Armodue harmony: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-09-10 04:38:22 UTC</tt>.<br>

: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-09-10 05:34:58 UTC</tt>.<br>

: The original revision id was <tt>~~161610451~~</tt>.<br>

: The original revision id was <tt>161614925</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>

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[[toc|flat]]

----

Chapter 1:

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"Modes of limited transposition" incorporates and expands - applying it in Armodue - the modal system designed by Olivier Messiaen ("Modes à transpositions limitées").

The basic idea of Messiaen is to divide the octave equally in two, three, four or six equal intervals (two tritones, three major thirds , four minors thirds or six wholetones) and then structure these intervals with further internal division in groups or modules (inserting notes in a specific order).

The particularity is to make the last note of one module coincide with the first note of the next group.

For example, dividing the octave in three equal parts: c - e; e - ab; as - c, Messiaen structures his third mode by establishing the form: tone - semitone - tone and thus the succession of notes:

**c** - d - eb - **e** - f# - g - **ab** - bb - b - **c**

The notes marked in bold c, e, ab delimit the three parts that make up the octave and constitute at the same time the first and last note of each module of the type tone - semitone - semitone.

In Armodue the tenth of sixteen eka can be divided into 2, 4, or 8 parts.

But apart from these subdivisions of the tenth, the modal system of Messiaen can be extended to all intervals with sizes between 3 and 8 eka.

XXX

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Chapter 4:

="Geometric" harmonic constructions with Armodue=

The composer Scriabin has based much of his compositions on nucleopolar accordances.

To understand the nature of such accordances we should refer to specular reflections of light in a crystal, according to a tidy symmetry.

XXX

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="Elastic" chords=

~~XXX~~</pre></div>

With "elastic chords" I mean all those chords constructed in a ways that all parts are equally spaced, except one or two intervals that are variable.

An example will illustrate the principle better. If you construct a chord putting six intervals on top of each other each of which is 7 eka in size, you get an equally-spaced chord composed of the notes:

1 - 5 - 8# - 3# - 7# - 2# - 6#

If you now provide that one of the intervals does not have to be necessarily 7 eka but may vary at will, you get what I call an "elastic" chord: that is a chord where all the "rings" (intervals) are rigid (the equidistant intervals) except for one elastic "ring" likely to be shortened or extended.

In the example chord, if you choose the second interval to be the one with the elastic propertiy, the original chord can vary and become for example:

1 - 5 - 8 - 3 - 7 - 2 - 6

(shortening the original interval 5-8# to 6 eka instead of 7) or

1 - 5 - 1# - 5# - 9 - 4 - 8

(extending the original interval 5-8# to 10 eka instead of 7).

There arise many effective progressions of chords varied by the selection of one or two "elastic" intervals.</pre></div>

<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>Armodue armonia</title></head><body><span style="font-size: 140%;">Armodue: basic elements of harmony</span><br />

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<br />

<a href="#Two theses supporting the system">Two theses supporting the system</a> | <a href="#The interval table">The interval table</a> | <a href="#Creating scales with Armodue: modal systems">Creating scales with Armodue: modal systems</a> | <a href="#x&quot;Geometric&quot; harmonic constructions with Armodue">&quot;Geometric&quot; harmonic constructions with Armodue</a> | <a href="#x&quot;Elastic&quot; chords">&quot;Elastic&quot; chords</a>

~~<br />~~

<hr />

<hr />

Chapter 1: <br />

<h1 id="toc0"><a name="Two theses supporting the system"></a>Two theses supporting the system</h1>

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<br />

&quot;Modes of limited transposition&quot; incorporates and expands - applying it in Armodue - the modal system designed by Olivier Messiaen (&quot;Modes à transpositions limitées&quot;).<br />

<br />

The basic idea of Messiaen is to divide the octave equally in two, three, four or six equal intervals (two tritones, three major thirds , four minors thirds or six wholetones) and then structure these intervals with further internal division in groups or modules (inserting notes in a specific order).<br />

