Armodue harmony: Difference between revisions
Wikispaces>hstraub **Imported revision 621663793 - 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>2017-11- | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2017-11-29 14:56:37 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>622731897</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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==The supremacy of the fifth and the seventh harmonic in Armodue== | ==The supremacy of the fifth and the seventh harmonic in Armodue== | ||
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 tradition - the cycle of fifths - is based on the perfect fifth and hence on the same frequency ratio 3:2. | 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 tradition - the cycle of fifths - is based on the perfect fifth and hence on the same frequency ratio 3:2. | ||
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==The triple mean of the double diagonal / side of the square== | ==The triple mean of the double diagonal / side of the square== | ||
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). | 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). | ||
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The arithmetic mean is exactly the interval of nine eka, the geometric mean exactly twelve eka and finally the harmonic mean exactly equal to fifteen eka. | The arithmetic mean is exactly the interval of nine eka, the geometric mean exactly twelve eka and finally the harmonic mean exactly equal to fifteen eka. | ||
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==Qualitative categories of intervals== | ==Qualitative categories of intervals== | ||
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: | 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: | ||
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==1 eka and 15 eka== | ==1 eka and 15 eka== | ||
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. | 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. | ||
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==2 eka and 14 eka== | ==2 eka and 14 eka== | ||
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 of the overtone series. It is the interval that is obtained by dividing the Tenth of Armodue (the classic Octave of 2/1) into eight equal parts, to form the [[8edo|8-equal tempered]] 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. | 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 of the overtone series. It is the interval that is obtained by dividing the Tenth of Armodue (the classic Octave of 2/1) into eight equal parts, to form the [[8edo|8-equal tempered]] 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. | ||
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==3 eka and 13 eka== | ==3 eka and 13 eka== | ||
The interval of 3 eka corresponds to the "wholetone" 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 eighth and ninth). If you build scales using successions of the "wholetone" of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor. | The interval of 3 eka corresponds to the "wholetone" 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 eighth and ninth). If you build scales using successions of the "wholetone" of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor. | ||
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==4 eka and 12 eka== | ==4 eka and 12 eka== | ||
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 of 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 Armodue is a crucial point in the inevitable interaction that the ear of a western listener will establish between the two different tempered systems. 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 dodecatonic system. | 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 of 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 Armodue is a crucial point in the inevitable interaction that the ear of a western listener will establish between the two different tempered systems. 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 dodecatonic system. | ||
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==5 eka and 11 eka== | ==5 eka and 11 eka== | ||
The interval of 5 eka is 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 (400 cents) is not so close to the natural major third (386,31 cents) as the interval of 5 eka (375 cents). | The interval of 5 eka is 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 (400 cents) is not so close to the natural major third (386,31 cents) as the interval of 5 eka (375 cents). | ||
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==6 eka and 10 eka== | ==6 eka and 10 eka== | ||
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 10 eka is the average between a perfect fifth and a minor sixth. | 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 10 eka is the average between a perfect fifth and a minor sixth. | ||
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==7 eka and 9 eka== | ==7 eka and 9 eka== | ||
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 sharpened fourth (525 cents), the second in a slightly flattened fifth (675 cents). | 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 sharpened fourth (525 cents), the second in a slightly flattened fifth (675 cents). | ||
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==8 eka== | ==8 eka== | ||
The interval of 8 eka divides the Tenth of Armodue in half, just as the tritone halves the typical octave. Indeed 8 eka correspond exactly to three wholetones, hence to an augmented fourth or diminished fifth. | The interval of 8 eka divides the Tenth of Armodue in half, just as the tritone halves the typical octave. Indeed 8 eka correspond exactly to three wholetones, hence to an augmented fourth or diminished fifth. | ||
