I’ve just finished a canon — #56 “Sugilite” — which explores the possibility of making music under two special constraints.  First, the piece uses strict contrary motion, which means that the follower is an upside-down version of the leader: wherever the leader descends, the follower ascends by the exact same melodic distance, and vice versa. (This technique is cumbersome enough that I’ve only explored it occasionally in my canon series so far: I had used strict contrary motion in Canon #37 “Graphite” and non-strict contrary motion in Canons #12 “Amber” and #13 “Carnelian”.) Second, the piece uses a major third on the first beat of every measure, following in the steps of other restricted interval canons: #52 “Pyrite” (fourths), #54 “Tektite” (major seconds) and #55 “Magnetite” (minor sevenths). There are a few short connecting passages that don’t participate in the canonic imitation.

Here is a visualization of the beginning of the piece.  You can see how the top voice (red) is a mirror image of the bottom voice (green) with a skew or time lag between them:

If we eliminate the skew between the voices, the mirror relationship becomes even clearer:

The challenge of working in contrary motion is that many melodic gestures are orientation-specific: they “make sense” when played in their upright form but sound confusing or unconvincing when played in their mirrored form.  The composer has to search for those gestures that are musically persuasive in both orientations.

The further challenge of working in strict contrary motion where interval sizes are preserved exactly in the mirrored voice (a major third up translates into a major third down, never a minor third) is that it’s nearly impossible to stay within the notes of a given key.  The technique lends itself best to pieces with a free or floating kind of tonality (as in Canon #56) or to pieces that are entirely non-tonal.

Why bother with the difficulties of strict contrary motion?  I think it leads to a fascinating relationship between voices, one that can be challenging to perceive at times, but one which is definitely accessible to the focused ear — a relationship where the voices seem to be “the same” in one sense, and yet in another sense they seem to be saying different things: sort of like twins with unmistakable and deep similarities, but with their own personalities and directions.  As they move in these contrary directions it is interesting to notice how similar they remain, or how different they become.

Here is Canon #56:

Premier performance of Canon #46 “Palladium” with the wonderful Matthew McConnell on harpsichord.  A new exploration of the idea of vertical shifting counterpoint as first described by theorist and composer Sergei Taneyev (1856-1915).  Part 1 is an initial statement of the canon with imitation at the octave.  Part 2 is a reprise with the top voice transposed up a semitone while remaining consonant with the bass (imitation is now at the minor ninth).  Part 3 is a reprise with a shifting interval of imitation.

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.

 

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.