Armodue harmony: Difference between revisions
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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:hstraub|hstraub]] and made on <tt>2010-09-10 04: | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-09-10 04:38:22 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>161610451</tt>.<br> | ||
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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="font-size: 140%;">Armodue: basic elements of harmony</span> | ||
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This is a translation of an article by Luca Attanasio. Original page in italian: [[http://www.armodue.com/armonia.htm]] | This is a translation of an article by Luca Attanasio. Original page in italian: [[http://www.armodue.com/armonia.htm]] | ||
//Note: This is a preliminary tranlsation. Parts that are still "under construction" are marked with "XXX".// | //Note: This is a preliminary tranlsation. Parts that are still "under construction" are marked with "XXX".// | ||
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==Other modal systems and various considerations on the scales in Armodue== | ==Other modal systems and various considerations on the scales in Armodue== | ||
The combination possibilities of trichords, tetrachorda, pentachords, hexachords and heptachords sum up to a very large number: at least in theory, different types of intervals (5, 6, 7, 8, 9, 10 or more eka) can function as pivotal intervals whose interior is organized in trichords, tetrachords etc. - in turn freely combined to form scales. | |||
Listing all the possibilities is beyond the scope of this brief treatise. At least in the initial phase, however, it will be advisable to use few types of scales, maybe with frequent tonal modulations (transposing the limited number of chosen scales to different tonics). | |||
The listener should acquire some familiarity with few well-chosen tetrachords, pentachords etc. rather than getting disoriented by a continuous change of intervallic structures. | |||
==A peculiarity of Armodue: the symmetrical "neo-diminished" scales== | |||
A very particular sound in the Armodue environment is achieved by the symmetrical scale with the structure 1 + 3 + 1 + 3 + 1 + 3 + 1 + 3 eka or its variant 3 + 1 + 3 + 1 + 3 + 1 + 3 + 1 eka. | |||
These two scales strongly evoke the sound of the "diminished" scales of the tempered system with their alternating tone-semitone pattern (widely used in jazz and blues): c - c# - d# - e - f# g - a - a#- - c (symmetrical scale of type semitone-tone) or c - d - eb - f - gb - g# - a - h - c (symmetrical scale of type tone-semitone). In Armodue, these scales are named for their special quality "neo-diminished" scales. | |||
==Modes of limited transposition== | |||
"Modes of limited transposition" incorporates and expands - applying it in Armodue - the modal system designed by Olivier Messiaen ("Modes à transpositions limitées"). | |||
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XXX</pre></div> | XXX</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>Armodue armonia</title></head><body>< | <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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This is a translation of an article by Luca Attanasio. Original page in italian: <a class="wiki_link_ext" href="http://www.armodue.com/armonia.htm" rel="nofollow">http://www.armodue.com/armonia.htm</a><br /> | This is a translation of an article by Luca Attanasio. Original page in italian: <a class="wiki_link_ext" href="http://www.armodue.com/armonia.htm" rel="nofollow">http://www.armodue.com/armonia.htm</a><br /> | ||
<em>Note: This is a preliminary tranlsation. Parts that are still &quot;under construction&quot; are marked with &quot;XXX&quot;.</em><br /> | <em>Note: This is a preliminary tranlsation. Parts that are still &quot;under construction&quot; are marked with &quot;XXX&quot;.</em><br /> | ||
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<!-- ws:start:WikiTextTocRule:46:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><a href="#Two theses supporting the system">Two theses supporting the system</a><!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --> | <a href="#The interval table">The interval table</a><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --> | <a href="#Creating scales with Armodue: modal systems">Creating scales with Armodue: modal systems</a><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --> | <a href="#x&quot;Geometric&quot; harmonic constructions with Armodue">&quot;Geometric&quot; harmonic constructions with Armodue</a><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --> | <a href="#x&quot;Elastic&quot; chords">&quot;Elastic&quot; chords</a><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --> | |||
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<hr /> | <hr /> | ||
Chapter 1: <br /> | Chapter 1: <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Two theses supporting the system"></a><!-- ws:end:WikiTextHeadingRule:0 -->Two theses supporting the system</h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Two theses supporting the system-The supremacy of the fifth and and the seventh harmonic in Armodue"></a><!-- ws:end:WikiTextHeadingRule:2 -->The supremacy of the fifth and and the seventh harmonic in Armodue</h2> | ||
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The twelve note system that has been ruling for several centuries is based on the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth) with the first harmonic or fundamental. Also, the pythagorean tradion - the cicle of fifths - is based on the perfect fifth and hence on the same frequency ratio 3:2.<br /> | The twelve note system that has been ruling for several centuries is based on the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth) with the first harmonic or fundamental. Also, the pythagorean tradion - the cicle of fifths - is based on the perfect fifth and hence on the same frequency ratio 3:2.<br /> | ||
