Re: Descending aug and dim intervals

"paramucho" <paramucho@[EMAIL PROTECTED]
> wrote in message 
news:4804985f.36174195@[EMAIL PROTECTED]
> 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...).

I think the common or garden (singly) diminished 2nd is the perhaps the 
pivotal interval to worry about.

C to Dbb

or perhaps F# to Gb makes it seem more "ordinary".

I think it resonable to assume a system exists in which Gb is a lower
pitch 
than F#.
In which case moving *up* a letter F->G corresponds, in this case, to
moving 
down in pitch.

If you think of intervals as absolute distances, then is this diminished
2nd 
larger than a perfect unison?  Well it can't be smaller!

The answer is that intervals are *not* absolute distances.   If they were,

you could compare them and always say whether one was bigger than the
other 
or whether they were equal.    But in fact saying  (eg) which of an 
augmented 4th or a perfect 5th is bigger is not really possible (without 
further definitions and/or data).    The diminished 2nd vs perfect unison 
just highlights the problem more.

Mathematically what is needed is a mapping of intervals onto numbers, such

that when intervals are added, the numbers onto which they map also add
up.

As I have said, two sensible mappings are
     (a) the semitone count
     (b) the logarithm of the frequency ratio

In both cases a perfect unison gives zero, and an undisputed downward 
interval gives a negative result.  The absolute value of the number can be

defined to be a measure of the interval.   These two only agree with each 
other (on which intervals are bigger or equal) in equal temperament.

In the end it is all a matter of *definition*.    Intuitive musical ideas 
start to come unstuck with unintuitive intervals (like the diminished 2nd)

and we (those of us who are mathematically inclined) need a definition
which 
makes sense.    For the frequency definition F# to Gb may generate a 
negative number in some temperaments, but that does not imply an 
inconsistency - just a different mapping from the semitone count in which
it 
generates zero.

Once one realises that it *is* a matter of definition one can *define* a 
"diminished unison" as going from F# to F (down one semitone) even though 
the concept sounds funny.    If going from F# to Gb can go down in pitch,
on 
has more or less divorced pitch ordering form intuitive ordering anyway,
and 
so it is no big leap (sorry!) to define going up by a diminished unison as

the same as going down by an augmented one.   And it is all completely 
self-consistent.  Just not intuitive.     And because of the latter, I 
appreciate why people don't like the idea :-)

Dave
-- 
David Webber
Author of 'Mozart the Music Processor'
http://www.mozart.co.uk
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