Sample Four Element Permutation List

(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   B   C   D   E                      E   B   C   D   A

 



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 !!