*Simon on Lewin*

##### This page is "under construction," but will in the future feature an ongoing
discussion of
David Lewin's *Generalized Musical Intervals and Transformations*, a fairly detailed
application of mathematics to the subjects of intervals and pitch class sets. As well as assisting
in my own understanding of his work, I hope that my observations will be of interest to others
and perhaps generate group discussion or possible avenues of analysis for term
papers.

[1] Lewin's first chapter is an attempt to devise a Generalized
Interval System, which
in effect, uses basic algebra to organize a given pitch space. Lewin treats each musical
space (major key, minor key, ocatonic, 12-tone, etc..) as an individual, unique set of
notes to which the same notation can be applied. Diatonic intervals are treated equally
regardless of tone and semitone, so in the c-major scale the interval from c to c is O, c to
d is 1, c to e is 2, c to f is 3 etc...

[2] Lewin's notation for intervals is int(s,t) where s and t are pitches in
a given
system. Therefore, using the above example of Diatonic C major: int(c,c)=0, int(c,d)=1,
int(c,e)=2, int(c,f)=3 etc... Given that the pitch space has octave equivalence, this has the
effect of working in numeric systems other than base 10. The major scale is base 7, the
octatonic is base 8, the pentatonic is base 5 etc.

[3] The system's first obvious strong point is the ability to equate
different pitch
spaces. Play or sing the following melodies:

At a glance there is an obvious similarity to the melodic contour, however, the
intervals don't make sense with each other. If we apply Lewin the discrepancies become
clear. Melody a is in C Major, b is in G Octatonic, and c is in D minor. Lewin's interval
formula int(s,t) gives us the following sets of interval class numbers:

a=(2,1,-2,3,1,2,-1,1) b=(2,1,-2,3,1,2,-1,1)
c=(2,1,-2,3,1,2,-1,1)
In other words they are the same melody transposed into three different pitch spaces.
One of the nicest things about this system is it eliminates the counterintuitive results that the
current system of musical intervals creates. For example a third plus a third equals a fifth
(3+3=5!!??). In Lewin's system this example becomes 2+2=4 where int(c,e)=2, int(e,g)=2, and
int(c,g)=4.

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