B. Beat-Level Four-Against-Three

Four-Against-Three in Simple Meter

Beat-level four-against-three arises in one of two ways: (1) with beat-level triplets against a four-way subdivision of the beat in simple meter (such as sixteenths against eighth-note triplets in 4/4) and (2) with quadruplets against a three-way division of the beat in compound meter (such as eight- or sixteenth-note quadruplets against eighth notes in 6/8). This section is divided into two parts based on these two cases.

The case of four-against-three in simple meter will be notationally more familiar, since it involves sustaining rhythmic patterns against one another that are familiar from Chapter 4. Yet, the underlying skill is the same in both cases. One must first maintain an accurate 3:2 pattern (a skill developed in Chapter 4) and then subdivide the 2 into 4.

Exercise 8B-1 demands this skill right away. Prior to their entrance, the clarinetist should imagine duplets against the triplets in the strings, which they will subdivide into sixteenths at their entrance. Exercise 8B-2 work similarly, except it is the pianist themself who much must imagine duplets in the opening measure against their own triplets.

Exercise 8B-3 revisits the full two-hand texture of an exercise encountered in the last chapter and provides an instance of four-against-three at the level of the eighth-note subdivision rather than the notated quarter-note beat. Regardless of whether the quarter note or eighth note is felt as the main beat, the skill of performing four-against-three at the level of a subdivision is essentially the same as at the beat level.

Exercise 8B-1: Joseph Küffner (previously misattributed to Carl Maria von Weber), Clarinet Quintet/Theme and Variations, Opus 32

Exercise 8B-2: Ludwig van Beethoven, Piano Concerto Number 5, Movement 2

Exercise 8B-3: Wolfgang Amadeus Mozart, Violin Sonata in G Major, K. 27, Movement 1

Four-Against-Three in Compound Meter

Once four-against-three is mastered in simple meter, the skill transfers relatively easily to quadruplets in compound meter.

Exercise 8B-4 contextualizes quadruplets as an outgrowth of duplets. Note that this is an instance where both duplets and quadruplets are notated as eighth notes, despite the fact that the quadruplets are a subdivision of the duplets.

Exercise 8B-5 is slightly more challenging insofar as it demands both four-against-three and immediate switching between the compound grouping of eighth notes and quadruplets with no intervening duplets.

Exercise 8B-6 presents the fuller passage of Exercise 8A-1. It is effective as a trio exercise or as a duet exercise, in which case the right-hand line of the piano may go to with the cello line (easier) or with the left-hand line of the piano (harder).

Exercise 8B-4: Ralph Vaughan Williams, The Lark Ascending

Exercise 8B-5: Zoltán Kodály, Duo for Violin and Cello, Opus 7, Movement 2

Exercise 8B-6: Leoš Janáček, Pohádka for Cello and Piano, Movement 2