Wellformedness, Myhill’s Property and Maximal Evenness – How to
Generalize Them for NonPythagorean Scales?
The three
properties mentioned in the title (Carey and Clampitt 1989, Clough and Myerson
1985, Clough and Douthett 1991) comprise a logical system that provides an
exhaustive theoretical description of ‘Pythagorean’ scales – scales generated
by a single interval of the fifth.
However, many musically important structures cannot be modeled as
Pythagorean scales. Clampitt’s 1997
concept of ‘pairwise wellformed (PWWF)’ scales is a very powerful
generalization of the previous logically neat but applicabilitywise more
limited properties and is able to accommodate various important tone
systems. In this
paper, I study two alternative generalizations of the three properties. Consider the A harmonic minor scale in just
intonation: A – B – C^{1} – D – E – F^{1} – Gsharp^{1}
(the superscripts trace the syntonic comma corrections). Two different perspectives can be adopted to
analyze its stepinterval structure.
The first perspective focuses on the generating patterns of the intervals. Under such perspective, the step pattern of
the scale contains four different steps: bacbada. It is not PWWF but it has a weaker property
that I call ‘quasi pairwise wellformedness (QPWWF)’: only two out of three
possible pairwise projections result in wellformed words. The second perspective preassumes an
embracing chromatic space and investigates the steps through their chromatic
sizes. In our example we can assume the
usual chromatic scale in just intonation, which leads us to the step pattern
2122131. Under the other perspective, I
propose the property of ‘quasi maximal evenness (QME)’ that requires generic
intervals to have chromatic lengths of up to three consecutive integers. (This is a natural generalization of the
original Clough and Douthett’s 1991 definition of maximal evenness that
required generic intervals to have chromatic lengths of up to two consecutive
integers.) It turns out that QPWWF and
QME are tightly connected in the case of fundamental generated tone systems,
which is a category that embraces a very wide range of musically relevant
scales. The theoretical system is
illustrated on several analytical examples taken from the music of Mozart,
Chopin, and Debussy.
