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. The layout-mappings described in the occur primarily for harmonic intervals,namely, those section “Button-Lattices and Layouts” translate formed from simultaneously sounded notes. It pre- the generating intervals of the tuning to the key- sumes that just intervals that are perceived as con- board surface. It is shown that transpositional sonant act as perceptual and cognitive landmarks (a invariance is identical to linearity of the layout- template) against which sounded intervals can be mapping. Successive sections then provide ex- mentally compared and identified. This is reason- amples of keyboard layouts that are (and others ably uncontroversial for orchestral instruments

Milne et al. 17

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 Figure 2. The syntonic ra- continua (compiled from tional continuum and information in Barbour MOS:5+2 ordinal contin- 1951, Lorentz 2001, uum (both defined later). Sethares 2004, and the The figure also shows the Huygens-Fokker Founda- historical and ethnic us- tion Web site at ages of many notable tun- www.xs4all.nl/huygensf). ings found within these

is not just, but is within its range of rational identi- fication, it is called a tempered interval.

Ordinal Identification

The ordinal mode of identification is presumed to occur primarily for melodic intervals (formed from successively sounded notes). It presumes that when an interval is played as part of an aesthetically con- sistent or conventionalized scale, it is identified by the number of scale notes (or steps) it spans. For ex- ample, Wilson (1975) writes that “our perception of Fourth-ness is not just acoustic, i.e., 4/3-determined, it is melodic and/or rhythmic-influenced to a high degree” (p. 1). Like rational identification, this is an indexical signification but one that requires the presence of a scalic background (i.e., context) to give meaning to “second-ness,” “third-ness,” “fourth- ness,” etc. For a scale to serve as a background, it must have intrinsic aesthetic consistency (i.e., be perceived as somehow complete and “correct”) and/or be conventionalized (i.e., made familiar with harmonic spectra. For inharmonic, computer- through repetitive use). generated sounds, inharmonic bells, and non- The diatonic scale can serve as a scalic back- Western instruments such as the metallophones of ground for common-practice music. For example, the Indonesian gamelan, these sensory consonances against the background of a C-major (diatonic) scale, may occur at different intervals (Sethares 2004), and the intervals C–D, D–E, E–F, F–G, and so on, are so other templates may be more appropriate. heard as “seconds” because they each span two ad- For sounds with harmonic spectra, when a sounded jacent scale notes; the intervals C–E, D–F, E–G, F–A, interval is tuned close to a small-integer-ratio JI in- and so forth, are heard as “thirds” because they each terval (such as 3:2 or 5:4), it may be heard as a repre- span three successive scale notes. sentation of that ratio. Using semiotic terminology, the sounded interval is an indexical signifier of the just ratio it approximates. For example, when an in- Dual Identification terval is tuned to 702 cents (the closest integer value to 3:2), it is likely to be heard as a representa- The two modes of identification overlap: the ra- tion of the 3:2 just fifth. As the interval’s tuning is tional mode plays a part in the identification of moved away from 702 cents, it gradually moves to a melodic intervals, the ordinal in the identification state where it is likely to be heard as an imperfect of harmonic intervals. For example, a melodic inter- representation of 3:2. (It will sound more or less val of approximately 3:2 will usually be heard as “in “out of tune.”) As the tuning is moved still further tune” or “out of tune” according to its proximity to from 702 cents, the perceived interval will eventu- this just interval; similarly, a harmonic interval ap- ally no longer represent 3:2 (and not even an out-of- proximating 3:2 will, in a traditional diatonic con- tune 3:2). At this point, the just interval is no longer text, be heard as spanning five notes of the scale. signified by the sounded interval. When an interval Furthermore, in real-world music it is not always

18 Computer Music Journal

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 possible to make a strict distinction between har- interval can be identified and therefore discrimi- monic intervals and melodic intervals. An arpeggio nated from other intervals has “fuzzy” boundaries is at least partially a harmonic structure. The bass that are context-dependent. note of a “stride bass” pattern, which is sounded only for the first and third beats of a bar, cognitively grounds the rest of the bar. Counterpoint, which is Tuning Ranges of Invariant Identification the interweaving of many melodies into a coherent harmonic structure, blurs the distinction between The previous discussion suggests that it may be ad- melody and harmony. vantageous to define the tuning range over which an Thus the rational and ordinal modes of identifica- interval preserves both its rational and ordinal iden- tion are intertwined. Indeed, in Western tonal mu- tity as that interval’s tuning range of invariant sic, there is a consistent linkage between the number identification. Because there are two modes by of steps an interval contains and its harmonic ratio. which an interval can be identified, there are two For example, the interval that spans three scale relevant tuning ranges to be considered that them- notes (i.e., the third) is commonly an interval that is selves depend on a listener’s innate abilities, expe- close to 5:4 or 6:5 (hence the names major third and riences, and training. Musical context is also minor third). Similarly, the interval that spans four important; the tuning range of invariant rational scale notes (i.e., the fourth) is typically an interval identity may also change based on the spectrum that is close to 4:3. Furthermore, where these links and/or timbre of the sounds (Sethares 2004). For differ, the interval often has a tonally dissonant these reasons, the final judgment as to the specific function that requires resolution to a more stable tuning boundaries may best lie in the hands of the interval. (For example, an augmented second com- artist and not the theorist. monly resolves to a perfect fourth; a diminished Nonetheless, in the presence of a scalic back- fifth commonly resolves to a major third.) ground, invariant ordinal identification can be pre- In conventionalized musical systems such as cisely bounded; furthermore, given a scale in a Western tonal music, where particular step sizes regular tuning system (as defined subsequently), the and harmonic ratios are consistently associated, size of every interval in the scale is determined by ordinality can (by association) symbolically signify the values of the generating intervals. This implies ratio, and ratio can (by association) symbolically that the range of generator tunings over which all of signify ordinality. This means that, within such a that scale’s intervals can be retuned but still main- conventionalized context, the tuning ratio over tain their ordinal identity—that scale’s ordinal con- which an interval can still signify a given just ratio tinuum—is limited by the points at which one or may be wider than expected if it were judged by more of that scale’s steps shrink in size to zero (or analyzing harmonies isolated from the musical con- equivalently, the points at which two of the scale text. In common practice, for example, the conven- steps “cross”). tional association of “5:4-ness” and “third-ness” In a regular temperament (as defined in the next means that a melodic third can be tuned very wide section), the sizes of all the intervals that are ca- but still signify 5:4. For example, the ultra-sharp pable of being rationally identified are controlled by supra-Pythagorean major thirds (greater than 408 the values of its generating intervals. This implies cents) that are sometimes used by string players for that the range over which all of that temperament’s expressive intonation (Sundberg et al. 1989) may be identifiable intervals can be retuned but still main- harmonically uncomfortable, but they still signify tain their rational identity—that temperament’s the same musical interval as the just quarter- rational continuum—is delimited by context and comma meantone thirds of 386 cents. (They are, subject-dependent “fuzzy” boundaries. after all, different expressions of the same notated Figure 2 shows the precisely bounded continuum interval.) Thus the tuning range over which a given for the twelve-note chromatic scale generated by

