The twelve equally-spaced tones of the Equal
Tempered Scale happen to contain a number
of intervals that approximate integer ratios
in pitch. The following table shows these
approximations:
INTERVAL
CENTS
NEARBY
RATIO IN
RATIO
CENTS
minor second
100
16/15
111.75
major second
200
9/8
203.91
minor third
300
6/5
315.64
major third
400
5/4
386.31
perfect fourth
500
4/3
498.04
tritone
600
perfect fifth
700
3/2
701.65
minor sixth
800
8/5
813.69
major sixth
900
5/3
884.36
minor seventh
1000
16/9
996.09
major seventh
1100
15/8
1088.27
octave
1200
2
1200.00
As you can see, the intervals in the Equal
Tempered Scale are NOT equal to the
harmonious integer ratios. Rather, the Equal
Tempered Scale is a compromise. It became
widely used because once a harpsichord or
piano is tuned to that scale, any composition
in any key could be played and no one chord
would sound better or worse than that same
chord in another key.
exactly, as in the periodic waveform shown in
the diagram below:
Because of its periodic nature, this sound’s
pitch can be easily identified and processed by
Auto-Tune 5.
Other sounds are more complex. This
waveform:
is of a violin section playing a single note in
unison. Our ears still sense a specific pitch,
but the waveform does not repeat itself. This
waveform is a summation of a number of
individually periodic violins. The summation
is non-periodic because the individual violins
are slightly out of tune with respect to one
another. Because of this lack of periodicity,
Auto-Tune 5 would not be able to process this
sound.
Some pitch terminology
The pitch of a periodic waveform is defined
as the number of times the periodic element
repeats in one second. This is measured in
Hertz (abbreviated Hz.). For example, the
pitch of A3 (the A above middle C on a piano)
is traditionally 440Hz (although that standard
varies by a few Hz. in various parts of the
world).
Pitches are often described relative to one
another as intervals, or ratios of frequency. For
example, two pitches are said to be one octave
apart if their frequencies differ by a factor of
two. Pitch ratios are measured in units called
cents. There are 1200 cents per octave. For
example, two tones that are 2400 cents apart
are two octaves apart. The traditional twelve-
tone Equal Tempered Scale that is used (or
rather approximated) in 99.9% of all Western
tonal music consists of tones that are, by
definition, 100 cents apart. This interval of 100
cents is called a semitone.
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