Two musical forms, the
simple canon and the mensuration canon, can be described in terms of geometrical
transformations. The forms are described mathematically, classical examples
are given along with a discussion of some of the musical and mathematical
limitations and considerations imposed by the form and a new mensuration
canon composed by co-author Shumays is presented and analyzed.
A canon in music can be described in terms of one melody line (which may be sung or played on an instrument) repeated in a linear translation,resulting in multiple voices performing simultaneously thereby creating harmony from the melody. While a typical mathematical translation is considered in space, in a canon the translation takes place in time. In the simplest canon, performer j produces the note n, which was first played (or sung) at time t, now at time t+(j-1)r where r is the time between successive entrances of new voices. When an entire canon is k measures long and there are a total of j distinct voices, the entries often occur after k/j measures.
The simplest
example of a canon is a round in which a (usually short) melody is repeated
several times with new voices starting and ending at different times, but
each new voice enters at the same point in the melody line relative to
the preceding voice. The translation vector in a round is a unit fraction
of the length of the initial line, where the denominator of the fraction
equals the number of voices. Harmonic structure is created because all
notes at integer multiples of the translation vector distance apart are
sounded simultaneously when the round is performed. One can think of the
translation as being modulo the length of the melody.
Perhaps the most famous of all canons is Pachelbel's Kanon in D, a "perfect" canon, in which each voice exactly mimics the one before. The harmony in this particular canon "works" because each phrase is constructed over the same progression of harmonies, which are repeated throughout the canon in pairs of measures, any of which could be played simultaneously without sounding dissonant. The trade-off for this easy way to assure that the chords created by the translation in the melody sound consonant is that the piece never "goes anywhere" harmonically. The only change in sound as the piece continues is in rhythmic and melodic variation.
Every two-bar
phrase in the Kanon in D utilizes the following sequence of chords.
|
Chords:
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I | V | vi | iii | IV | I | IV | V |
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Root:
|
D | A | B | F# | G | D | G | A |
This particular canon is so well known
and the chord structure so distinctive that it has become known as the
Pachelbel progression.
A different type of musical/mathematical creation is called a mensuration canon. Mensuration is an early musical term which is analogous to our notion of time signature. A mensuration canon can be described as a mathematical dilation, where each "voice" carries the same melodic line but at a different speed. In some of the early examples of mensuration canons from the Fifteenth and Sixteenth Centuries, the melody line is denoted once, along with several time signatures and each musician is expected to perform the line at one of the time signatures. The faster performers might be instructed to repeat their line several times or else the piece might stop short of the slower performers completing the entire melody line.
According to the Harvard Dictionary of Music [1 p. 125], "A landmark of canonic art, comparable to the canons in Bach's Goldberg Variations, is Ockeghem's Missa Prolationem, each movements consists of a double mensuration canon." In particular, this means there are two melody lines each of which are sung simultaneously at two different speeds by a total of four voices.
According
to Timothy A. Smith [2], "In 1974 the autograph manuscript of the Goldberg
Variations was unexpectedly discovered in private possession in France.
Accompanying the manuscript was an appendix containing a cycle of fourteen
canons upon the first eight notes of the Goldberg ground. The discovery
of the hitherto unknown manuscript was immediately hailed as the most important
addition of a Bach source in recent decades. Of the fourteen canons, only
numbers 11 and 13 had been known before 1974. Ö In J. S. Bachís
Goldberg Variations Canon 14, labeled Canon a 4 per Augmentationem et
Diminutionem, the last canon is for four voices in rhythmic proportions
(mensuration canon). Bach labeled it ëcanon for 4 voices in augmentation
and diminutioní." This canon may be heard by visiting the web site
listed at the start of this paragraph or at [3].
Mensuration
Canons can be characterized by the ratios of the speed of the various voices.
For example, Le Ray Au Soleyl by Johannes Ciconia (ca. 1335-1411)
is a 4:3:1 mensuration canon. In modern musical notation it has been transcribed
so that the fastest line is in 
time and the remaining two lines are in 
.
Four quarter notes of the fastest line equal three in the second fastest
line and both equal one in the slowest line. All three voices stop when
the first line has reached the end of the piece. The first two measures
of a transcription in modern notation are shown below. [4]
- LE RAY AU SOLEYL
and sums of consecutive powers of .
The basic powers of
are musically denoted by whole notes, half notes, quarter notes, eight
notes and occasional sixteenth or even thirty-second notes, where two half
notes equal one whole note, etc. Consecutive powers are indicated by a
dot following a note where, for example, a dotted quarter note has a value
equal to 
of a whole note. While triples do occur, they are essentially described
by a special notation that means "play these three notes in the time that
would ordinarily be devoted to two notes!" This means that the majority
of mensuration canons are in ratios of 1:2 or 1:4 or 1:2:4, depending on
the number of voices.
Another
limitation on mensuration canon music is a result of how fast a note can
be produced and how long a note can be held. Thus in a mensuration canon
with a fastest to slowest ratio of n:1, the slowest voice cannot include
any note that is too fast for the fastest voice to produce, i.e., one where
the duration d of any note in the slowest voice has the property that 
,
where a note of duration 
is the shortest feasible note. Likewise, if the longest note a performer
can reasonably be expected to produce without running out of wind (or performing
too boring a part) is a note of duration y, then this gives rise to the
restriction that in the fastest voice all notes must be of duration at
least 
.
In other words, the longest notes in a piece canít be too long or
the slowest voice is boring and the shortest notes canít be too
fast or they canít be performed in the fastest voice.
