TILING CANONS: Aural Curiosities and Illusions

Since 2021, I've been exploring special musical patterns called tiling rhythmic canons. They're like puzzles where different parts of the pattern fit perfectly together without any gaps or overlaps. During this study, I uncovered intriguing auditory phenomena, such as music that can be simultaneously dynamic and static, or music that suddenly shifts from polyphony (many voices) to monody (a single voice), resulting in a curious listening experience. I've also been fascinated by the "fractal" nature of these patterns, where smaller versions mimic larger sections, adding another layer of complexity.

The outcome of my study is the Book of Canons, a collection of percussion pieces based on 7-beat rhythmic patterns. These compositions attempt to showcase the versatility and intricacy of tiling canons in music.

But before we dive into discussing these canons in my music, let's clarify some definitions.

Tiling
Tiling in music happens when two or more statements of a rhythmic pattern fit together exactly, leaving no gaps or overlaps in the rhythmic grid. It's like creating a mosaic where a geometric shape is repeated over and over, covering every inch of space.

Figure 1 lists the four related tiling rhythmic patterns used in The Book of Canons. Each one is 14 beats long and contains seven attacks (represented by black squares). White squares in Figure 1 indicate silence.

Figure 2 shows how two statements of Canon 1 combine to achieve tiling. Over the 14 beats, either voice 1 or 2 is heard (but never both) on every beat. Both play the same

rhythmic pattern, with the red square indicating where the rhythmic pattern begins. (In voice 2, the pattern begins on the 8th beat of the grid.) Each of the four patterns in Figure 1 has this tiling property.

Canons at different speeds
Canons at different speeds are also possible with these patterns. They can, for instance, produce canons that move a third or a fifth slower than the original. Figure 3 shows three statements from Canon 1 (voice 1) in the top row. Below are three voices (labelled 1a, 1b, and 1c) that combine to match the rhythm of voice 1. Furthermore, 1a, 1b, and 1c play the identical rhythmic pattern of voice 1, but at one-third the speed stretched over 14 x 3 = 42 beats. (The red squares in Figure 2 indicate the beginning of the pattern.)

It follows that just as voice 1 in Figure 2 may be subdivided into three slower-moving canons, so can voice 2. Canon 1, therefore, may be grouped into six parts over 42 beats in addition to a two-part canon over 14 beats as shown in Figure 2. The same property exists for canons moving at a 5th slower over 5 x 14 = 70 beats, as outlined in Figure 4.

In the first row of Figure 4 are five statements of Canon 1 (voice 1) over 70 beats. Below that are five versions of Canon 1 that create a composite rhythm that matches the first row. In addition, each voice (voice 1a, b, c, d, e) plays the same rhythmic pattern as voice 1 but at one-fifth the speed.

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Tiling Rhythmic Canons in my music

Let’s talk about the music. I'll discuss three pieces from the Book of Canons (Many-One, Slowly Turning, and Cross-Rhythms) that are based on Canon 1. The Book of Canons are for untuned percussion and for these arrangements I use wood blocks. As a coda, I will introduce Zooming In, a longer piece for tuned percussion, which is not part of Book of Canons.

Many-One
The curious fluid relationship between polyphony and monody is investigated in the Many-One pieces. In Many-One (10 parts): Canon 1, voices are gradually added, creating a more intricate contrapuntal texture until the tenth and final voice is introduced (at 3’41 in the recording below). At that point, tiling occurs. The polyphony disappears, replaced by a single line. This curious aural phenomenon is a feature of tiling. The recording includes each statement played twice to help the listener hear the progression. Therefore, the piece opens with the first voice playing the 7-note rhythmic pattern twice before the second voice enters at 0’24.)

Figure 5
Many-One (10 parts): Canon 1

Slowly Turning
The Slowly Turning pieces showcase music that is simultaneously dynamic and static. In contrast to Many-Ones, Slowly Turning (6 parts): Canon 1 has a fixed number of voices - five out of six possible canons. (Slowly Turning exists maximum contrapuntal density just before tiling happens.) Each section introduces a new sound and removes one systematically - like a slowly rotating hexagonal object where one side is hidden from view. The listener's attention is drawn to the new sound, which is heard as a 'downbeat' before fading into the background in the following section.