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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 seventh harmonic.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Two theses supporting the system-The triple mean of the double diagonal / side of the square"></a><!-- ws:end:WikiTextHeadingRule:4 -->The triple mean of the double diagonal / side of the square</h2> | ||
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From a philosophical point of view, the system of twelve notes was justified in the past by this mathematical property: the arithmetic mean and the harmonic mean of the octave (interval ratio 2:1) correspond to the perfect fifth (ratio 3:2) and the perfect fourth (ratio 4:3), while the geometric mean divides the octave exatcly into two tritone intervals (ratio: square root of 2).<br /> | From a philosophical point of view, the system of twelve notes was justified in the past by this mathematical property: the arithmetic mean and the harmonic mean of the octave (interval ratio 2:1) correspond to the perfect fifth (ratio 3:2) and the perfect fourth (ratio 4:3), while the geometric mean divides the octave exatcly into two tritone intervals (ratio: square root of 2).<br /> | ||
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<hr /> | <hr /> | ||
Chapter 2: <br /> | Chapter 2: <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="The interval table"></a><!-- ws:end:WikiTextHeadingRule:6 -->The interval table</h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="The interval table-Qualitative categories of intervals"></a><!-- ws:end:WikiTextHeadingRule:8 -->Qualitative categories of intervals</h2> | ||
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Armodue consists of sixteen types of intervals, which can be grouped two by two (by complementarity: each two intervals are the reverse of the other and add up to the tenth (interval sum of 16 eka) in eight categories that will be to be analysed individually:<br /> | Armodue consists of sixteen types of intervals, which can be grouped two by two (by complementarity: each two intervals are the reverse of the other and add up to the tenth (interval sum of 16 eka) in eight categories that will be to be analysed individually:<br /> | ||
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8 eka<br /> | 8 eka<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="The interval table-The intervals of 1 eka and 15 eka"></a><!-- ws:end:WikiTextHeadingRule:10 -->The intervals of 1 eka and 15 eka</h2> | ||
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The interval of one eka, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents.<br /> | The interval of one eka, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents.<br /> | ||
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In Armodue the intervals of 1 and 15 eka eka are considered harsh dissonances and as such should be used with caution in chords. However, all rules may be applied that already govern the treatment of harsh dissonances in the dodecatonic system.<br /> | In Armodue the intervals of 1 and 15 eka eka are considered harsh dissonances and as such should be used with caution in chords. However, all rules may be applied that already govern the treatment of harsh dissonances in the dodecatonic system.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="The interval table-The intervals of 2 eka and 14 eka"></a><!-- ws:end:WikiTextHeadingRule:12 -->The intervals of 2 eka and 14 eka</h2> | ||
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The interval of 2 eka corresponds to 3/2 of a tempered semitone (quantifies in 150 cents) and is found exactly between the eleventh and twelfth harmonic ov ther overtone series. It is the interval that is obtained by dividing the tenth of Armodue (the tempered octave) into eight equal parts, to form the <a class="wiki_link" href="/8edo">8-equal tempered</a> scale. Since only harmonics of higher number come close to it, it sounds particularly unnatural to the ear. This makes it suitable for geometric constructions where symmetry and artificialness prevail, for example in fractal sound structures.<br /> | The interval of 2 eka corresponds to 3/2 of a tempered semitone (quantifies in 150 cents) and is found exactly between the eleventh and twelfth harmonic ov ther overtone series. It is the interval that is obtained by dividing the tenth of Armodue (the tempered octave) into eight equal parts, to form the <a class="wiki_link" href="/8edo">8-equal tempered</a> scale. Since only harmonics of higher number come close to it, it sounds particularly unnatural to the ear. This makes it suitable for geometric constructions where symmetry and artificialness prevail, for example in fractal sound structures.<br /> | ||
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The interval of 2 eka and its complement of 14 eka are defined as neutral dissonances of Armodue.<br /> | The interval of 2 eka and its complement of 14 eka are defined as neutral dissonances of Armodue.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="The interval table-The intervals of 3 eka and 13 eka"></a><!-- ws:end:WikiTextHeadingRule:14 -->The intervals of 3 eka and 13 eka</h2> | ||
