Re: Descending aug and dim intervals

"Tom K." <tkorth1@[EMAIL PROTECTED]
> wrote in message 
news:E8-dnVay7JzSh2PanZ2dnUVZ_qmlnZ2d@[EMAIL PROTECTED]
> Not following you here, Ian.  An interval is just a measured distance 
> whether reckoned up or down.  By the same token, if I'm driving from New

> York to Miami, am I supposed to measure the mileage from south to north 
> instead of the north-south route I'm actually traveling?  And will the 
> results differ depending on which method I use?

The really interesting thing is that and interval is not "just a measured 
distance" :-)

One can associate a distance measure with an interval, not in a unique
way. 
In fact two such measures are in common use:

The number of semitones in the interval.
The logarithm of the frequency ratio of the notes forming the interval.

Both are additive but they are only equivalent in equal temperament.

And give that they are additive (with 0 being associated with a unison)
they 
are also directional: nothing breaks down if you associate a negative 
distance with a downward interval.

Intervals themselves add in a much more interesting way: a minor 7th plus
a 
minor 7th is a minor 13th and so on.   And in adding intervals both the 
distance measures above also add.    (But while you can find either
distance 
measure from the interval, you can't get the interval from the distance 
measure: there is enharmonic ambiguity.)

In driving from New York to Miami and back again you may be interested in 
the net distance achieved.  It is zero.   This is found additively using a

signed measure of the distance.  You may also be interested in the fuel 
consumption, in which case you need to add the absolute distances moved.

In intervals the absolute distances are actually even less interesting
than 
in driving.   If you go down an octave and up a 4th, then the net interval

moved is down a 5th.    This is something one may validly wish to know. 
The 
sum of the absolute intervals is meaningless.

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