(determines the order of events for a given voice,
using the concept of 2-transposition
to maintain “continuity” between permutations)
(Four Voice Canons)
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)
(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:
Sv = 1 - (b/a)
·
where Sv 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):
S2 = 1/3; S3 = 3/5; S4 = 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 !!