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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. | 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. | ||
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==Arrangement of tones in the chords== | ==Arrangement of tones in the chords== | ||
Chords can be realized in any way, placing the notes that constitute it more or less distributed (narrow or 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). | Chords can be realized in any way, placing the notes that constitute it more or less distributed (narrow or 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). | ||
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==Modal systems based on tetrachords and pentachords== | ==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 [[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. | 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. | ||
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==Modal systems based on hexachords== | ==Modal systems based on hexachords== | ||
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). | 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). | ||
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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, tetrachords, 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. | The combination possibilities of trichords, tetrachords, 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. | ||
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==A peculiarity of Armodue: the symmetrical "neo-diminished" scales== | ==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 dodecatonic 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. (Actually, they are scales of [[Diminished|diminished temperament]], in 12edo as well as in 16edo!) | |||
|| [[media type="file" key="htgt12edo.mp3"]] || [[media type="file" key="htgt16edo.mp3"]] || | |||
|| Diminished ocatatonic scale in 12edo || Diminished octatonic scale in 16edo || | |||
==Modes of limited transposition== | ==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"). | "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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<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><span style="font-size: 140%;">Armodue: basic elements of harmony</span><br /> | <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 /> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:48:&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:48 --><!-- ws:start:WikiTextTocRule:49: --><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:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 2em;"><a href="#Two theses supporting the system-The supremacy of the fifth and the seventh harmonic in Armodue">The supremacy of the fifth and the seventh harmonic in Armodue</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><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:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 1em;"><a href="#The interval table">The interval table</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><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:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 2em;"><a href="#The interval table-1 eka and 15 eka">1 eka and 15 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><div style="margin-left: 2em;"><a href="#The interval table-2 eka and 14 eka">2 eka and 14 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><div style="margin-left: 2em;"><a href="#The interval table-3 eka and 13 eka">3 eka and 13 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 2em;"><a href="#The interval table-4 eka and 12 eka">4 eka and 12 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><div style="margin-left: 2em;"><a href="#The interval table-5 eka and 11 eka">5 eka and 11 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><div style="margin-left: 2em;"><a href="#The interval table-6 eka and 10 eka">6 eka and 10 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><div style="margin-left: 2em;"><a href="#The interval table-7 eka and 9 eka">7 eka and 9 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><div style="margin-left: 2em;"><a href="#The interval table-8 eka">8 eka</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><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:62 --><!-- ws:start:WikiTextTocRule:63: --><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:63 --><!-- ws:start:WikiTextTocRule:64: --><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:64 --><!-- ws:start:WikiTextTocRule:65: --><div style="margin-left: 2em;"><a href="#Creating scales with Armodue: modal systems-Modal systems based on tetrachords and pentachords">Modal systems based on tetrachords and pentachords</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><div style="margin-left: 2em;"><a href="#Creating scales with Armodue: modal systems-Modal systems based on hexachords">Modal systems based on hexachords</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><div style="margin-left: 2em;"><a href="#Creating scales with Armodue: modal systems-Other modal systems and various considerations on the scales in Armodue">Other modal systems and various considerations on the scales in Armodue</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><div style="margin-left: 2em;"><a href="#Creating scales with Armodue: modal systems-A peculiarity of Armodue: the symmetrical &quot;neo-diminished&quot; scales">A peculiarity of Armodue: the symmetrical &quot;neo-diminished&quot; scales</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><div style="margin-left: 2em;"><a href="#Creating scales with Armodue: modal systems-Modes of limited transposition">Modes of limited transposition</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><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:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 1em;"><a href="#x&quot;Elastic&quot; chords">&quot;Elastic&quot; chords</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:72 -->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 /> | ||
