Armodue harmony: Difference between revisions

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>2010-09-13 08:31:04 UTC</tt>.<br>

: This revision was by author [[User:guest|guest]] and made on <tt>2010-09-14 13:39:03 UTC</tt>.<br>

: The original revision id was <tt>~~162156543~~</tt>.<br>

: The original revision id was <tt>162577475</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 24:

In the tempered system the fifth and the seventh harmonic appear as major third and minor seventh intervals formed to the fundamental (equating all intervals to their octave-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.

In Armodue, in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.

In Armodue (see [[16edo]], in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.

For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the seventh harmonic.

For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the [[seventh harmonic]].

==The triple mean of the double diagonal / side of the square==

Line 160:

==Modal systems based on tetrachords and pentachords==

There are many systems how to generate various scales in Armodue. The most important ones, however, are based on the formation of tetrachords or pentachords and their subsequent unions. By analogy with the tetrachords (modes) of the interval system that was already developed in ancient Greek music, the tenth of Armodue (the tempered octave) is divided into two intervals of seven eka each (seven eka have a width close to five semitones or a perfect fourth of the tempered system): a lower interval of seven eka and an upper interval of also seven eka.

There are many systems how to generate various scales in Armodue. The most important ones, however, are based on the formation of [[tetrachord|tetrachords]] or pentachords and their subsequent unions. By analogy with the tetrachords (modes) of the interval system that was already developed in ancient Greek music, the tenth of Armodue (the tempered octave) is divided into two intervals of seven eka each (seven eka have a width close to five semitones or a perfect fourth of the tempered system): a lower interval of seven eka and an upper interval of also seven eka.

The two intervals constituted that way will limit the tetrachords or pentachords that will be constituted inside of them and will be disjoint with a central interval of two eka between - to sum up to 16 eka or the entire scope of the tenth of Armodue.

Line 350:

In the tempered system the fifth and the seventh harmonic appear as major third and minor seventh intervals formed to the fundamental (equating all intervals to their octave-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.<br />

<br />

In Armodue, in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.<br />

In Armodue (see <a class="wiki_link" href="/16edo">16edo</a>, in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.<br />

<br />

For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the seventh harmonic.<br />

For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the <a class="wiki_link" href="/seventh%20harmonic">seventh harmonic</a>.<br />

<br />

<h2 id="toc2"><a name="Two theses supporting the system-The triple mean of the double diagonal / side of the square"></a>The triple mean of the double diagonal / side of the square</h2>

Line 486:

<h2 id="toc16"><a name="Creating scales with Armodue: modal systems-Modal systems based on tetrachords and pentachords"></a>Modal systems based on tetrachords and pentachords</h2>

<br />

There are many systems how to generate various scales in Armodue. The most important ones, however, are based on the formation of tetrachords or pentachords and their subsequent unions. By analogy with the tetrachords (modes) of the interval system that was already developed in ancient Greek music, the tenth of Armodue (the tempered octave) is divided into two intervals of seven eka each (seven eka have a width close to five semitones or a perfect fourth of the tempered system): a lower interval of seven eka and an upper interval of also seven eka.<br />

There are many systems how to generate various scales in Armodue. The most important ones, however, are based on the formation of <a class="wiki_link" href="/tetrachord">tetrachords</a> or pentachords and their subsequent unions. By analogy with the tetrachords (modes) of the interval system that was already developed in ancient Greek music, the tenth of Armodue (the tempered octave) is divided into two intervals of seven eka each (seven eka have a width close to five semitones or a perfect fourth of the tempered system): a lower interval of seven eka and an upper interval of also seven eka.<br />

<br />

The two intervals constituted that way will limit the tetrachords or pentachords that will be constituted inside of them and will be disjoint with a central interval of two eka between - to sum up to 16 eka or the entire scope of the tenth of Armodue.<br />

