(determines the order of events for a given voice,

using the concept of 2-transposition

to maintain “continuity” between permutations)

** **

** **

**(Four Voice Canons)**

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** **

** CBAD BCAD BACD BADC**

** BDAC DBAC DABC DACB**

** DCAB CDAB CADB CABD**

** ACBD ABCD ADCB ACDB**

** BCDA BDCA DBCA DCBA**

** ABDC ADBC CDBA CBDA**

** **

(this list used in *FVC #6*)

**An example of a
2-transposition**

** **

** **

** **

**A Typical Mensuration Canon
(time structure)**

**(Four Voice Canons)**

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** **

** **

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(duration of voice 2 = 2/3 * duration of voice 1, etc.)

The equation for the the starting point of a voice, based on its duration
ratio, is:

S_{v} = 1 - (b/a)

·
where S_{v }is the starting
point in time for voice *v*; *a/b* is the duration ratio for voice *v* (*a/b*
> 1); 1 is the duration of the piece.

For example (above):

S_{2} = 1/3; S_{3} = 3/5; S_{4} = 3/4

**Tempo Ratios Used in the
Canons (so far)**

#4, #5, #7, #11 (for instruments), whole number ratios are used, based on
the Fibbonacci sequence (2, 3, 5, 8).

*#*3, #6, and #8, one irrational ratio (the golden mean).

*#9* (“Anna Canon”), there are four versions, three with a different set of
superparticular ratios (a: 4:5:6:7, b: 6:7:8:9, c: 8:9:10:11), one with the
golden mean (d: f).

#10 (“Four Boys Mannin’”), seven voices, with the superparticular ratios
6:7:8:9:10:11:12.

#11 (“Parsons Canon” from *Tetherball*),
5 voices, with the ratios 4:5:6:8:12

#12 (“Doggerel” from *3 New Hampshire
Songs*), four voices, with the ratios 4:5:6:7.

#7, #13, #14, #15, #17, #19 do not specify the tempo ratios for their
realizations. On the Cold Blue CD set, #7 (gamelan) uses: 12:17:29:43; #14
(“Kid Canon”) uses 1: 6/5 : 7/5 : 5/3: 11/6 : 2/1 (or 30:36:42:50:55:60); #17
(“Guitar Canon”) uses 4:5:6:7:9:11; #18 (“Trio Canon”) uses 6:7:8:10.

**A word processor example of a
mensuration canon**

!! Caution: Do not try this at home without strict parental supervision !!