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Wikispaces>acousticsoftombak **Imported revision 304675414 - Original comment: ** |
Wikispaces>acousticsoftombak **Imported revision 304675644 - Original comment: ** |
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<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>2012-02-24 00: | : 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> | : The original revision id was <tt>304675644</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"> | <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"><span style="color: #000000; font-family: arial,sans-serif; font-size: 140%;">**Arithmetic rational** **divisions of octave** </span> | ||
<span style="text-align: left;"><span style="color: black; font-family: | <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> | ||
<span style="text-align: left;"><span style="color: black; font-family: | <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> | ||
<span style=" | <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 __**R1**__ (wich is ratio of C/C) and the last , __**Rn**__ </span><span style="color: black; font-size: 15px;">(wich is ratio of 2C/C), with common difference of </span>__<span style="color: black; font-size: 15px;">**d**</span>__</span></span> | ||
<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> | |||
<span style="text-align: left;"><span style="font-family: | <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">**R2 = R1+d** </span></span> | ||
<span style="text-align: | <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">**R3= R1+2d** </span></span> | ||
<span style="text-align: left;"> | <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> | ||
<span style=" | <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">**………**</span></span> | ||
<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> | |||
<span style=" | |||
<span | |||
<span style="text-align: | |||
<span style="text-align: | |||
<span style="font-family: | <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> | ||
<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> | |||
<span style="display: block; text-align: center;">(C+m/C)</span> | |||
<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> | |||
<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> | |||
<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 >........> R3 > R2 > R1** so : </span></span> | |||
<span style="display: block; text-align: left;">**<span style="color: black; font-family: Arial; font-size: 13px;">L1 > L2 > L3 > …… > Ln</span>**</span> | |||
[[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"]] | |||
<span style=" | <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> | ||
[[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"]] | |||
<span style="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></pre></div> | [[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> | |||
<span style="display: block; text-align: center;">**<span style="color: black; font-family: Arial; font-size: 13px;">__Relation between harmonics and ADO system__</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> | |||
[[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="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;">__Relation between Otonality and ADO system__</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>__<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>__**</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;">**__<span style="color: windowtext; font-family: arial,sans-serif; font-size: 16px;">Related to ADO</span>__**</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></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; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>ADO</title></head><body><span style="color: | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>ADO</title></head><body><span style="color: #000000; font-family: arial,sans-serif; font-size: 140%;"><strong>Arithmetic rational</strong> <strong>divisions of octave</strong> </span><br /> | ||
< | <br /> | ||
<span style=" | <span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;"><strong>ARDO</strong> (which is simplified as <strong><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticrationaldivisionsofoctave" rel="nofollow">ADO</a>)</strong> is an intervallic system <span style="color: black; font-family: arial,sans-serif; font-size: 15px;">considered as </span></span></span><br /> | ||
<span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;"><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 <span style="color: black; font-family: arial,sans-serif; font-size: 15px;">terms of sequence. </span></span></span><br /> | |||
< | <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><strong>R1</strong></u> (wich is ratio of C/C) and the last , <u><strong>Rn</strong></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;"><strong>d</strong></span></u></span></span><br /> | ||
<span style="text-align: | <span style="display: block; text-align: center;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">(which is <strong>1/C</strong>), we have : </span></span><br /> | ||
<span style=" | <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;"><strong>R2 = R1+d</strong> </span></span><br /> | ||
