Armodue harmony
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[[toc]] =**Armodue: basic elements of harmony**= This is a translation of an article by Luca Attanasio. Original page in italian: [[http://www.armodue.com/armonia.htm]] //Note: This is a preliminary tranlsation. Parts that are still "under construction" are marked with "XXX".// ---- =Chapter 1: Two theses supporting the system= ==The supremacy of the fifth and 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 tradion - the cicle of fifths - is based on the perfect fifth and hence on the same frequency ratio 3:2. 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. In the tempered system the fifth and the seventh harmonic appear as major third and minor seventh intervals formed to the fundamental (equating all intervals to their octaver-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals. In Armodue, in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness. For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the seventh harmonic. ==The triple mean of the double diagonal / side of the square== 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). 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. 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. =Chapter 2: The interval table= XXX =Chapter 3: Creating scales with Armodue: modal systems= XXX =Chapter 4: "Geometric" harmonic constructions with Armodue= XXX =Chapter 5: "elastic" chords= XXX
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<html><head><title>Armodue armonia</title></head><body><!-- ws:start:WikiTextTocRule:16:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 1em;"><a href="#Armodue: basic elements of harmony">Armodue: basic elements of harmony</a></div> <!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 1em;"><a href="#Chapter 1: Two theses supporting the system">Chapter 1: Two theses supporting the system</a></div> <!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><div style="margin-left: 2em;"><a href="#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</a></div> <!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><div style="margin-left: 2em;"><a href="#Chapter 1: 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:20 --><!-- ws:start:WikiTextTocRule:21: --><div style="margin-left: 1em;"><a href="#Chapter 2: The interval table">Chapter 2: The interval table</a></div> <!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><div style="margin-left: 1em;"><a href="#Chapter 3: Creating scales with Armodue: modal systems">Chapter 3: Creating scales with Armodue: modal systems</a></div> <!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><div style="margin-left: 1em;"><a href="#Chapter 4: "Geometric" harmonic constructions with Armodue">Chapter 4: "Geometric" harmonic constructions with Armodue</a></div> <!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><div style="margin-left: 1em;"><a href="#Chapter 5: "elastic" chords">Chapter 5: "elastic" chords</a></div> <!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --></div> <!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Armodue: basic elements of harmony"></a><!-- ws:end:WikiTextHeadingRule:0 --><strong>Armodue: basic elements of harmony</strong></h1> <br /> This is a translation of an article by Luca Attanasio. Original page in italian: <a class="wiki_link_ext" href="http://www.armodue.com/armonia.htm" rel="nofollow">http://www.armodue.com/armonia.htm</a><br /> <em>Note: This is a preliminary tranlsation. Parts that are still "under construction" are marked with "XXX".</em><br /> <br /> <br /> <hr /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Chapter 1: Two theses supporting the system"></a><!-- ws:end:WikiTextHeadingRule:2 -->Chapter 1: Two theses supporting the system</h1> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Chapter 1: 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 /> 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 /> <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 /> <br /> In the tempered system the fifth and the seventh harmonic appear as major third and minor seventh intervals formed to the fundamental (equating all intervals to their octaver-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.<br /> <br /> In Armodue, in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.<br /> <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 /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Chapter 1: 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 /> 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 /> 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 /> 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 /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Chapter 2: The interval table"></a><!-- ws:end:WikiTextHeadingRule:8 -->Chapter 2: The interval table</h1> <br /> XXX<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Chapter 3: Creating scales with Armodue: modal systems"></a><!-- ws:end:WikiTextHeadingRule:10 -->Chapter 3: Creating scales with Armodue: modal systems</h1> <br /> XXX<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h1> --><h1 id="toc6"><a name="Chapter 4: "Geometric" harmonic constructions with Armodue"></a><!-- ws:end:WikiTextHeadingRule:12 -->Chapter 4: "Geometric" harmonic constructions with Armodue</h1> <br /> XXX<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="Chapter 5: "elastic" chords"></a><!-- ws:end:WikiTextHeadingRule:14 -->Chapter 5: "elastic" chords</h1> <br /> XXX</body></html>