Digital Sound & Music: Concepts, Applications, & Science, Chapter 3, last updated 6/25/2013
Beethoven and that continues as the historical and theoretic foundation of music in the United
States, Europe, and Western culture. The major and minor scales and chords are taken from this
context, which we refer to as Western music. Many other types of note progressions and
intervals have been used in other cultures and time periods, leading to quite different
characteristic sounds: the modes of ancient Greece, the Gregorian chants of the Middle Ages,
the pentatonic scale of ancient Oriental music, the Hindu 22 note octave, or the whole tone scale
of Debussy, for example. While we won’t cover these, we encourage the reader to explore these
other musical traditions.
To give us a common language for understanding music, we focus our discussion on the
musical notation used for keyboards like the piano. Keyboard music expressed and notated in
the Western tradition provides a good basic knowledge of music and gives us a common
vocabulary when we start working with MIDI in Chapter 6.
3.1.2 Tones and Notes
Musicians learn to sing, play instruments, and compose music using a symbolic language of
music notation. Before we can approach this symbolic notation, we need to establish a basic
In the vocabulary of music, a sound with a single fundamental frequency is called a tone.
The fundamental frequency of a tone is the frequency that gives the tone its essential pitch. A
piccolo plays tones with higher fundamental frequencies than the frequencies of a flute, and thus
it is higher pitched.
A tone that has an onset and a duration is called a note. The onset of the note is the
moment when it begins. The duration is the length of time that the note remains audible. Notes
can be represented symbolically in musical notation, as we’ll see in the next section. We will
also use the word “note” interchangeably with “key” when referring to a key on a keyboard and
the sound it makes when struck.
As described in Chapter 2, tones created by musical instruments, including the human
voice, are not single-frequency. These tones have overtones at frequencies higher than the
fundamental. Overtones add a special quality to the sound, but they don’t change our overall
perception of the pitch. When the frequency of an overtone is an integer multiple of the
fundamental frequency, it is a harmonic overtone. Stated mathematically for frequencies and
, if and n is a positive integer, then is a harmonic frequency relative to
fundamental frequency . Notice that every frequency is a harmonic frequency relative to itself.
It is called the first harmonic, since . The second harmonic is the frequency where .
For example, the second harmonic of 440 Hz is 880 Hz; the third harmonic of 440 Hz is 3*440
Hz = 1320 Hz; the fourth harmonic of 440 Hz is 4*440 Hz = 1760 Hz; and so forth. Musical
instruments like pianos and violins have harmonic overtones. Drums beats and other non-
musical sounds have overtones that are not harmonic.
Another special relationship among frequencies is the octave. For frequencies and , if
where n is a positive integer, then and “sound the same,” except that is higher
pitched than . Frequencies and are separated by n octaves. Another way to describe the
octave relationship is to say that each time a frequency is moved up an octave, it is multiplied by
2. A frequency of 880 Hz is one octave above 440 Hz; 1760 Hz is two octaves above 440 Hz;
3520 Hz is three octaves above 440 Hz; and so forth. Two notes separated by one or more
octaves are considered equivalent in that one can replace the other in a musical composition
without disturbing the harmony of the composition.