# Chapter 2 – Simple Operations on Pitch Classes and Pitch Class Sets

## 2.1“Clock” Math or Modulo Math

• When manipulating pitch classes, you will use a special operator, called the “modulo” operator.
• The “modulo” operator takes the remainder of an integer divided by some other integer.
• For example:  19 modulo 12  =  7 (i.e. 12 goes into 19 once, with 7 left over)
• Pitch class sets use “modulo 12”. Any number above 12 should be reduced, using “mod 12”, to a number from 0 to 11.
• The modulo operator can be visualized using a clock face: • Some interesting characteristics of the clock face:

o       A tritone is made up of two notes which are opposite of each other (for example:  C – 0 and F – 6)

o       The notes of a cross make up a doubly-diminished 7th chord (for example:  C – 0, D – 3, F – 6, A – 9)

o       The augmented triad (C – 0, E – 4, G – 8) is also pleasingly symmetric.

## 2.2Transposing Pitch Class Sets

• To transpose a pitch class set, add (or subtract) the same number to all elements of the list:
[0,1,4]  =>  (transpose up a major third)  [0+4, 1+4, 4+4]  =>  [4,5,8]
In this example, the chord “C D E” is transposed up to “E F G “.
• Remember to use “Module 12” when numbers are greater than or equal to 12:
[0,1,4]  =>  (transpose up a major 7th)  [0+11, 1+11, 4+11]  =>  [11, 12, 15]  =>  [11, 0, 3]

## 2.3Inverting Pitch Class Sets

• To invert a PC Set, subtract each element of the list from 12:
[0,1,4]  =>  [12 – 0, 12 – 1, 12 – 4]  =>  [12, 11, 8]  =>  [0, 11, 8]
(don’t forget to use Mod 12 if any of the numbers are greater than 11)
For example:  The chord “C D E” becomes “C B A “.
• By convention, simple inversion is always around Pitch Class C (0). Therefore, any note of the chord which is N half-steps above C, will be flipped to be come a note N half-steps below C. In the above example, the note “E” (4 half-steps above C) was flipped to become “A ” (4 half-steps below C).
• Very often you will want to invert and transpose at the same time:
[0,1,4] => [ (12-0) + 4, (12-1) + 4, (12-4) + 4]  =>  [16, 15, 12]  =>  [4, 3, 0]
This has a special notation:  T4I  (invert and then transpose up 4 half steps)
• Examples of the PC Sets shown above: 