<br />

The particularity is to make the last note of one module coincide with the first note of the next group.<br />

<br />

For example, dividing the octave in three equal parts: c - e; e - ab; as - c, Messiaen structures his third mode by establishing the form: tone - semitone - tone and thus the succession of notes:<br />

<br />

<br />

The notes marked in bold c, e, ab delimit the three parts that make up the octave and constitute at the same time the first and last note of each module of the type tone - semitone - semitone.<br />

<br />

In Armodue the tenth of sixteen eka can be divided into 2, 4, or 8 parts.<br />

<br />

But apart from these subdivisions of the tenth, the modal system of Messiaen can be extended to all intervals with sizes between 3 and 8 eka.<br />

<br />

XXX<br />

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<h1 id="toc21"><a name="x&quot;Geometric&quot; harmonic constructions with Armodue"></a>&quot;Geometric&quot; harmonic constructions with Armodue</h1>

<br />

The composer Scriabin has based much of his compositions on nucleopolar accordances.<br />

<br />

To understand the nature of such accordances we should refer to specular reflections of light in a crystal, according to a tidy symmetry.<br />

<br />

XXX<br />

<br />

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<h1 id="toc22"><a name="x&quot;Elastic&quot; chords"></a>&quot;Elastic&quot; chords</h1>

<br />

~~XXX~~</body></html></pre></div>

With &quot;elastic chords&quot; I mean all those chords constructed in a ways that all parts are equally spaced, except one or two intervals that are variable.<br />

<br />

An example will illustrate the principle better. If you construct a chord putting six intervals on top of each other each of which is 7 eka in size, you get an equally-spaced chord composed of the notes:<br />

<br />

1 - 5 - 8# - 3# - 7# - 2# - 6#<br />

<br />

If you now provide that one of the intervals does not have to be necessarily 7 eka but may vary at will, you get what I call an &quot;elastic&quot; chord: that is a chord where all the &quot;rings&quot; (intervals) are rigid (the equidistant intervals) except for one elastic &quot;ring&quot; likely to be shortened or extended.<br />

<br />

In the example chord, if you choose the second interval to be the one with the elastic propertiy, the original chord can vary and become for example:<br />