Milne et al. 19

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 fifths and octaves. Also shown on this chart is a magnitude of one or more of the generating inter- conjectured “fuzzy” boundary for the rational con- vals to a control interface provides a convenient tinuum of the regular temperament defined by the means to “navigate” the tuning continuum. Partic- syntonic comma (which is explained in the next ular values within this continuum may produce section). This conjectured range has been estimated some intervals that approximate JI intervals and so by assuming that only the common practice conso- are rationally identifiable; we might consider, there- nances are rationally identifiable, and that their ra- fore, that there has been a mapping of JI intervals to tional identification switches from one common that tuning system. practice consonance to another at the tuning that is For such a temperament-mapping to be transposi- midway between their just tunings. In common- tionally invariant, it must be linear, though it need practice music, these two continua are convention- not be invertible (i.e., it need not be one-to-one). ally associated, though the ordinal continuum is The embodiment of such a temperament-mapping wider than the rational. in a suitable tuning system is called a regular tem- perament, and it can be characterized by the small JI intervals called commas that are tempered to Tuning Systems and Temperaments unison (Smith 2006). This means that a regular temperament is characterized by its temperament- A tuning system is defined here to be a collection of mapping, not its tuning, so any given temperament precisely tuned musical intervals. There are many has a range of suitable tunings. To be concrete, two

ways in which the intervals may be chosen: A “bou- or more intervals a1, a2,...,an are said to be multi-

tique” tuning system might have all of its intervals plicatively dependent if there are integers z1,

z1 z2 zn chosen arbitrarily, and another tuning system might z2,...,zn, not all zero, such that a1 a2 ... an = 1.

be generated by a predefined mathematical proce- If there are no such zi, then the ai are said to be mul- dure (McLaren 1991). The regular tunings form one tiplicatively independent. The rank of a tuning sys- class of tuning systems in which all of the intervals tem is the number of multiplicatively independent are generated multiplicatively from a finite number intervals needed to generate it. A regular tempera- of generating intervals (or generators). Such tuning ment typically has lower rank than the JI system systems ensure that every given note has the same that is temperament-mapped to it (i.e., the mapping set of intervals above and below it as every other is non-invertible). When the temperament-mapping note in the system; this means that regular tuning loses rank, all intervals can no longer be just. How- systems are inherently transpositionally invariant. ever, as long as there is a range of generator values An example regular tuning system is 3-limit JI (also over which the intervals are correctly rationally known as Pythagorean tuning), which has two gen- identified, the regular temperament can be consid-

erators G1 = 2 and G2 = 3, and consists of all prod- ered to be valid. i j i × j ucts of the form G1 G2 = 2 3 , where i and j are This is analogous to the way a projection of the integers. Thus the intervals of 3-limit JI can all be three-dimensional surface of a globe to a two- found in a series of stacked just perfect fifths, allow- dimensional map inevitably distorts distance, area, ing for octave equivalence. In general, a regular tun- and angle. However, so long as the countries have

ing is characterized by n generators G1 to Gn and identifiable shapes, the projection can be considered

i1 i2 in consists of all intervals G1 G2 ... Gn , where the i1, valid. Different map projections result in different

i2,...,in are integer-valued exponents. distortions, and some map projections are more or Altering the tuning of a generator affects the tun- less suitable to specific purposes. Some projections ing of the system in a predictable way. For example, (such as the Mercator Projection) have the virtue of –1 –1 × the perfect fifth in 3-limit JI is G1 G2 (i.e., 2 3). If wide familiarity; so it is also with temperament- a (non-JI) regular tuning is created by changing the mappings (such as those that lead to 12-TET).

value of G2, the value of the fifth, and all other in- For example, 3-limit JI can be temperament- tervals, changes in a patterned way. Assigning the mapped to a one-dimensional system using an ap-

20 Computer Music Journal

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 a b T T propriate comma G1 G2 = 1, where a and b are and any note (i, j, k) is mapped to (i – 4k, j + 4k) . integers. One notable tempering retains the octave For example, a just major third 5/4 = 2–2 30 51 is 19/12 ≈ T G1 = 2 and tempers G2 to 2 3, which requires represented by the vector (–2, 0, 1) . This is that a = 19 and b = –12, resulting in the familiar 12- temperament-mapped by G to (–6, 4)T, which repre- TET. This comma can be interpreted musically by sents the tempered interval α–6β4. If α were tem- saying that in this temperament, 19 octaves minus pered to 2 (no change in the octave) and β were twelve equal-tempered perfect twelfths equals a tempered to 230/19 (the 19-TET twelfth), then α–6β4 = unison. A second example is in 5-limit JI, which 2–62120/19 ≈ 1.24469 ≈ 5/4 is the 19-TET approxima- consists of all intervals of the form 2i 3j 5k, where i, tion to the just major third. Similarly, if α were tem- j, and k are integers. This can be reduced to a two- pered to 2 and β were tempered to 3 × (81/80)–1/4 (the α–6β4 dimensional regular temperament by choosing G1 quarter-comma meantone fifth), then = 5/4 is

