WINGATE: PIANO SONATA No. 1
“The Transcendental”
or
The First 3000 Digits of Euler’s Number for Piano Solo

Part of Wingate’s ‘Irrationals Trilogy’ (including the Symphony No. 3 ‘Pi’ and the String Quartet No. 2 ‘Phi’), the Transcendental Sonata’s subtitle is inspired by the mathematical status of Euler’s number as both an irrational number and a transcendental one (i.e. a number that is not a root of any non-zero polynomial with rational coefficients). For inveterate pianisti, the ‘Transcendental’ subtitle also impudently serves to evoke Franz Liszt’s famous ‘Transcendental Études’ or Études d'exécution transcendante, S.139 for piano, an astonishingly difficult set of pieces published in 1852. Also not without its Herculean difficulties, Wingate’s sonata contains four movements bearing titles inspired by the many marvelous and profound mathematical properties of the number e, which—as part of Euler’s formula especially—has often been described as an epitome of mathematical beauty. The dynamic result of this numerico-melodic union is a sonata unique in the solo piano repertoire.

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The composer’s procedure to create the sonata allowed any pitch event to be doubled at any octave ad libitum (of course up to a maximum of two sets of octaves at once, compelled by the maximum reach of the human hand), and this yielded a particular and pervasive type of sonority throughout the piece’s four movements, often resulting in soundscapes of an open, welcoming character due to the ubiquitous presence of these open octaves as the only possible ‘chords’ (and this despite the music’s ostensibly atonal character). The piece is often, in effect, a de facto octave étude for the piano, as the pianist must accomplish dramatic and often heavy-handed keyboard gymnastics to accommodate the multiple-octave voicings of consecutive single pitches across the keyboardscape. Pedaled conglomerates of successive tones or exactingly-rolled near-simultaneities create the only quasi-chordal effects, with these agglomerate entities often functioning as melodic units or thematic fragments, as, for example, in the first movement’s forceful D-G-C#-G# opening motive, or the gently rolled ‘chords’ throughout the sonata’s affable Adagio. (Pedaling, in fact, takes on a role of outsized technical importance throughout the work, as it is the only enabler of gathered pitch co-occurrences in this as-it-were ‘non-chordal’ compositional strategy on the piano.) Moto perpetuo interludes of rapid-fire non-aggregates often serve as a mitigating textural contrast to these pedal clusters. Additionally, the peculiar order of digits in various realms of Euler’s number sometimes served to suggest the recurring musical material itself, as in the case of the proliferation of repeated double numbers (in digits 2001-3000) that inspired the declamatory repeated-note motives in the work’s final movement.

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The first movement (‘Derangements’) uses three contrasting thematic ideas throughout its 1000 notes, and these musical themes maintain their identities thanks to happenstantial occurrences of repeated groups of intervals embedded in e’s numerical sequence. In fact, Euler’s number has inadvertently provided places for both a development section and recapitulation section in its first 1000 digits, enabling this first movement to express a somewhat traditional Classical sonata form. The second movement (‘Exponential Decay’) brings forth a series of quietly disconcerting chord-like gestures, intermingled with hollow octave interplays between the left and right hands, all linked together by subtly menacing processions of bass-register notes, which ultimately lead to what sounds like a final half-cadence on the 1500th note/digit. The third movement (‘Asymptotics’) is a piano-thrashing, toccata-like scherzo with multiple personalities, by turns bombastic and then quietly relentless. This movement’s perpetuum mobile left-hand figures drive and delineate right-hand melodies which happen to be anchored to important beats, suggesting melodic inevitability. The frenzied and implusive fourth movement (‘Convergent Complexities’), as mentioned above, uses e’s many groups of double digits to launch powerful fanfare-like gestures into the piece’s finale. Episodes of excessive pounding and pedaling proceed almost unabated throughout the movement’s rondo-like structure, until the sonata at last comes to a thunderous close as the 2973rd through 3000th of e’s digits just so happen to provide some last-moment restatements of the movement’s major themes, followed by a curt, but oddly normal-sounding C-major-like final cadence.​ [7-0 = G-C.]

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The three pieces constituting Wingate’s ‘Irrationals Trilogy’ may provoke some unsettling theoretical questions concerning the relative importance of pitch sequence in music, including fundamental ones such as: how much does pitch—and by extension, tonality—actually matter to musical meaning and perception? But also, how much does composerly taste and intervention consequentially serve to enhance or obscure any pitch sequence’s possible tonal- or atonal-sounding outcomes? And what if these numbers (pi, phi, and e) had not turned out to have been so serendipitously overly-obliging and abundant in useful musical material? The composer did not explore their possibilities before starting the projects. But beyond these ruminations lie the Irrationals Trilogy pieces themselves—strange embodiments of number clothed in musical raiment, yet demanding of the ear no knowledge whatsoever of numerals or set theory to hear the manifested music.

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The sonata’s score cover image is a digitally-rendered visual representation of Euler’s number by French-Canadian mathematician Simon Plouffe.