Gawboy, Anna: The Wheatstone Concertina and Symmetrical Arrangements of Tonal Space

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Anna Gawboy The Wheatstone Concertina

Figure 9a. Ellis’s derivation of just chords and scales from the harmonic series (1877):

“Diagram of Harmonics of B-flat”

Figure 9b. Ellis’s “God Save the Queen” demonstrated on concertina in just intonation.

The notation underneath each chord refers to the diagram in Figure 9a.

Figure 9c. Pitches of Figure 9a arranged in a lattice in order to demonstrate chord formation

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tuning, meantone temperament, and equal temperament to convince his

audience of the superiority of just intonation. “Just Intonation ensures real

harmony,” concluded ellis (1877, 34).

As Figure 9c suggests, ellis’s just concertina had two different Ds, one

tuned as a perfect fifth up from G, and the other, labeled “D

,” tuned a perfect

fifth down from A

. This enabled each triad of “God Save the Queen” to sound

pure. ellis’s just concertina was made by lachenal and Company (an offshoot

and sometime rival of Wheatstone and Company) according to his particular

specifications (ellis 1877, 17). ellis could have requested that lachenal rear-

range the button layout of his just demonstration concertina in order to make

it correspond to the Tonnetz in Figure 9c, somehow adapted to accommodate

octave duplication for selected pitches. however, ellis evidently preferred to

rely on the muscle memory he had developed during his many years playing

concertina with the layout given in Figure 2. ellis’s description of his just instru-

ment, included as an appendix to his translation of helmholtz’s treatise, indi-

cated that he modified the treble arrangement by eliminating the accidentals

A≤ and e≤ in order to place the pitches A and D next to A

and D

(helmholtz

[1863/1877] 1912, 470). Figure 10 shows the result of these modifications.

Figure 10. Layout of Ellis’s just demonstration concertina

So far, we have only seen a loose correlation between the concertina’s

button layouts and graphic representations of its tuning. however, the con-

certina’s potential to schematically represent the acoustic derivation of its

pitches on its fingerboards did not escape the notice of ellis and his peers. In

1874, robert Bosanquet gave papers before the royal Society and the royal

Musical Association in which he described the generalized keyboard layout

that now bears his name, as well as several symmetrical arrangements showing

the derivation of pitches according to various tuning methods. Bosanquet’s

work explicitly linked symmetrical arrangements of tonal space and instru-

ment layouts. his generalized keyboard design, shown in Figure 11, allowed

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Anna Gawboy The Wheatstone Concertina

Figure 11. Layout of Bosanquet’s enharmonic harmonium

a performer to navigate the enharmonic harmonium’s hundreds of keys

using identical fingerings for any transposition,

a feat anticipated—but left

incomplete—by Wheatstone’s “self-transposing” Double concertinas.

19 Oettingen viewed Bosanquet’s enharmonic harmonium

when he attended the Loan Exhibition of Scientific Appa-

ratus at the South Kensington Museum in 1876. He lightly

adapted Bosanquet’s keyboard design for his own Orthoto-

nophonium of 1916 (Goldbach 2007).

20 In 1896, the Swiss mathematician Kaspar Wicki invented

a layout for a concertina that allowed isomorphic fingerings

for all scales (Gaskins 2003). The arrangement was inde-

pendently reinvented by the physicist Brian Hayden in 1967

(Hayden 1986) and has recently been adapted by computer

programmer James Plamondon to the Thummer, an elec-

tronic instrument capable of producing a wide variety of tim-

bres and tunings (Milne, Sethares, and Plamondon 2008).

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Figure 12 shows Bosanquet’s “Complete Symmetrical Arrangement,” in

which equally tempered half-steps progress horizontally and pure fifths pro-

gress vertically. Bosanquet calculated that it was possible on this array to obtain

an interval that deviated from a pure major third by only 1/1,000 of an equally

tempered semitone, allowing nearly just triads to coexist with equal interval

cycles built of tempered half-steps. Bosanquet then rather casually remarked,

“It would be possible, if desired, to construct a keyboard for such a system, the

dots . . . being replaced by concertina-keys. It would then be possible to play

fifths and thirds perfect to the above degree of approximation” (1874, 146).