<span style="text-align: left;"><span style=" | <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;"><strong>R3= R1+2d</strong> </span></span><br /> | ||
<span style="display: block; text-align: left;"><span style="font-family: arial,sans-serif;"><strong><span style="color: black; font-size: 15px;">R4 = R1+3d </span></strong></span></span><br /> | |||
<span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;"><strong>………</strong></span></span><br /> | |||
<span style="display: block; text-align: left;"><span style="font-family: Times New Roman;"><strong><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">Rn = R1+(n-1)d</span></strong></span></span><br /> | |||
<br /> | <br /> | ||
<span style="text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">Each consequent divisions like <strong>R4</strong> and <strong>R3</strong> have a difference of <strong>d</strong> with each other.The concept of division here is a bit different from <strong>EDO</strong> and other systems (which is the difference of cents of two consequent degree). In <strong>ADO</strong>, 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><br /> | <span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">Each consequent divisions like <strong>R4</strong> and <strong>R3</strong> have a difference of <strong>d</strong> with each other.The concept of division here is a bit different from <strong>EDO</strong> and other systems (which is the difference of cents of two consequent degree). In <strong>ADO</strong>, 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><br /> | ||
<span style="text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">For any <strong>C-ADO</strong> system with <a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/c/cardinality.aspx" rel="nofollow">**cardinality**</a> of <strong>C</strong>, we have ratios related to different degrees of <strong>m</strong> as : </span></span><br /> | <span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">For any <strong>C-ADO</strong> system with <a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/c/cardinality.aspx" rel="nofollow">**cardinality**</a> of <strong>C</strong>, we have ratios related to different degrees of <strong>m</strong> as : </span></span><br /> | ||
<span style="text-align: center;">(C+m/C)</span><br /> | <span style="display: block; text-align: center;">(C+m/C)</span><br /> | ||
<span style="text-align: left;"><span style="font-family: arial,sans-serif;">For example , in <strong>12-ADO</strong> the ratio related to the first degree is 13/12 .</span></span><br /> | <span style="display: block; text-align: left;"><span style="font-family: arial,sans-serif;">For example , in <strong>12-ADO</strong> the ratio related to the first degree is 13/12 .</span></span><br /> | ||
<span style="text-align: left;"><span style="font-family: arial,sans-serif;"><strong>12-ADO</strong> can be shown as series like: </span><strong><span style="color: black; font-family: Arial; font-size: 13px;">12:13</span></strong><strong><span style="color: black; font-family: Arial; font-size: 13px;">:14:15:16:17:18:19:20:21:22:23:24</span></strong><span style="color: black; font-family: arial; font-size: 13px;"> or </span><strong><span style="color: black; font-family: arial; font-size: 13px;">12 13 </strong><span style="color: black; font-family: Arial; font-size: 13px;">14 15 16 17 18 19 20 21 22 23 24</span><strong> < | <span style="display: block; text-align: left;"><span style="font-family: arial,sans-serif;"><strong>12-ADO</strong> can be shown as series like: </span><strong><span style="color: black; font-family: Arial; font-size: 13px;">12:13</span></strong><strong><span style="color: black; font-family: Arial; font-size: 13px;">:14:15:16:17:18:19:20:21:22:23:24</span></strong><span style="color: black; font-family: arial; font-size: 13px;"> or </span><strong><span style="color: black; font-family: arial; font-size: 13px;">12 13 </span></strong><span style="color: black; font-family: Arial; font-size: 13px;">14 15 16 17 18 19 20 21 22 23 24</span> <strong><span style="color: black; font-family: Arial; font-size: 13px;">.</span></strong></span><br /> | ||
< | <span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">For an <strong>ADO</strong> intervallic system with <strong>n</strong> divisions we have <span style="font-family: arial,sans-serif;">unequal divisions of length </span>by dividing string length to<strong><span style="color: black; font-family: Arial; font-size: 13px;">n</span></strong> unequal divisions based on each degree ratios.If the first division has ratio of <strong>R1</strong> and length of <strong><span style="color: black; font-family: Arial; font-size: 13px;">L1</span></strong> and the last, <strong>Rn</strong> and <strong><span style="color: black; font-family: Arial; font-size: 13px;">Ln</span></strong> , we have: <strong>Ln = 1/Rn</strong> and if <strong>Rn &gt;........&gt; R3 &gt; R2 &gt; R1</strong> so : </span></span><br /> | ||
& | <span style="display: block; text-align: left;"><strong><span style="color: black; font-family: Arial; font-size: 13px;">L1 &gt; L2 &gt; L3 &gt; …… &gt; Ln</span></strong></span><br /> | ||
