reply to Bill Alves

>I wrote,
>
>}>>In MT, the two German augmented sixth chords (usually Bb D F G# and Eb G
>Bb
>}>>C#) have a particularly concordant quality, resembling the seventh chords
>}>>sometimes encountered in barbershop singing. The two incomplete French
>}>>augmented sixth chords (Bb E G# and Eb A C#) also have a unique resonance.
>}>>Huygens, in advocating meantone tempermant, pointed out (back in the 17th
>}>>century, I believe) that these latter chords are used in compositions and
>in
>}>>meantone are tuned very nearly 1/7:1/5:1/4. This may have been the first
>}>>attempt to use the number 7 in explaining musical practice.
>
>Bill Alves wrote,
>
>}I haven't read Huygens (apart from what Barbour says about him), but this
>}seems a curious justification. Given that the minor seventh of the dominant
>}seventh chord resolves inward and the augmented sixth of the German
>}augmented sixth chord resolves outward, wouldn't one expect that, ideally,
>}the augmented sixth to be a larger interval than the minor seventh? (If,
>}indeed, one considers the seventh harmonic to be in need of resolution at
>}all.)
>
I'm not sure why one would expect that. Is there some size of minor
seventh that, to you, is a dividing line, such that a slightly smaller
one resolves inward, while a slightly larger one resolves outward? Try
answering this question:

"Given that the diminished fifth resolves inward and the augmented
fourth resolves outward, wouldn't one expect that, ideally, the
augmented fourth to be a larger interval than the diminished fifth?"

If you answer that in the affirmative, then we just have to be content
saying that meantone temperament does not have all the properties of an
"ideal" tuning. If you answer in the negative, then tell me why you
think this case is different from that of the minor seventh/augmented
sixth.

I do think that, since in many cases the melodic interval of resolution
is a diatonic semitone, for example in the resolution of a tritone, the
resolution of the augmented sixth will sound particularly convincing if
that same diatonic semitone is used in both voices. In meantone
temperament, the diatonic semitone happens to be larger than the
chromatic semitone.

I also think that resolution has much more to do with position within a
scale (in particular, a tonal mode) than with absolute interval size.
WIthout a scalar context, I don't think intervals have any particular
>tendency to resolve one way or the other.