Isomorphic Controllers and Dynamic Tuning: Invariant Fingering over a Tuning Continuum

In the Western musical tradition, two pitches are generally considered the "same" if they have nearly equal fundamental frequencies. Likewise, two pitches are in the "same" pitch class if the frequency of one is a power-of-two multiple of the other. Two intervals are the "same" (in one sense, at least) if they are an equal number of cents wide, even if their constituent pitches are different. Two melodies are the "same" if their sequences of intervals, in rhythm, are identical, even if they are in different keys. Many other examples of this kind of "sameness" exist.

It can be useful to "gloss over" obvious differences if meaningful similarities can be found. This article introduces the idea of tuning invariance, by which relationships among the intervals of a given scale remain the "same" over a range of tunings. This requires that the frequency differences between intervals that are considered the "same" are "glossed over" to expose underlying similarities. This article shows how tuning invariance can be a musically useful property by enabling (among other things) dynamic tuning, that is, real-time changes to the tuning of all sounded notes as a tuning variable changes along a smooth continuum. On a keyboard that is (1) tuning invariant and (2) equipped with a device capable of controlling one or more continuous parameters (such as a slider or joystick), one can perform novel real-time polyphonic musical effects such as tuning bends and temperament modulations—and even new chord progressions—all within the time-honored framework of tonality. Such novel musical effects are discussed briefly in the section on dynamic tuning, but the bulk of this article deals with the mathematical and perceptual abstractions that are their prerequisite.

How can one identify those note layouts that are tuning invariant? What does it mean for a given interval to be the "same" across a range of tunings? How is such a "range of tunings" to be defined for a given temperament? The following sections answer these questions in a concrete way by examining two ways of organizing the perception of intervals (the rational and the ordinal), by defining useful methods of mapping an underlying just intonation (JI) template to a simple tuning system and scalic structure, and by describing the isomorphic mapping of that tuning system to a keyboard layout so that the resulting system is capable of both transpositional and tuning invariance.

Background

On the standard piano-style keyboard, intervals and chords often have different shapes in different keys. For example, the geometric pattern of the major third C–E is different from the geometric pattern of the major third D–F-sharp. Similarly, the major scale is fingered differently in each of the twelve keys. (In this article, the term "fingering" is used to denote the geometric pattern, without regard to which digits of the hand press which keys.) Other playing surfaces, such as the keyboards of Bosanquet (1877) and Wicki (1896) have the property that each interval, chord, and scale type have the same geometric shape in every key. Such keyboards are said to be transpositionally invariant (Keislar 1987).

There are many possible ways to tune musical intervals and scales, and the introduction of computer and software synthesizers makes it possible to realize any sound in any tuning (Carlos 1987). Typically, however, keyboard controllers are designed primarily for the familiar 12-tone equal temperament (12-TET), which divides the octave into twelve logarithmically equal pieces. Is it possible to create a keyboard surface that is capable of supporting many possible tunings? Is it possible to do so in a way that analogous musical intervals are fingered the same throughout the various tunings, so that (for example) the 12-TET fifth is fingered the same as the just fifth and the 17-TET fifth? (Just intervals are those consisting of notes whose constituent frequencies are related by ratios of small integers; for example, the just fifth is given by the ratio 3:2, and the just major third is 5:4.)

This article answers this question by presenting examples of...