Exercise 2-1. Modulo 12 arithmetic and pc intervals

Because the pitch classes form a modulo 12 system, you need to become familiar with modulo 12 arithmetic. (You've likely been using it for years in measuring time intervals on the twelve-hour clock.) Just remember that 12 is the modulus of the system: 12 = 0 in any arithmetical operation, and all numbers in the system fall in the range 0 - 11. When the result of any operation would lie outside this range you add or subtract 12 (or a multiple of 12) to normalize the result to within the range.

4 + 2 = 6	4 + 10 =14 (-12) = 2
8 - 3 = 5	3 - 8 = -5 (+12) = 7

1. We use modulo 12 arithmetic in measuring the intervals between two pitch classes. There are usually two such pc intervals: "ascending" (clockwise on the pc clockface) and "descending" (counter-clockwise). Arithmetically, they are measured as follows:

   "ascending":	1st pc + x = 2nd pc
   "descending":	1st pc - x = 2nd pc

   Below are some pairs of pitch classes. In your mind, name the pc intervals, both "ascending" and "descending," from the first pitch class to the second. (You will find that these two intervals are complementary: their numbers sum to 12.) The pitch-class clock face is provided to help you.

   Click on the blank lines to reveal the correct interval names. Pressing the Reload button will remove the answers.

    

   1st pc		2nd pc	"ascending" interval	"descending" interval
   4		7
   2		8
   10		2
   8		7
   5		0
   9		11
   3		10
   6		0
   4		2
   11		3

   
   
   

2. We also use modulo 12 arithmetic when we deal with the transposition of pitch classes. By convention, such transposition is always measured in the "ascending" direction only. To describe transposition of pcs, we can use a mathematical shorthand: T_n, where"n" is the number of semitones by which a pc is to be transposed. To find the result of T_n, the equation is simply

   1st pc + n = 2nd pc

   The integers in the left-hand column below represent pitch classes. In your mind, transpose each pitch class by the required amount. Click on the blank lines to reveal the correct interval names. Pressing the Reload button will remove the answers.

   1st pc	transposition	2nd pc
   4	T5
   3	T4
   11	T2
   7	T3
   6	T8
   9	T1
   5	T10
   10	T7
   10	T5
   9	T3

   
   

Page last modified 17 July 2001 / GRT

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