Armodue theory
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[[toc|flat]] ---- =Armodue Theory of 16EDO= //(summary translation from the italian site// [[http://www.armodue.com/ricerche.htm|Armodue]] ) Referring not only to the [[16edo|16-edo equal temperament]], but also to half-equal and [[Lou Harrison]]'s [[JustIntonation|Just intonation]] 16 note scale, the natural octave division by <span style="font-family: Verdana; font-size: small; line-height: normal;">[[http://www.pertout.com/|Andrián Pertout]] and the 16-to-31 [[OverToneSeries|overtone scale]], </span>Armodue has been proposed as a new notation and theory system. Desiring to make the approach to Armodue as easy as possible, but conscious that they had to give new names to the notes that constitute the system, the Italian creators of the <span style="background-position: 100% 50%; cursor: pointer; padding-right: 10px;">[[@http://armodue.com/|Armodue]]</span> system named them numbering from 1 to 9: 1, 1#, 2, 2#, 3, 3#, 4, 5, 5#, 6, 6#, 7, 7#, 8, 8#, 9 Consequently, the interval between a note at frequency n and other at frequency 2n is called a //tenth//. The basic (micro-)tone of Armodue, whatever concrete temperament is used, is always called //eka// (from Sanskrit eka: one, unit). In the chromatic Armodue scale, one eka always corresponds to the interval between any two consecutive notes. For composing in Armodue it's useful to use a //tetragram// (staff with 4 lines) || [[image:http://www.armodue.com/TETR-%5B1%5D.jpg caption="copyright Armodue, used with permission"]] || || copyright Armodue, used with permission || If for the execution of a musical piece we need to write on two or more tetragrams, the notes will be written in the same way for every tetragram. In other words, the "1" note will be written immediately under the first line __in every tenth__. In Armodue we have only a numeric clef, that show us the tenth: || [[image:http://www.armodue.com/Chiave.gif caption="copyright Armodue, used with permission"]] || || copyright Armodue, used with permission || The clefs 1,2,3... refers to the tenths: first, second, third... So, in the illustrated example above, the first tetragram (from top) refers to the 3rd tenth (central tenth, corresponding to the octave C3-C4), the second tetragram to the 5th tenth and the third to the 2nd. If we need to write simultaneously on several staves, we will draws normal braces. The keyboard conceived by the Armodue authors has the same disposition as Goldsmith's one (except the curvature): || [[image:http://www.armodue.com/Tastiera.jpg caption="copyright Armodue, used by permission"]] || || copyright Armodue, used by permission || [[image:Armodue_Neck.PNG width="640" height="126" caption="Dot-Pattern inlays, by Armodue"]] =Armodue just intonation - connection to Lou Harrison and Andrián Pertout= Armodue Just Intonation is nothing else than the 16-note scale by[[Lou Harrison| Lou Harrison]] based on simple ratios and pure intervals. In this scale there are practically the same twelve intervals of the "natural" or "Zarlinian" scale (the semitone 16/15, the minor wholetone 10/9, the minor third 6/5, the major third 5/4, the pure forth and fifth 4/3 and 3/2, the minor sixth 8/5, the major sixth 5/3, the minor seventh that is the complement of the minor wholetone 9/5, the major seventh 15/8) in addtion to the tritone and four other intervals that are based on the harmonic seventh (the ratios 8/7, 7/6, 12/7, 7/4). The composer Andrián Pertout used a very similar scale in his composition "Sonus dulcis" ([[http://web.archive.org/web/20040511172650/http://www.users.bigpond.com/apertout/Sonus.htm|article on archive.org]]), namely the following (according to [[http://www.armodue.com/risorse.htm|Armodue]]): ||~ Armodue note ||~ Interval ||~ Ratio ||~ Cents || || 1 || unison || 1/1 || 0 || || 1# || major half-tone || 16/15 || 112 || || 2 || minor tone || 10/9 || 182 || || 2# || major tone || 9/8 || 204 || || 3 || minor third || 6/5 || 316 || || 3# || major third || 5/4 || 386 || || 4 || fourth || 4/3 || 498 || || 5 || harmonic tritone || 45/32 || 590 || || 5# || cyclic tritone || 64/45 || 610 || || 6 || fifth || 3/2 || 702 || || 6# || diminished sixth || 8/5 || 814 || || 7 || harmonic sixth || 5/3 || 884 || || 7# || harmonic minor seventh || 7/4 || 969 || || 8 || minor seventh || 16/9 || 996 || || 8# || minor seventh || 9/5 || 1018 || || 9 || major seventh || 15/8 || 1088 || XXX =Semi-equalized Armodue= One step of 16edo (75 cents) is nearly equal to two steps (2\31) of [[31edo]] (77.42 cents). If we take the latter as a base, we get semi-equalized Armodue. In this temperament there is inevitably a smaller microtone (eka between the notes '7#' and '8'). leading to the 16 note MOS Valentine[16] of [[Starling temperaments#Valentine temperament|valentine temperament]]. Similarly we might use three steps of [[46edo]], 3\46, 78.26 cents, or five steps of [[77edo]], 77.92 cents. Semi-equalized Armodue provides a balance between the symmetry of the equalized system and the purity of natural intervals (intervals of semi-equalized Armodue are very pure, and at the same time it preserves the symmetry of the equalized system and its interval sizes almost unchanged). ||~ Armodue note ||~ cents (16edo) ||~ cents (semi-equalized Armodue based on [[31edo]]) || || 1 || 0 || 0 || || 1# || 75 || 77.42 || || 2 || 150 || 154.84 || || 2# || 225 || 232.26 || || 3 || 300 || 309.68 || || 3# || 375 || 387.10 || || 4 || 450 || 464.52 || || 5 || 525 || 541.94 || || 5# || 600 || 619.35 || || 6 || 675 || 696.77 || || 6# || 750 || 774.19 || || 7 || 825 || 851.61 || || 7# || 900 || 929.03 || || 8 || 975 || 967.74 || || 8# || 1050 || 1045.16 || || 9 || 1125 || 1122.58 ||
Original HTML content:
<html><head><title>Armodue theory</title></head><body><!-- ws:start:WikiTextTocRule:6:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Armodue Theory of 16EDO">Armodue Theory of 16EDO</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Armodue just intonation - connection to Lou Harrison and Andrián Pertout">Armodue just intonation - connection to Lou Harrison and Andrián Pertout</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Semi-equalized Armodue">Semi-equalized Armodue</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: -->
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<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Armodue Theory of 16EDO"></a><!-- ws:end:WikiTextHeadingRule:0 -->Armodue Theory of 16EDO</h1>
<br />
<em>(summary translation from the italian site</em> <a class="wiki_link_ext" href="http://www.armodue.com/ricerche.htm" rel="nofollow">Armodue</a> )<br />
<br />
Referring not only to the <a class="wiki_link" href="/16edo">16-edo equal temperament</a>, but also to half-equal and <a class="wiki_link" href="/Lou%20Harrison">Lou Harrison</a>'s <a class="wiki_link" href="/JustIntonation">Just intonation</a> 16 note scale, the natural octave division by <span style="font-family: Verdana; font-size: small; line-height: normal;"><a class="wiki_link_ext" href="http://www.pertout.com/" rel="nofollow">Andrián Pertout</a> and the 16-to-31 <a class="wiki_link" href="/OverToneSeries">overtone scale</a>, </span>Armodue has been proposed as a new notation and theory system.<br />
<br />
Desiring to make the approach to Armodue as easy as possible, but conscious that they had to give new names to the notes that constitute the system, the Italian creators of the <span style="background-position: 100% 50%; cursor: pointer; padding-right: 10px;"><a class="wiki_link_ext" href="http://armodue.com/" rel="nofollow" target="_blank">Armodue</a></span> system named them numbering from 1 to 9:<br />
<br />
1, 1#, 2, 2#, 3, 3#, 4, 5, 5#, 6, 6#, 7, 7#, 8, 8#, 9<br />
<br />
Consequently, the interval between a note at frequency n and other at frequency 2n is called a <em>tenth</em>.<br />
<br />
The basic (micro-)tone of Armodue, whatever concrete temperament is used, is always called <em>eka</em> (from Sanskrit eka: one, unit). In the chromatic Armodue scale, one eka always corresponds to the interval between any two consecutive notes.<br />
<br />
For composing in Armodue it's useful to use a <em>tetragram</em> (staff with 4 lines)<br />
<br />
<table class="wiki_table">
<tr>
<td><!-- ws:start:WikiTextRemoteImageRule:353:<img src="http://www.armodue.com/TETR-%5B1%5D.jpg" alt="copyright Armodue, used with permission" title="copyright Armodue, used with permission" /> --><table class="captionBox"><tr><td class="captionedImage"><img src="http://www.armodue.com/TETR-%5B1%5D.jpg" alt="copyright Armodue, used with permission" title="copyright Armodue, used with permission" /></td></tr><tr><td class="imageCaption">copyright Armodue, used with permission</td></tr></table><!-- ws:end:WikiTextRemoteImageRule:353 --><br />