Bosanquet hoped that his efforts to design instruments capable of pro-

ducing ever more elaborate divisions of the octave would not only advance

“musical science” but also encourage the development of alternative com-

positional systems based on the newly available pitch resources. “Different

[intonational] systems have their different advantages,” he wrote, “and we

may contemplate the possibility of music being written for, or adapted to, one

system or another, just as hitherto music has been written for performance

in one key or another of the equal temperament” (1874, 5). euler, too, had

held out hope that his calculations would “lead to new genera and to new and

unusual musical compositions” (Smith 1960, 170).

Bosanquet’s enharmonic

harmonium and his proposal for a generalized concertina may be seen as

manifestations of the mutual interdependence euler described between tun-

ing systems, instruments, and musical compositions:

Since the tones used in producing music are offered to the ear by means of the

human voice or instruments, then, before anything else, the voice and the

instruments must be adapted for the production of all tones required for ren-

dering a musical work. Since the exponent of a musical work contains all the

necessary tones, it should indicate the number and nature of the tones produc-

ible by musical instruments. Thus, construction of the musical instruments

depends on the exponent of the musical work which is to be presented to the

hearing by means of those instruments. Thus as we wish to perform musical

works of various exponents, we may have to have musical instruments which are

adapted to those exponents. (Smith 1960, 164)

As we have seen, the english concertina began its history as a realiza-

tion of this idea. euler had described his diatonic-chromatic genus as “most

perfectly in accord with the true principles of harmony” (Smith 1960, 31), and

Wheatstone designed the concertina to express these principles. The concer-

tina’s early tuning and button layouts were premised on a belief that just into-

nation was the true expression of harmony and harmony was the true basis of

music. however, by the time ellis was using the concertina in public lectures

21 Interestingly, these acousticians’ hopes would be

directly fulfilled by Harry Partch (Gilmore 1998, 48–49,

176–77). Partch was greatly inspired by his reading of Helm-

holtz, particularly Ellis’s appendix describing Bosanquet’s

enharmonic harmonium, the just concertina, and other new

instruments capable of performing various tuning systems.

Partch’s marimba based on the eleven-limit tonality diamond

is a particularly vivid realization of the idea of isomorphism

between instrument layouts and pitch relationships. My

thanks to Julian Hook for suggesting this connection.

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Anna Gawboy The Wheatstone Concertina

Figure 12. Bosanquet’s “Complete Symmetrical Arrangement”

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intended to popularize this argument, Wheatstone and Company had already

succumbed to the practical allure of equal temperament for their commercial

instruments, and the layout of the treble concertina had made the perfor-

mance of harmony according to standard textures and voice-leading norms

almost a virtuosic feat.

Yet, the english concertina’s story provides further insight into the man-

ner in which physics, instrument building, and music theory overlapped in

nineteenth-century popular culture, as the recent work of Jackson (2006) and

Steege (2007) has explored. Shortly after Wheatstone’s death, one of his col-

leagues delivered a eulogy before the royal Musical Association. “There is

one characteristic, at least, which runs through all of [Wheatstone’s] investiga-

tions: he always sought to embody the results of his study of the laws of nature

in a practical form, so that they might be beneficial to mankind” (Adams 1876,

85). Wheatstone summarized his own philosophy as follows: “The application

of the principles of science to ornamental and amusing purposes contributes,

in a great degree, to render them extensively popular; for the exhibition of

striking experiments induces the observer to investigate their causes with addi-

tional interest, and enables him more permanently to remember their effects”

([1827] 1879, 21). In some ways, the english concertina parallels another

popular Wheatstone invention: the stereoscope, a handheld device that com-

bined two parallel images into the appearance of a three-dimensional com-

posite (Wheatstone [1838] 1879). Peering through the stereoscope quickly

became a favorite pastime in Victorian parlors and something of a craze after

the invention of photography. At the same time, however, Wheatstone’s con-

clusions regarding binocular vision based on his stereoscopic experiments

overturned prevailing optical theory (Turner 1993), especially after his

research was championed and elaborated by helmholtz (1867). Although the

stereoscope’s popular success and scientific implications were both more far-

reaching, the english concertina exhibited a similar interrelated duality of

purpose. For Victorian home music makers, Wheatstone’s invention provided

an opportunity to cultivate “an elevating recreation that will enlarge the mind,

purify the affections, and strengthen the intellect” (Cawdell 1865, 11), while

offering British musical scientists the possibility of making ideal musical rela-

tionships real in their efforts to narrow the ancient gap between speculative

harmonics and musical practice.

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Anna Gawboy The Wheatstone Concertina

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Anna Gawboy is assistant professor of music at Ohio State University. She plays English concertina

in several folk ensembles in Michigan’s Upper Peninsula.