 19 20 21 22 23 24&lt;/span&gt; &lt;strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;.&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
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&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;For an &lt;strong&gt;ADO&lt;/strong&gt; intervallic system with &lt;strong&gt;n&lt;/strong&gt; divisions we have &lt;span style="font-family: arial,sans-serif;"&gt;unequal divisions of length &lt;/span&gt;by dividing string length to&lt;strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;n&lt;/span&gt;&lt;/strong&gt; unequal divisions based on each degree ratios.If the first division has ratio of &lt;strong&gt;R1&lt;/strong&gt; and length of &lt;strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;L1&lt;/span&gt;&lt;/strong&gt; and the last, &lt;strong&gt;Rn&lt;/strong&gt; and &lt;strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;Ln&lt;/span&gt;&lt;/strong&gt; , we have: &lt;strong&gt;Ln = 1/Rn&lt;/strong&gt; and if &lt;strong&gt;Rn &amp;gt;........&amp;gt; R3 &amp;gt; R2 &amp;gt; R1&lt;/strong&gt; so : &lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
External image: http://sites.google.com/site/240edo/ADO-4-custom-size-350-238.jpg<br>
&lt;span style="display: block; text-align: left;"&gt;&lt;strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;L1 &amp;gt; L2 &amp;gt; L3 &amp;gt; …… &amp;gt; Ln&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
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: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;This lengths are related to reverse of ratios in system.The above picture shows the differences between divisions of length in 12-ADO system . On the contrary , we have equal divisios of length in &lt;a class="wiki_link_ext" href="http://sites.google.com/site/240edo/equaldivisionsoflength(edl)" rel="nofollow"&gt;**EDL system**&lt;/a&gt;:&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
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&lt;!-- ws:start:WikiTextRemoteImageRule:5:&amp;lt;div style=&amp;quot;text-align: center&amp;quot;&amp;gt;&amp;lt;a href=&amp;quot;http://sites.google.com/site/240edo/ADO-3.jpg&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://sites.google.com/site/240edo/ADO-3-custom-size-604-289.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;/div&amp;gt; --&gt;&lt;div style="text-align: center"&gt;&lt;a href="http://sites.google.com/site/240edo/ADO-3.jpg" rel="nofollow"&gt;&lt;img src="http://sites.google.com/site/240edo/ADO-3-custom-size-604-289.jpg" alt="external image ADO-3-custom-size-604-289.jpg" title="external image ADO-3-custom-size-604-289.jpg" /&gt;&lt;/a&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:5 --&gt;&lt;br /&gt;
 
&lt;span style="display: block; text-align: center;"&gt;&lt;br /&gt;
]]
&lt;/span&gt;&lt;br /&gt;
 
&lt;span style="display: block; text-align: center;"&gt;&lt;strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;&lt;u&gt;Relation between harmonics and ADO system&lt;/u&gt;&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
<span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;">This lengths are related to reverse of ratios in system.The above picture shows the differences between divisions of length in 12-ADO system . On the contrary , we have equal divisios of length in [http://sites.google.com/site/240edo/equaldivisionsoflength(edl) **EDL system**]:</span></span>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;&lt;strong&gt;ADO&lt;/strong&gt; (like &lt;strong&gt;EDL)&lt;/strong&gt; is based on &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow"&gt;**Superparticular ratios**&lt;/a&gt; and &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harmonic_series_%28music%29" rel="nofollow"&gt;**harmonic series**&lt;/a&gt;. Have a look at 12-ADO in this picture:&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextRemoteImageRule:7:&amp;lt;div style=&amp;quot;text-align: center&amp;quot;&amp;gt;&amp;lt;a href=&amp;quot;http://sites.google.com/site/240edo/ADO-2.jpg&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://sites.google.com/site/240edo/ADO-2-custom-size-378-270.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;/div&amp;gt; --&gt;&lt;div style="text-align: center"&gt;&lt;a href="http://sites.google.com/site/240edo/ADO-2.jpg" rel="nofollow"&gt;&lt;img src="http://sites.google.com/site/240edo/ADO-2-custom-size-378-270.jpg" alt="external image ADO-2-custom-size-378-270.jpg" title="external image ADO-2-custom-size-378-270.jpg" /&gt;&lt;/a&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:7 --&gt;&lt;br /&gt;
[http://sites.google.com/site/240edo/ADO-5.jpg
&lt;br /&gt;
 
