Whilst structural properties of musical scales have been extensively studied within the field of diatonic theory, they have only recently been specifically discussed in the context of non-octave-repeating scales. Given that many scale properties rely on the existence of a generative interval, these are only revealed in non-octave-repeating scales when considering the full octave-span of the scale as the interval of repetition (Crowley & Gómez, 2023). Yet, due to the manner in which most non-octave-repeating scales employed in musical practice are constructed, these properties do not, generally, seem to apply to them.

Since the inclusion of a fifth-repeating scale in the 9th century Musica Enchiriadis treatise to present day, the most common method of constructing non-octave-repeating scales has been by linking together similar scale-fragments, themselves smaller than the total range of the scale –a process which can be viewed as analogous to Messiaen's modes of limited transposition–, resulting in symmetrical scales of a varying octave-span. Thus, being of a symmetrical nature, several properties attributed to scales such as the diatonic collection do not apply in this context.

The present study shows that considering the span of the actual scale-fragments that constitute a given non-octave-repeating scale as the interval of periodicity, as opposed to that of the total scale, reveals that many widely employed non-octave-repeating scales are indeed endowed with relevant scale properties found in octave-repeating scales, such as the major scale. This is formalized and a specific interval class vector is developed for such collections, revealing, for instance, how scales such as the Enchiriadis scale or that of the Znamenny Rospev employed in Byzantine chant –as well as in the context of Jewish Steiger or Persian Dastgah– are, in fact, maximally even, deep, or well-formed.

In the work of contemporary composers, symmetric non-octave-repeating scales have been favoured for offering the potential of working within a highly chromatic context –most contain the full chromatic aggregate– without forgoing certain advantages inherent in traditional scales, such as modulation. Considering common pitches as opposed to pitch-classes, Weston (2012) regards non-octave-repeating scales which show a high variety regarding common tones under transposition as especially advantageous in this sense. The present study shows that Weston's scales are in fact deep scales and, being therefore endowed with unique multiplicity, do not only show a high variety, but a but a maximal variety of common tones under transposition, thus providing a systematic way of arriving at such constructs.

Analysing symmetric non-octave-repeating scales in this way not only reveals the existence of properties not previously considered, but also shows a direct link between these theoretical properties and musical practice, revealing how scales which predate the diatonic collection share many of its properties –an argument in favour of their intuitive appeal– and how these properties may prove interesting or advantageous in the context of contemporary composition.

References

Crowley, E., & Gómez-Martín, F. (2023). Structural properties of multi-octave scales. Journal of Mathematics and Music, 17(2), 291-318.

Weston, C. A. (2012). Some properties of non-octave-repeating scales, and why composers might care.