source file: m1370.txt
Date: Mon, 30 Mar 1998 15:44:01 -0800

Subject: Re: reply to Bill Alves

From: alves@orion.ac.hmc.edu (Bill Alves)

>>I 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.)
>>
Paul answered:
>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?"
>
Yes.

>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.

Yes, though there are different sized tritones in a given meantone tuning
of 12 keys. The example you gave from Huygens was specific in that the
augmented sixths in the keys of D and G (common keys, to be sure) would be
close to 7/4 ratios, while the dominant sevenths of those keys would be
larger intervals. What struck me is not that a 7/4 augmented sixth would be
unusable, but that Huygens would use it as a justification for the tuning.

>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.

Of course, music is an art, and there are many different things in musical
context that contribute to the feeling of, what shall we call it, the
resolution tendency or naturalness of resolution. However, I do believe
that one of these criteria in music of this period is that the shortest
path to resolution tends to be the most "natural" -- thus the importance of
the leading tone and its sharpening in minor keys.

By the way, I talked to someone the other day who argued for equal
temperament because of the importance of the ambiguities possible in such
chords (the favorite Romantic trick of interpreting a dominant seventh as a
German augmented sixth or vice-versa). The ability to reinterpret such
chord structures multiple ways, he felt, was at the heart of our tonal
system. (I wonder how Huygens would have felt about that -- do you think
he's arguing for a distinction between the sounds of the two chords?)

I explained, recalling the thread here some time ago on the possibility of
musical puns in just intonation, that such reinterpretations were not
impossible in JI, especially if the number of tones in the tuning system
were greater than twelve. Finally, though, we failed to convince each other
to cross over what was ultimately a difference is aesthetic preference.
Perhaps he felt a greater attachment to the tonal-egalitarian traditions of
late romantic chromaticism, atonality, and serialism than do I.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves                                      email: alves@hmc.edu ^
^ Harvey Mudd College                 URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St.                            (909)607-4170 (office) ^
^ Claremont CA 91711 USA                           (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^