(typically near 2), G2 (near 3), and G3 (near 5) so that the justly tuned major third found in the quarter- comma meantone tuning. abc= GGG123 1 (1) where a, b, and c are specified integers. Equation 1 defines a comma that is tempered to unison. The Syntonic Rational Continuum well known syntonic comma (which has an untem- pered tuning of 81/80) is the special case where a = The rational continuum of a rank-r regular tem- –4, b = 4, and c = –1; tempering the generators so perament is here defined as that (r-dimensional) that Equation 1 holds, produces various tunings of range of generator tunings within which the ra- the syntonic temperament (as illustrated in Figure 2). tional identity of all rationally identifiable intervals The comma can be solved for one of its terms as is maintained. The rational continuum of the syn- –a/c –b/c G3 = G1 G2 , and a typical interval of the regular tonic temperament is therefore called the syntonic temperament can be written in terms of two gener- rational continuum. As described previously, the α GCD(a,c)/|c| β GCD(b,c)/|c| ating intervals = G1 and = G2 tuning range of this continuum is context- where GCD(n, m) is the greatest common divisor of dependent and not easily specified a priori. α|c|/GCD(a,c) β|c|/GCD(b,c) n and m. Thus G1 = , G2 = , G3 = The syntonic rational continuum extends beyond α–aSIGN(c)/|GCD(a,c)| β–b/SIGN(c)|GCD(b,c)|, and a typical inter- the narrower range implied by “meantone,” which i j k val G1 G2 G3 of the JI is temperament-mapped to usually refers to the range of syntonic tunings pro- α(ci–ak)/|GCD(a,c)| β(cj–bk)/|GCD(b,c)|. In matrix notation, the viding reasonably pure-sounding tunings (for sounds vector (i, j, k)T is temperament-mapped by with harmonic spectra) and/or which have an estab- lished historical use—a range of approximately 19- ⎡ c −⋅acSIGN()⎤ TET to 12-TET. (See Figure 2.) ⎢ 0 ⎥ ac R = ⎢GCD(,)ac GCD(,) ⎥ (2) ⎢ c −⋅bcSIGN()⎥ 0 ⎣⎢ GCD(,)bc GCD(,)bc ⎦⎥ Scales to A scale is here defined to be a subset of a tuning sys- T ⎡i c−− akSIGN() c j c bkSIGN() c ⎤ tem used for a specific musical purpose. When using ⎢ ⎥ a rank-2 tuning system with generators α and β, a ⎢ ⎥ ⎣ GCD(ac , ) GCD(bc , ) ⎦ scale can be simply constructed by stacking integer powers of β and then reducing (dividing or multiply- where T is the transpose operator. Thus, the ing by integer powers of α) so that every term lies temperament-mapping is linear and (typically) between 1 and α. This is called an α-reduced β-chain, loses rank. For the syntonic comma, and it produces a scale that repeats at intervals of α; ⎡1 0 −4⎤ α = 2, representing repetition at the octave, is the R = ⎢ ⎥ (3) ⎣0 1 4 ⎦ most common value. Any arbitrary segment of an

Milne et al. 21

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 Figure 3. Sizes of the major second (M2), minor second (m2), and augmented uni- son (AU), over a range of β generator tunings.

α-reduced β-chain can be used to form a scale, and the number of notes it contains is called its cardi- nality. For a given tuning of α and β, α-reduced β- chains with certain cardinalities are called moment of symmetry (MOS) scales (Wilson 1975; also known as well-formed scales after Carey and Clampitt 1989). Such scales have a number of musically ad- vantageous properties: they are distributionally even, which means that the scale has two step sizes that are distributed as evenly as possible (Clough et al. 1999); they have constant structure, which means that every given interval always spans the same number of notes (Grady 1999). Two familiar MOS scales generated by α = 2 and β≈3/2 are the five-note pentatonic scale (e.g., D, E, G, A, C) and (half tones). As β is increased, the whole tones grow the seven-note diatonic scale (e.g., C, D, E, F, G, A, larger and the half tones grow smaller. When β B); other β-values generate MOS scales with quite reaches 23/5, the semitones disappear completely; different intervallic structures. These provide fertile above 23/5, there is no seven-note MOS scale avail- resources for non-standard microtonal scales. able, because the seven-note scale contains three Every MOS scale with specified generators and different step sizes. On the other hand, as β is de- cardinality has a valid tuning range over which it creased below 27/12, the tones get smaller and the can exist; beyond this range, notes must be added to semitones get larger. When β reaches 24/7, the tones or removed from the scale to regain distributional and semitones become the same size; below 24/7, the evenness and constant structure. The boundaries of MOS scale’s internal structure changes so that it this tuning range mark the tuning points at which has five small intervals and two large intervals, a some of the scale’s steps shrink to unison, and scale structure that is the inverse of the familiar di- therefore the tuning points at which the ordinal atonic scale. These changing step sizes are illus- identity of intervals in that scale changes. An MOS trated in Figure 3. This tuning range is equivalent to scale generated by α and β with cardinality c has a Blackwood’s “range of recognizable diatonic tun- two-dimensional valid tuning range of αp/q < β < αr/s, ings” (1985). The valid tuning ranges for MOS scales where p/q and r/s are adjacent members of a Farey are, therefore, a generalization of the concept of sequence of order c–1. (A Farey sequence of order n range of recognizability beyond the familiar dia- is the set of irreducible fractions between 0 and 1 tonic/chromatic context. with denominators less than n, arranged in increas- The MOS scale that is recognizably diatonic is ing order.) called MOS:5+2 (i.e., it contains five large steps and For example, the 12-note MOS scale generated by two small steps). The valid tuning range of this α = 2 and β≈3/2, which is the chromatic scale that MOS scale is 24/7 < β < 23/5 (see Figure 2) and it pre- provides the background for common practice mu- serves, therefore, the ordinal identity of the diatonic sic, has a valid tuning range of 24/7 < β < 23/5 (4/7 and and chromatic intervals used in common practice. 3/5 being adjacent members of the Farey sequence When intervals are played as a part of a conven- of order 11). Expressing this range in simpler terms, tionalized or aesthetically consistent scale, they can the “perfect fifth” is between 1200 × 4/7 = 685.7 and be ordinally identified by the number of notes or in- 1200 × 3/5 = 720 cents in size. Consider the familiar tervals they span. Owing to their distributional seven-note diatonic scale as the tuning traverses evenness and constant structure, MOS scales are this range: starting with a generating interval β likely to be heard as intrinsically aesthetically con- tuned to the 12-TET fifth 27/12, the scale consists of sistent, so the valid tuning range of any given MOS five large steps (whole tones) and two small steps scale is equivalent to the range within which all of

