Armodue harmony: Difference between revisions
Wikispaces>hstraub **Imported revision 161375309 - 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:hstraub|hstraub]] and made on <tt>2010-09- | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-09-10 03:15:27 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>161604825</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> | ||
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---- | ---- | ||
Chapter 1: | |||
=Two theses supporting the system= | |||
==The supremacy of the fifth and and the seventh harmonic in Armodue== | ==The supremacy of the fifth and and the seventh harmonic in Armodue== | ||
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---- | |||
Chapter 2: | |||
=The interval table= | |||
==Qualitative categories of intervals== | ==Qualitative categories of intervals== | ||
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==Gradation of harmonic tensions== | ==Gradation of harmonic tensions== | ||
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. | |||
The progression order (forward or backwards) will be as follows: | |||
Open consonances - neutral consonances - sweet consonances - sweet dissonances - neutral dissonances - harsh dissonances - unstable dissonances. | |||
If, for example, a progression is to be built containing a chord "A", where sweet dissonances prevail, a chord "B", where open consonances prevail, and a "C", with prevalence of neutral consonances, possible progressions are B - C - A (order of growing tension) or A - C - B (order of decreasing tension). | |||
==Arrangement of tones in the chords== | |||
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). | |||
Compound intervals (i.e. one or tenths added to the the simple interval), in chords with more widely distributed notes, come out clearer and more resonant than the corresponding simple (octavr-reduced) intervals in the case of sweet or neutral consonances, more powerful in the case of open consonances - while the intervals of dissonant character, when appearing as compound intervals, lose much of their sharp character and gain brilliance. | |||
When in a chord the intervals between adjacent notes decrease from the bass to the melody note, we have the pyramid form, which increases the clarity and resonance of the chord (since the ear associates this to the series of the harmonics); the contrary happens stacking the intervals in increasing size (upside down pyramid form). | |||
If we want to emphasize the dissonant quality of a chord, we can simply double or triple the notes that form the dissonant intervals, and we will proceed similarly with the notes that form consonances to increase the consonance of the chord. | |||
---- | |||
Chapter 3: | |||
=Creating scales with Armodue: modal systems= | |||
XXX | XXX | ||
---- | |||
Chapter 4: | |||
="Geometric" harmonic constructions with Armodue= | |||
XXX | XXX | ||
---- | |||
Chapter 5: | |||
="Elastic" chords= | |||
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><!-- ws:start:WikiTextTocRule: | <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><!-- ws:start:WikiTextTocRule:38:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Armodue: basic elements of harmony">Armodue: basic elements of harmony</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 1em;"><a href="#Two theses supporting the system">Two theses supporting the system</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 2em;"><a href="#Two theses supporting the system-The supremacy of the fifth and and the seventh harmonic in Armodue">The supremacy of the fifth and and the seventh harmonic in Armodue</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 2em;"><a href="#Two theses supporting the system-The triple mean of the double diagonal / side of the square">The triple mean of the double diagonal / side of the square</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 1em;"><a href="#The interval table">The interval table</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 2em;"><a href="#The interval table-Qualitative categories of intervals">Qualitative categories of intervals</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 1 eka and 15 eka">The intervals of 1 eka and 15 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 2 eka and 14 eka">The intervals of 2 eka and 14 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 3 eka and 13 eka">The intervals of 3 eka and 13 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 4 eka and 12 eka">The intervals of 4 eka and 12 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 5 eka and 11 eka">The intervals of 5 eka and 11 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 6 eka and 10 eka">The intervals of 6 eka and 10 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 2em;"><a href="#The interval table-The intervals of 7 eka and 9 eka">The intervals of 7 eka and 9 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 2em;"><a href="#The interval table-The interval of 8 eka">The interval of 8 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 2em;"><a href="#The interval table-Gradation of harmonic tensions">Gradation of harmonic tensions</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 2em;"><a href="#The interval table-Arrangement of tones in the chords">Arrangement of tones in the chords</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><div style="margin-left: 1em;"><a href="#Creating scales with Armodue: modal systems">Creating scales with Armodue: modal systems</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><div style="margin-left: 1em;"><a href="#x&quot;Geometric&quot; harmonic constructions with Armodue">&quot;Geometric&quot; harmonic constructions with Armodue</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 1em;"><a href="#x&quot;Elastic&quot; chords">&quot;Elastic&quot; chords</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --></div> | ||
<!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Armodue: basic elements of harmony"></a><!-- ws:end:WikiTextHeadingRule:0 --><strong>Armodue: basic elements of harmony</strong></h1> | |||
<br /> | <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 /> | 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 /> | ||
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<br /> | <br /> | ||
<hr /> | <hr /> | ||
<br /> | Chapter 1: <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name=" | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Two theses supporting the system"></a><!-- ws:end:WikiTextHeadingRule:2 -->Two theses supporting the system</h1> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name=" | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Two theses supporting the system-The supremacy of the fifth and and the seventh harmonic in Armodue"></a><!-- ws:end:WikiTextHeadingRule:4 -->The supremacy of the fifth and and the seventh harmonic in Armodue</h2> | ||
