## 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 7^{th} 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.

## 2.2 Transposing 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 7^{th}) [0+11, 1+11, 4+11] => [11, 12, 15] => [11, 0, 3]

## 2.3 Inverting 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:

Copyright © 2004-2017 by Paul Nelson, all rights reserved.