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The interval of 3 eka corresponds to the &quot;wholetone&quot; of Armodue (it is slightly wider than the tempered wholetone). This interval is particularly pleasing to the ear because it is very close to the natural tone that is formed with the seventh and the eighth harmonic (the tempered wholetone, by comparison, sounds less natural to the ear because it is formed with higher harmonics: the ninth and tenth). If you build scales using successions of the &quot;wholetone&quot; of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor.<br /> | The interval of 3 eka corresponds to the &quot;wholetone&quot; of Armodue (it is slightly wider than the tempered wholetone). This interval is particularly pleasing to the ear because it is very close to the natural tone that is formed with the seventh and the eighth harmonic (the tempered wholetone, by comparison, sounds less natural to the ear because it is formed with higher harmonics: the ninth and tenth). If you build scales using successions of the &quot;wholetone&quot; of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor.<br /> | ||
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The intervals of 3 and 13 eka are among the most suggestive intervals in Armodue and should be classified as sweet dissonances, as the tempered major second and minor seventh.<br /> | The intervals of 3 and 13 eka are among the most suggestive intervals in Armodue and should be classified as sweet dissonances, as the tempered major second and minor seventh.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="The interval table-The intervals of 4 eka and 12 eka"></a><!-- ws:end:WikiTextHeadingRule:16 -->The intervals of 4 eka and 12 eka</h2> | ||
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With the intervals of 4 and 12 eka we have two intervals that are very popular and familiar to the ear translated into Armodue. Indeed 4 eka and 12 eka correspond exactly to the minor third and the major sixth in the dodecatonic system. Therefore, there is the evident possibility of evoking major and minor triads with Armodue (the minor triad is created in stacking 4 eka and 5 eka on a base tone, the major triad stacking 5 eka and 4 eka). The perfect equivalence of the two considered intervals in the dodecatonic system and in Armodue is a crucial point in the inevitable interaction that the ear of a western listener will establish between the two different tunings. In fact, listening to the intervals of 4 and 12 eka, the ear will immediately associate these Armodue intervals to two already familiar ones (the minor third and the major sixth). For this reason, many of the other intervals present - in an Armodue environment - in a context to those of 4 and 12 eka are likely to be felt by the ear as abnormal and unknown. The composers will give much attention every time they use one of these two intervals, trying to predict the reactions of an ear used to the dodecaphonic system.<br /> | With the intervals of 4 and 12 eka we have two intervals that are very popular and familiar to the ear translated into Armodue. Indeed 4 eka and 12 eka correspond exactly to the minor third and the major sixth in the dodecatonic system. Therefore, there is the evident possibility of evoking major and minor triads with Armodue (the minor triad is created in stacking 4 eka and 5 eka on a base tone, the major triad stacking 5 eka and 4 eka). The perfect equivalence of the two considered intervals in the dodecatonic system and in Armodue is a crucial point in the inevitable interaction that the ear of a western listener will establish between the two different tunings. In fact, listening to the intervals of 4 and 12 eka, the ear will immediately associate these Armodue intervals to two already familiar ones (the minor third and the major sixth). For this reason, many of the other intervals present - in an Armodue environment - in a context to those of 4 and 12 eka are likely to be felt by the ear as abnormal and unknown. The composers will give much attention every time they use one of these two intervals, trying to predict the reactions of an ear used to the dodecaphonic system.<br /> | ||
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The intervals of 4 and 12 eka belong without doubt - being of equivalent size - to the same category that includes the minor third and the major sixth: that of sweet consonances.<br /> | The intervals of 4 and 12 eka belong without doubt - being of equivalent size - to the same category that includes the minor third and the major sixth: that of sweet consonances.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="The interval table-The intervals of 5 eka and 11 eka"></a><!-- ws:end:WikiTextHeadingRule:18 -->The intervals of 5 eka and 11 eka</h2> | ||
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The interval of 5 eka is very close to the natural major third that appears as the ratio between the fourth and fifth harmonic of the overtone series. The major third of the dodecatonic system is not so close to the natural major third as the interval of 5 eka.<br /> | The interval of 5 eka is very close to the natural major third that appears as the ratio between the fourth and fifth harmonic of the overtone series. The major third of the dodecatonic system is not so close to the natural major third as the interval of 5 eka.<br /> | ||
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The complement of 5 eka is the interval of 11 eka, very close to the natural minor sixth that occurs between the fifth and eighth harmonic. Since the intervals of 5 and 11 eka are associated to the tempered third and sixth, the same considerations hold that were made in the previous paragraph about the intervals of 4 and 12 eka. They too are classified as sweet consonances.<br /> | The complement of 5 eka is the interval of 11 eka, very close to the natural minor sixth that occurs between the fifth and eighth harmonic. Since the intervals of 5 and 11 eka are associated to the tempered third and sixth, the same considerations hold that were made in the previous paragraph about the intervals of 4 and 12 eka. They too are classified as sweet consonances.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="The interval table-The intervals of 6 eka and 10 eka"></a><!-- ws:end:WikiTextHeadingRule:20 -->The intervals of 6 eka and 10 eka</h2> | ||
