What Some Dead White Guy Did: A Statistical Topology Of Keyboard Usage In Beethoven's Sonatas For Pianoforte
by David Pocknee
This paper takes a statistical approach to keyboard usage in Beethoven's pianoforte sonatas through the analysis of MIDI files of the works. The data obtained is used to draw conclusions about Beethoven's usage of his piano keyboards over this repertoire. It is shown that Beethoven's key usage closely approximates a normal distribution, and the data is used to examine possible biomechanical, aesthetic, pedagogic and topological reasons for this and the implications this has for the predictability of composers and improvisers who compose or improvise at a piano. Two solutions to this statistical bias are proposed: the creation of alternatively shaped keyboards which compensate for the aforementioned biases by inverting the size of keys relative to their statistical probability of usage, and the creation of a pure data patch that remaps MIDI keyboards through the random reassignment of pitches. This research is part of an ongoing collaboration with Alex Grimes, looking at the role of topology in piano music.
This paper deals with both piano “keys” and the “keys” related to harmony (such as “C major”). In order to avoid confusion, whenever keys related to harmony are being addressed, they will be referred to as “harmonic keys”, despite the clumsiness of this formulation.
In this paper, note names will always be in the form of a capitalized note-letter, followed by a superscripted number representing the octave (e.g. C4, where this represents the middle C on the piano - A0 would represent the lowest A on the piano). Where quotations use a different system, a translation will be placed in square brackets afterwards.
All bar numbers refer to the 1958 Harold Craxton edition of the collected sonatas, mentioned in the reference list.
Between 1795 and 1822, Beethoven wrote thirty-two sonatas for pianoforte. This repertoire has been the subject of much analysis, forming, as it does, the backbone of the classical piano repertoire. However, as far as the author knows, this is the first study in which statistical techniques have been applied to the analysis of these works.
The aim of the present study is to provide a general overview of Beethoven's usage of the piano keyboard during the twenty-seven years in which these works were written, as well as to use this data to draw conclusions about the effect of the size of piano keyboard upon his compositions. This data will also be used for speculating on how other parameters, such as aesthetics, biomechanics and pedagogy, may have influenced the statistical spread of notes over the piano keyboard in these thirty-two sonatas.
1.1 Beethoven's Pianos
Beethoven lived during a time in which the construction of the piano underwent rapid changes, particularly in regards to the compass of the keyboard. In order to understand how the compass of the keyboard may have impacted upon his compositional style, a brief summary of the existing writing on the subject is presented:
The pianos which Beethoven played on, and wrote his sonatas for, is a hotly debated topic in certain circles (see Sadie, 1971, Melville, 1971a, 1971b, & 1972 and Newsome, 1970 & 1971), with frustratingly little evidence to allow concrete conclusions to be drawn. Melville disappointedly states that “Unfortunately, there is no actual evidence, save in this one instance [the use of his 1803 Erard piano for the C minor piano concerto], that I could discover to illustrate what actual piano Beethoven was using at any given time” (Melville, 1971b, p. 757). Newman (1971) points out that one of the main problems in this endeavour is that much of the literature has centred around the three existing pianos that Beethoven owned and ignores the fact that he appears to have had frequent loans of other pianos, which are only sparsely and incompletely documented.
[Beethoven] may never have owned any piano that was not presented to him as a gift. In fact, he seems never to have needed to buy one of his own...he appears only too happy to accept loans of pianos that the dealers were only to happy to have so illustrious a figure borrow (Newman, 1971, p. 501).
The three Beethoven pianos still in existence are an Erard (received in 1803 and reaching from F1 to C7), a Broadwood (received in 1818 and reaching from C1 to C7) and a Graf (received in 1825, and thus after the completion of the sonatas).
The difficulty in ascertaining the precise piano which Beethoven was using for the composition or performance of a particular sonata is frustrating and, as will be seen later on from the data, there are many instances in which the range of a particular sonata does not fit within the range of any piano which Beethoven was documented to have owned. However, educated guesses can be made into the compass of these instruments and, in some cases, the documented instruments can be matched to a particular sonata with some degree of certainty.
1.1.1 Sonatas 1-20
Newman states that:
Starting with range, it is essential to recall that all twenty sonatas prior to Op. 53 – that is prior to 1803, using composition dates for the present discussion-are confined to the five-octave range or less, from contra-F [F1] to F3 [F6], that prevailed in the harpsichord and early piano throughout the 18th century. The one exception being a single f# [F#6] in Op. 14/I/i/41 [1798/99], which still appears unremarked in all the editions, but may have been a slip. (Newman, 1970, 491)
As can be seen from the data presented later on in this paper, this is correct, although Newman neglects to mention a high Gb6 which occurs in bar 128 of the first movement of Sonata No. 5 (Op. 10, No. 1) (1796/98). Although these two notes may be anomalous, they could also indicate the use of a pianoforte with a range of five octaves and two notes, from F1 to G6, as it is known that the Viennese piano-maker Anton Walter was building keyboards with this compass at around this time (1785) (Melville, 1971a, p. 42)1. Given the close time-span during which these two pieces were being written, this could indicate that they were both composed for a piano of this range, perhaps even one built by the same maker, given that “[an Anton Walter piano] was found in his dwelling about 1801, when Czerny first played for him at the age of ten” (Newman, 1971, p. 487).
1.1.2 Sonatas 21-27
Sonatas 21-27 appear to be written for a variety of different-sized pianos. It is known that, from 1803, Beethoven was in the possession of an Erard piano with a compass from F1 to C7 (Melville, 1971a, p. 42) and, it seems from the data, that out of the seven works written in this period, five of them seem designed for this instrument (Sonatas 21, 23, 24, 25, and 27). Sonata 22 appears to be written for a piano with the same five octave compass as the majority of the twenty earlier sonatas, and Sonata 26 (Das Lebewohl) extends over a full six octaves, from F1 to F7, which indicates this was written for another, larger piano.
1.1.3 Sonatas 28-32
As Tovey points out, the last five sonatas seem written for instruments of different ranges (Tovey, 1958, p. 6). It is known that from 1818 Beethoven owned a Broadwood piano that encompassed six octaves (C1 to C7)(Newman, 1971, p. 493), yet from the data, it can be observed that only one of the last five sonatas fit within this range, Sonata 31. The last sonata fits, with the exception of a single high Eb7, which at first looks like an anomaly in the data, lasting only for 96ms. However, as Melville points out:
Beethoven wrote an ossia for this one note, simply to accommodate the Broadwood (see the facsimile of the Berlin autograph of the first and second movements). Beethoven obviously liked to play Op. 111 on both pianos [his Broadwood and a Streicher piano], and considered it worthwhile to insert an alternative note at this point.” (Melville, 1972, p. 362).
Sonata 30 (Op. 109), seems similarly to lie almost completely within the range of the Broadwood except for three errant high C#s,. Here, however, there is no evidence of an ossia, so we must presume that a different piano was used, or else this was a similarly ossia-ed passage for which the documentation has been lost.
As Tovey notes: “The finale of Op. 101 [Sonata 28] is evidently for an instrument of wider compass than the other movements” (Tovey, 1958, p. 5). Whilst the other movements fit within the range of Beethoven's Erard, the finale may well be written for the same six-octaves-and-a-fourth (C1 to F7) sized instrument as that used for Sonata 29.
1 This range was a also popular for harpsichords.