<br /> | <br /> | ||
For terminology see the <a class="wiki_link" href="/Armodue%20theory">Armodue overview page</a>.<br /> | For terminology see the <a class="wiki_link" href="/Armodue%20theory">Armodue overview page</a>.<br /> | ||
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<hr /> | <hr /> | ||
Chapter 1:<br /> | Chapter 1:<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc0"><a name="Two theses supporting the system"></a><!-- ws:end:WikiTextHeadingRule:2 -->Two theses supporting the system</h1> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc1"><a name="Two theses supporting the system-The supremacy of the fifth and the seventh harmonic in Armodue"></a><!-- ws:end:WikiTextHeadingRule:4 -->The supremacy of the fifth and the seventh harmonic in Armodue</h2> | ||
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 tradition - the cycle 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 tradition - the cycle of fifths - is based on the perfect fifth and hence on the same frequency ratio 3:2.<br /> | |||
<br /> | <br /> | ||
But, if in the twelve note system the pitch of the third harmonic - hence the perfect fourth and the perfect fifth - are almost perfectly respected (the tempered fourth and fifth differ only of the fiftieth part of a semitone from the natural fifth and fourth), this cannot be said about the odd harmonics (even harmonics are not counted because they are simply duplicates in the octave of odd harmonics) immediately above the third one: the fifth and seventh harmonic.<br /> | But, if in the twelve note system the pitch of the third harmonic - hence the perfect fourth and the perfect fifth - are almost perfectly respected (the tempered fourth and fifth differ only of the fiftieth part of a semitone from the natural fifth and fourth), this cannot be said about the odd harmonics (even harmonics are not counted because they are simply duplicates in the octave of odd harmonics) immediately above the third one: the fifth and seventh harmonic.<br /> | ||
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For this reason, especially important in Armodue are the intervals 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="/7_4">seventh</a> harmonic (7/4 Ratio).<br /> | For this reason, especially important in Armodue are the intervals 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="/7_4">seventh</a> harmonic (7/4 Ratio).<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&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:6 -->The triple mean of the double diagonal / side of the square</h2> | ||
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 /> | |||
<br /> | <br /> | ||
Analogously, the philosophical foundation of Armodue and esadecafonia can be shown by calculating the three means of frequency geometrically equivalent to the ratio between the double of the diagonal (square root of 2 multiplied by 2) and the side length (of measure: 1) of a square.<br /> | Analogously, the philosophical foundation of Armodue and esadecafonia can be shown by calculating the three means of frequency geometrically equivalent to the ratio between the double of the diagonal (square root of 2 multiplied by 2) and the side length (of measure: 1) of a square.<br /> | ||
<br /> | <br /> | ||
The arithmetic mean is exactly the interval of nine eka, the geometric mean exactly twelve eka and finally the harmonic mean exactly equal to fifteen eka.<br /> | The arithmetic mean is exactly the interval of nine eka, the geometric mean exactly twelve eka and finally the harmonic mean exactly equal to fifteen eka.<br /> | ||
<br /> | <br /> | ||
<hr /> | <hr /> | ||
Chapter 2:<br /> | Chapter 2:<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc3"><a name="The interval table"></a><!-- ws:end:WikiTextHeadingRule:8 -->The interval table</h1> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc4"><a name="The interval table-Qualitative categories of intervals"></a><!-- ws:end:WikiTextHeadingRule:10 -->Qualitative categories of intervals</h2> | ||
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 /> | |||
<br /> | <br /> | ||
1 eka - 15 eka<br /> | 1 eka - 15 eka<br /> | ||
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8 eka<br /> | 8 eka<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc5"><a name="The interval table-1 eka and 15 eka"></a><!-- ws:end:WikiTextHeadingRule:12 -->1 eka and 15 eka</h2> | ||
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 /> | |||
<br /> | <br /> | ||
This property of the eka makes it particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism can be applied in Armodue considering the eka as equivalent to a tempered semitone. The complement of one eka is the interval of 15 eka, comparable to a slightly enlarged major seventh of the dodecatonic system. The small size of the eka also makes it appropriate to evoke oriental sounds and atmospheres. The small intervals of 1 eka, 2 eka and 3 eka in Armodue lend themselves magnificiently to the design of melodies and scales of exquisite modal and arabic flavour.<br /> | This property of the eka makes it particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism can be applied in Armodue considering the eka as equivalent to a tempered semitone. The complement of one eka is the interval of 15 eka, comparable to a slightly enlarged major seventh of the dodecatonic system. The small size of the eka also makes it appropriate to evoke oriental sounds and atmospheres. The small intervals of 1 eka, 2 eka and 3 eka in Armodue lend themselves magnificiently to the design of melodies and scales of exquisite modal and arabic flavour.<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: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc6"><a name="The interval table-2 eka and 14 eka"></a><!-- ws:end:WikiTextHeadingRule:14 -->2 eka and 14 eka</h2> | ||