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The interval of 6 eka is particularly striking, with its complement of 10 eka. 6 eka is located exactly at the point of equidistance between the major third and the perfect fourth of the tempered system, while 12 eka is the average between a perfect fifth and a minor sixth.<br /> | The interval of 6 eka is particularly striking, with its complement of 10 eka. 6 eka is located exactly at the point of equidistance between the major third and the perfect fourth of the tempered system, while 12 eka is the average between a perfect fifth and a minor sixth.<br /> | ||
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The intervals of 6 and 10 eka are classified as neutral consonances.<br /> | The intervals of 6 and 10 eka are classified as neutral consonances.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="The interval table-The intervals of 7 eka and 9 eka"></a><!-- ws:end:WikiTextHeadingRule:22 -->The intervals of 7 eka and 9 eka</h2> | ||
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The intervals corresponding to the perfect fourth and and the perfect fifth in Armodue are the intervals of 7 and 9 eka, the first one quantifies in a slightly shaprened fourth, the second in a slightly flattened fifth.<br /> | The intervals corresponding to the perfect fourth and and the perfect fifth in Armodue are the intervals of 7 and 9 eka, the first one quantifies in a slightly shaprened fourth, the second in a slightly flattened fifth.<br /> | ||
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As for the classification, 7 and 9 eka belong to the open consonances.<br /> | As for the classification, 7 and 9 eka belong to the open consonances.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="The interval table-The interval of 8 eka"></a><!-- ws:end:WikiTextHeadingRule:24 -->The interval of 8 eka</h2> | ||
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The interval of 8 eka divides the Armodue tenth in half, just as the tritone halves the tempered octave. Indeed 8 eka correspond exactly to three wholetones, hence to an augmented fourth or diminished fifth.<br /> | The interval of 8 eka divides the Armodue tenth in half, just as the tritone halves the tempered octave. Indeed 8 eka correspond exactly to three wholetones, hence to an augmented fourth or diminished fifth.<br /> | ||
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The interval of 8 eka is the first axis of symmetry in Armodue: indeed in this system we proceed with progressive splittings of the tenth: 16 eka, 8 eka, 4 eka, 2 eka and 1 eka. As the tritone of the tempered system, the interval of 8 eka is an unstable dissonance.<br /> | The interval of 8 eka is the first axis of symmetry in Armodue: indeed in this system we proceed with progressive splittings of the tenth: 16 eka, 8 eka, 4 eka, 2 eka and 1 eka. As the tritone of the tempered system, the interval of 8 eka is an unstable dissonance.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="The interval table-Gradation of harmonic tensions"></a><!-- ws:end:WikiTextHeadingRule:26 -->Gradation of harmonic tensions</h2> | ||
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When a chord is followed by another chord, we must carefully evaluate the consonance/dissonance quality of every interval contained - in addition to dealing with the fluidity of the movement of the voices. If different chords must be put into a sequence, we recommend to let them proceed so that there is a gradation in the distribution of the interval tensions.<br /> | When a chord is followed by another chord, we must carefully evaluate the consonance/dissonance quality of every interval contained - in addition to dealing with the fluidity of the movement of the voices. If different chords must be put into a sequence, we recommend to let them proceed so that there is a gradation in the distribution of the interval tensions.<br /> | ||
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If, for example, a progression is to be built containing a chord &quot;A&quot;, where sweet dissonances prevail, a chord &quot;B&quot;, where open consonances prevail, and a &quot;C&quot;, with prevalence of neutral consonances, possible progressions are B - C - A (order of growing tension) or A - C - B (order of decreasing tension).<br /> | If, for example, a progression is to be built containing a chord &quot;A&quot;, where sweet dissonances prevail, a chord &quot;B&quot;, where open consonances prevail, and a &quot;C&quot;, with prevalence of neutral consonances, possible progressions are B - C - A (order of growing tension) or A - C - B (order of decreasing tension).<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="The interval table-Arrangement of tones in the chords"></a><!-- ws:end:WikiTextHeadingRule:28 -->Arrangement of tones in the chords</h2> | ||
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Chords can be realized in any way, placing the notes that constitute it more or less distributed (narrow ore wide) on a higher or a lower register (more or less clarity), in pyramid form or pyramid upside down ( more or less harmonic force) or in another form, with doubles, repetitions and omissions of one or more parties (emphasis on the intervals involved in the doubled or tripled notes).<br /> | Chords can be realized in any way, placing the notes that constitute it more or less distributed (narrow ore wide) on a higher or a lower register (more or less clarity), in pyramid form or pyramid upside down ( more or less harmonic force) or in another form, with doubles, repetitions and omissions of one or more parties (emphasis on the intervals involved in the doubled or tripled notes).<br /> | ||