& | <!-- ws:start:WikiTextRemoteImageRule:1:&lt;div style=&quot;text-align: center&quot;&gt;&lt;a href=&quot;http://sites.google.com/site/240edo/ADO-4.jpg&quot; rel=&quot;nofollow&quot;&gt;&lt;img src=&quot;http://sites.google.com/site/240edo/ADO-4-custom-size-350-238.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt;&lt;/div&gt; --><div style="text-align: center"><a href="http://sites.google.com/site/240edo/ADO-4.jpg" rel="nofollow"><img src="http://sites.google.com/site/240edo/ADO-4-custom-size-350-238.jpg" alt="external image ADO-4-custom-size-350-238.jpg" title="external image ADO-4-custom-size-350-238.jpg" /></a></div><!-- ws:end:WikiTextRemoteImageRule:1 --><br /> | ||
<span style="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 <a class="wiki_link_ext" href="http://sites.google.com/site/240edo/equaldivisionsoflength(edl)" rel="nofollow">**EDL system**</a>:</span></span><br /> | <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 <a class="wiki_link_ext" href="http://sites.google.com/site/240edo/equaldivisionsoflength(edl)" rel="nofollow">**EDL system**</a>:</span></span><br /> | ||
<!-- ws:start:WikiTextRemoteImageRule:3:&lt;div style=&quot;text-align: center&quot;&gt;&lt;a href=&quot;http://sites.google.com/site/240edo/ADO-5.jpg&quot; rel=&quot;nofollow&quot;&gt;&lt;img src=&quot;http://sites.google.com/site/240edo/ADO-5-custom-size-346-235.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt;&lt;/div&gt; --><div style="text-align: center"><a href="http://sites.google.com/site/240edo/ADO-5.jpg" rel="nofollow"><img src="http://sites.google.com/site/240edo/ADO-5-custom-size-346-235.jpg" alt="external image ADO-5-custom-size-346-235.jpg" title="external image ADO-5-custom-size-346-235.jpg" /></a></div><!-- ws:end:WikiTextRemoteImageRule:3 --><br /> | |||
< | <!-- ws:start:WikiTextRemoteImageRule:5:&lt;div style=&quot;text-align: center&quot;&gt;&lt;a href=&quot;http://sites.google.com/site/240edo/ADO-3.jpg&quot; rel=&quot;nofollow&quot;&gt;&lt;img src=&quot;http://sites.google.com/site/240edo/ADO-3-custom-size-604-289.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt;&lt;/div&gt; --><div style="text-align: center"><a href="http://sites.google.com/site/240edo/ADO-3.jpg" rel="nofollow"><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" /></a></div><!-- ws:end:WikiTextRemoteImageRule:5 --><br /> | ||
<span style="text-align: center;">< | <span style="display: block; text-align: center;"><br /> | ||
<span style="text-align: center;"><strong><span style="color: black; font-family: Arial; font-size: 13px;"><u>Relation between harmonics and ADO system</u></span></strong></span><br /> | </span><br /> | ||
<span style="text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;"><strong>ADO</strong> (like <strong>EDL)</strong> is based on <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">**Superparticular ratios**</a> and <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harmonic_series_%28music%29" rel="nofollow">**harmonic series**</a>. Have a look at 12-ADO in this picture:</span></span><br /> | <span style="display: block; text-align: center;"><strong><span style="color: black; font-family: Arial; font-size: 13px;"><u>Relation between harmonics and ADO system</u></span></strong></span><br /> | ||
< | <span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;"><strong>ADO</strong> (like <strong>EDL)</strong> is based on <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">**Superparticular ratios**</a> and <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harmonic_series_%28music%29" rel="nofollow">**harmonic series**</a>. Have a look at 12-ADO in this picture:</span></span><br /> | ||
<!-- ws:start:WikiTextRemoteImageRule:7:&lt;div style=&quot;text-align: center&quot;&gt;&lt;a href=&quot;http://sites.google.com/site/240edo/ADO-2.jpg&quot; rel=&quot;nofollow&quot;&gt;&lt;img src=&quot;http://sites.google.com/site/240edo/ADO-2-custom-size-378-270.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt;&lt;/div&gt; --><div style="text-align: center"><a href="http://sites.google.com/site/240edo/ADO-2.jpg" rel="nofollow"><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" /></a></div><!-- ws:end:WikiTextRemoteImageRule:7 --><br /> | |||
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<span style="color: black; font-family: arial; font-size: 13px;">The above picture shows that <strong>ADO</strong> system is classified as :</span><br /> | |||
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<span style="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;"><a class="wiki_link_ext" href="http://tonalsoft.com/enc/e/epimorios.aspx" rel="nofollow" target="_blank">**epimorios**</a></span><span style="color: black; font-family: arial; font-size: 13px;"> <strong>(</strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">**Superparticular**</a><strong>)</strong> divisions.</span></span></span><br /> | <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;"><a class="wiki_link_ext" href="http://tonalsoft.com/enc/e/epimorios.aspx" rel="nofollow" target="_blank">**epimorios**</a></span><span style="color: black; font-family: arial; font-size: 13px;"> <strong>(</strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">**Superparticular**</a><strong>)</strong> divisions.</span></span></span><br /> | ||
<span style="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><br /> | <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><br /> | ||
