Re: Descending aug and dim intervals

"Tom K." <tkorth1@[EMAIL PROTECTED]
> wrote in message 
news:Z-edncsoOfh7JWLanZ2dnUVZ_s2tnZ2d@[EMAIL PROTECTED]
>> 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.
>>
>
> Sorry, being a keyboard player, I tend to think more in ET.
>
>> 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.
>
> What is a "negative distance" in terms of musical intervals, Dave?

Sorry - I'm talking very loosely - a more accurate term would be "negative

displacement".

>  Sounds a bit like like our late, unlamented "Diminished Prime"!

Not late, not lamented - but let that one rest for the moment :-)

> I guess my point is that intervals may be undirected (harmonic) or 
> directed (melodic), and directed intervals may be measured either up or 
> down, forwards or backwards without any difference in the result.  Of 
> course, with some performers, there may also be misdirected 
> intervals...:-)
>
>> 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.)
>
> Unless you are assuming ET and measuring in semitones.  Then your 
> m7+m7=m13 becomes 10+10=20.

In terms of semitone-count that works in any temperament.   The semitone 
measure of m7+m7=m13 is 10+10=20.    But that's a many to one mapping, not

an equivalent statement.   For example:

maj7+dim7=min13
maj7+aug6=aug12

also both map on to 10+10=20 in semitones.  The structure of intervals is 
much richer than just counting semitones.

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