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 of the overtone series. It is the interval that is obtained by dividing the Tenth of Armodue (the classic Octave of 2/1) 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 of the overtone series. It is the interval that is obtained by dividing the Tenth of Armodue (the classic Octave of 2/1) 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 /> | |||
<br /> | <br /> | ||
In the dodecatonic system, the octave (1200 cents) divides up into an augmented fourth and a diminished fifth (600 cents), the tritone thus obtained can be divided into two minor thirds (300 cents), but the minor third may not further be subdivided into two parts. It is here that, where the possibilities of the dodecatonic system end, the possibilities of Armodue start, and we can continue in progressive subdivisions: the minor third, redefined as four eka (300 cents), is two eka plus two eka (150 cents); in turn, two eka is made up of two intervals of one eka each (75 cents). These algebraic/geometric properties of the initially considered interval of two eka make it particularly suitable for symmetric harmonic constructions - hence &quot;speculative harmony&quot;. In practice, you can build speculative chords using only intervals of 16, 8, 4 or 2 eka between a voice and the adjacent one. One or more notes thus obtained can later be altered one eka up or down; in this way, a harmonic construction that is rigidly squared first gains a new harmonic coloration particularly significant in the context.<br /> | In the dodecatonic system, the octave (1200 cents) divides up into an augmented fourth and a diminished fifth (600 cents), the tritone thus obtained can be divided into two minor thirds (300 cents), but the minor third may not further be subdivided into two parts. It is here that, where the possibilities of the dodecatonic system end, the possibilities of Armodue start, and we can continue in progressive subdivisions: the minor third, redefined as four eka (300 cents), is two eka plus two eka (150 cents); in turn, two eka is made up of two intervals of one eka each (75 cents). These algebraic/geometric properties of the initially considered interval of two eka make it particularly suitable for symmetric harmonic constructions - hence &quot;speculative harmony&quot;. In practice, you can build speculative chords using only intervals of 16, 8, 4 or 2 eka between a voice and the adjacent one. One or more notes thus obtained can later be altered one eka up or down; in this way, a harmonic construction that is rigidly squared first gains a new harmonic coloration particularly significant in the context.<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: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc7"><a name="The interval table-3 eka and 13 eka"></a><!-- ws:end:WikiTextHeadingRule:16 -->3 eka and 13 eka</h2> | ||
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 eighth and ninth). 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 eighth and ninth). 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 /> | |||
<br /> | <br /> | ||
The complement of the interval of 3 eka is the interval of 13 eka, which has a huge importance in Armodue as it corresponds to the <a class="wiki_link" href="/7_4">natural minor seventh</a> - the interval given by the ratio of the fourth harmonic with the seventh harmonic. Who has delved deeper into harmony topics is aware of how much a dominant seventh chord (example: C-E-G-Bb) played in 12-tone equal temperament differs from the corresponding natural chord found by overlapping the fourth, the fifth, the sixth and the seventh harmonic. This is due mainly to the non-negligible difference in pitch of the tempered minor seventh (1000 cents) and the seventh harmonic (968,83 cents)(in this example the Bb). In Armodue, the minor seventh is returned to its natural pitch, the interval of 13 eka is perceived as very natural and euphonious (975 cents).<br /> | The complement of the interval of 3 eka is the interval of 13 eka, which has a huge importance in Armodue as it corresponds to the <a class="wiki_link" href="/7_4">natural minor seventh</a> - the interval given by the ratio of the fourth harmonic with the seventh harmonic. Who has delved deeper into harmony topics is aware of how much a dominant seventh chord (example: C-E-G-Bb) played in 12-tone equal temperament differs from the corresponding natural chord found by overlapping the fourth, the fifth, the sixth and the seventh harmonic. This is due mainly to the non-negligible difference in pitch of the tempered minor seventh (1000 cents) and the seventh harmonic (968,83 cents)(in this example the Bb). In Armodue, the minor seventh is returned to its natural pitch, the interval of 13 eka is perceived as very natural and euphonious (975 cents).<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:18:&lt;h2&gt; --><h2 id="toc8"><a name="The interval table-4 eka and 12 eka"></a><!-- ws:end:WikiTextHeadingRule:18 -->4 eka and 12 eka</h2> | ||
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 of 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 Armodue is a crucial point in the inevitable interaction that the ear of a western listener will establish between the two different tempered systems. 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 dodecatonic 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 of 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 Armodue is a crucial point in the inevitable interaction that the ear of a western listener will establish between the two different tempered systems. 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 dodecatonic system.<br /> | |||
