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dominant-7th (chord) / dom7 / x7

[Joe Monzo]

Traditional diatonic analysis

The dominant-7th chord is formally a chord containing 4 notes, built upward in pitch from the root with the intervals of a major-3rd, minor-3rd, and another minor-3rd, resulting in the chord-members identified as root, 3rd, 5th, and 7th. The actual intervals above the root are thus the major-3rd, perfect-5th, and minor-7th.

The dominant-7th chord arises naturally as the chord whose root is the fifth note, designated by Roman numeral "V" and called the "dominant", of the diatonic major-scale -- its common abbreviation is thus "V7". For example:

     key of C-major:

     C   D   E F   G   A   B C   D   E F   G
                   |       |     |     |
                  root    3rd   5th    7th
                   \______/ \___/ \____/
                   maj-3rd min-3rd min-3rd

The dominant-7th chord plays an important role in musical harmony, especially at divisions of the formal structure, because of its use in cadences. In traditional music-theory it is considered to be a dissonant chord in need of resolution, specifically the resolution of the tritone which is included within the chord, as in the B:F interval in the chord illustrated above; note that the use of the word "tritone" here is in the loose sense, which designates both the true tritone (augmented-4th) and its inversion (diminished-5th). The normal resolution is for the 3rd of the chord to move upward one diatonic-semitone to the tonic of the key (I), and for the 7th of the chord to move downward one diatonic-semitone to the 3rd of the tonic chord (III), so in the above example B-->C and F-->E. Thus the most dissonant interval normally used in tonal music is resolved to the most consonant, and by the shortest distance in voice-leading, and additionally by contrary motion, which is felt to be a stronger resolution than oblique or similar motion.

        key of C-major:

     	G   A   B C   D   E F   G
        *       *     *     *   *    G dominant-7th chord (V)
                 \         /    |
                  *       *     *    C major chord (I)

Traditionally, in the meantone tunings which have been standard in Western theory and practice, the dominant-7th actually has been composed of discordant intervals, particularly the minor-7th. But in more recent times, many composers working in just-intonation (JI) have come to think of the dominant-7th chord as a consonant entity with the 7th of the chord tuned as the harmonic-7th of ratio 7:4. This usage parallels the use of the dominant-7th chords in blues music, in which every chord is a dominant-7th, but where they are normally tuned in 12-edo as played on a guitar.

12-edo pc-set analysis

In 12-edo pc-set theory, the dominant-7th chord is classified as follows:

interval-string    prime  interval-vector   Solomon     Forte
     (is)           form        (iv)        catalog #   set-name

     3324           0368       012111         065       4-27[b]

Below is a diagram illustrating the prime-form and interval-string:

0 1 2 3 4 5 6 7 8 9 A B 0   =  12-edo degree integers (pcs)
B C   D   E F   G   A   B   =  diatonic C-major scale, rotated to most compact arrangement
*     *     *   *       *   =  chord notes
0     3     6   8           =  prime-form (pcs)
 \_3_/ \_3_/ \2/ \__4__/    =  interval-string (ics)

The interval-vector is self-evident:

0     3     6   8       0   =  pcs (prime-form)
 \_3_/ \_3_/ \2/ \__4__/   \
       \___5___/            =  instances of intervals
 \____6____/               /

interval  interval-
semitones  vector

   1         0      none
   2         1      F:G = 6,8
   3         2      B:D = 0,3 and D:F = 3,6
   4         1      G:B = 8,0
   5         1      D:G = 3,8
   6         1      B:F = 0,6
. . . . . . . . .

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