<br />

1 - 5 - 8 - 3 - 7 - 2 - 6<br />

(shortening the original interval 5-8# to 6 eka instead of 7) or<br />

<br />

1 - 5 - 1# - 5# - 9 - 4 - 8<br />

(extending the original interval 5-8# to 10 eka instead of 7).<br />

<br />

There arise many effective progressions of chords varied by the selection of one or two &quot;elastic&quot; intervals.</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-09-10 04:38:22 UTC</tt>.<br>
+: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-09-10 05:34:58 UTC</tt>.<br>
-: The original revision id was <tt>161610451</tt>.<br>
+: The original revision id was <tt>161614925</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>
@@ Line 12: / Line 12: @@
 [[toc|flat]]
 ----
 Chapter 1:
@@ Line 254: / Line 253: @@
 "Modes of limited transposition" incorporates and expands - applying it in Armodue - the modal system designed by Olivier Messiaen ("Modes à transpositions limitées").
+The basic idea of Messiaen is to divide the octave equally in two, three, four or six equal intervals (two tritones, three major thirds , four minors thirds or six wholetones) and then structure these intervals with further internal division in groups or modules (inserting notes in a specific order).
+The particularity is to make the last note of one module coincide with the first note of the next group.
+For example, dividing the octave in three equal parts: c - e; e - ab; as - c, Messiaen structures his third mode by establishing the form: tone - semitone - tone and thus the succession of notes:
+**c** - d - eb - **e** - f# - g - **ab** - bb - b - **c**
+The notes marked in bold c, e, ab delimit the three parts that make up the octave and constitute at the same time the first and last note of each module of the type tone - semitone - semitone.
+In Armodue the tenth of sixteen eka can be divided into 2, 4, or 8 parts.
+But apart from these subdivisions of the tenth, the modal system of Messiaen can be extended to all intervals with sizes between 3 and 8 eka.
 XXX
@@ Line 261: / Line 274: @@
 Chapter 4:
 ="Geometric" harmonic constructions with Armodue=
+The composer Scriabin has based much of his compositions on nucleopolar accordances.
+To understand the nature of such accordances we should refer to specular reflections of light in a crystal, according to a tidy symmetry.
 XXX
@@ Line 269: / Line 286: @@
 ="Elastic" chords=
-XXX</pre></div>
+With "elastic chords" I mean all those chords constructed in a ways that all parts are equally spaced, except one or two intervals that are variable.
+An example will illustrate the principle better. If you construct a chord putting six intervals on top of each other each of which is 7 eka in size, you get an equally-spaced chord composed of the notes:
+- 5 - 8# - 3# - 7# - 2# - 6#
+If you now provide that one of the intervals does not have to be necessarily 7 eka but may vary at will, you get what I call an "elastic" chord: that is a chord where all the "rings" (intervals) are rigid (the equidistant intervals) except for one elastic "ring" likely to be shortened or extended.
+In the example chord, if you choose the second interval to be the one with the elastic propertiy, the original chord can vary and become for example:
+- 5 - 8 - 3 - 7 - 2 - 6
+(shortening the original interval 5-8# to 6 eka instead of 7) or
+- 5 - 1# - 5# - 9 - 4 - 8
+(extending the original interval 5-8# to 10 eka instead of 7).
+There arise many effective progressions of chords varied by the selection of one or two "elastic" intervals.</pre></div>
 <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;Armodue armonia&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="font-size: 140%;"&gt;Armodue: basic elements of harmony&lt;/span&gt;&lt;br /&gt;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextTocRule:46:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;a href="#Two theses supporting the system"&gt;Two theses supporting the system&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt; | &lt;a href="#The interval table"&gt;The interval table&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt;&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt; | &lt;a href="#Creating scales with Armodue: modal systems"&gt;Creating scales with Armodue: modal systems&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt;&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt;&lt;!-- ws:end:WikiTextTocRule:64 --&gt;&lt;!-- ws:start:WikiTextTocRule:65: --&gt;&lt;!-- ws:end:WikiTextTocRule:65 --&gt;&lt;!-- ws:start:WikiTextTocRule:66: --&gt;&lt;!-- ws:end:WikiTextTocRule:66 --&gt;&lt;!-- ws:start:WikiTextTocRule:67: --&gt;&lt;!-- ws:end:WikiTextTocRule:67 --&gt;&lt;!-- ws:start:WikiTextTocRule:68: --&gt; | &lt;a href="#x&amp;quot;Geometric&amp;quot; harmonic constructions with Armodue"&gt;&amp;quot;Geometric&amp;quot; harmonic constructions with Armodue&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:68 --&gt;&lt;!-- ws:start:WikiTextTocRule:69: --&gt; | &lt;a href="#x&amp;quot;Elastic&amp;quot; chords"&gt;&amp;quot;Elastic&amp;quot; chords&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:69 --&gt;&lt;!-- ws:start:WikiTextTocRule:70: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:70 --&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:70 --&gt;&lt;hr /&gt;
