Compositional structure
Multicultural Mathematics
Dr. Rachel Hall

Compositional structure refers to the way patterns are organized within a musical piece.  Arithmetic series and the tihai ending are both compositional structures found in Indian classical music.

Repeats.  The simplest structure consists of repeating the same pattern a number of times.  In abcdrums, if you want to repeat pattern A 12 times, you create pattern A, and then assign A12 to your desired drum, like this:

Exercise 1:  repeats

1.  Snare drum:  A12  (substitute your favorite drum if you want)
A:  dddd dddz
The space in pattern A doesn't change anything--it just makes it easier to check that pattern A is 8 beats long.

Common forms.  Compositional structure is similar to rhyme scheme in poetry.  A basic structure, found often in Western music, is AABA form.  This means you play pattern A twice, then pattern B, then finally pattern A again.  To get an idea of what AABA form sounds like, try this abcdrums exercise:

Exercise 2:  AABA form

1. Snare drum:  AABA
A:  dddd dddz
B:  dddz dddz
Other forms common in Western music are
AABB,
ABAC (``Mary had a little lamb'' has this structure),
AABC (``The Star Spangled Banner'' has this structure),
AABBCC,
ABBA (``Twinkle, twinkle, little star'' has this structure).
Exercise 3:  ABAC form
1. Snare drum:  ABAC
C:  dzdz dzzz   (leave A and B as above)
Exercise 4:  nested repeats
Parentheses allow you to repeat combinations of patterns.  To repeat a form such as AABA 4 times, you would write (AABA)4.  The total length of (AABA)4 is 4 times the length of AABA.  In the exercises we've been doing, A and B have length 8 beats, so AABA has length 32 beats and (AABA)4 has length 128 beats.

Leaving patterns A, B, and C alone, try the following structures:

• 1.  Snare drum:  (AABA)4
• 1.  Snare drum:  (AB)3 C   What is the total length of this pattern?
• 1.  Snare drum:  ((AB)3 C)4   What is the total length of this pattern?
Mathematically, compositional structure is a little like algebra:  you have variables (A, B, etc.) that can be assigned values (in this case, patterns).

Adding instruments and rests.  There is one rule to remember:  every instrument must be assigned the same number of beats.  This doesn't mean that each instrument has the same number of rests or hits--just the same total number of beats.  Using the 8-beat patterns A, B, and C, and rests indicated by z (for 8 beats rest, type z8), the following is a legitimate composition:

1.  Low Woodblock:   A  z8 B  z8
2.  High Woodblock:  z8 A  z8 C
Exercise 5:  Try this on abcdrums.

However, the following is not a legitimate composition, because the number of beats in part 1 is 24, but there are 32 beats in part 2.

1.  Low Woodblock:   A  z8 B
2.  High Woodblock:  z8 A  z8 C
This form (i.e. when one instrument ``answers'' another) is called call-and-response.

Delaying the entry of one instrument.  Sometimes you don't want every instrument to start at once!  To do this, you must assign rests to some instruments at the beginning of the piece:

Exercise 6:  Delayed entry

1.  Low Woodblock:  (A  z8 B  z8)4
2.  High Woodblock: (z8 A  z8 C )4
3.  Bass Drum:      z32  (DDDC)3
4.  High Bongo:     z32  (EEEC)3
Pattern D:  dzzz dzdz
Pattern E:  zddd zdzd
Patterns D and E are called complementary--whenever D has a hit, E has a rest, and vice-versa.