22 Computer Music Journal

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 its intervals are ordinally invariant. Furthermore, generating intervals (1, 0) and (0, 1), and the layout- these ranges hold for many musically useful alter- mapping L is the 2 × 2 matrix ations of MOS scales such as the harmonic minor ⎡ψ ω ⎤ = x x scale (A, B, C, D, E, F, G-sharp). L ⎢ψ ω ⎥ (4) ⎣ y y ⎦ which transforms the temperament’s generating in- ψ ψ MOS:5+2 Ordinal Continuum tervals into the lattice’s basis vectors ψ = ( x, y) ω ω and ω = ( x, y). The elements of L must be integers The ordinal continuum of an MOS:a+b scale, (or else some intervals will be layout-mapped to lo- which has a large steps and b small steps, is here cations without buttons), L must be invertible (that defined as the (two-dimensional) range of generator is, the determinant of L is nonzero, or else either tunings that gives this scale form. It is equivalent some buttons would have no assigned note or some to αp/q < β < αr/s, where p/q and r/s are adjacent notes would have no corresponding button), and the members of a Farey sequence of order 2a + b – 1. determinant of L must be ±1 (or else the inverse The ordinal continuum of MOS:5+2 is, therefore, will not be integer-valued). This mapping provides called the MOS:5+2 ordinal continuum. the mathematical setting for two results. First, if the layout-mapping L is linear, the layout is trans- positionally invariant (Theorem 1). Second, the Button Lattices and Layouts converse holds as well: if the keyboard layout is transpositionally invariant, the layout-mapping L Instruments capable of playing a number of discrete must be linear (Theorem 2). pitches may use many buttons (or keys). For an in- Theorem 2 justifies the use of linear (matrix) no- strument to have transpositional invariance, it is tation for the layout-mapping. Proof of the theo- necessary that the buttons are arranged in a regular rems is given in Appendices A and B. The theorems and spatially repeating pattern; such a structure is presume an infinitely sized button-lattice; physical called a lattice (Insall, Rowland, and Weisstein keyboards will necessarily have finite size, and the 2007). If the lattice has one dimension, it may be linearity will be violated at the edges. The finite called a button row; if it has two dimensions, it may size also limits the number of octaves that can be be called a button field. A layout is here defined as realized. There are many possible layout matrices L, the physical embodiment of an invertible, but not and several concrete examples are given in the next necessarily linear, layout-mapping from a regular section. The choice of L impacts the ease with which temperament to an integer-valued button lattice. In particular scales and chords can be fingered, as well the same way that a regular temperament has a fi- as the balance between the number of octave- nite number of generating intervals (e.g., α and β) reduced intervals and overall octave range that can that generate all its intervals, a lattice can be repre- fit on a keyboard of a given geometry. For some lay- sented with a finite number of basis vectors that outs, such as that of Fokker, the playability-related generate all its vectors. The logical means to layout- metrics vary greatly as the tuning is changed, map from a temperament to a button lattice is to whereas these metrics are more stable across a wide map the temperament’s generating intervals to the range of tunings on some other layouts, such as the lattice’s basis vectors. Wicki. Identifying and quantifying such metrics is Let L : Zn → Zn map from the n generating inter- an important area for future investigation. vals of the temperament to the n-dimensional but- Because a layout-mapping is invertible, by defini- ton lattice. For example, with n = 2, the temperament tion, it can be linear only if it does not lose rank. contains two generating intervals α and β. Any in- Linear and invertible mappings are called isomor- terval of the temperament can be expressed as a phic, and henceforth all layouts with this type of two-vector (j, k) representing the interval αjβk. The layout-mapping will be referred to as isomorphic standard basis for the temperament consists of the layouts. The theorems show that given a regular