&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;The above picture shows that &lt;strong&gt;ADO&lt;/strong&gt; system is classified as :&lt;/span&gt;&lt;br /&gt;
<div class='external-image-warning' style='background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block'>
&lt;br /&gt;
External image: http://sites.google.com/site/240edo/ADO-5-custom-size-346-235.jpg<br>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="font-family: Times New Roman;"&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;- System with unequal &lt;/span&gt;&lt;span style="color: blue; font-family: arial; font-size: 13px;"&gt;&lt;a class="wiki_link_ext" href="http://tonalsoft.com/enc/e/epimorios.aspx" rel="nofollow" target="_blank"&gt;**epimorios**&lt;/a&gt;&lt;/span&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt; &lt;strong&gt;(&lt;/strong&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow"&gt;**Superparticular**&lt;/a&gt;&lt;strong&gt;)&lt;/strong&gt; divisions.&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;- System based on ascending series of superparticular ratios with descending sizes.&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial; font-size: 13px;"&gt;- System which covers superparticular ratios between harmonic of number C (in this example 12)to harmonic of Number 2C(in this example 24).&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
</div>
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial,sans-serif; font-size: 15px;"&gt;&lt;strong&gt;- &lt;span style="font-family: arial,sans-serif;"&gt;&lt;a class="wiki_link_ext" href="http://sites.google.com/site/240edo/ADO-EDL.XLS" rel="nofollow"&gt;An spreadsheet showing relation between harmonics , superparticular ratios and ADO system&lt;/a&gt;&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
[[Category:IMPORTDEBUG - Change External Images]
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: arial; font-size: 15px;"&gt;&lt;strong&gt;-&lt;/strong&gt; &lt;span style="font-family: arial,sans-serif;"&gt;&lt;a class="wiki_link_ext" href="http://www.music.sc.edu/fs/bain/software/BainTheOvertoneSeries.pdf" rel="nofollow"&gt;The Overtone Series&lt;/a&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 
&lt;span style="display: block; text-align: center;"&gt;&lt;span style="color: black; font-family: arial; font-size: 15px;"&gt;&lt;strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;&lt;u&gt;Relation between Otonality and ADO system&lt;/u&gt;&lt;/span&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
]]
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;We can consider &lt;/span&gt;&lt;strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;ADO&lt;/span&gt;&lt;/strong&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt; system as &lt;/span&gt;&lt;span style="color: blue; font-family: Arial; font-size: 13px;"&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Otonal" rel="nofollow" target="_blank"&gt;**Otonal system**&lt;/a&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt; .&lt;strong&gt;Otonality&lt;/strong&gt; is a term introduced by &lt;/span&gt;&lt;span style="color: blue; font-family: Arial; font-size: 13px;"&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harry_Partch" rel="nofollow" target="_blank"&gt;**Harry Partch**&lt;/a&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt; to describe chords whose notes are the overtones (multiples) of a given fixed tone.Considering ADO , an Otonality is a collection of pitches which can be expressed in ratios that have equal denominators. For example, 1/1, 5/4, and 3/2 form an Otonality because they can be written as 4/4, 5/4, 6/4. Every Otonality is therefore part of the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harmonic_series_%28music%29" rel="nofollow"&gt;**harmonic series**&lt;/a&gt;. nominator here is called &amp;quot;&lt;/span&gt;&lt;span style="color: blue; font-family: Arial; font-size: 13px;"&gt;&lt;a class="wiki_link_ext" href="http://tonalsoft.com/enc/n/nexus.aspx" rel="nofollow" target="_blank"&gt;**Numerary nexus**&lt;/a&gt;&lt;/span&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;&amp;quot;.An Otonality corresponds to an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arithmetic_series" rel="nofollow"&gt;**arithmetic series**&lt;/a&gt; of frequencies or a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29" rel="nofollow"&gt;**harmonic series**&lt;/a&gt; of wavelengths or distances on a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/String_instrument" rel="nofollow"&gt;**string instrument**&lt;/a&gt;.&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;&lt;strong&gt;&lt;span style="color: black; font-family: 'Times New Roman'; font-size: 13px;"&gt;- &lt;/span&gt;&lt;u&gt;&lt;span style="color: windowtext; font-family: 'times new roman'; font-size: 16px;"&gt;&lt;a class="wiki_link_ext" href="http://240edo.googlepages.com/ADO-EDL.XLS" rel="nofollow"&gt;Fret position calculator (excel sheet ) based on EDL system and string length&lt;/a&gt;&lt;/span&gt;&lt;/u&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
[http://sites.google.com/site/240edo/ADO-3.jpg
&lt;span style="display: block; text-align: left;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;&lt;span style="color: #0000ff; font-family: arial,sans-serif; font-size: 16px;"&gt;&lt;a class="wiki_link_ext" href="http://sites.google.com/site/240edo/ADOandEDO.xls" rel="nofollow"&gt;- How to approximate EDand ADO systems with each other?Download this file&lt;/a&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 
&lt;span style="display: block; text-align: center;"&gt;&lt;span style="color: black; font-family: Arial; font-size: 13px;"&gt;&lt;strong&gt;&lt;u&gt;&lt;span style="color: windowtext; font-family: arial,sans-serif; font-size: 16px;"&gt;Related to ADO&lt;/span&gt;&lt;/u&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
<div class='external-image-warning' style='background-color:#f8f9fa; border: 1px solid #eaecf0; padding-left: 0.5em; padding-right:0.5em; display:inline-block'>
&lt;br /&gt;
External image: http://sites.google.com/site/240edo/ADO-3-custom-size-604-289.jpg<br>
&lt;span style="display: block; text-align: center;"&gt;&lt;span style="color: black; font-family: arial; font-size: 24px;"&gt;&lt;a class="wiki_link_ext" href="http://www.soundtransformations.btinternet.co.uk/Danerhudyarthemarmonicseries.htm" rel="nofollow"&gt;**Magic of Tone and the Art of Music by the late Dane Rhudyar**&lt;/a&gt;&lt;/span&gt;&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
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]]
 
