Digital Sound & Music: Concepts, Applications, & Science, Chapter 3, last updated 6/25/2013
3
In Western music, an octave is separated into 12 frequencies corresponding to notes on a
piano keyboard, named as shown in Figure 3.1. From C to B we have 12 notes, and then the next
octave starts with another C, after which the sequence of letters repeats. An octave can start on
any letter, as long as it ends on the same letter. (The sequence of notes is called an octave
because there are eight notes in a diatonic scale, as is explained below.) The white keys are
labeled with the letters. Each of the black keys can be called by one of two names. If it is named
relative to the white key to its left, a sharp symbol is added to the name, denoted C#, for
example. If it is named relative to the white key to its right, a flat symbol is added to the name,
denoted D♭, for example.
Figure 3.1 Keyboard showing octave and key labels
Each note on a piano keyboard corresponds to a physical key that can be played. There
are 88 keys on a standard piano keyboard. MIDI keyboards are usually smaller. Since the notes
from A through G are repeated on the keyboard, they are sometimes named by the number of the
octave that they're in, as shown in Figure 3.2.
Figure 3.2 Piano or MIDI keyboard
Middle C on a standard piano has a frequency of approximately 262 Hz. On a piano with
88 keys, middle C is the fourth C, so it is called C4. Middle C (C3 on the keyboard shown) is
the central position for playing the piano, with regard to where the right and left hands of the
pianist are placed. The standard reference point for tuning a piano is the A above middle C,
which has a frequency of 440 Hz. This means that the next A going up the keys to the right has a
frequency of 880 Hz. A note of 880 Hz is one octave away from 440 Hz, and both are called A
on a piano keyboard.
The interval between two consecutive keys (also called notes) on a keyboard, whether the
keys are black or white, is called a semitone. A semitone is the smallest frequency distance
between any two notes. Neighboring notes on a piano keyboard (and equivalently, two
neighboring notes on a chromatic scale) are separated by a frequency factor of approximately
1.05946. This relationship is described more precisely in the equation below.
Let f be the frequency of a note k. Then the note one octave above f has a
frequency of . Given this octave relationship and the fact that there are 12
notes in an octave, the frequency of the note after k on a chromatic scale is

.
Equation 3.1
b b b b b
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