<span style="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><br /> | <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><br /> | ||
<span style="text-align: left;"><span style="color: black; font-family: arial; font-size: 15px | <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;"><strong>- <span style="font-family: arial,sans-serif;"><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/ADO-EDL.XLS" rel="nofollow">An spreadsheet showing relation between harmonics , superparticular ratios and ADO system</a></span></strong></span></span><br /> | ||
<span style="text-align: left;"><span style="color: black; font-family: arial; font-size: 15px;"><strong>- </strong | <span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 15px;"><strong>-</strong> <span style="font-family: arial,sans-serif;"><a class="wiki_link_ext" href="http://www.music.sc.edu/fs/bain/software/BainTheOvertoneSeries.pdf" rel="nofollow">The Overtone Series</a></span></span></span><br /> | ||
<span style="display: block; text-align: center;"><span style="color: black; font-family: arial; font-size: 15px;"><strong><span style="color: black; font-family: Arial; font-size: 13px;"><u>Relation between Otonality and ADO system</u></span></strong></span></span><br /> | |||
<span style="text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">We can consider </span><strong><span style="color: black; font-family: Arial; font-size: 13px;">ADO</span></strong><span style="color: black; font-family: Arial; font-size: 13px;"> system as </span><span style="color: blue; font-family: Arial; font-size: 13px;"><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Otonal" rel="nofollow" target="_blank">**Otonal system**</a></span><span style="color: black; font-family: Arial; font-size: 13px;"> .<strong>Otonality</strong> is a term introduced by </span><span style="color: blue; font-family: Arial; font-size: 13px;"><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harry_Partch" rel="nofollow" target="_blank">**Harry Partch**</a></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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harmonic_series_%28music%29" rel="nofollow">**harmonic series**</a>. nominator here is called &quot;</span><span style="color: blue; font-family: Arial; font-size: 13px;"><a class="wiki_link_ext" href="http://tonalsoft.com/enc/n/nexus.aspx" rel="nofollow" target="_blank">**Numerary nexus**</a></span><span style="color: black; font-family: Arial; font-size: 13px;">&quot;.An Otonality corresponds to an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arithmetic_series" rel="nofollow">**arithmetic series**</a> of frequencies or a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29" rel="nofollow">**harmonic series**</a> of wavelengths or distances on a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/String_instrument" rel="nofollow">**string instrument**</a>.</span></span><br /> | <span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">We can consider </span><strong><span style="color: black; font-family: Arial; font-size: 13px;">ADO</span></strong><span style="color: black; font-family: Arial; font-size: 13px;"> system as </span><span style="color: blue; font-family: Arial; font-size: 13px;"><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Otonal" rel="nofollow" target="_blank">**Otonal system**</a></span><span style="color: black; font-family: Arial; font-size: 13px;"> .<strong>Otonality</strong> is a term introduced by </span><span style="color: blue; font-family: Arial; font-size: 13px;"><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harry_Partch" rel="nofollow" target="_blank">**Harry Partch**</a></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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harmonic_series_%28music%29" rel="nofollow">**harmonic series**</a>. nominator here is called &quot;</span><span style="color: blue; font-family: Arial; font-size: 13px;"><a class="wiki_link_ext" href="http://tonalsoft.com/enc/n/nexus.aspx" rel="nofollow" target="_blank">**Numerary nexus**</a></span><span style="color: black; font-family: Arial; font-size: 13px;">&quot;.An Otonality corresponds to an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arithmetic_series" rel="nofollow">**arithmetic series**</a> of frequencies or a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29" rel="nofollow">**harmonic series**</a> of wavelengths or distances on a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/String_instrument" rel="nofollow">**string instrument**</a>.</span></span><br /> | ||
<span style="text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;"><strong><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;"><a class="wiki_link_ext" href="http://240edo.googlepages.com/ADO-EDL.XLS" rel="nofollow">Fret position calculator (excel sheet ) based on EDL system and string length</a></span></u></strong></span></span><br /> | <span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;"><strong><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;"><a class="wiki_link_ext" href="http://240edo.googlepages.com/ADO-EDL.XLS" rel="nofollow">Fret position calculator (excel sheet ) based on EDL system and string length</a></span></u></strong></span></span><br /> | ||