<span style="display: block; text-align: center;">
 
</span>
 
<span style="display: block; text-align: center;">'''<span style="color: black; font-family: Arial; font-size: 13px;"><u>Relation between harmonics and ADO system</u></span>'''</span>
 
<span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;">'''ADO''' (like '''EDL)''' is based on [http://en.wikipedia.org/wiki/Superparticular_number **Superparticular ratios**] and [http://en.wikipedia.org/wiki/Harmonic_series_%28music%29 **harmonic series**]. Have a look at 12-ADO in this picture:</span></span>
 
[http://sites.google.com/site/240edo/ADO-2.jpg
 
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External image: http://sites.google.com/site/240edo/ADO-2-custom-size-378-270.jpg<br>
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
</div>
[[Category:IMPORTDEBUG - Change External Images]
 
]]
 
<span style="color: black; font-family: arial; font-size: 13px;">The above picture shows that '''ADO''' system is classified as :</span>
 
<span style="display: block; text-align: left;"><span style="font-family: Times New Roman;"><span style="color: black; font-family: arial; font-size: 13px;">- System with unequal </span><span style="color: blue; font-family: arial; font-size: 13px;">[http://tonalsoft.com/enc/e/epimorios.aspx **epimorios**]</span><span style="color: black; font-family: arial; font-size: 13px;"> '''('''[http://en.wikipedia.org/wiki/Superparticular_number **Superparticular**]''')''' divisions.</span></span></span>
 