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In a particular and interesting case, a composer could also decide not to use the intervals giving the color to the harmonies - minor and major thirds and sixths transferred in Armodue: intervals of 4, 5, 11 and 12 eka. Excluding these four types of intervals in the texture of the chords, the ear probably will realize at once that it is in a new and unknown musical environment.<br /> | In a particular and interesting case, a composer could also decide not to use the intervals giving the color to the harmonies - minor and major thirds and sixths transferred in Armodue: intervals of 4, 5, 11 and 12 eka. Excluding these four types of intervals in the texture of the chords, the ear probably will realize at once that it is in a new and unknown musical environment.<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:20:&lt;h2&gt; --><h2 id="toc9"><a name="The interval table-5 eka and 11 eka"></a><!-- ws:end:WikiTextHeadingRule:20 -->5 eka and 11 eka</h2> | ||
The interval of 5 eka is 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 (400 cents) is not so close to the natural major third (386,31 cents) as the interval of 5 eka (375 cents).<br /> | |||
The interval of 5 eka is 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 (400 cents) is not so close to the natural major third (386,31 cents) as the interval of 5 eka (375 cents).<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:22:&lt;h2&gt; --><h2 id="toc10"><a name="The interval table-6 eka and 10 eka"></a><!-- ws:end:WikiTextHeadingRule:22 -->6 eka and 10 eka</h2> | ||
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 10 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 10 eka is the average between a perfect fifth and a minor sixth.<br /> | |||
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This property constitutes the most interesting point of the intervals examined so far. The interval of 6 eka joins the concise color of the major third with the propulsion and the dynamism of the perfect fourth, the interval of 10 eka combines the twilight character of the minor sixth with the staticity and the balance of the perfect fifth.<br /> | This property constitutes the most interesting point of the intervals examined so far. The interval of 6 eka joins the concise color of the major third with the propulsion and the dynamism of the perfect fourth, the interval of 10 eka combines the twilight character of the minor sixth with the staticity and the balance of the perfect fifth.<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:24:&lt;h2&gt; --><h2 id="toc11"><a name="The interval table-7 eka and 9 eka"></a><!-- ws:end:WikiTextHeadingRule:24 -->7 eka and 9 eka</h2> | ||
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 sharpened fourth (525 cents), the second in a slightly flattened fifth (675 cents).<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 sharpened fourth (525 cents), the second in a slightly flattened fifth (675 cents).<br /> | |||
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These Armodue intervals, however, are incompatible with the concept of the cycle of the fifth that is on the base of the dodecatonic system. Especially with the intervals of 7 and 9 eka, Armodue shows an entirely new and different system, compared to the ruling dodecatonic system.<br /> | These Armodue intervals, however, are incompatible with the concept of the cycle of the fifth that is on the base of the dodecatonic system. Especially with the intervals of 7 and 9 eka, Armodue shows an entirely new and different system, compared to the ruling dodecatonic system.<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:26:&lt;h2&gt; --><h2 id="toc12"><a name="The interval table-8 eka"></a><!-- ws:end:WikiTextHeadingRule:26 -->8 eka</h2> | ||
The interval of 8 eka divides the Tenth of Armodue in half, just as the tritone halves the typical 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 Tenth of Armodue in half, just as the tritone halves the typical octave. Indeed 8 eka correspond exactly to three wholetones, hence to an augmented fourth or diminished fifth.<br /> | |||
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Being exactly half of 16 eka, the interval of 8 eka is its own complement; this feature makes it particularly uneasy and unstable - just like the tritone among the dodecatonic tempered system.<br /> | Being exactly half of 16 eka, the interval of 8 eka is its own complement; this feature makes it particularly uneasy and unstable - just like the tritone among the dodecatonic tempered system.<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:28:&lt;h2&gt; --><h2 id="toc13"><a name="The interval table-Gradation of harmonic tensions"></a><!-- ws:end:WikiTextHeadingRule:28 -->Gradation of harmonic tensions</h2> | ||
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 /> | |||
<br /> | <br /> | ||
The progression order (forward or backwards) will be as follows:<br /> | The progression order (forward or backwards) will be as follows:<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:30:&lt;h2&gt; --><h2 id="toc14"><a name="The interval table-Arrangement of tones in the chords"></a><!-- ws:end:WikiTextHeadingRule:30 -->Arrangement of tones in the chords</h2> | ||
Chords can be realized in any way, placing the notes that constitute it more or less distributed (narrow or 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 or 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 /> | |||
<br /> | <br /> | ||