Milne et al. 23

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 temperament, transpositional invariance requires fifth contains seven, and a major third contains four. an isomorphic layout; and given a regular tempera- The layout-mapping can be either L = 1 or L = –1. If ment and an MOS-scalic background, tuning invari- the 12-TET generating interval is layout-mapped to ance requires that the tunings of the generating +1, the notes progress sequentially higher in pitch intervals (as embodied in the layout) remain within from left to right, which corresponds to a linear key- that temperament’s and MOS scale’s tuning range board as shown in Figure 4. for invariant identification. This means that trans- The tuning range of rational invariance for this positional invariance on a button-row is only pos- temperament is small. Increasing the size of the sible for a rank-one (i.e., equal) temperament. To get generator by just two cents increases the size of the transpositional invariance across the tuning range octave by 24 cents. Though the piano is frequently of rank-two temperaments (like the non-equal tuned with stretched octaves, this stretching is typi- meantone tunings), a button field of at least two di- cally less than about half a cent on the generating mensions is required. Because it is difficult to con- interval (about six cents per octave). The tuning ceive of a button lattice operating effectively in range of ordinal invariance is unbounded, because more than two dimensions, only rank-one and rank- no amount of retuning of the generator changes the two temperaments are considered in this article. order of the notes. For this case, the ordinal range is The following examples illustrate these points by not a useful measure. demonstrating concrete isomorphic layouts that are Owing to the isomorphic layout-mapping, this invariant in both transposition and tuning. layout has the property that any rationally identifi- able interval or chord (such as the major triad) is fin- gered the same for any position on the button-row. Examples of Layout-Mappings Starting on any note (for instance C), a major triad is played using the buttons four and seven steps to the The examples in this section use the syntonic tem- right. The geometric shape of a major triad is, there- perament tuned to quarter-comma meantone, 12- fore, always 0–4–7 for all transpositions. Similarly, TET, and 19-TET, and assume an MOS:5+2 scalic a melody is fingered the same wherever it is located background—design choices that are compatible on the button-row. Starting on any note (for in- with common-practice tonal music. Two types of stance C), the melody Re–Mi–Fa–Fi–Sol is played layouts are shown. One-dimensional button rows using a button, then a button two steps to the right, provide the simplest setting; drawings such as Fig- then one more step to the right, then one more step ures 4–6 should be interpreted as consisting of a to the right, then one more step to the right. The single row of identical buttons. For two-dimensional geometric shape of this melody is, therefore, always button fields, there are several ways in which the 0–2–3–4–5 for all transpositions. buttons can be arranged: in a rectangular grid, in off- A different rank-one temperament that also tem- set rows (like brickwork), in a hexagonal grid, or as a pers out the syntonic comma (among others) is 19- tiling of parallelograms. The Wicki layout provides TET, as shown in Figure 5. This temperament has a our primary two-dimensional example, though ob- single generating interval α = 21/19 (approximately viously other layouts could be used. 63.2 cents), and the octave consists of 19 of these generating intervals. The perfect fifth is eleven gen- erating intervals wide, and the major third is six Layout-Mapping from a Rank-One Temperament to generating intervals wide. Any isomorphic layout- a Button Row mapping of 19-TET (e.g., L = 1) has, like the above The familiar 12-TET has a single generating interval 12-TET example, a small range of rational tuning α of 21/12 (100 cents) and tempers out the syntonic invariance and a similar transpositional invariance. comma (among other commas). An octave consists However, the fingering of both the major triad 0–6– of twelve of these generating intervals, a perfect 11 and the Re–Mi–Fa–Fi–Sol melody 0–3–5–6–8 are

24 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 4. The one- Figure 5. Note positions of Figure 6. Nonlinear piano- dimensional keyboard lay- 19-TET using L = 1. style layout for a rank-2 out L = 1 in 12-TET. temperament, such as quarter-comma meantone.

Figure 4

Figure 5

Figure 6

different from the 12-TET design in Figure 4. This F). In the nonlinear layout of Figure 6, ascending shows concretely how the linear one-dimensional four steps sometimes produces a major third and design fails to be tuning invariant. sometimes produces a diminished fourth, thus breaking transpositional invariance. For example, starting from C and proceeding four steps to E pro- Layout-Mapping from a Rank-Two Temperament to duces a just major third of size 386 cents, which is a Button Row clearly identifiable as 5:4. On the other hand, start- A layout-mapping is invertible by definition, so a ing at C-sharp and ascending four steps to F pro- layout-mapping from a rank-two regular tempera- duces a diminished fourth of 427 cents. This is a ment to a one-dimensional button-row must be so-called “wolf” interval that is unlikely, in this nonlinear. As established previously, a nonlinear context (where there exist more closely tuned major (non-isomorphic) layout-mapping “breaks” transpo- thirds), to be rationally identified as 5:4. sitional invariance. A common example of a nonlin- The only way to gain transpositional invariance ear layout-mapping is a piano-style layout, which for this layout is to tune the fifths to 27/12, which is takes a finite subset of the notes produced by a the only (octave-reduced) tuning that does not dif- rank-two meantone (such as quarter-comma), and ferentiate between major thirds and diminished layout-maps them in pitch order to a twelve-note- fourths. This is consistent with the theorem be- per-octave keyboard, as illustrated in Figure 6. The cause when the fifth is tuned to 27/12 and the octave physical/geometrical irregularity of the piano key- is tuned to 2, the two generating intervals are no board, interpreted as a row of black keys inter- longer multiplicatively independent. In this case, spersed with a row of white keys, is collapsed here the rank of the temperament has collapsed to one to a single row for the sake of simplicity. and the tuning has become 12-TET. A similar argu- Such layout-mappings (as applied to standard, ment shows that for a button-row with N notes per geometrically irregular keyboards) were used in the octave, any tuning that produces an n-TET where n 17th century (Barbour 1951), though sometimes A- = N will have transpositional invariance, but for flat was used in place of G-sharp. In quarter-comma this tuning only. (or any other meantone tuning requiring two gener- To summarize: On a button-row with keys of ating intervals), major thirds (e.g., C–E) are tuned fixed size, the physical size of any given interval is differently than diminished fourths (e.g., C-sharp– different in 12-TET, 17-TET, 19-TET, quarter-

Milne et al. 25

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 Figure 7. The position of Figure 8. The position of notes using the Wicki notes using the Wicki layout on a Thummer layout on a QWERTY keyboard. keyboard.