The musical considerations to decide what "works" harmonically are clearly much more complex for mensuration canons than for simple canons. We suggest a combinatorial view of what the composer must consider for sounding acceptable when performed together. Each phrase in a piece will be labeled with a successive letter of the alphabet and the phrases that are to be performed simultaneously are written under one another. Thus for the first sixth of the Ciconia piece, the following diagram would be appropriate.
Here the A phrase consists of the three dotted quarter notes A-F-D filling the first measure of the slowest voice. Note that the first nine notes of the fastest voice (which take five measures to be sing in the slowest voice) must musically match the entire A phrase in the slowest voice as well as three fourths of the same phrase in the middle voice. Similar considerations exist for all phrases of this piece.
Clearly,
ideas of what is musically acceptable change from one musical period to
the next. A recording of the Ciconia piece can be heard on the compact
disk "Homage to Johannes Ciconia." [5] While the music clearly sounds unusual
to the contemporary ear, it fits within the bounds of Fourteenth Century
musical style.
Several contemporary versions of a mensuration canon have been composed. Co-author Sami Shumays composed one especially for this paper. Simply entitled "Mensuration Canon," is a 16 measure piece which follows a 4:2:1 pattern with the slowest voice finishing the entire 4-phrase melody once, the middle voice completing the piece twice and the fastest voice completing it 4 times. Thus his piece has characteristics of a round combined with those of a mensuration canon.
Since the music is otherwise
unpublished, a copy of it is included here.
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The following diagram shows the relation among the three voices in this piece.
Some of the musical considerations that may be inferred from this diagram, and which were considered by the composer, include that the first fourth and first half of the A-phrase had to sound well with all of that phrase. Likewise the second half of the A-phrase had to sound well with all of the B-phrase, as did the third fourth of the A phrase. Shumays was interested in the pairwise compatibility of the phrases, since when all pairs work together, then so does the whole piece. In particular this was helpful in the sense that once the A-phrase had been composed it was going to appear 8 times in conjunction with one or another of the voices. It had to match with itself in two different contexts. Note also that there is a basic symmetry in the piece where one can view it from the end where each D phrase corresponds to an A phrase from the beginning with the roles of B and C also interchanged. Thus there are also 8 distinct pairings involving D and something else, although two of the 16 total involved both A and D for a total of 14 involving either A and/or D.
Initially it might have seemed that there were 48 pairings that had to be considered for acceptable pairs of sounds, namely, each of the 16 measures of the fastest repetition along with each of the three combinations of the portions of the slower voices performed simultaneously with one of the fastest measures. However the above analysis shows that there are only 14 pairing involving A and/or D. To complete the count we only need to consider the number of pairing involving only the 6 pairing with portions of the B and C phrase with one another and the 2 instances where B is played with a portion of itself and C with a portion of itself. This means only 24 distinct pairings occur rather than the 48 theoretically possible.
A performance of Shumays Mensuration Canon performed on the piano by Albert Kim may be accessed on the world-wide web. [6]
A contemporary collection of mensuration canons is Four Voice Canons #3-6 Music for computer, voice and instruments by Larry Polansky with Phil Burk, Jody Diamond, Tom Erbe, Chris Mann, and William Winant. The following information is taken from their web site. [7]
In #3 (1975), for digitally synthesized sounds, the permutation lists are applied to spatial location, envelope, amplitude, pitch, duration and several aspects of timbre. In #3 the duration ratio between all voices is the golden mean; in the other pieces, it is an approximation of that ratio.
#4 (1978-79), written for the choreographer Andrea Smith, uses four successively higher notes from the first 16 harmonics on C, making use of the complete range of one marimba. The rhythmic relationships of the four voices are 2:3:5:8.
#5 (1983), written for percussionist William Winant, uses four families of percussion instruments -- wood, tambourine, metal, and skin. Each voice uses one instrument from each family, with the first (slowest) the lowest and the fourth (fastest) the highest. The durations of the slowest voice are a half-, quarter-, dotted-quarter- and eighth-note, adding up to one measure of 5/4. The three other voices scale these values by factors of 3, 5 and 8.
#6 (1986) was written for a homebrew sampler, built at the Mills College Center for Contemporary Music from a single board 68000 computer running a prototype of HMSL, and analog-to-digital and digital-to-analog converters. The sound material is four five-second samples: frogs in the Mills frog pond, a Javanese rebab (played by Jody Diamond), a baritone sax (played by Anthony Braxton), and the sine wave preset of a Kurzweil 250 sampler. The sampling procedure in this work is intentionally : no filters, zero-crossing algorithms, deglitching or smoothing techniques were used. All of the sounds, except the frogs, rise in pitch over the five second span, so the computer program, stochastically choosing starting memory locations later and later in the sample memory, tends to play higher pitches for faster voices. The rhythmic values, derived from the permutation lists and the mensuration canon, are determined simply as a number of bytes of memory for a given part. The piece is generated and performed in real time by the computer.
The
authors hope that you, the reader, will enjoy listening to mensuration
canons and will be intrigued by the mathematics of their existence, composition
and performance.
References
1. Harvard Dictionary of Music, Second Edition, Willi Apel, Belknap Presss of Harvard University Press, Cambridge, MA. 1969.
2. http://jan.ucc.nau.edu/~tas3/fourteencanonsgg.html
3. http://prs.net/cgi-bin/n.cgi/m/1/bwv1087.mid
4. The works of Johannes Ciconia, in Polyphonic music of the fourteenth century; v. 24, Monaco: Editions de l'Oiseau-Lyre, c1985, p 177-8.
5. "Homage to Johannes Ciconia." by the Ensemble Pan, Project Ars Nova, New Albion Records CD number NA048.