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 (octave-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 /> | 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 (octave-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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<hr /> | <hr /> | ||
Chapter 3:<br /> | Chapter 3:<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc15"><a name="Creating scales with Armodue: modal systems"></a><!-- ws:end:WikiTextHeadingRule:32 -->Creating scales with Armodue: modal systems</h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&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:34 -->Modal systems based on tetrachords and pentachords</h2> | ||
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 /> | |||
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 /> | |||
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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 /> | 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 /> | ||
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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:36:&lt;h2&gt; --><h2 id="toc17"><a name="Creating scales with Armodue: modal systems-Modal systems based on hexachords"></a><!-- ws:end:WikiTextHeadingRule:36 -->Modal systems based on hexachords</h2> | ||
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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In Armodue, however, there is another very significant interval that deserves to function as Pivotal interval: that of 13 eka - whose width is very close to the natural minor seventh (the frequency ratio between the seventh and the fourth harmonic of the overtone series [7/4]).<br /> | In Armodue, however, there is another very significant interval that deserves to function as Pivotal interval: that of 13 eka - whose width is very close to the natural minor seventh (the frequency ratio between the seventh and the fourth harmonic of the overtone series [7/4]).<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:38:&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:38 -->Other modal systems and various considerations on the scales in Armodue</h2> | ||
The combination possibilities of trichords, tetrachords, 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 /> | |||
The combination possibilities of trichords, tetrachords, 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 /> | 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 /> | 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: | <!-- ws:start:WikiTextHeadingRule:40:&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:40 -->A peculiarity of Armodue: the symmetrical &quot;neo-diminished&quot; scales</h2> | ||
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 /> | |||
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 /> | <br /> | ||
These two scales strongly evoke the sound of the &quot;diminished&quot; scales of the dodecatonic 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. (Actually, they are scales of <a class="wiki_link" href="/Diminished">diminished temperament</a>, in 12edo as well as in 16edo!)<br /> | |||
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<td><!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/htgt12edo.mp3?h=20&amp;w=240&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;htgt12edo.mp3&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;20&quot; width=&quot;240&quot;/&gt; --><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fhtgt12edo.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:0 --><br /> | |||
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<td><!-- ws:start:WikiTextMediaRule:1:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/htgt16edo.mp3?h=20&amp;w=240&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;htgt16edo.mp3&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;20&quot; width=&quot;240&quot;/&gt; --><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fhtgt16edo.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:1 --><br /> | |||
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<td>Diminished ocatatonic scale in 12edo<br /> | |||
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<td>Diminished octatonic scale in 16edo<br /> | |||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h2&gt; --><h2 id="toc20"><a name="Creating scales with Armodue: modal systems-Modes of limited transposition"></a><!-- ws:end:WikiTextHeadingRule:42 -->Modes of limited transposition</h2> | ||
&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 /> | |||
&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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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 /> | 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 /> | ||
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Chapter 4:<br /> | Chapter 4:<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc21"><a name="x&quot;Geometric&quot; harmonic constructions with Armodue"></a><!-- ws:end:WikiTextHeadingRule:44 -->&quot;Geometric&quot; harmonic constructions with Armodue</h1> | ||
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The composer Scriabin has based much of his compositions on &quot;nucleopolar accordances&quot; or &quot;chord centers&quot; (Klangzentrum).<br /> | The composer Scriabin has based much of his compositions on &quot;nucleopolar accordances&quot; or &quot;chord centers&quot; (Klangzentrum).<br /> | ||
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Chapter 5:<br /> | Chapter 5:<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h1&gt; --><h1 id="toc22"><a name="x&quot;Elastic&quot; chords"></a><!-- ws:end:WikiTextHeadingRule:46 -->&quot;Elastic&quot; chords</h1> | ||
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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 /> | 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 /> | ||