comma, or any alternative tuning. In each tuning, Wicki layout allows a wide range of pitches to be the performer’s fingers have to press different keys played with a single hand, and all notes of any MOS to produce intervals that are the same (within the scale are always clustered in a simple vertical band. definition of a given temperament). A keyboard that (Proof of the latter property is beyond the scope of is tuning-invariant is preferable, so that the per- this article.) However, it has a less obvious relation- former’s fingers press the same keys to produce the ship between pitch and button position than is same intervals (within a given temperament), inde- found, for instance, in the design of Bosanquet. (For pendent of the current tuning. syntonic tunings, Bosanquet has an “easterly” pitch axis, like the standard keyboard, while the Wicki layout has a less obvious “north-northeasterly” Layout-Mapping from a Rank-Two Temperament to pitch axis.) The basic principles illustrated in the a Button Field following examples may also be applied to other One solution to these problems is to use a two- regular temperaments, MOS scales, and layouts. dimensional button-field so that the layout- mapping from the rank-two syntonic temperament to the button-field is isomorphic. The simplicity af- Tuning Ranges of Invariance forded by the invariant fingering of an isomorphic layout is illustrated in the following examples using Assuming that the generating intervals are α = 2 (an the Wicki layout. octave), and β = F (an alterable tempered perfect To be concrete, Wicki is a layout-mapping to a fifth), the pitch order of buttons on a Wicki layout hexagonal button-field (which can be approximated that is ten buttons wide has seven different configu- by rotating an integer valued button-field by 45°), rations as the tuning of F moves across the MOS:5+2 such that the two generating intervals α and β are ordinal continuum of 24/7 < F < 23/5. Figure 9 shows layout-mapped to the basis vectors ψ = (1, 1)T and the pitch order of the buttons, starting from D3 (num- ω = (1, 0)T, i.e., bered 0) up to D4 (an octave above), as F traverses this range. Despite these different configurations, ⎡1 1⎤ L = ⎢ ⎥ every melodic interval that is “legal” to common ⎣1 0⎦ practice has its order preserved. (Intervals such as as illustrated in Figure 7. G-sharp–A-flat, F-flat–E-sharp, or C-flat–B-sharp This layout provides a good balance of octave- typically have no melodic function in common- reduced intervals versus overall octave range for rel- practice music.) For example, observe that although atively small button fields (such as the QWERTY the pitch order changes as the tuning is raised from computer keyboard as illustrated in Figure 8) when 24/7 to 23/5, the diatonic notes (colored white in this using syntonic tunings (α≈3/2), and it also func- figure) remain in the same order, and so do the “le- tions well over many non-syntonic tuning continua gal” chromatic intervals. The shape of the chromatic (where β may take any value). Additionally, the melody Re–Mi–Fa–Fi–Sol is indicated by crosses,

26 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 9. Pitch order of 24/7 (7-TET); (b) 24/7 < F < notes across the MOS:5+2 27/12; (c) F = 27/12 (12-TET); ordinal continuum’s tun- (d) 27/12 < F < 210/17; (e) F = ing range. The notes within 210/17 (17-TET); (f) 210/17 < F the D3 to D4 octave have < 23/5; (g) F = 23/5 (5-TET). been highlighted. (a) F =

(a) (b)

(c) (d)

(e) (f)

(g)

Milne et al. 27

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 Figure 10. The step posi- this button-field. (a) Mean- tions for a selection of tone n-TET (19-TET); (b) meantone n-TETs. The meantone n-TET (50-TET); notes within the D3 to D4 (c) meantone n-TET (31- octave have been high- TET); (d) meantone n-TET lighted. For n-TETs where (43-TET); (e) meantone n > 19, some of the notes n-TET (12-TET). fall beyond the edges of

(a) (b)

(c) (d)

(e) not easily specified a priori. Figure 2 shows the MOS:5+2 ordinal tuning range, which corresponds roughly with a reasonable (though inexact) syntonic rational tuning range. Using an isomorphic keyboard, a performer can maintain the same fingerings of chords and melodies throughout the relevant range of this tuning contin- uum. Note that Figure 2 indicates only the syntonic temperament, but a given tuning can belong to more than one regular temperament. For example, and the pitch order within this shape remains the it is likely that 53-TET tuning falls within the tun- same, only breaking down when F ≤ 24/7 or F ≥ 23/5. ing ranges of invariant rational identity for both the Another natural consequence of invariant finger- syntonic and the schismatic temperament (which ing is that the steps of the various n-TETs falling tempers the 5-limit generators of Equation 1 with within any given continuum automatically line up a = –15, b = 8, and c = 1). to the correct button position. This is illustrated by Figure 10, which shows a selection of meantone n-TETs. Dynamic Tuning Figure 11 shows the positions on the Wicki layout of the common practice harmonic consonances that Keyboard designs that are invariant in both transpo- can be signified over a range of values of the fifth F. sition and tuning offer several performance possibil- This example is again centered on the note D3 (an ities to the computer-based musician and composer. arbitrary choice). All the tunings of the syntonic (or MOS:5+2) con- As discussed previously, the limits for invariant tinuum can be characterized by the width of their rational identification are context-dependent and tempered fifth. The exact value of this fifth can be

28 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 11. Position of the Figure 12. A simple chord progression can be played signified common practice progression annotated with identical fingerings harmonic consonances us- with the scale-step num- in all keys and all tunings ing the Wicki layout and bers in 31-TET, 12-TET, of the syntonic rational the syntonic temperament. and 17-TET. On an isomor- continuum. phic keyboard, this chord

linked to a single slider, modulation wheel, joystick axis, or other control device. As the performer moves this control, thereby changing the width of the tempered perfect fifth, the pitches of all sounded notes will change correspondingly to match the cur- rent tuning. If the extremes of the control’s range correspond to the extremes of the MOS:5+2 ordinal continuum, then sliding that control from one end to the other will result in the continuum of tunings shown in Figures 2 and 9. For example, one might choose to perform a simple I–IV–V–I pattern such as shown in Figure 12. The three staves show the score as it would be writ- ten in conventional notation; the numerical annota- tions show the scale steps in each of three tunings (31-TET, 12-TET, and 17-TET) as the parameter β moves from 218/31 to 27/12 to 210/17. As shown in Fig- ure 9, the fingering remains the same throughout keyboard players greater flexibility in mimicking the continuum. Thus, players of a tuning-invariant the kinds of expressive pitch deviations that string keyboard may be able to transfer competence in one (and aerophone) players reveal when altering their tuning (such as 12-TET) to others; hard-won manual intonation phrase by phrase (Sundberg et al. 1989). dexterity can be transferred directly to other tun- Moreover, tuning bends are not limited to mono- ings within the continuum. This is not possible on phonic implementations. Rocking the controller the piano keyboard, guitar fretboard, or other com- back-and-forth describes a kind of (polyphonic) vi- mon musical interfaces. brato where each note might have a different The tuning can also be changed dynamically dur- amount and direction of pitch deviation. ing performance. By analogy with pitch bend, this might be called “tuning bend,” where the exact tun- ing of each note in each interval is changed in re- Discussion sponse to the physical motion of the controller. For example, the performer might push up into a supra- Music education, performance, and composition Pythagorean tuning (like 17-TET) to give melodies can benefit from the transpositional and tuning in- more expressive power, drop down to 12-TET to variance provided by isomorphic button fields. The perform a smooth enharmonic modulation, and piano-style keyboard is incapable of achieving the then drop down to a quasi-meantone tuning (like advantages of an isomorphic keyboard for three rea- 31-TET) for more pleasing triads. This may give sons. First, it is essentially one-dimensional, and