-&lt;hr /&gt;
 Chapter 1: &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Two theses supporting the system"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Two theses supporting the system&lt;/h1&gt;
@@ Line 519: / Line 551: @@
   &lt;br /&gt;
 &amp;quot;Modes of limited transposition&amp;quot; incorporates and expands - applying it in Armodue - the modal system designed by Olivier Messiaen (&amp;quot;Modes à transpositions limitées&amp;quot;).&lt;br /&gt;
+&lt;br /&gt;
+The basic idea of Messiaen is to divide the octave equally in two, three, four or six equal intervals (two tritones, three major thirds , four minors thirds or six wholetones) and then structure these intervals with further internal division in groups or modules (inserting notes in a specific order).&lt;br /&gt;
+&lt;br /&gt;
+The particularity is to make the last note of one module coincide with the first note of the next group.&lt;br /&gt;
+&lt;br /&gt;
+For example, dividing the octave in three equal parts: c - e; e - ab; as - c, Messiaen structures his third mode by establishing the form: tone - semitone - tone and thus the succession of notes:&lt;br /&gt;
+&lt;br /&gt;
+&lt;strong&gt;c&lt;/strong&gt; - d - eb - &lt;strong&gt;e&lt;/strong&gt; - f# - g - &lt;strong&gt;ab&lt;/strong&gt; - bb - b - &lt;strong&gt;c&lt;/strong&gt;&lt;br /&gt;
+&lt;br /&gt;
+The notes marked in bold c, e, ab delimit the three parts that make up the octave and constitute at the same time the first and last note of each module of the type tone - semitone - semitone.&lt;br /&gt;
+&lt;br /&gt;
+In Armodue the tenth of sixteen eka can be divided into 2, 4, or 8 parts.&lt;br /&gt;
+&lt;br /&gt;
+But apart from these subdivisions of the tenth, the modal system of Messiaen can be extended to all intervals with sizes between 3 and 8 eka.&lt;br /&gt;
 &lt;br /&gt;
 XXX&lt;br /&gt;
@@ Line 527: / Line 573: @@
 &lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc21"&gt;&lt;a name="x&amp;quot;Geometric&amp;quot; harmonic constructions with Armodue"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;&amp;quot;Geometric&amp;quot; harmonic constructions with Armodue&lt;/h1&gt;
   &lt;br /&gt;
+The composer Scriabin has based much of his compositions on nucleopolar accordances.&lt;br /&gt;
+&lt;br /&gt;
+To understand the nature of such accordances we should refer to specular reflections of light in a crystal, according to a tidy symmetry.&lt;br /&gt;
+&lt;br /&gt;
 XXX&lt;br /&gt;
 &lt;br /&gt;
@@ Line 534: / Line 584: @@
 &lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc22"&gt;&lt;a name="x&amp;quot;Elastic&amp;quot; chords"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;&amp;quot;Elastic&amp;quot; chords&lt;/h1&gt;
   &lt;br /&gt;
-XXX&lt;/body&gt;&lt;/html&gt;</pre></div>
+With &amp;quot;elastic chords&amp;quot; I mean all those chords constructed in a ways that all parts are equally spaced, except one or two intervals that are variable.&lt;br /&gt;
+&lt;br /&gt;
+An example will illustrate the principle better. If you construct a chord putting six intervals on top of each other each of which is 7 eka in size, you get an equally-spaced chord composed of the notes:&lt;br /&gt;
+&lt;br /&gt;
+- 5 - 8# - 3# - 7# - 2# - 6#&lt;br /&gt;
+&lt;br /&gt;
+If you now provide that one of the intervals does not have to be necessarily 7 eka but may vary at will, you get what I call an &amp;quot;elastic&amp;quot; chord: that is a chord where all the &amp;quot;rings&amp;quot; (intervals) are rigid (the equidistant intervals) except for one elastic &amp;quot;ring&amp;quot; likely to be shortened or extended.&lt;br /&gt;
+&lt;br /&gt;
+In the example chord, if you choose the second interval to be the one with the elastic propertiy, the original chord can vary and become for example:&lt;br /&gt;
+&lt;br /&gt;
+- 5 - 8 - 3 - 7 - 2 - 6&lt;br /&gt;
+(shortening the original interval 5-8# to 6 eka instead of 7) or&lt;br /&gt;
+&lt;br /&gt;
+- 5 - 1# - 5# - 9 - 4 - 8&lt;br /&gt;
+(extending the original interval 5-8# to 10 eka instead of 7).&lt;br /&gt;
+&lt;br /&gt;
+There arise many effective progressions of chords varied by the selection of one or two &amp;quot;elastic&amp;quot; intervals.&lt;/body&gt;&lt;/html&gt;</pre></div>