<span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;">- System based on ascending series of superparticular ratios with descending sizes.</span></span>
 
<span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;">- System which covers superparticular ratios between harmonic of number C (in this example 12)to harmonic of Number 2C(in this example 24).</span></span>
 
<span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">'''- <span style="font-family: arial,sans-serif;">[http://sites.google.com/site/240edo/ADO-EDL.XLS An spreadsheet showing relation between harmonics , superparticular ratios and ADO system]</span>'''</span></span>
 
<span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 15px;">'''-''' <span style="font-family: arial,sans-serif;">[http://www.music.sc.edu/fs/bain/software/BainTheOvertoneSeries.pdf The Overtone Series]</span></span></span>
 
<span style="display: block; text-align: center;"><span style="color: black; font-family: arial; font-size: 15px;">'''<span style="color: black; font-family: Arial; font-size: 13px;"><u>Relation between Otonality and ADO system</u></span>'''</span></span>
 
<span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">We can consider </span>'''<span style="color: black; font-family: Arial; font-size: 13px;">ADO</span>'''<span style="color: black; font-family: Arial; font-size: 13px;"> system as </span><span style="color: blue; font-family: Arial; font-size: 13px;">[http://en.wikipedia.org/wiki/Otonal **Otonal system**]</span><span style="color: black; font-family: Arial; font-size: 13px;"> .'''Otonality''' is a term introduced by </span><span style="color: blue; font-family: Arial; font-size: 13px;">[http://en.wikipedia.org/wiki/Harry_Partch **Harry Partch**]</span><span style="color: black; font-family: Arial; font-size: 13px;"> to describe chords whose notes are the overtones (multiples) of a given fixed tone.Considering ADO , an Otonality is a collection of pitches which can be expressed in ratios that have equal denominators. For example, 1/1, 5/4, and 3/2 form an Otonality because they can be written as 4/4, 5/4, 6/4. Every Otonality is therefore part of the [http://en.wikipedia.org/wiki/Harmonic_series_%28music%29 **harmonic series**]. nominator here is called "</span><span style="color: blue; font-family: Arial; font-size: 13px;">[http://tonalsoft.com/enc/n/nexus.aspx **Numerary nexus**]</span><span style="color: black; font-family: Arial; font-size: 13px;">".An Otonality corresponds to an [http://en.wikipedia.org/wiki/Arithmetic_series **arithmetic series**] of frequencies or a [http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29 **harmonic series**] of wavelengths or distances on a [http://en.wikipedia.org/wiki/String_instrument **string instrument**].</span></span>
 
<span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">'''<span style="color: black; font-family: 'Times New Roman'; font-size: 13px;">- </span><u><span style="color: windowtext; font-family: 'times new roman'; font-size: 16px;">[http://240edo.googlepages.com/ADO-EDL.XLS Fret position calculator (excel sheet ) based on EDL system and string length]</span></u>'''</span></span>
 
<span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;"><span style="color: #0000ff; font-family: arial,sans-serif; font-size: 16px;">[http://sites.google.com/site/240edo/ADOandEDO.xls - How to approximate EDand ADO systems with each other?Download this file]</span></span></span>
 
<span style="display: block; text-align: center;"><span style="color: black; font-family: Arial; font-size: 13px;">'''<u><span style="color: windowtext; font-family: arial,sans-serif; font-size: 16px;">Related to ADO</span></u>'''</span></span>
 
<span style="display: block; text-align: center;"><span style="color: black; font-family: arial; font-size: 24px;">[http://www.soundtransformations.btinternet.co.uk/Danerhudyarthemarmonicseries.htm **Magic of Tone and the Art of Music by the late Dane Rhudyar**]</span></span>      [[Category:ADO]]
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