@@ 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>2010-09-13 08:31:04 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2010-09-14 13:39:03 UTC</tt>.<br>
-: The original revision id was <tt>162156543</tt>.<br>
+: The original revision id was <tt>162577475</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 24: / Line 24: @@
 In the tempered system the fifth and the seventh harmonic appear as major third and minor seventh intervals formed to the fundamental (equating all intervals to their octave-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.
-In Armodue, in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.
+In Armodue (see [[16edo]], in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.
-For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the seventh harmonic.
+For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the [[seventh harmonic]].
 ==The triple mean of the double diagonal / side of the square==
@@ Line 160: / Line 160: @@
 ==Modal systems based on tetrachords and pentachords==
-There are many systems how to generate various scales in Armodue. The most important ones, however, are based on the formation of tetrachords or pentachords and their subsequent unions. By analogy with the tetrachords (modes) of the interval system that was already developed in ancient Greek music, the tenth of Armodue (the tempered octave) is divided into two intervals of seven eka each (seven eka have a width close to five semitones or a perfect fourth of the tempered system): a lower interval of seven eka and an upper interval of also seven eka.
+There are many systems how to generate various scales in Armodue. The most important ones, however, are based on the formation of [[tetrachord|tetrachords]] or pentachords and their subsequent unions. By analogy with the tetrachords (modes) of the interval system that was already developed in ancient Greek music, the tenth of Armodue (the tempered octave) is divided into two intervals of seven eka each (seven eka have a width close to five semitones or a perfect fourth of the tempered system): a lower interval of seven eka and an upper interval of also seven eka.
 The two intervals constituted that way will limit the tetrachords or pentachords that will be constituted inside of them and will be disjoint with a central interval of two eka between - to sum up to 16 eka or the entire scope of the tenth of Armodue.
@@ Line 350: / Line 350: @@
 In the tempered system the fifth and the seventh harmonic appear as major third and minor seventh intervals formed to the fundamental (equating all intervals to their octave-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.&lt;br /&gt;
 &lt;br /&gt;
-In Armodue, in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.&lt;br /&gt;
+In Armodue (see &lt;a class="wiki_link" href="/16edo"&gt;16edo&lt;/a&gt;, in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.&lt;br /&gt;
 &lt;br /&gt;
-For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the seventh harmonic.&lt;br /&gt;
+For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the &lt;a class="wiki_link" href="/seventh%20harmonic"&gt;seventh harmonic&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Two theses supporting the system-The triple mean of the double diagonal / side of the square"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;The triple mean of the double diagonal / side of the square&lt;/h2&gt;
@@ Line 486: / Line 486: @@
 &lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Creating scales with Armodue: modal systems-Modal systems based on tetrachords and pentachords"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Modal systems based on tetrachords and pentachords&lt;/h2&gt;
   &lt;br /&gt;
-There are many systems how to generate various scales in Armodue. The most important ones, however, are based on the formation of tetrachords or pentachords and their subsequent unions. By analogy with the tetrachords (modes) of the interval system that was already developed in ancient Greek music, the tenth of Armodue (the tempered octave) is divided into two intervals of seven eka each (seven eka have a width close to five semitones or a perfect fourth of the tempered system): a lower interval of seven eka and an upper interval of also seven eka.&lt;br /&gt;
+There are many systems how to generate various scales in Armodue. The most important ones, however, are based on the formation of &lt;a class="wiki_link" href="/tetrachord"&gt;tetrachords&lt;/a&gt; or pentachords and their subsequent unions. By analogy with the tetrachords (modes) of the interval system that was already developed in ancient Greek music, the tenth of Armodue (the tempered octave) is divided into two intervals of seven eka each (seven eka have a width close to five semitones or a perfect fourth of the tempered system): a lower interval of seven eka and an upper interval of also seven eka.&lt;br /&gt;
 &lt;br /&gt;
 The two intervals constituted that way will limit the tetrachords or pentachords that will be constituted inside of them and will be disjoint with a central interval of two eka between - to sum up to 16 eka or the entire scope of the tenth of Armodue.&lt;br /&gt;