Keys, Strings, and Valves: Theorizing Instrumental Spaces

By Jonathan De Souza (University of Western Ontario)

Author’s note:  This blog post builds on my book Music at Hand: Instruments, Bodies, and Cognition (Oxford University Press, 2017).  The book examines body-instrument interaction in various musical styles, combining music theory, psychology, and phenomenology.

The hall is still.  The pianist rubs his hands, then raises them, inhaling as he prepares to strike the opening chord.  His hands drop and… laughter ripples through the audience.  Clearly, this is no ordinary recital.  The “pianist” is the comedian Rowan Atkinson.  Starting over, he launches into Beethoven’s “Pathétique” Sonata.  The performance is full of energy, his fingers flying and his face full of expression.  Only one thing is missing:  the piano.

Atkinson is miming.  His exaggerated gestures and facial contortions supplement the sound, emphasizing theatrical or visual aspects of instrumental performance.  Yet this routine—like his invisible drum kit sketch—might also call attention to the absent instrument.  It’s easy to take instruments for granted, especially familiar ones like the piano.  But as Martin Heidegger argues, awareness of a tool can be heightened when the tool is missing or broken, when everyday expectations are interrupted.  Paradoxically, then, making instruments invisible helps show how they mediate instrumentalists’ actions.  With the “air piano,” for example, Atkinson’s hands travel along a horizontal line, sweeping left and right.  Even this virtual keyboard constitutes a space for embodied performance.

How are instrumental spaces organized?  How are they traversed in performance?  How do they present pitches in particular locations, or according to particular dimensions?

Transformational theory offers one way to approach such questions.  In general, transformational theory uses mathematical groups to model diverse musical “spaces.”  These spaces might involve pitches or chords, but also rhythmic patterns, timbral spectra, contrapuntal permutations, textural streams, banjo picking patterns, and so on.  Near the beginning of Generalized Musical Intervals and Transformations, for example, David Lewin shows how chromatic pitches resemble the integers (Z)—that is, the infinite set of all positive and negative whole numbers.  We can imagine notes, like numbers, going up and down endlessly, into regions too high and too low to hear.  And we can measure the difference between any two pitches (or integers), counting the steps between them.  Alternatively, Lewin treats the chromatic pitch classes as a twelve-element cycle (Z12), like the numbers on a clock face or the months of the year.  The huge chromatic descent from the Pathétique’s introduction would represent smooth movement through either space, with each step corresponding to the interval –1.

De Souza-PAIG Ex 1

Example 1.  Beethoven, Piano Sonata no. 8 in C minor, “Pathétique,” op. 13, mvt. i, m. 10

Piano keys resemble the set of integers, too.  Any real keyboard—from a toy piano to a concert grand—has a finite number of keys.  But conceptually it could continue indefinitely.  (This is demonstrated by another quasi-Heideggerian comedy routine, where Victor Borge keeps reaching for nonexistent high keys.)  Because each octave has the same arrangement of keys, we might also imagine a cycle of twelve “key classes” (see Example 2).  Either way, keyboard space is further defined by an asymmetrical pattern of white and black notes.  At the piano, we might say, 12 is 7 + 5.  This supports a distinctive kind of stepwise motion, since a player might move one “step” in white-key or black-key space (e.g., G-flat to E-flat would be –1 in black-key space, as shown in Example 2b).  Of course, keyboard patterns typically correlate with pitch patterns—but these associations can come apart, with prepared piano or keyboard MIDI controllers, and we can model instrumental patterns apart from their expected sounds.

De Souza-PAIG Ex 2

Example 2.  (a) Key-class space, and (b) a transformation network showing same-color adjacencies (white-key and black-key space)

Key color relates closely to fingering.  The standard “French” fingering for the chromatic scale, for example, keeps the thumb on white notes, letting the longer index and middle fingers reach for the raised black keys (see fingering in Example 1).  These finger-key associations impose kinesthetic groupings (12 as 2 + 3 + 2 + 2 + 3), which pianists generally hide through an even touch.  Aspects of keyboard space, though, are central to certain pieces—for example, Chopin’s Étude in G-flat major, op. 10, no. 5, where the melody floats along the black keys (see Example 3).  (Lang Lang mischievously highlights the étude’s black-keyness by playing the right hand with an apple or orange.)

