On Fri, 11 Apr 2008 19:42:13 GMT, "Steve Latham" <llatham@[EMAIL PROTECTED]
>
wrote:
>
>"Tom K." <tkorth1@[EMAIL PROTECTED]
> wrote in message
>news:Z-edncsoOfh7JWLanZ2dnUVZ_s2tnZ2d@[EMAIL PROTECTED]
>
>> What is a "negative distance" in terms of musical intervals, Dave?
Sounds
>> a bit like like our late, unlamented "Diminished Prime"!
>
>You guys got me thinking about this again (oh brother).
>
>So:
>
>You have a Major 3rd.
>
>If you increase the distance of this interval by raising the upper note
an
>octave, or lowering the lower note an octave, you create a "compound
>interval", in this case, a Major 10th.
>
>If you consider a Perfect Unison to be a "simple" interval, than
performing
>the same operation will yield a Perfect 8ve.
>
>It stands to reason that, in order to simplify a compound interval, you
>would perform the opposite operation - decreasing the distance by an
octave.
>
>If we decrease Perfect 12th by a half step, thus making it a diminished
>12th, then "simplifying" it by decreasing the distance by an octave, we
are
>left with a diminished 5th.
>
>If we decrease a Perfect 8ve by a half step, thus making it a diminished
>8ve, then simplifying it makes - a diminished Unison.
The issue boils down to the definition of the various types of octave.
I use these definitions:
Any interval which exceeds a perfect octave is a compound interval.
Any interval which does not exceed a perfect octave is a simple
interval.
I won't deal with the perfect octave itself in this post -- it's a
whole other can of worms.
An augmented octave exceeds a perfect octave and thus is compound: a
compound augmented unison.
A diminished octave does not exceed a perfect octave and is thus a
simple interval. It inverts to an augmented unison.
>This makes a bunch of assumptions - for instance, that the unison itself
is
>considered a "distance". From a practical standpoint, two people can play
>the same note, and lines can cross, so unisons are at least practical
from
>that standpoint.
>
>Can you augment a unison? If so, it should allow for being diminished
too.
>The practicality of such would be pretty meaningless in almost any
context I
>can think of though. It's kind of like saying -1 = 1-2. It does, but
either
>way you're left in debt!
You can augment a unison.
You can't diminish a unison unless you redefine the rules to admit
negative interval values. Clearly, you can define an interesting,
useful and internally consistent system with negative values, but it
is a different system.
Here's a thought experiment: what is the inverse of a (mythic)
diminished unison? The result has to be a simple interval. By the
rules above, an augmented octave is not a simple interval -- it's a
compound augmented unison.
I handle cases such as you mention by always measuring from the lowest
*pitch* (clearly, I could have chosen to always measure from the
highest pitch, it's the consistency that counts). Thus, the interval
C-Cb is measured from Cb to C, which is an augmented unison. (I seem
to recall Dave objecting to this procedure, and I'm starting to think
about it again in terms of the doubly diminished 2nd...).
(Clearly, choosing the lowest (or highest) pitch of a perfect unison
might take some time :-))
Ian
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