I’ve just taken another look at my exploration of rhythmic tiling canons from 2015. I’ve corrected a few errors and given the piece a new name. I’ve also updated the video to include captions that explain what’s happening in each section of the piece.
https://www.youtube.com/watch?v=wmxTGi60DGo
From the notes on YouTube:
This piece is named after the artist M. C. Escher (1898-1972) because of his fascination with tessellations: the way a carefully crafted shape can be repeated across a surface in an interlocking way, so as to fully cover the surface without any gaps or overlaps. In music, there is a related concept called a rhythmic tiling canon, where a special rhythm is repeated by some number of musicians in a staggered fashion, so that no two players ever coincide on the same beat, but every possible beat in every measure is hit by someone. Such rhythms are akin to “tessellations” of musical time.
This piece explores eight rhythmic tiling canons that were described in the paper “Asymmetric Rhythms and Tiling Canons” by Rachel W. Hall and Paul Klingsberg (2006). These are the only tiling canons that exist under certain constraints – where the rhythmic cycle consists of twelve beats and the entrances of the players are equally spaced.
The objectives of this piece are, first, to present the eight canons in a musically engaging way (by carefully choosing where each rhythm should begin, which beats to accent, which instruments to use, which order the instruments should enter, how to arrange the eight sections, etc.) and second, to progress from one canon to another in a connected fashion, where rhythms from neighboring canons are temporarily combined, and each section seems to morph into the next.
Captions have been added to the video to clarify the structure of the piece. As you watch, you will notice sections where one specific rhythmic pattern is played solo, or in duo, or in trio. Any section marked TRIO is a very special event where the full rhythmic tiling canon emerges. In these trio sections, every possible beat is struck without any gaps or overlaps – remember, most rhythmic patterns can’t be played this way, and it’s rare to find one that’s suitable. You will also notice connecting sections where different rhythms are juxtaposed. These juxtapositions are planned so that no two players ever coincide on the same beat (though some beats may be skipped) and they were chosen out of many possible combinations for their musical interest.
Ignoring the question of where accents should fall, each rhythmic pattern in the piece can be written as a sequence of ones and zeros where one represents a hit and zero represents a rest. Using this notation, the eight patterns are as follows:
Pattern A: 101001010000 Pattern B: 001001011000 Pattern C: 001011010000 Pattern D: 111100000000 Pattern E: 000100001110 Pattern F: 110000110000 Pattern G: 110100100000 Pattern H: 100100100100
Here is a rough outline of the piece:
Pattern A: SOLO, then DUO, then TRIO, then VARIATIONS Patterns A and B: JUXTAPOSITION Pattern B: TRIO Patterns B and C: JUXTAPOSITION Pattern C: TRIO Patterns A and C: JUXTAPOSITION Pattern A: DUO, then TRIO (reprise) Patterns A and D: JUXTAPOSITION Pattern D: TRIO Patterns D and E: JUXTAPOSITION Pattern E: TRIO, then DUO Patterns D and E: JUXTAPOSITION Patterns D and F: JUXTAPOSITION Pattern F: TRIO, then DUO Patterns F and G: JUXTAPOSITION Pattern G: TRIO, then DUO Patterns G and H: JUXTAPOSITION Pattern H: TRIO, then SOLO
The geometric pattern shown at the beginning of the video is NOT by M. C. Escher, though it’s safe to assume he would have admired it. The pattern is the fifteenth known way of tiling the plane using convex pentagons. It was discovered by Mann, McCloud, and Von Derau in 2015, the same year this piece was composed.
The eight individual rhythms used in this piece are illustrated here: https://www.youtube.com/watch?v=vBsCfl-MgwA
This piece is part of the album Canons by Rudi Seitz: https://rudiseitz.bandcamp.com/album/canons
An earlier version of this video was released in Jan 2015. The composition was given a new title in 2016, some small corrections to the music were made, and captions were added to the video. The visualization was made with MIDITrail software by Wada Masashi.
See also my followup piece, Fifteen Beats. ■