Marek Žabka

Well-formedness, Myhill’s Property and Maximal Evenness – How to Generalize Them for Non-Pythagorean 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 well-formed (PWWF)’ scales is a very powerful generalization of the previous logically neat but applicability-wise 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 – G-sharp1 (the superscripts trace the syntonic comma corrections).  Two different perspectives can be adopted to analyze its step-interval 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 well-formedness (QPWWF)’: only two out of three possible pairwise projections result in well-formed words.  The second perspective pre-assumes 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.