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Chapter 3: <br /> | Chapter 3: <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc15"><a name="Creating scales with Armodue: modal systems"></a><!-- ws:end:WikiTextHeadingRule:30 -->Creating scales with Armodue: modal systems</h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Creating scales with Armodue: modal systems-Modal systems based on tetrachords and pentachords"></a><!-- ws:end:WikiTextHeadingRule:32 -->Modal systems based on tetrachords and pentachords</h2> | ||
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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 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 /> | ||
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To build scales with full awareness, we must be familiar with the listed 23 types of tetrachords and pentachords and thoroughly study their sound and quality. Only when we have fully mastered the structures at the base of the scales - precisely the tetrachords and pentachords - we can proceed to their mutual combination in the formation of scales.<br /> | To build scales with full awareness, we must be familiar with the listed 23 types of tetrachords and pentachords and thoroughly study their sound and quality. Only when we have fully mastered the structures at the base of the scales - precisely the tetrachords and pentachords - we can proceed to their mutual combination in the formation of scales.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Creating scales with Armodue: modal systems-Modal systems based on hexachords"></a><!-- ws:end:WikiTextHeadingRule:34 -->Modal systems based on hexachords</h2> | ||
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The tetrachords and pentachords treated so far are focused on the characteristic interval of 7eka, the Armodue equivalent of the diatessaron in the greek modal system (the perfect fourth, five tempered semitones wide).<br /> | The tetrachords and pentachords treated so far are focused on the characteristic interval of 7eka, the Armodue equivalent of the diatessaron in the greek modal system (the perfect fourth, five tempered semitones wide).<br /> | ||
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For example, taking the formula 3, 1, 3, 3, 3 we get a scale formed by the notes: 1, # 2, 3, 5, 6 #, 8 to which we can add the note 9 as leading tone.<br /> | For example, taking the formula 3, 1, 3, 3, 3 we get a scale formed by the notes: 1, # 2, 3, 5, 6 #, 8 to which we can add the note 9 as leading tone.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Creating scales with Armodue: modal systems-Other modal systems and various considerations on the scales in Armodue"></a><!-- ws:end:WikiTextHeadingRule:36 -->Other modal systems and various considerations on the scales in Armodue</h2> | ||
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The combination possibilities of trichords, tetrachorda, pentachords, hexachords and heptachords sum up to a very large number: at least in theory, different types of intervals (5, 6, 7, 8, 9, 10 or more eka) can function as pivotal intervals whose interior is organized in trichords, tetrachords etc. - in turn freely combined to form scales.<br /> | |||
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Listing all the possibilities is beyond the scope of this brief treatise. At least in the initial phase, however, it will be advisable to use few types of scales, maybe with frequent tonal modulations (transposing the limited number of chosen scales to different tonics).<br /> | |||
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The listener should acquire some familiarity with few well-chosen tetrachords, pentachords etc. rather than getting disoriented by a continuous change of intervallic structures.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Creating scales with Armodue: modal systems-A peculiarity of Armodue: the symmetrical &quot;neo-diminished&quot; scales"></a><!-- ws:end:WikiTextHeadingRule:38 -->A peculiarity of Armodue: the symmetrical &quot;neo-diminished&quot; scales</h2> | |||
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A very particular sound in the Armodue environment is achieved by the symmetrical scale with the structure 1 + 3 + 1 + 3 + 1 + 3 + 1 + 3 eka or its variant 3 + 1 + 3 + 1 + 3 + 1 + 3 + 1 eka.<br /> | |||
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These two scales strongly evoke the sound of the &quot;diminished&quot; scales of the tempered system with their alternating tone-semitone pattern (widely used in jazz and blues): c - c# - d# - e - f# g - a - a#- - c (symmetrical scale of type semitone-tone) or c - d - eb - f - gb - g# - a - h - c (symmetrical scale of type tone-semitone). In Armodue, these scales are named for their special quality &quot;neo-diminished&quot; scales.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Creating scales with Armodue: modal systems-Modes of limited transposition"></a><!-- ws:end:WikiTextHeadingRule:40 -->Modes of limited transposition</h2> | |||
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&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 /> | |||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc22"><a name="x&quot;Elastic&quot; chords"></a><!-- ws:end:WikiTextHeadingRule:44 -->&quot;Elastic&quot; chords</h1> | ||
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