A Mathematical Model for Optimal Tuning Systems

A Mathematical Model for Optimal Tuning Systems Larry Polansky, Daniel Rockmore, Micah K Johnson, Douglas Repetto, and Wei Pan In this paper we propose a mathematical framework for the optimiza tion of tuning systems. We begin with an informal definition of "tuning system." We then propose five general constraints that seem common to their evolution. The central idea of this paper is the quan tification of those constraints in terms of a set of numerical parameters. Given a choice of parameter values we use appropriate optimization methods to produce an optimal tuning for a specific set of values. Finally, we consider some historical and Javanese tunings from this per spective, and use the framework to generate a few examples of novel tuning systems. 70 PerspectivesofNew Music Tuning systems A tuning system is a set of intonations for intervals or pitch classes. A tuning systemmight be used by a musical culture, group of musicians, or even a single composer. Such a system may also serve as an abstrac tion, or model, for the derivation of any number of related systems and sub-systems. Smaller, functional subsets of pitches, such as scales, are extracted from a larger tuning system for specific musical purposes. Examples include the formation of major and minor (and other) scales from 12 tone equal temperament (12-ET), or the various Javanese pathet (manyura, nem, sanga, etc.) which are subsets of slendro and pdog tuning (Perlman, 40^3). Tuning systems are neither static nor rigid. Although most musical cultures need some agreed-upon standard formusicians to tune their instruments and sing to, tuning systems evolve and fluctuate over time and in space (i.e., historically and geographically) and vary stylistically within musical practice. Most musical cultures have some standard or canonical tuning, articulated in either oral or inwritten traditions. Such a systemmay often be canonized in a specific instrument, like the piano, or the gender inCentral Javanese music which may hold the tuning for an entire gamelan. We are interested here in a formal framework for tuning systems themselves, not the intricate (but no less important) musical variations and manifestations of such a system.Musicians deviate freely and artis tically from standardized tunings in fluid, complex ways. For example, the many musical genres that share the nomenclature and intervallic template of 12-ET (like jazz and blues) are intonationally diverse. But the complexities and nuances of intonational usage associated with established tuning systems are beyond the scope of this paper. Culturally- and historically-specific constraints may influence the formation of tuning systems.A new system that resembles a pre-existing one isoften desirable, as inCentral Javanese gamelan tunings which ref erence well-known gamelans.1 A tuning systemmight adapt over time in the performance of an evolving body ofmusic. This latter consideration is an important factor in the historical evolution of tunings inEuropean music over the past millennium, including just intonations (JIs), mean tones, well-temperaments (WTs), equal temperaments (ETs), and twentieth-century experimental tuning systems. From a formal, abstract perspective tuning systems can be seen as specific attempts to solve certain problems, and understood as the resolution of a particular set of intonational constraints. The genesis of A Mathematical Model forOptimal Tuning Systems 71 these problems?whether they emanate from issues of culture, economy, convenience, aesthetics, or some complex combination of all of these?is another issue. Our focus is on a relatively small set of important factors common to the creation of tuning systems,whose natural quantitative formulation enables the use of optimization techniques for analysis.We believe this approach has important implications towards a deeper understanding of tuning and even musical style. Five Constraints Tuning systems through history and across cultures are the result of a set of complex compromises aimed at simultaneously incorporating some or all of the following structural constraints: 1. Pitch set. use of a fixed number of pitches (and consequently, a fixed number of intervals); 2. Repeat factor, use of a modulus, or repeatfactor1 for scales, and for the tuning system itself (e.g., an octave); 3. Intervals', an idea or set of ideas of correct or "ideal" intervals, defined in terms of frequency relationships; 4. Hierarchy, a ranking of importance for...