Chordal space: Difference between revisions
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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> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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Lower case letters indicate minor key, uppercase major. This was first proposed by Vial (1767) (later Weber, Riemann, Schoenberg), the advantage over the circle of fifths being that it represents both relative and parallel major. (Lerdahl, 2001) | Lower case letters indicate minor key, uppercase major. This was first proposed by Vial (1767) (later Weber, Riemann, Schoenberg), the advantage over the circle of fifths being that it represents both relative and parallel major. (Lerdahl, 2001) | ||
== | ==Principles of chordal space== | ||
The Vial/Weber chordal space depicts two different sorts of relationships: shared common tones and efficient voice leading. For example, the proximity of the C major and e minor chords reflects the fact that the two chords share two common tones, E and G. Moreover, one chord can be transformed into another by moving a single note by just one semitone: to transform a C major chord into an E minor chord, one need only move C to B. Furthermore, the Vial/Weber chordal space is closely related to the two-dimensional lattices described in the article on pitch space: every chord on the Vial/Weber chordal space can be associated with a triangle on the [[http://en.wikipedia.org/wiki/Tonnetz|Tonnetz]] or two-dimensional pitch class lattice. | The Vial/Weber chordal space depicts two different sorts of relationships: shared common tones and efficient voice leading. For example, the proximity of the C major and e minor chords reflects the fact that the two chords share two common tones, E and G. Moreover, one chord can be transformed into another by moving a single note by just one semitone: to transform a C major chord into an E minor chord, one need only move C to B. Furthermore, the Vial/Weber chordal space is closely related to the two-dimensional lattices described in the article on pitch space: every chord on the Vial/Weber chordal space can be associated with a triangle on the [[http://en.wikipedia.org/wiki/Tonnetz|Tonnetz]] or two-dimensional pitch class lattice. | ||
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We can now take the twenty-four major and minor triads of equal temperament and place them on the vertices of a regular 24-gon. We then draw lines from triads separated by one step, and also from each major triad to its parallel minor triad, and obtain a geometric picture of the regular graph in question, which satisfactorily models the triadic relationships in 12 equal temperament. | We can now take the twenty-four major and minor triads of equal temperament and place them on the vertices of a regular 24-gon. We then draw lines from triads separated by one step, and also from each major triad to its parallel minor triad, and obtain a geometric picture of the regular graph in question, which satisfactorily models the triadic relationships in 12 equal temperament. | ||
[[file:Chordal12.svg]] | [[file:Chordal12.svg]] | ||
We may obtain very similar pictures for any of the other equal temperaments supporting the use of [[meantone temperament]], including in particular [[19edo]] and [[31edo]], by drawing a 38-gon or a 62-gon respectively, and linking all chords separated by one step, and the correct half separated by seven steps. | We may obtain very similar pictures for any of the other equal temperaments supporting the use of [[meantone temperament]], including in particular [[19edo]] and [[31edo]], by drawing a 38-gon or a 62-gon respectively, and linking all chords separated by one step, and the correct half separated by seven steps. | ||
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*Mathieu, W. A. (1997). ''Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression''. Inner Traditions Intl Ltd. ISBN 0892815604. | *Mathieu, W. A. (1997). ''Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression''. Inner Traditions Intl Ltd. ISBN 0892815604. | ||
[[Category:Music theory]] | [[category:Music theory|Category : Music theory]] | ||
[[Category:Pitch]] | [[category:Pitch|Category : Pitch]] | ||
This article uses information from an old version of the Wikipedia article of the same name, removed not for being wrong but for being too "original".</pre></div> | This article uses information from an old version of the Wikipedia article of the same name, removed not for being wrong but for being too "original".</pre></div> | ||