While there is change, the progression is entirely predictable. There are no surprises here. This absolute regularity translates into a non-traditional listening experience (at least to my ears). The Slowly Turning pieces do not develop and move in any traditional sense; instead, any progression is heard as providing different perspectives of a fixed object.

Slowly Turning (6 parts): Canon 1 has six sections
Section a: 1, 2, 3, 4, 5
Section b: 6, 1, 2, 3, 4
Section c: 5, 6, 1, 2, 3
Section d: 4, 5, 6, 1, 2
Section e: 3, 4, 5, 6, 1
Section f: 2, 3, 4, 5, 6
Section a: 1, 2, 3, 4, 5
(and so on)

Figure 6: Slowly Turning (6 parts): Canon 1

In this version of Slowly Turning, I provide a 'key' to assist the listener hear the new element and therefore follow the progression. For the first three sections (a, b, and c above), the new sound is heard first as a solo and then as part of the 5-part ensemble. From section d, the piece maintains a constant 5-part texture with no more solos.

Dual
Dual: Canon 1 is a study of polymeter. Two pieces, each with a distinctive metric character, are superimposed. Dual: Canon 1 is striking since the two parts fit perfectly due to tiling* yet each part’s metrical nature is fully audible. Two metres can be heard at the same time.

The form of the piece is

Intro                          Section 1 (0’15)                                Section 2 (2’35)
2-part canon            2-part canon (2 sounds)                   2-part canon (2 sounds)
6-part canon (3 sounds)                 10-part canon (3 sounds)

A 2-part canon, with a 14-beat duration, comes and goes at regular intervals throughout Dual. In section 1, superimposed over the 2-part canon is a 6-part canon, which begins with one voice and slowly adds until all six voices are heard. Similarly, in section 2, a 10-part canon enters one voice at a time. The process of accumulation recalls the Many-One pieces.

While the 42-beat length of the 6-part canon is difficult to hear, the clear 3-beat feel of the 6-part canon clashes with the 7-beat groupings of the 2-part canon (divided into 3+2+2). Likewise, in section 2, the 5-beat of the 10-part canon is juxtaposed with the 7-beat groupings of the 2-part canon.

(Dual features two duos - 2 & 6 parts, and 2 & 10 parts. It is also possible to superimpose the 6-part and 10-part canons or even stack all three - 2-, 6- and 10-part canons - on top of each other! However, because of the complexity, I decided to not include these other possibilities in Dual. I wanted to keep the clashing meters as clear as possible.)

Figure 7 Dual: Canon 1

CODA
Zooming In - 'fractal' properties, in which parts of a figure are smaller-scale replicas of the whole
Finally, I’ll introduce Zooming In, a complex piece for three players on tuned percussion instruments. The piece is based on the Canon 1 pattern and includes 2-, 6-, and 18-part canons, creating a sense of wheels within wheels, where the 7-note pattern is played simultaneously at many different speeds.

In Section 1, the piece starts with a slow version of Canon 1 played over 126 quaver beats. Notes are gradually added until each of the three musicians play six statements of the pattern, resulting in an 18-part texture. The performers’ six notes can also be grouped into two sets of three (three lower and three upper notes), creating a faster version of Canon 1 over 42 quaver beats. Additionally, when 42-quaver patterns are combined between all three players, an even quicker version (14 quavers long) can be heard.

The intricate web of canons moving at various speeds offers different listening pathways. In this recording, the three players are separated spatially, allowing the listener to focus on the single-note slowest pattern (126 quaver length), the composite canons played by one player (42 quavers), or the canons of the three performers (14 quavers).

As the piece progresses, there is an intensification as the pitch is reorganised and the music "zooms in". The single-note version of the pattern becomes shorter in each of the three sections. Section 1 is built on the 142-quaver long pattern, section 2 (at 7’41) has a 42-quaver long pattern, and section 3 (at 12’45) features a 14-quaver long pattern.