Milne et al. 29

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 one-dimensional layouts can only have transposi- boards. Consider that accompanists are often asked tional invariance for equal temperaments. Second, to transpose a piece at sight to accompany the range its offset pattern of twelve keys per octave is only of a particular vocalist. Being able to transpose on well suited to tunings that use twelve (or fewer) the fly may be the mark of a master musician today, notes per octave. Third, the piano keyboard’s offset but with an isomorphic keyboard, the task becomes white and black keys hide the consistent patterns of significantly easier. With an isomorphic keyboard 12-TET—let alone any other tunings—forcing stu- and dynamic tuning, performers can also execute dents to learn each key’s scales, chords, and other real-time polyphonic tuning bends, temperament properties by rote. modulations, and other effects that are simply im- In contrast, with an isomorphic button-field, a possible on a piano keyboard or guitar fretboard. student need only learn the geometric shape of a Composers can also benefit from the transposi- given interval once (within a given temperament), tional and tuning invariance of isomorphic key- and thereafter apply that knowledge to all occur- boards. Just as Beethoven was the first to exploit the rences of that interval, independent of its location novel technology of the metronome to specify pre- within a key, across keys, or across tunings. This re- cise tempi (Stadlen 1967), composers writing for iso- duces rote memorization considerably and engages morphic keyboards can indicate appropriate tunings the student’s visual and tactile senses in discerning for pieces that are meant to imply a baroque (quarter- the consistency of music’s patterns. comma), medieval Ars Nova (Pythagorean), or mod- Furthermore, the MOS:5+2 ordinal continuum ern atonal (12-TET) feel. includes the tunings of many cultures and eras. Of course, there are also disadvantages to such At one of the continuum’s extremes, for example, isomorphic keyboards compared to that of the pi- 7-TET provides a close approximation to tradi- ano. The playing of diatonic intervals connected by tional Thai music (Morton 1980) and to tradition- parallel motion is more complex than on the white ally tuned African balafon music (Jessup 1983), keys of the piano, and with some layouts, there is a and at the other extreme, 5-TET is close to Indone- less obvious linear relationship between pitch sian slendro (Surjodiningrat, Sudarjana, and Susanto height and location on the keyboard. Although iso- 1993). Between these two extremes can be found morphic keyboards have been proposed and built in many tunings that have been explored in European the past, they do not have a large installed user base history, such as quarter-comma meantone, or a deep repertoire. This paper has shown that iso- Pythagorean tuning, and today’s ubiquitous 12-TET. morphic keyboards can be coupled with appropriate A student, having mastered the use of an isomor- computer technology to provide the deeper benefits phic button field in any one of these tunings, can of tuning invariance and dynamic tuning. apply the same motor skills and music theory to all the rest, making multi-cultural music education considerably easier. References Although this article has focused on the syntonic temperament and diatonic and chromatic scales, Barbour, J. M. 1951. Tuning and Temperament: A Histori- the benefits of tuning invariance are gained by all cal Survey. East Lansing, Michigan: Michigan State rank-two regular temperaments, such as schis- College Press. Reprint, New York: Da Capo Press, 1973. matic, porcupine, magic, and hanson (Tonalsoft Reprint, New York, Dover, 2004. 2007), and their respective MOS scales. The differ- Blackwood, E. 1985. The Structure of Recognizable Dia- tonic Tunings. Princeton, New Jersey: Princeton Uni- ent geometries of different temperaments and MOS versity Press. scales on the same keyboard make it easier for stu- Bosanquet, R. H. M. 1877. Elementary Treatise on Musi- dents to compare, contrast, and understand them, cal Intervals and Temperament. London: Macmillan. experiencing those differences through sight and Cited in Helmholtz (1877). touch in addition to hearing. Carey, N., and D. Clampitt. 1989. “Aspects of Well- Performers can also benefit from isomorphic key- Formed Scales.” Music Theory Spectrum 11:187–206.

