6.3: Modulo Arithmetic
What time is four hours later than 10 o’clock?
When we make calculations like this, we are doing modular arithmetic . Modular arithmetic is like regular arithmetic, except that the numbers “wrap around” or restart when they reach a certain value, called the modulus . In the case of our 12-hour clock, the modulus is 12.
Musical structures can often be best understood using this modular arithmetic. Think of the C-major scale. We begin on C, then D, E, F, G and back to A before B and returning to C. This is a modular system — we might call it modulo-G because after G we go back to the beginning (A).
In post-tonal music, once we assume octave and enharmonic equivalence, our pitch-class environment includes twelve unique pitch classes, just like the twelve hours on the clock. In this universe, modular arithmetic is a very useful way to imagine getting around.
Counting in this modulo 12 (or mod12 ) universe works just as in basic math [“1, 2, 3, …”], but after 11, we “begin again” at 0. (Note that while clocks start at 1 and end on 12, modular arithmetic always (re)starts with zero.) Conversely, when counting down [“10, 9, 8, …”], we follow 0 with 11.
While we are used to thinking of numbers on an infinite line, modular thinking wraps them into a finite number, generally represented by a circle. On this circle, all values are a number from 0 to 11.
Pitch-Class Space
Addition and Subtraction
To add or subtract in mod12, perform the calculation in the usual manner (7 + 15 = 22) and then add or subtract 12s until you get a number from 0 to 11 (22 - 12 = 10).
Adding and subtracting can represent many musical ideas: moving seven half steps above D takes you to A (2 + 7 = 9); combining 2 half steps and 11 half steps produces 1 half step (2 + 11 = 1; or starting with C, moving up 2 half steps reaches D, and 11 more C-sharp — 1 higher than the original C).
Modular arithmetic is a quick way to calculate various intervals between pitches or pitch classess. Some examples:
What is the interval class from pitch class 7 (G) to pitch class 10 (B-flat)? 10 - 7 = 3 What is the pitch class 5 semitones above B-natural (11)? 11 + 5 = 4. That is, E.