<span style="text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;" | <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;"><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/ADOandEDO.xls" rel="nofollow">- How to approximate EDand ADO systems with each other?Download this file</a></span></span></span><br /> | ||
<span style="display: block; text-align: center;"><span style="color: black; font-family: Arial; font-size: 13px;"><strong><u><span style="color: windowtext; font-family: arial,sans-serif; font-size: 16px;">Related to ADO</span></u></strong></span></span><br /> | |||
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<span style="text-align: center;"><span style="color: black; font-family: arial; font-size: 24px;"><a class="wiki_link_ext" href="http://www.soundtransformations.btinternet.co.uk/Danerhudyarthemarmonicseries.htm" rel="nofollow">**Magic of Tone and the Art of Music by the late Dane Rhudyar**</a></span></span></body></html></pre></div> | <span style="display: block; text-align: center;"><span style="color: black; font-family: arial; font-size: 24px;"><a class="wiki_link_ext" href="http://www.soundtransformations.btinternet.co.uk/Danerhudyarthemarmonicseries.htm" rel="nofollow">**Magic of Tone and the Art of Music by the late Dane Rhudyar**</a></span></span></body></html></pre></div> | ||
Revision as of 00:21, 24 February 2012
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<span style="color: #000000; font-family: arial,sans-serif; font-size: 140%;">**Arithmetic rational** **divisions of octave** </span> <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> <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> <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 __**R1**__ (wich is ratio of C/C) and the last , __**Rn**__ </span><span style="color: black; font-size: 15px;">(wich is ratio of 2C/C), with common difference of </span>__<span style="color: black; font-size: 15px;">**d**</span>__</span></span> <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> <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">**R2 = R1+d** </span></span> <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">**R3= R1+2d** </span></span> <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> <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">**………**</span></span> <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> <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> <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> <span style="display: block; text-align: center;">(C+m/C)</span> <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> <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> <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 >........> R3 > R2 > R1** so : </span></span> <span style="display: block; text-align: left;">**<span style="color: black; font-family: Arial; font-size: 13px;">L1 > L2 > L3 > …… > Ln</span>**</span> [[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"]] <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> [[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> <span style="display: block; text-align: center;">**<span style="color: black; font-family: Arial; font-size: 13px;">__Relation between harmonics and ADO system__</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> [[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="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;">__Relation between Otonality and ADO system__</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>__<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>__**</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;">**__<span style="color: windowtext; font-family: arial,sans-serif; font-size: 16px;">Related to ADO</span>__**</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>
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<html><head><title>ADO</title></head><body><span style="color: #000000; font-family: arial,sans-serif; font-size: 140%;"><strong>Arithmetic rational</strong> <strong>divisions of octave</strong> </span><br /> <br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;"><strong>ARDO</strong> (which is simplified as <strong><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/arithmeticrationaldivisionsofoctave" rel="nofollow">ADO</a>)</strong> is an intervallic system <span style="color: black; font-family: arial,sans-serif; font-size: 15px;">considered as </span></span></span><br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;"><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 <span style="color: black; font-family: arial,sans-serif; font-size: 15px;">terms of sequence. </span></span></span><br /> <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><strong>R1</strong></u> (wich is ratio of C/C) and the last , <u><strong>Rn</strong></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;"><strong>d</strong></span></u></span></span><br /> <span style="display: block; text-align: center;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">(which is <strong>1/C</strong>), we have : </span></span><br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;"><strong>R2 = R1+d</strong> </span></span><br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;"><strong>R3= R1+2d</strong> </span></span><br /> <span