TUNING digest 1370: Meantone

The various meantone routines compromised fifths in order to better tune
thirds. The augmented sixth that comes close to a 7:4 is a residual produ=
ct
of the tuning routine and its classical functional treatment - with
possibly a few late and dramatic exceptions, among them the larger Mozart=

Fantasie and certain places in the operas - does not appear to be relate=
d
to its specific intonation, that is, the augmented sixth is not a unique
feature of the keys of D and G or G and C. =


Nor would one expect it to be: if the seventh is so well in tune, why wou=
ld
it be heard as needing to resolve? Of course, this is a rhetorical point =
as
I think Paul Erlich is in the right direction when he locates resolution =
as
a property of an ordered collection (he says scale, but this description
opens it to other orderings, e.g. by fifths). I doubt, however, that
special preference would be given to movement by like intervals,
particularly in the _contrary_ motion resolution of an augmented sixth as=

this would seem to be extremely difficult for the ear to judge accurately=
=2E =

=


I contend that if a system and the music composed for it, like meantone a=
nd
its historical repertoire (with the aforementioned exceptions) clearly
articulates a five-limit lattice with limited equivalences, the
acoustically almost-harmonic seventh will just not get parsed as such,
particularly as the would be seventh just does not appear over the
fundamentals of the most useful pitches. Of course, new music composers a=
re
free to try to find ways to introduce such relationships (as I suspect
Mozart was doing), but I am unaware of any other than Douglas Leedy who
have done so. Leedy's meantone works include strikingly articulated
introductions of these intervals, and his _Lou Harrison's Round_ is
essentially a transcription to the black keys of the harpsichord of
Harrison's slendro _Gending Pak Tjokro_.

For those interested, Fallen Leaf Press in Berkeley has published a volum=
e
of Leedy's pieces for harpsichord in Just Intonation. The harpsichord
volume in meantone has not yet been published, but Leedy did contribute a=

small article about composing in meantone to the _Perspectives_ forum on
microtonality.
=

Paul wrote:
>>}>"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?"

I answered:
>>}Yes.
>>
>>}You really do believe that? So now tell me, what is, ideally, the dividing
>>line between minor sevenths }which tend to resolve inwards and those that
>>tend to resolve outwards?

As I said, I think that the shortest path to resolution tends to be the
most "natural." A resolution of a leading tone to the tonic is journey of a
semitone, and more natural than the resolution of the sub-tonic to the
tonic, a whole tone. By extension, a resolution by 76 cents, say, is
slightly more natural than one by 117 cents (though I do agree that is just
one factor in the musical context). I believe that string players, for
example, have a natural tendency to slightly sharpen or flatten notes in
the direction of the resolution.

>>}Yes, though there are different sized tritones in a given meantone tuning
>>}of 12 keys.
>>
>>I'm not sure what you mean. In a given meantone tuning, there is only one
>>size of augmented fourth, and one size of diminished fifth. Enharmonic
>>equivalents (in the 12-tone sense) cannot be used in meantone temperament.

Handel's 16-key organ was the exception, not the rule, and the vast
majority of meantone keyboards had 12 keys. Thus musicians had to decide
whether to tune a given tritone to an augmented fourth or diminished fifth,
and so they might encounter music where they had to play a Db when they had
tuned an C#. Thus enharmonics, while theoretically distinct, had to be
treated as the same for practical purposes on most keyboards.

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) ^
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>}>>}You really do believe that? So now tell me, what is, ideally, the
>dividing
>}>>line between minor sevenths }which tend to resolve inwards and those that
>}>>tend to resolve outwards?
>
>}As I said, I think that the shortest path to resolution tends to be the
>}most "natural." A resolution of a leading tone to the tonic is journey of a
>}semitone, and more natural than the resolution of the sub-tonic to the
>}tonic, a whole tone. By extension, a resolution by 76 cents, say, is
>}slightly more natural than one by 117 cents (though I do agree that is just
>}one factor in the musical context). I believe that string players, for
>}example, have a natural tendency to slightly sharpen or flatten notes in
>}the direction of the resolution.
>
>You didn't answer my question.
>
>}Handel's 16-key organ was the exception, not the rule, and the vast
>}majority of meantone keyboards had 12 keys. Thus musicians had to decide
>}whether to tune a given tritone to an augmented fourth or diminished fifth,
>}and so they might encounter music where they had to play a Db when they had
>}tuned an C#. Thus enharmonics, while theoretically distinct, had to be
>}treated as the same for practical purposes on most keyboards.
>
>Do you have an example where more than two different augmented sixths appear
>in the notation of a piece written for a twelve-tone keyboard in meantone
>temperament?