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Operations on Pitch Classes and Pitch Class Sets

# 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 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 by Paul Nelson, all rights reserved.