style="display: block; text-align: left;"><span style="font-family: arial,sans-serif;"><strong><span style="color: black; font-size: 15px;">R4 = R1+3d </span></strong></span></span><br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;"><strong>………</strong></span></span><br /> <span style="display: block; text-align: left;"><span style="font-family: Times New Roman;"><strong><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">Rn = R1+(n-1)d</span></strong></span></span><br /> <br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">Each consequent divisions like <strong>R4</strong> and <strong>R3</strong> have a difference of <strong>d</strong> with each other.The concept of division here is a bit different from <strong>EDO</strong> and other systems (which is the difference of cents of two consequent degree). In <strong>ADO</strong>, 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><br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">For any <strong>C-ADO</strong> system with <a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/c/cardinality.aspx" rel="nofollow">**cardinality**</a> of <strong>C</strong>, we have ratios related to different degrees of <strong>m</strong> as : </span></span><br /> <span style="display: block; text-align: center;">(C+m/C)</span><br /> <span style="display: block; text-align: left;"><span style="font-family: arial,sans-serif;">For example , in <strong>12-ADO</strong> the ratio related to the first degree is 13/12 .</span></span><br /> <span style="display: block; text-align: left;"><span style="font-family: arial,sans-serif;"><strong>12-ADO</strong> can be shown as series like: </span><strong><span style="color: black; font-family: Arial; font-size: 13px;">12:13</span></strong><strong><span style="color: black; font-family: Arial; font-size: 13px;">:14:15:16:17:18:19:20:21:22:23:24</span></strong><span style="color: black; font-family: arial; font-size: 13px;"> or </span><strong><span style="color: black; font-family: arial; font-size: 13px;">12 13 </span></strong><span style="color: black; font-family: Arial; font-size: 13px;">14 15 16 17 18 19 20 21 22 23 24</span> <strong><span style="color: black; font-family: Arial; font-size: 13px;">.</span></strong></span><br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">For an <strong>ADO</strong> intervallic system with <strong>n</strong> divisions we have <span style="font-family: arial,sans-serif;">unequal divisions of length </span>by dividing string length to<strong><span style="color: black; font-family: Arial; font-size: 13px;">n</span></strong> unequal divisions based on each degree ratios.If the first division has ratio of <strong>R1</strong> and length of <strong><span style="color: black; font-family: Arial; font-size: 13px;">L1</span></strong> and the last, <strong>Rn</strong> and <strong><span style="color: black; font-family: Arial; font-size: 13px;">Ln</span></strong> , we have: <strong>Ln = 1/Rn</strong> and if <strong>Rn >........> R3 > R2 > R1</strong> so : </span></span><br /> <span style="display: block; text-align: left;"><strong><span style="color: black; font-family: Arial; font-size: 13px;">L1 > L2 > L3 > …… > Ln</span></strong></span><br /> <!-- ws:start:WikiTextRemoteImageRule:1:<div style="text-align: center"><a href="http://sites.google.com/site/240edo/ADO-4.jpg" rel="nofollow"><img src="http://sites.google.com/site/240edo/ADO-4-custom-size-350-238.jpg" alt="" title="" /></a></div> --><div style="text-align: center"><a href="http://sites.google.com/site/240edo/ADO-4.jpg" rel="nofollow"><img src="http://sites.google.com/site/240edo/ADO-4-custom-size-350-238.jpg" alt="external image ADO-4-custom-size-350-238.jpg" title="external image ADO-4-custom-size-350-238.jpg" /></a></div><!-- ws:end:WikiTextRemoteImageRule:1 --><br /> <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 <a class="wiki_link_ext" href="http://sites.google.com/site/240edo/equaldivisionsoflength(edl)" rel="nofollow">**EDL system**</a>:</span></span><br /> <!-- ws:start:WikiTextRemoteImageRule:3:<div style="text-align: center"><a href="http://sites.google.com/site/240edo/ADO-5.jpg" rel="nofollow"><img src="http://sites.google.com/site/240edo/ADO-5-custom-size-346-235.jpg" alt="" title="" /></a></div> --><div style="text-align: center"><a href="http://sites.google.com/site/240edo/ADO-5.jpg" rel="nofollow"><img src="http://sites.google.com/site/240edo/ADO-5-custom-size-346-235.jpg" alt="external image ADO-5-custom-size-346-235.jpg" title="external image ADO-5-custom-size-346-235.jpg" /></a></div><!-- ws:end:WikiTextRemoteImageRule:3 --><br /> <!-- ws:start:WikiTextRemoteImageRule:5:<div style="text-align: center"><a href="http://sites.google.com/site/240edo/ADO-3.jpg" rel="nofollow"><img src="http://sites.google.com/site/240edo/ADO-3-custom-size-604-289.jpg" alt="" title="" /></a></div> --><div style="text-align: center"><a href="http://sites.google.com/site/240edo/ADO-3.jpg" rel="nofollow"><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" /></a></div><!