</td>
</tr>
<tr>
<td>copyright Armodue, used with permission<br />
</td>
</tr>
</table>
<br />
<br />
If for the execution of a musical piece we need to write on two or more tetragrams, the notes will be written in the same way for every tetragram.<br />
In other words, the "1" note will be written immediately under the first line <u>in every tenth</u>.<br />
<br />
In Armodue we have only a numeric clef, that show us the tenth:<br />
<br />
<table class="wiki_table">
<tr>
<td><!-- ws:start:WikiTextRemoteImageRule:354:<img src="http://www.armodue.com/Chiave.gif" alt="copyright Armodue, used with permission" title="copyright Armodue, used with permission" /> --><table class="captionBox"><tr><td class="captionedImage"><img src="http://www.armodue.com/Chiave.gif" alt="copyright Armodue, used with permission" title="copyright Armodue, used with permission" /></td></tr><tr><td class="imageCaption">copyright Armodue, used with permission</td></tr></table><!-- ws:end:WikiTextRemoteImageRule:354 --><br />
</td>
</tr>
<tr>
<td>copyright Armodue, used with permission<br />
</td>
</tr>
</table>
<br />
<br />
<br />
<br />
The clefs 1,2,3... refers to the tenths: first, second, third...<br />
So, in the illustrated example above, the first tetragram (from top) refers to the 3rd tenth (central tenth, corresponding to the octave C3-C4),<br />
the second tetragram to the 5th tenth and the third to the 2nd. If we need to write simultaneously on several staves, we will draws normal braces.<br />
<br />
The keyboard conceived by the Armodue authors has the same disposition as Goldsmith's one (except the curvature):<br />
<table class="wiki_table">
<tr>
<td><!-- ws:start:WikiTextRemoteImageRule:355:<img src="http://www.armodue.com/Tastiera.jpg" alt="copyright Armodue, used by permission" title="copyright Armodue, used by permission" /> --><table class="captionBox"><tr><td class="captionedImage"><img src="http://www.armodue.com/Tastiera.jpg" alt="copyright Armodue, used by permission" title="copyright Armodue, used by permission" /></td></tr><tr><td class="imageCaption">copyright Armodue, used by permission</td></tr></table><!-- ws:end:WikiTextRemoteImageRule:355 --><br />
</td>
</tr>
<tr>
<td>copyright Armodue, used by permission<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextLocalImageRule:352:<img src="/file/view/Armodue_Neck.PNG/215944122/640x126/Armodue_Neck.PNG" alt="Dot-Pattern inlays, by Armodue" title="Dot-Pattern inlays, by Armodue" style="height: 126px; width: 640px;" /> --><table class="captionBox"><tr><td class="captionedImage"><img src="/file/view/Armodue_Neck.PNG/215944122/640x126/Armodue_Neck.PNG" alt="Armodue_Neck.PNG" title="Armodue_Neck.PNG" style="height: 126px; width: 640px;" /></td></tr><tr><td class="imageCaption">Dot-Pattern inlays, by Armodue</td></tr></table><!-- ws:end:WikiTextLocalImageRule:352 --><br />
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<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Armodue just intonation - connection to Lou Harrison and Andrián Pertout"></a><!-- ws:end:WikiTextHeadingRule:2 -->Armodue just intonation - connection to Lou Harrison and Andrián Pertout</h1>
<br />
Armodue Just Intonation is nothing else than the 16-note scale by<a class="wiki_link" href="/Lou%20Harrison"> Lou Harrison</a> based on simple ratios and pure intervals. In this scale there are practically the same twelve intervals of the "natural" or "Zarlinian" scale (the semitone 16/15, the minor wholetone 10/9, the minor third 6/5, the major third 5/4, the pure forth and fifth 4/3 and 3/2, the minor sixth 8/5, the major sixth 5/3, the minor seventh that is the complement of the minor wholetone 9/5, the major seventh 15/8) in addtion to the tritone and four other intervals that are based on the harmonic seventh (the ratios 8/7, 7/6, 12/7, 7/4).<br />
<br />
The composer Andrián Pertout used a very similar scale in his composition "Sonus dulcis" (<a class="wiki_link_ext" href="http://web.archive.org/web/20040511172650/http://www.users.bigpond.com/apertout/Sonus.htm" rel="nofollow">article on archive.org</a>), namely the following (according to <a class="wiki_link_ext" href="http://www.armodue.com/risorse.htm" rel="nofollow">Armodue</a>):<br />