De Souza-PAIG Ex 3

Example 3.  Chopin, Étude in G-flat major, “Black Keys,” op. 10, no. 5, mm. 1–4

If keyboard space is linear, other instrumental spaces can be multidimensional.  String instruments like the violin or guitar juxtapose two dimensions:  players can move along or across the strings.  When playing a chromatic scale on the violin, most finger moves travel along a single string but with occasional cross-string breaks.  For a chromatic scale in Bruch’s first Violin Concerto, the soloist’s fingers ascend along the D string and the A string, then climb further up the E string.  Soon after this scalar passage, though, the solo part features rapid, repeated string crossing.  With these two orthogonal dimensions, the fingerboard might be understood as a space of finger/string coordinates, analogous to the Cartesian plane (Z × Z, see Example 4).  Again, the topology of this space is conceptually independent of any particular tuning.

De Souza-PAIG Ex 4

Example 4.  A partial map of fingerboard/fretboard space.  Each node’s label combines a finger-position number and a string number.  Horizontal arrows move along strings, while vertical arrows move across strings.  Though real instruments have a finite number of finger-positions and strings, the theoretical space remains unbounded (for further discussion, see my forthcoming article in the Journal of Music Theory 62/1).

Trumpet valves offer a different kind of space.  While piano keys activate a sound when pressed, valves—whether up or down—help facilitate sounds produced by breath and lips.  And where each piano key is associated with a single pitch, a valve combination opens up a field of sonic possibilities.  Mathematically, this valve space combines three two-element cycles (Z2 × Z2 × Z2).  There are eight possible valve patterns here, in four inversionally related pairs (e.g., ●●● inverts to ◦◦◦, ●◦◦ inverts to ◦●●, etc.).  Moreover, these patterns also parallel eight valve-changing operations, moves or intervals in valve space.  Each operation can be represented by three plus or minus signs, which either keep (+) or change (−) each valve’s position:  (+ + +) keeps all valve positions the same, (− − −) changes all of them, (+ − +) changes only the middle valve, and so on.  These possibilities can be laid out in a table (Example 5), or a spatial network (Example 6).

De Souza-PAIG Ex 5

Example 5.  Table of valve combinations and operations.  While this is an exhaustive list of valve patterns, it is possible to define other transformations in valve space (e.g., a rotation transformation could take ●◦◦ to ◦●◦, etc.; or a retrograde transformation could take ●●◦ to ◦●●).

De Souza-PAIG Ex 6

Example 6.  A network for valve space, showing selected operations.  Dashed arrows correspond to inversion (− − −).

Again, formalizing this space can help us analyze instrumental patterns that might be inaudible or deliberately concealed.  The open valve position, somewhat like a violin’s open strings, often has a distinctive position here.  For example, a descending chromatic scale from a high G (◦◦◦) involves an interesting additive process (12 as 3 + 4 + 5).

De Souza-PAIG Ex 7

Example 7.  Transformation network showing valve positions (and transformations) for a descending chromatic scale on trumpet

This pattern appears without the A-flat in Jean-Baptiste Arban’s variations on “The Carnival of Venice” (Var. 1, m. 12), making the sequence of valve operations slightly more consistent (see Example 8).  The passage uses four valve patterns, but (with one early exception) only two valve operations:  change-2nd-valve (+ − +) and keep-3rd-valve (− − +).  Alternating between (+ − +) and (− − +) creates a four-element cycle (see Example 9).  Repeating both operations twice, that is, returns to the starting valve pattern (i.e., (+ − +)(− − +)(+ − +)(− − +) = (+ + +)).

De Souza-PAIG Ex 8

Example 8.  Transformation network for Arban, “The Carnival of Venice,” Var. 1, m. 12

De Souza-PAIG Ex 9

Example 9.  Network showing two (+ − +)(− − +) cycles, related by inversion (dashed arrows).  The cycle on the left relates to the preceding example.  Such cycles can be created by alternating any two valve operations (not including (+ + +)).  For example, (+ − +)(− − −) and (− − +)(− − −) cycles are also implicit in this network.

Such patterns help make Arban’s showpiece highly idiomatic, despite its difficulty.  In the final variation, every other note has the open valve position (◦◦◦).  Each operation is repeated twice in a row, immediately undoing itself.  The variation, in fact, can be played with a single finger!  This break with conventional technique directs attention to the valves, to the instrument itself.

A transformational approach to instrumental space, of course, has its limits.  It is productively supplemented by ethnographic, organological, or phenomenological methods, since it models instruments and performative actions in a relatively abstract or idealized way.  Formalized keyboard space, in a sense, is just as imaginary as Atkinson’s air piano.  Still, transformational thinking can support analysis of characteristic instrumental moves and stimulate reflection on instrumental topology.  And at the same time, these explorations continue Lewin’s own interest in bringing performative perspectives into music theory, and his desire to theorize musical space from the inside.

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