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Lower case letters indicate minor key, uppercase major. This was first proposed by Vial (1767) (later Weber, Riemann, Schoenberg), the advantage over the circle of fifths being that it represents both relative and parallel major. (Lerdahl, 2001)<br /> | Lower case letters indicate minor key, uppercase major. This was first proposed by Vial (1767) (later Weber, Riemann, Schoenberg), the advantage over the circle of fifths being that it represents both relative and parallel major. (Lerdahl, 2001)<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x- | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Principles of chordal space"></a><!-- ws:end:WikiTextHeadingRule:2 -->Principles of chordal space</h2> | ||
<br /> | <br /> | ||
The Vial/Weber chordal space depicts two different sorts of relationships: shared common tones and efficient voice leading. For example, the proximity of the C major and e minor chords reflects the fact that the two chords share two common tones, E and G. Moreover, one chord can be transformed into another by moving a single note by just one semitone: to transform a C major chord into an E minor chord, one need only move C to B. Furthermore, the Vial/Weber chordal space is closely related to the two-dimensional lattices described in the article on pitch space: every chord on the Vial/Weber chordal space can be associated with a triangle on the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tonnetz" rel="nofollow">Tonnetz</a> or two-dimensional pitch class lattice.<br /> | The Vial/Weber chordal space depicts two different sorts of relationships: shared common tones and efficient voice leading. For example, the proximity of the C major and e minor chords reflects the fact that the two chords share two common tones, E and G. Moreover, one chord can be transformed into another by moving a single note by just one semitone: to transform a C major chord into an E minor chord, one need only move C to B. Furthermore, the Vial/Weber chordal space is closely related to the two-dimensional lattices described in the article on pitch space: every chord on the Vial/Weber chordal space can be associated with a triangle on the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tonnetz" rel="nofollow">Tonnetz</a> or two-dimensional pitch class lattice.<br /> | ||
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We can now take the twenty-four major and minor triads of equal temperament and place them on the vertices of a regular 24-gon. We then draw lines from triads separated by one step, and also from each major triad to its parallel minor triad, and obtain a geometric picture of the regular graph in question, which satisfactorily models the triadic relationships in 12 equal temperament.<br /> | We can now take the twenty-four major and minor triads of equal temperament and place them on the vertices of a regular 24-gon. We then draw lines from triads separated by one step, and also from each major triad to its parallel minor triad, and obtain a geometric picture of the regular graph in question, which satisfactorily models the triadic relationships in 12 equal temperament.<br /> | ||
<br /> | <br /> | ||
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We may obtain very similar pictures for any of the other equal temperaments supporting the use of <a class="wiki_link" href="/meantone%20temperament">meantone temperament</a>, including in particular <a class="wiki_link" href="/19edo">19edo</a> and <a class="wiki_link" href="/31edo">31edo</a>, by drawing a 38-gon or a 62-gon respectively, and linking all chords separated by one step, and the correct half separated by seven steps.<br /> | We may obtain very similar pictures for any of the other equal temperaments supporting the use of <a class="wiki_link" href="/meantone%20temperament">meantone temperament</a>, including in particular <a class="wiki_link" href="/19edo">19edo</a> and <a class="wiki_link" href="/31edo">31edo</a>, by drawing a 38-gon or a 62-gon respectively, and linking all chords separated by one step, and the correct half separated by seven steps.<br /> | ||
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*Mathieu, W. A. (1997). ''Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression''. Inner Traditions Intl Ltd. ISBN 0892815604.<br /> | *Mathieu, W. A. (1997). ''Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression''. Inner Traditions Intl Ltd. ISBN 0892815604.<br /> | ||
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<a class="wiki_link" href="http://category.wikispaces.com/Music%20theory">Category | <a class="wiki_link" href="http://category.wikispaces.com/Music%20theory">Category : Music theory</a><br /> | ||
<a class="wiki_link" href="http://category.wikispaces.com/Pitch">Category | <a class="wiki_link" href="http://category.wikispaces.com/Pitch">Category : Pitch</a><br /> | ||
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This article uses information from an old version of the Wikipedia article of the same name, removed not for being wrong but for being too &quot;original&quot;.</body></html></pre></div> | This article uses information from an old version of the Wikipedia article of the same name, removed not for being wrong but for being too &quot;original&quot;.</body></html></pre></div> | ||