<br /> | <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 /> | 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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name=" | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Two theses supporting the system-The triple mean of the double diagonal / side of the square"></a><!-- ws:end:WikiTextHeadingRule:6 -->The triple mean of the double diagonal / side of the square</h2> | ||
<br /> | <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 /> | 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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<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name=" | <hr /> | ||
Chapter 2: <br /> | |||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="The interval table"></a><!-- ws:end:WikiTextHeadingRule:8 -->The interval table</h1> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name=" | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="The interval table-Qualitative categories of intervals"></a><!-- ws:end:WikiTextHeadingRule:10 -->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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name=" | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="The interval table-The intervals of 1 eka and 15 eka"></a><!-- ws:end:WikiTextHeadingRule:12 -->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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name=" | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="The interval table-The intervals of 2 eka and 14 eka"></a><!-- ws:end:WikiTextHeadingRule:14 -->The intervals of 2 eka and 14 eka</h2> | ||
<br /> | <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 /> | 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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name=" | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="The interval table-The intervals of 3 eka and 13 eka"></a><!-- ws:end:WikiTextHeadingRule:16 -->The intervals of 3 eka and 13 eka</h2> | ||
<br /> | <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 /> | 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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name=" | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="The interval table-The intervals of 4 eka and 12 eka"></a><!-- ws:end:WikiTextHeadingRule:18 -->The intervals of 4 eka and 12 eka</h2> | ||
<br /> | <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 /> | 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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name=" | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="The interval table-The intervals of 5 eka and 11 eka"></a><!-- ws:end:WikiTextHeadingRule:20 -->The intervals of 5 eka and 11 eka</h2> | ||
<br /> | <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 /> | 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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name=" | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="The interval table-The intervals of 6 eka and 10 eka"></a><!-- ws:end:WikiTextHeadingRule:22 -->The intervals of 6 eka and 10 eka</h2> | ||
<br /> | <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 /> | 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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name=" | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="The interval table-The intervals of 7 eka and 9 eka"></a><!-- ws:end:WikiTextHeadingRule:24 -->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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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:28:&lt;h2&gt; --><h2 id="toc14"><a name=" | <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="The interval table-Gradation of harmonic tensions"></a><!-- ws:end:WikiTextHeadingRule:28 -->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 /> | |||
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The progression order (forward or backwards) will be as follows:<br /> | |||
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Open consonances - neutral consonances - sweet consonances - sweet dissonances - neutral dissonances - harsh dissonances - unstable dissonances.<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 /> | |||
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<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="The interval table-Arrangement of tones in the chords"></a><!-- ws:end:WikiTextHeadingRule:30 -->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 /> | |||
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Compound intervals (i.e. one or tenths added to the the simple interval), in chords with more widely distributed notes, come out clearer and more resonant than the corresponding simple (octavr-reduced) intervals in the case of sweet or neutral consonances, more powerful in the case of open consonances - while the intervals of dissonant character, when appearing as compound intervals, lose much of their sharp character and gain brilliance.<br /> | |||
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When in a chord the intervals between adjacent notes decrease from the bass to the melody note, we have the pyramid form, which increases the clarity and resonance of the chord (since the ear associates this to the series of the harmonics); the contrary happens stacking the intervals in increasing size (upside down pyramid form).<br /> | |||
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If we want to emphasize the dissonant quality of a chord, we can simply double or triple the notes that form the dissonant intervals, and we will proceed similarly with the notes that form consonances to increase the consonance of the chord.<br /> | |||
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Chapter 3: <br /> | |||
<!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Creating scales with Armodue: modal systems"></a><!-- ws:end:WikiTextHeadingRule:32 -->Creating scales with Armodue: modal systems</h1> | |||
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XXX<br /> | XXX<br /> | ||
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Chapter 4: <br /> | |||
<!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="x&quot;Geometric&quot; harmonic constructions with Armodue"></a><!-- ws:end:WikiTextHeadingRule:34 -->&quot;Geometric&quot; harmonic constructions with Armodue</h1> | |||
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XXX<br /> | XXX<br /> | ||
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Chapter 5: <br /> | |||
<!-- ws:start:WikiTextHeadingRule:36:&lt;h1&gt; --><h1 id="toc18"><a name="x&quot;Elastic&quot; chords"></a><!-- ws:end:WikiTextHeadingRule:36 -->&quot;Elastic&quot; chords</h1> | |||
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XXX</body></html></pre></div> | XXX</body></html></pre></div> | ||