WINGATE: STRING QUARTET NO. 2
“Phi” (“Φ”), or The Golden Ratio Quartet
[a.k.a. The First 2000 Digits of Phi for String Quartet]

Movements:

I. Euclidian Counterpoint I (Digits 1-342) - Grave

II. Fibonacci’s Hares (Digits 343-1139) - Allegro

III. Toccata of the Gnomons (Digits 1140-1835) - Molto Allegro

IV. Euclidian Counterpoint II (Digits 1836-2000) - Grave (attacca)

Date:
2022

Duration:
19' 30"

Notes:

Wingate’s Second String Quartet (‘Phi’ or The Golden Ratio Quartet) is built from the superstar irrational number of mystical repute known as Phi (Φ), the so-called ‘Golden Ratio’ celebrated by mathematicians since the time of Pythagoras and Euclid, and allegedly bestowing its golden gifts upon both human civilization and the natural world in perpetual and ubiquitous geometric glory. For the purposes of this quartet, each numeral 0-9 is first assigned one of ten pitches following pitch-class integer notation (0=C, 1=C#, 2=D, etc.), and then the first 2000 digits of Phi become an ordered series of notes spread out over twenty minutes of music. For example, Phi’s first five digits (1.6180) become the first five notes of the quartet: C#, F#, C#, G#, C, and so on, until all of the first 2000 digits are ‘heard’. This numeral-to-note procedure was similarly used with the number Pi (π) to build the composer’s Third Symphony, giving these two works a shared irrational pedigree. But being a musical embodiment of an irrational number does not necessitate some kind of absurd musical result, as both of these numerically-derived works put forward the impression of purposeful musical motion, perhaps raising interesting theoretical questions concerning ‘random’ pitch occurrence and musical perception.

 

The first movement of the Phi Quartet sets Phi’s first 342 digits as a dense contrapuntal adagio of falling gestures and angular string meanderings, titled in honor of Euclid, who first described Phi‘s irrational qualities around 300 B.C.E. The quartet’s second movement, using digits 343 through 1139, is a frolicsome gigue made of jumping rhythmical figures, with its title referring to a passage from the thirteenth-century book known as the Liber Abaci by Leonardo of Pisa (a.k.a. Fibonacci), in which he illustrates his now-eponymous, Phi-trending numerical sequence via the example of libidinous rabbits and their mathematically-derived progeny. The quartet’s third movement uses digits 1140 through 1835 as nimble strands of doubled-note figurations, traversing pointed dynamic contrasts and moments of tonal chaos mixed with ethereal string harmonic congruences. The title of this movement refers to the geometric entity known as the ‘Golden Gnomon’ (an obtuse isosceles triangle whose side-to-base ratio is the reciprocal of Phi), and to its role in the structure of the regular pentagram — here a geometric arabesque of the five fingers a toccare. This scattered toccata crashes without pause into the quartet’s fourth and final movement, a grave reiteration of the piece’s opening material but with different pitches, slowly winding its tenacious counterpoint around the 1836th through 2000th digits of Phi, and closing the quartet as a serialist paean to the mathematical sublime.

 

Some technical notes: As with the Pi Symphony, there are no B-flats or B-naturals in the Phi Quartet, as the ten Arabic numerals in mathematics only go up to the ‘A‘ in the Western musical scale. But the total absence of these eleventh and twelfth pitches tends to go strangely unnoticed by the ear, perhaps due to the relative strength of the limited array of ‘keys’ remaining when no leading tone to C (or to B) is possible, and when the A will never resolve upwards, thus somewhat curtailing three possible harmonic centricities. It will also be noted that the composer’s rules for creating the piece allowed any pitch to be sustained past the point at which any of its following pitches may have already stopped sounding, sometimes causing a feeling of renewed prominence to an old pitch similar to the sounding of a new pitch, and thus creating a de facto reordering of the pitches in one’s actual perception. Also, the employment of the pitches in the third movement may be seen to take some liberty against purism by doubling the tones in repeated sautillé sixteenth-notes instead of having the order sounded singly as elsewhere. But the strict order of Phi’s numerals-cum-notes in the score nevertheless remains ardently and irrationally sequential in occurrence all the way to the quartet’s 2000th pitch event, the viola’s delicate and resolution-like F-natural in the final configuration.