<br />
<table class="wiki_table">
<tr>
<th>Armodue note<br />
</th>
<th>Interval<br />
</th>
<th>Ratio<br />
</th>
<th>Cents<br />
</th>
</tr>
<tr>
<td>1<br />
</td>
<td>unison<br />
</td>
<td>1/1<br />
</td>
<td>0<br />
</td>
</tr>
<tr>
<td>1#<br />
</td>
<td>major half-tone<br />
</td>
<td>16/15<br />
</td>
<td>112<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>minor tone<br />
</td>
<td>10/9<br />
</td>
<td>182<br />
</td>
</tr>
<tr>
<td>2#<br />
</td>
<td>major tone<br />
</td>
<td>9/8<br />
</td>
<td>204<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>minor third<br />
</td>
<td>6/5<br />
</td>
<td>316<br />
</td>
</tr>
<tr>
<td>3#<br />
</td>
<td>major third<br />
</td>
<td>5/4<br />
</td>
<td>386<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>fourth<br />
</td>
<td>4/3<br />
</td>
<td>498<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>harmonic tritone<br />
</td>
<td>45/32<br />
</td>
<td>590<br />
</td>
</tr>
<tr>
<td>5#<br />
</td>
<td>cyclic tritone<br />
</td>
<td>64/45<br />
</td>
<td>610<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>fifth<br />
</td>
<td>3/2<br />
</td>
<td>702<br />
</td>
</tr>
<tr>
<td>6#<br />
</td>
<td>diminished sixth<br />
</td>
<td>8/5<br />
</td>
<td>814<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>harmonic sixth<br />
</td>
<td>5/3<br />
</td>
<td>884<br />
</td>
</tr>
<tr>
<td>7#<br />
</td>
<td>harmonic minor seventh<br />
</td>
<td>7/4<br />
</td>
<td>969<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>minor seventh<br />
</td>
<td>16/9<br />
</td>
<td>996<br />
</td>
</tr>
<tr>
<td>8#<br />
</td>
<td>minor seventh<br />
</td>
<td>9/5<br />
</td>
<td>1018<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>major seventh<br />
</td>
<td>15/8<br />
</td>
<td>1088<br />
</td>
</tr>
</table>
<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Semi-equalized Armodue"></a><!-- ws:end:WikiTextHeadingRule:4 -->Semi-equalized Armodue</h1>
<br />
One step of 16edo (75 cents) is nearly equal to two steps (2\31) of <a class="wiki_link" href="/31edo">31edo</a> (77.42 cents). If we take the latter as a base, we get semi-equalized Armodue. In this temperament there is inevitably a smaller microtone (eka between the notes '7#' and '8'). leading to the 16 note MOS Valentine[16] of <a class="wiki_link" href="/Starling%20temperaments#Valentine temperament">valentine temperament</a>. Similarly we might use three steps of <a class="wiki_link" href="/46edo">46edo</a>, 3\46, 78.26 cents, or five steps of <a class="wiki_link" href="/77edo">77edo</a>, 77.92 cents.<br />
<br />
Semi-equalized Armodue provides a balance between the symmetry of the equalized system and the purity of natural intervals (intervals of semi-equalized Armodue are very pure, and at the same time it preserves the symmetry of the equalized system and its interval sizes almost unchanged).<br />
<br />
<table class="wiki_table">
<tr>
<th>Armodue note<br />
</th>
<th>cents (16edo)<br />
</th>
<th>cents (semi-equalized Armodue based on <a class="wiki_link" href="/31edo">31edo</a>)<br />
</th>
</tr>
<tr>
<td>1<br />
</td>
<td>0<br />
</td>
<td>0<br />
</td>
</tr>
<tr>
<td>1#<br />
</td>
<td>75<br />
</td>
<td>77.42<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>150<br />
</td>
<td>154.84<br />
</td>
</tr>
<tr>
<td>2#<br />
</td>
<td>225<br />
</td>
<td>232.26<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>300<br />
</td>
<td>309.68<br />
</td>
</tr>
<tr>
<td>3#<br />
</td>
<td>375<br />
</td>
<td>387.10<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>450<br />
</td>
<td>464.52<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>525<br />
</td>
<td>541.94<br />
</td>
</tr>
<tr>
<td>5#<br />
</td>
<td>600<br />
</td>
<td>619.35<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>675<br />
</td>
<td>696.77<br />
</td>
</tr>
<tr>
<td>6#<br />
</td>
<td>750<br />
</td>
<td>774.19<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>825<br />
</td>
<td>851.61<br />
</td>
</tr>
<tr>
<td>7#<br />
</td>
<td>900<br />
</td>
<td>929.03<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>975<br />
</td>
<td>967.74<br />
</td>
</tr>
<tr>
<td>8#<br />
</td>
<td>1050<br />
</td>
<td>1045.16<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>1125<br />
</td>
<td>1122.58<br />
</td>
</tr>
</table>
</body></html>