Chordal space: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-20 19:22:32 UTC</tt>.<br>

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One of the earliest graphical models of chord-relationships was devised by [[http://en.wikipedia.org/wiki/Johann_David_Heinichen|Johann David Heinichen]] in 1728; he proposed placing the major and minor chords in a circular arrangement of twenty-four chords arranged according to the circle of fifths; reading clockwise, ... F, d, C, a, G, ... (Lerdahl, 2001). The currently more popular major on the outside relative minor on the inside format was proposed by [[http://en.wikipedia.org/wiki/David_Kellner|David Kellner]]  (1737).  

[[http://en.wikipedia.org/wiki/Gottfried_Weber|Gottfried Weber]]  and F. G. Vial suggested a [[http://en.wikipedia.org/wiki/Lattice_graph|grid graph]] or [[http://en.wikipedia.org/wiki/Square_lattice|square lattice]] model of chordal space; Weber's lattice centered on C major, is:

 

 

However, when constructing a chordal space, several general principles are useful. The first is that chordal space should be or define a [[regular graph]], whose regularity is linked to the regularity of the corresponding modulatory space. The second is that to start out with at least, only the most basic chords of the tonal system in question should be considered. The third is that two chords should be linked if and only if they share a common interval. Chords sharing a common note can then be reached via these closer connections.

In the circular arrangement F - d - C - a ..., the chords F and d share two common tones, and can be linked by efficient voice leading. However, the chords d and C do not share any common tones, and cannot be linked by very efficient voice leading. By contrast in the series d - F - a - C - e - G ..., every chord shares two notes with its neighbors and can be transformed into them by moving one note by one or two semitones. The resulting pattern of chords can be generated in the Vial/Weber space, by moving upward along adjacent columns in the space.

Any two adjacent chords in this chain are now linked by two intervals, so that the two chords adjacent to a given chord are strongly linked to that chord. Next we note (under the assumption of meantone temperament) that d is also linked by a shared interval (this time the fifth) with D. We therefore draw a line ahead seven steps from the minor triad to the major triad on the same root, or behind seven steps from a major triad to its associated minor triad. We do not, however, draw a line ahead from C major to c# minor, or behind from e minor to Eb major, because these share only one note.

One of the earliest graphical models of chord-relationships was devised by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Johann_David_Heinichen" rel="nofollow"&gt;Johann David Heinichen&lt;/a&gt; in 1728; he proposed placing the major and minor chords in a circular arrangement of twenty-four chords arranged according to the circle of fifths; reading clockwise, ... F, d, C, a, G, ... (Lerdahl, 2001). The currently more popular major on the outside relative minor on the inside format was proposed by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/David_Kellner" rel="nofollow"&gt;David Kellner&lt;/a&gt;  (1737). &lt;br /&gt;

&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gottfried_Weber" rel="nofollow"&gt;Gottfried Weber&lt;/a&gt;  and F. G. Vial suggested a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_graph" rel="nofollow"&gt;grid graph&lt;/a&gt; or &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Square_lattice" rel="nofollow"&gt;square lattice&lt;/a&gt; model of chordal space; Weber's lattice centered on C major, is:&lt;br /&gt;

However, when constructing a chordal space, several general principles are useful. The first is that chordal space should be or define a &lt;a class="wiki_link" href="/regular%20graph"&gt;regular graph&lt;/a&gt;, whose regularity is linked to the regularity of the corresponding modulatory space. The second is that to start out with at least, only the most basic chords of the tonal system in question should be considered. The third is that two chords should be linked if and only if they share a common interval. Chords sharing a common note can then be reached via these closer connections.&lt;br /&gt;

In the circular arrangement F - d - C - a ..., the chords F and d share two common tones, and can be linked by efficient voice leading. However, the chords d and C do not share any common tones, and cannot be linked by very efficient voice leading. By contrast in the series d - F - a - C - e - G ..., every chord shares two notes with its neighbors and can be transformed into them by moving one note by one or two semitones. The resulting pattern of chords can be generated in the Vial/Weber space, by moving upward along adjacent columns in the space.&lt;br /&gt;

Any two adjacent chords in this chain are now linked by two intervals, so that the two chords adjacent to a given chord are strongly linked to that chord. Next we note (under the assumption of meantone temperament) that d is also linked by a shared interval (this time the fifth) with D. We therefore draw a line ahead seven steps from the minor triad to the major triad on the same root, or behind seven steps from a major triad to its associated minor triad. We do not, however, draw a line ahead from C major to c# minor, or behind from e minor to Eb major, because these share only one note.&lt;br /&gt;

*[&lt;!-- ws:start:WikiTextUrlRule:129:http://66.98.148.43/~xenharmo/sevlat.htm --&gt;&lt;a class="wiki_link_ext" href="http://66.98.148.43/~xenharmo/sevlat.htm" rel="nofollow"&gt;http://66.98.148.43/~xenharmo/sevlat.htm&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:129 --&gt; Seven-limit modulatory and chordal space]&lt;br /&gt;