# minor-7th / minor seventh / min7 / m7

[Joe Monzo]

The smaller of the two diatonic intervals which encompass 7 degrees of the of the diatonic scale. The minor-7th is composed of 4 tones (i.e., whole-tones) and 2 diatonic semitones, and is one chromatic semitone smaller than the major-7th.

_______ minor-7th _______
|                         |
C    D    E  F    G    A    B  C
t   s   t    t     t  s
M2  m2  M2   M2    M2 m2

minor-7th   =   4t    + 2s      =   4(t-s) + 6s
=   4(M2) + 2(m2)   =   4(+1)  + 6(m2)

Thus, the minor-7th contains 4 chromatic semitones and 6 diatonic semitones (or equivalently, 4 augmented-primes and 6 minor-2nds). In 12-edo, the minor-7th encompasses 10 equal semitones.

Narrowing the minor-7th by one more chromatic semitone results in the diminished-7th.

. . . . . . . . .

Aside from the strict diatonic usage defined above, "minor-7th" may denote any interval in the range of 1000 cents.

Some minor 7ths commonly found in just intonation are:

type               ratio          ~cents
just minor-7th           9/5   1017.596288  (~= 10173/5)
pythagorean minor-7th   16/9    996.0899983 (~=  9961/11)
harmonic 7th             7/4    968.8259065 (~=  9685/6)

The standard 12-edo minor-7th is 2(10/12) = exactly 1000 cents.

All chain-of-5th type tunings have a true minor-7th of -2 generators (5ths), and also an augmented-6th of +10 generators. In all tunings of the meantone family, the true minor-7th is quite wider than the harmonic-7th, but in most meantones the augmented-6th is a good approximation of the harmonic-7th. This closeness of approximation is probably responsible for the popularity of the various "augmented 6th chords" ("Italian", "German", "French") in "common-practice" harmony, as is renders chords which are very concordant and approach the 4:5:6:7 proportion (or subsets or variations of it). Here are some typical examples:

-- minor-7th (= -2 generators) --   -- augmented-6th (= +10 generators) --

meantone       2,3,5-monzo         ~cents            2,3,5-monzo        ~cents

1/4-comma   [ 2    0  , -1/2 >   1006.843143      [- 5    0  , 5/2 >  965.7842847
1/5-comma   [12/5 -2/5, -2/5 >   1004.692514      [- 7    2  , 2   >  976.5374295
1/6-comma   [ 8/3 -2/3, -1/3 >   1003.258761      [-25/3 10/3, 5/3 >  983.7061927

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