-- ws:end:WikiTextRemoteImageRule:5 --><br /> <span style="display: block; text-align: center;"><br /> </span><br /> <span style="display: block; text-align: center;"><strong><span style="color: black; font-family: Arial; font-size: 13px;"><u>Relation between harmonics and ADO system</u></span></strong></span><br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;"><strong>ADO</strong> (like <strong>EDL)</strong> is based on <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">**Superparticular ratios**</a> and <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harmonic_series_%28music%29" rel="nofollow">**harmonic series**</a>. Have a look at 12-ADO in this picture:</span></span><br /> <!-- ws:start:WikiTextRemoteImageRule:7:<div style="text-align: center"><a href="http://sites.google.com/site/240edo/ADO-2.jpg" rel="nofollow"><img src="http://sites.google.com/site/240edo/ADO-2-custom-size-378-270.jpg" alt="" title="" /></a></div> --><div style="text-align: center"><a href="http://sites.google.com/site/240edo/ADO-2.jpg" rel="nofollow"><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" /></a></div><!-- ws:end:WikiTextRemoteImageRule:7 --><br /> <br /> <span style="color: black; font-family: arial; font-size: 13px;">The above picture shows that <strong>ADO</strong> system is classified as :</span><br /> <br /> <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;"><a class="wiki_link_ext" href="http://tonalsoft.com/enc/e/epimorios.aspx" rel="nofollow" target="_blank">**epimorios**</a></span><span style="color: black; font-family: arial; font-size: 13px;"> <strong>(</strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">**Superparticular**</a><strong>)</strong> divisions.</span></span></span><br /> <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><br /> <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><br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;"><strong>- <span style="font-family: arial,sans-serif;"><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/ADO-EDL.XLS" rel="nofollow">An spreadsheet showing relation between harmonics , superparticular ratios and ADO system</a></span></strong></span></span><br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 15px;"><strong>-</strong> <span style="font-family: arial,sans-serif;"><a class="wiki_link_ext" href="http://www.music.sc.edu/fs/bain/software/BainTheOvertoneSeries.pdf" rel="nofollow">The Overtone Series</a></span></span></span><br /> <span style="display: block; text-align: center;"><span style="color: black; font-family: arial; font-size: 15px;"><strong><span style="color: black; font-family: Arial; font-size: 13px;"><u>Relation between Otonality and ADO system</u></span></strong></span></span><br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">We can consider </span><strong><span style="color: black; font-family: Arial; font-size: 13px;">ADO</span></strong><span style="color: black; font-family: Arial; font-size: 13px;"> system as </span><span style="color: blue; font-family: Arial; font-size: 13px;"><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Otonal" rel="nofollow" target="_blank">**Otonal system**</a></span><span style="color: black; font-family: Arial; font-size: 13px;"> .<strong>Otonality</strong> is a term introduced by </span><span style="color: blue; font-family: Arial; font-size: 13px;"><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harry_Partch" rel="nofollow" target="_blank">**Harry Partch**</a></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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harmonic_series_%28music%29" rel="nofollow">**harmonic series**</a>. nominator here is called "</span><span style="color: blue; font-family: Arial; font-size: 13px;"><a class="wiki_link_ext" href="http://tonalsoft.com/enc/n/nexus.aspx" rel="nofollow" target="_blank">**Numerary nexus**</a></span><span style="color: black; font-family: Arial; font-size: 13px;">".An Otonality corresponds to an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arithmetic_series" rel="nofollow">**arithmetic series**</a> of frequencies or a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29" rel="nofollow">**harmonic series**</a> of wavelengths or distances on a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/String_instrument" rel="nofollow">**string instrument**</a>.</span></span><br /> <span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;"><strong><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;"><a class="wiki_link_ext" href="http://240edo.googlepages.com/ADO-EDL.XLS" rel="nofollow">Fret position calculator (excel sheet ) based on EDL system and string length</a></span></u></strong></span></span><br /> <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;"><a class="wiki_link_ext" href="http://sites.google.com/site/240edo/ADOandEDO.xls" rel="nofollow">- How to approximate EDand ADO systems with each other?Download this file</a></span></span></span><br /> <span style="display: block; text-align: center;"><span style="color: black; font-family: Arial; font-size: 13px;"><strong><u><span style="color: windowtext; font-family: arial,sans-serif; font-size: 16px;">Related to ADO</span></u></strong></span></span><br /> <br /> <span style="display: block; text-align: center;"><span style="color: black; font-family: arial; font-size: 24px;"><a class="wiki_link_ext" href="http://www.soundtransformations.btinternet.co.uk/Danerhudyarthemarmonicseries.htm" rel="nofollow">**Magic of Tone and the Art of Music by the late Dane Rhudyar**</a></span></span></body></html>