Andrew Milne,* William Sethares,** and James Isomorphic Controllers and Plamondon† *Department of Music Dynamic Tuning: Invariant University of Jyväskylä Finland Fingering over a Tuning [email protected] **Department of Electrical and Computer Engineering Continuum University of Wisconsin-Madison Madison, WI 53706 USA [email protected] †Thumtronics Inc. 6911 Thistle Hill Way Austin, TX 78754 USA [email protected] In the Western musical tradition, two pitches are all within the time-honored framework of tonality. generally considered the “same” if they have nearly Such novel musical effects are discussed briefly in equal fundamental frequencies. Likewise, two the section on dynamic tuning, but the bulk of this pitches are in the “same” pitch class if the frequency article deals with the mathematical and perceptual of one is a power-of-two multiple of the other. Two abstractions that are their prerequisite. intervals are the “same” (in one sense, at least) if How can one identify those note layouts that are they are an equal number of cents wide, even if tuning invariant? What does it mean for a given in- their constituent pitches are different. Two melodies terval to be the “same” across a range of tunings? are the “same” if their sequences of intervals, in How is such a “range of tunings” to be defined for a rhythm, are identical, even if they are in different given temperament? The following sections answer keys. Many other examples of this kind of “same- these questions in a concrete way by examining two ness” exist. ways of organizing the perception of intervals (the It can be useful to “gloss over” obvious differ- rational and the ordinal), by defining useful meth- ences if meaningful similarities can be found. This ods of mapping an underlying just intonation (JI) article introduces the idea of tuning invariance, by template to a simple tuning system and scalic struc- which relationships among the intervals of a given ture, and by describing the isomorphic mapping of scale remain the “same” over a range of tunings. that tuning system to a keyboard layout so that the This requires that the frequency differences be- resulting system is capable of both transpositional tween intervals that are considered the “same” are and tuning invariance. “glossed over” to expose underlying similarities. This article shows how tuning invariance can be a musically useful property by enabling (among other Background things) dynamic tuning, that is, real-time changes to the tuning of all sounded notes as a tuning vari- On the standard piano-style keyboard, intervals and able changes along a smooth continuum. On a key- chords often have different shapes in different keys. board that is (1) tuning invariant and (2) equipped For example, the geometric pattern of the major with a device capable of controlling one or more third C–E is different from the geometric pattern of continuous parameters (such as a slider or joystick), the major third D–F-sharp. Similarly, the major one can perform novel real-time polyphonic musi- scale is fingered differently in each of the twelve cal effects such as tuning bends and temperament keys. (In this article, the term “fingering” is used to modulations—and even new chord progressions— denote the geometric pattern, without regard to which digits of the hand press which keys.) Other Computer Music Journal, 31:4, pp. 15–32, Winter 2007 playing surfaces, such as the keyboards of Bosan- © 2007 Massachusetts Institute of Technology. quet (1877) and Wicki (1896) have the property that Milne et al. 15 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 Figure 1. Thumtronics’ clude two thumb-operated forthcoming USB-MIDI joysticks and optional in- controller, the Thummer, ternal motion sensors. contains two Wicki-layout The Thummer can be held keyboards, each with 57 like a concertina (a) or note-controlling buttons. laid flat, like most key- A variety of controllers in- board instruments (b). each interval, chord, and scale type have the same (a) geometric shape in every key. Such keyboards are said to be transpositionally invariant (Keislar 1987). There are many possible ways to tune musical in- tervals and scales, and the introduction of computer and software synthesizers makes it possible to real- ize any sound in any tuning (Carlos 1987). Typi- cally, however, keyboard controllers are designed primarily for the familiar 12-tone equal tempera- ment (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 fre- quencies 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 two related tuning continua (parameter- other keyboard layouts such as those of Fokker ized families of tunings where each specific tuning (1955) or Bosanquet could have been used instead. corresponds to a particular value of the parameter) The Wicki layout can be conveniently mapped to a that exhibit tuning invariance (where, on an appro- standard computer keyboard, facilitating the explo- priate instrument, all intervals and chords within a ration of the ideas presented in this article. specified set have the same geometric shape in all of Thumtronics’ forthcoming Thummer music con- the tunings of the continuum). A keyboard that is troller (see www.thummer.com), shown in Figure 1, transpositionally invariant, tuning-invariant, and uses the Wicki note layout by default. has a continuous controller has three advantages. There are several technical, musical, and percep- First, having a single set of fingerings within and tual questions that must be addressed to realize a across all keys of any given tuning makes it easier keyboard that is both transpositionally and tuning to visualize the underlying structure of the music. invariant. First, there must be a range of tunings Second, having this same single set of interval over which pitch intervals—and therefore their fin- shapes across the tuning continuum makes it easier gerings—remain in some sense the “same.” This re- for musicians to explore the use of alternative tun- quires that differently tuned intervals be identified ings such as the various meantones, Pythagorean, as serving the same role; for instance, the 12-TET 17-TET, and beyond. Third, assigning the continu- fifth must be identified with the just fifth and the ous parameter to a control interface enables a 19-TET fifth. Said differently, tuning invariance re- unique form of expression, for example, dynami- quires that there be a number of distinguishable in- cally tuning (or retuning) all sounded notes in real tervals by which the invariance can be measured, time, where the scalic function of the notes remains because to say that two numerically different inter- the same, even as the tuning changes. vals are both “perfect fifths,” it is necessary to iden- In this article, the Wicki layout is used to con- tify a perfect fifth as an interval distinguishable cretely demonstrate the formation of the pitches from a major third, or a perfect fourth, or a dimin- and notes on a practical keyboard surface, though ished fifth, and so forth. This issue of the identity of 16 Computer Music Journal Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 Figure 1—continued (b) musical intervals is discussed in detail in the next that are not) invariant in both transposition and section (“Intervals”) by contrasting rational and or- tuning. The tuning continuum pictured in Figure 2 dinal modes of interval identification. The rational provides the primary example of this article. It be- mode is determined by the correspondence of an in- gins at 7-TET, and by varying the size of the perfect terval to a low-ratio JI interval, and the ordinal fifth, it moves continuously through various mean- mode is determined by the number of scale notes an tone tunings, 12-TET, Pythagorean, and many other interval spans. tunings, ending at 5-TET, while retaining fingering Second, there must be a tuning system that is itself invariance throughout. Finally, a musical example transpositionally invariant with regard to both forms illustrates static snapshots of the dynamic retuning of identification. This requires that each and every process. note in the system has identical intervals above and below, and that the presumed temperament-mapping of JI to this tuning system is consistent. Such a tun- Intervals ing system is called a regular tuning system, and the embodiment of a such a temperament-mapping This section investigates how intervals are identi- in a regular tuning system is called a regular tem- fied and distinguished, and it discusses criteria by perament. These are defined more fully in the sec- which two numerically different intervals may be tion “Tuning Systems and Temperaments.” said to play analogous roles in different tuning sys- Third, given such a tuning system, it is necessary tems. Intervals may be identified—and therefore to define useful sets of scales. The section entitled discriminated—in at least two ways, here defined as “Scales” focuses on those scales known as MOS or the rational and the ordinal. well-formed, which have a number of musically advantageous properties. Fourth, it is necessary to layout-map the regular Rational Identification temperament to a keyboard or button-field in a manner that maintains transpositional and tuning The rational mode of identification is presumed to invariance.