[Smt-talk] Altered pitch, preserved contour

[Thomas Noll]

Dear List,
A useful mathematical means to study contour might be diastematic  
matrices or sub-structures thereof.
A diastematic matrix of a melody with n tones is an nxn-matrix whose  
entries at index (i,k) are +1, 0, -1 depending on the direction of  
the interval between the ith and kth note of the melody.
The matrix is antisymmetric by construction. A sub-structure of  
interest is the upper secondary diagonal whose entries refer to the  
"surface diastematic contour", i.e. to the up/down pattern of the  
successive intervals only. The full matrix considers all internal  
intervals of the melody in question. Chantal Buteau has  
systematically used these diastematic contour paradigms in her  
computational approach to some of Rudolph Reti's ideas.
best wishes
Thomas Noll

Am 20.05.2009 um 20:02 schrieb Stephen Guy Soderberg:

> Over-mathematizing can be a fatal error, Dmitri (or at least  
> unnecessarily limiting).  In my opinion, multiplication is simply a  
> subset of the (compositionally) much more interesting general idea  
> being discussed.  But since you bring up the math, I'm reminded of  
> some wise words from Hermann Weyl in his little book, Symmetry:
>
> "If nature were all lawfulness then every phenomenon would share  
> the full symmetry of the universal laws of nature....  The mere  
> fact that this is not so proves that _contingency_ is an essential  
> feature of the world."
>
> I would only add:  ... and what makes it at all worthwhile to be  
> alive.
>
> Cheers,
>
> Steve
>
>
> Stephen Soderberg
> Music Division
> Library of Congress
>
>
>>>> Dmitri Tymoczko <dmitri at Princeton.EDU> 5/20/2009 9:46 AM >>>
>
> You're actually talking about something a bit more specific than just
> "preserving contour" -- as music theorists use the term "contour,"
> this usually refers to just the ordinal ranking of notes in pitch
> space.  So (C4, C#4, B3) and (C4, B6, Db3) have the same contour,
> even though the ratio of their intervals is different.  This is
> because they both exhibit the sequence (middle, high, low).
>
> Interestingly, the relevant geometrical space is the (n-1)-
> dimensional sphere, which represents equivalence classes of n-note
> pitch sequences under positive pitch-space multiplication.
>
>
>
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*********************************************************
Thomas Noll
http://flp.cs.tu-berlin.de/~noll
noll at cs.tu-berlin.de
Escola Superior de Musica de Catalunya, Barcelona
Departament de Teoria i Composició
Tel (priv.):   +34 93 268 75 19
Tel (mobil): +34 66 368 12 02

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