30 Computer Music Journal

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 Carlos, W. 1987. “Tuning: At the Crossroads.” Computer Tonalsoft. 2007. “Equal Temperament.” Encyclopedia of Music Journal 11(1):29–43. Microtonal Music Theory. Available online at Clough, J., N. Engebretsen, and J. Kochavi. 1999. “Scales, tonalsoft.com/enc/e/equal-temperament.aspx. Sets, and Interval Cycles.” Music Theory Spectrum (Accessed 26 March 2007.) 10:19–42. Wicki, K. 1896. “Tastatur für Musikinstrumente.” Swiss Fokker, A. D. 1955. “Equal Temperament and the Thirty- patent 13329. One-Keyed Organ.” The Scientific Monthly 81(4):161– Wilson, E. 1975. “Letter to Chalmers Pertaining to Mo- 166. Available online at www.xs4all.nl/~huygensf/doc/ ments Of Symmetry/Tanabe Cycle.” Available online fokkerorg.html. (Accessed 26 March 2007.) at www.anaphoria.com/mos.PDF. (Accessed 26 March Grady, K. 1999. “CS.” Message posted to Alternate Tun- 2007.) ings Mailing List. Available online at launch.groups .yahoo.com/group/tuning/message/5244. (Accessed 11 October 2006.) Appendix A: Proof of Theorem 1 Helmholtz, H. L. F. 1877. On the Sensations of Tone. Trans. A.J. Ellis. Reprint, New York: Dover, 1954. Let f : Z2 → Z2 be an invertible mapping from the Insall, M., T. Rowland, and E. Weisstein. 2007. “Point two-dimensional generators of a rank-2 tempera- Lattice.” MathWorld—A Wolfram Web Resource. ment to the two-dimensional button-lattice. Any Available online at mathworld.wolfram.com/ PointLattice.html. (Accessed 4 June 2007.) note N in the temperament can be represented in α β αn βm Jessup, L. 1983. The Mandinka Balafon: An Introduction terms of its generators and as N = , which with Notation for Teaching. La Mesa, California: Xylo can be written as the vector (n, m) for n, m ∈ Z. The Publications. note is located on the button-lattice at f(N) = f(n, m). Keislar, D. 1987. “History and Principles of Microtonal An interval I is a ratio of two notes Keyboards.” Computer Music Journal, 11(1):18–28. N nm11 Published with corrections in 1988 as “History and 1 = (A.1) nm22 Principles of Microtonal Keyboard Design,” Report N2 STAN-M-45, Stanford University. Available online at which can be written (j, k), where j = n –n and k = ccrma.stanford.edu/STANM/stanms/stanm45. (Ac- 1 2 m m transpositionally invariant cessed 13 June 2007.) 1– 2. A layout is if Lorentz, R. 2001. “19-TET in a Renaissance Chanson by every fixed interval I is fingered in the same man- Guillaume Costeley.” Paper presented at Microfest, ner, i.e., if Claremont, California, April 6. fN()−=− fN () fN () fN () (A.2) McLaren, B. 1991. “General Methods for Generating Mu- 12 34 sical Scales.” Xenharmonikon 13:20–44. whenever Morton, D. 1980. “The Music of Thailand.” In E. May, ed. Music of Many Cultures. Berkeley, California: Univer- N N 1 ==3 I (A.3) sity of California Press, pp.63–82. N2 N4 Sethares, W. A. 2004. Tuning, Timbre, Spectrum, Scale, 2nd ed. London: Springer-Verlag. Thus, transpositional invariance requires that the Smith, G.W. 2006. “Xenharmony.” Available online at difference in locations between notes on the key- bahamas.eshockhost.com/xenharmo. (Accessed Octo- board depends only on the interval (the j and k) and ber 2006.) not on the particular notes (the n1 and m1). Expand- Stadlen, P. 1967. “Beethoven and the Metronome.” Music ing Equation A.2 yields and Letters 48:330–349. Sundberg, J., F. Anders, and L. Frydén. 1989. “Rules for ⎛ n ⎞ ⎛ nj+ ⎞ ⎛ n ⎞ ⎛ nj+ ⎞ f⎜ 1 ⎟ − f⎜ 1 ⎟ = f⎜ 3 ⎟ − f⎜ 3 ⎟ (A.4) ⎝ ⎠ ⎝ + ⎠ ⎝ ⎠ ⎝ + ⎠ Automated Performance of Ensemble Music.” Con- m1 mk1 m3 mk3 temporary Music Review 3(1):89–109. Surjodiningrat, W., P. J. Sudarjana, and A. Susanto. 1993. where we used the equality of the intervals from Tone Measurements of Outstanding Javanese Gamelan Equation A.3, i.e., that n2 = n1 + j, m2 = m1 + k, n4 =

in Yogyakarta and Surakarta. Yogyakarta, Indonesia: n3 + j, and m4 = m3 + k. If the mapping f is linear, Gadjah Mada University Press. then this can be rewritten

Milne et al. 31

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021 ⎛ n ⎞ ⎛ n ⎞ ⎛ j⎞ ⎛ n ⎞ ⎛ n ⎞ ⎛ j⎞ Additivity f⎜ 1 ⎟ − f⎜ 1 ⎟ + f⎜ ⎟ = f⎜ 3 ⎟ − f⎜ 3 ⎟ + f⎜ ⎟ (A.5) ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ m1 m1 k m3 m3 k Because Equation B.1 holds for all x and y, it must which collapses to an identity for any ni and mi. hold for x = j and y = 0. Substituting into Equation This demonstrates that linear layout mappings are B.1 shows f(j + i) – f(j) = f(0 + i) – f(0). Because f(0) = 0, transpositionally invariant. An analogous argument this can be rearranged to show f(i + j) = f(i) + f(j). works for any invertible rank-r mapping f : Zr → Zr.

Homogeneity Appendix B: Proof of Theorem 2 Since Equation B.1 holds for all x and y, it must The converse of Theorem 1 is demonstrated by hold for x = i and y = 0. Substituting into Equation showing that transpositional invariance implies lin- B.1 shows f(i + i) – f(i) = f(0 + i) – f(0). Because f(0) = 0, earity of f. This will first be shown in the scalar this can be rearranged to show f(2i) = 2f(i). Induction case, where the defining Equation (A.1) for transpo- can be used to show the general case. Suppose that sitional invariance is that for all fixed intervals i, f((k – 1)i) = (k – 1)f(i). Let x = (k – 1)i and y = 0. Substi- tuting into Equation B.1 shows f((k – 1)i + i) – f((k – fx()()()()+− i fx = fy +− i fy (B.1) 1)i) = f(i). Using the inductive hypothesis, this can for every x, y ∈ Z. We also assume that f(0) = 0. Lin- be rearranged to show f(ki) = (k – 1)f(i) + f(i) = kf(i). earity of f is shown by demonstrating additivity and The generalization to two (or n) dimensions is homogeneity. straightforward.

32 Computer Music Journal

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 by guest on 24 September 2021