Introduction to Acoustics

To provide a greater discrimination in the measurements
of frequency, the semitone is divided into 100
further logarithmic increments called cents. The octave
is therefore equivalent to 1200 cents and a quartertone
to ≈ 50 cents, with the exact relationship between
frequencies given by
interval = 1200
ln 2 ln( f2/ f1) cents (15.20)
corresponding to ∼ 1.73 × 10 3 (∆ f/ f ) cents for small
fractional changes ∆ f/ f .
Early musical scales were based on various variants
of the natural harmonic series of frequencies, fn = nf1,
where n is an integer (e.g. 200, 400, 600, ... 1600 Hz),
illustrated by the audio . These notes correspond
to the harmonics produced when lightly touching
a bowed string at integral subdivisions of its length (1/2,
1/3, 1/4, etc.) . These simple divisions give
successive musical intervals of the octave, perfect fifth,
perfect fourth, major third and minor third, with frequency
ratios 2/1, 3/2, 4/3, 5/4 and 6/5, respectively.
The seventh member of the harmonic sequence has no
counterpart in traditional western classical music, although
it is sometimes used by modern composers for
special effect [15.12].
Just temperament corresponds to musical scales
based on these integer fraction intervals. The Pythagorean
scale is based on the particularly consonant
intervals of the octave (2/1) and perfect fifth (3/2),
which can be used to generate individual intervals of
the form 3 p /2 q or 2 p /3 q ,wherep and q are positive
integers. Although the Pythagorean and just-tempered
scales coincide for the octave, perfect fifth and fourth,
there are musically significant differences in the tuning
for all other defined intervals, and all intervals
other than the octave differ slightly from those of
the equally tempered scale. A comparison between
the musical intervals of just and equal temperament
Table 15.1 Principal intervals and differences between justand
equal-temperament intervals
Interval Just Equal ∆ f/ f
Just-equal cents
Octave 2/1 2.00 0
Perfect fifth 3/2 27/12 = 1.498 +2
Perfect fourth 4/3 25/12 = 1.334 −2
Major third 5/4 24/12 = 1.260 −13
Minor third 6/5 23/12 = 1.189 +15
Tone 9/8 22/12 = 1.122 −4
Semitone 16/15 21/12 = 1.066 +1
Musical Acoustics 15.1 Vibrational Modes of Instruments 543
tuning is shown in Table 15.1, with the fractional mistuning
indicated in cents. Because of the differences
in tunings of the musical intervals, music transposed
from one key to another will generally sound badly
out of tune (particularly for commonly used intervals
like the major and minor third) – unlike those
played on a modern equal-tempered keyboard. Prior
to the now almost universal practice of tuning keyboard
instruments to a equal-tempered scale, many
tuning schemes were devised which partially overcame
the problems of tuning when playing in a succession
of different keys (see Fletcher and Rossing [15.5],
Sect. 17.6, and Barbour [15.13] for further discussion).
Singers, stringed and wind instrument players
can adjust the pitch of the notes they produce to optimise
the tuning with other performers and for musical
effect.
Figure 15.5 and audio illustrate the difference
in the sounds of a major triad formed from the
just intervals (1, 5/4, 3/2) and the equivalent equaltempered
scale intervals (1 : 1.260 : 1.498). The rational
Pythagorean intervals give a repetitive waveform of
constant amplitude, while the less-consonant, inharmonic,
equal-tempered intervals have a non-repetitive
waveform with an easily discernable periodic beat in
amplitude resulting from the departures in harmonicity
of its component frequencies, as illustrated in Fig. 15.5.
Interestingly, the pitch of the equally tempered intervals
also sounds slightly higher, though both share the same
fundamental.
A sequence of upward fifths (frequency ratio 3/2)
and downward octaves (ratio 1/2) can be used to fill
in all the semitones of an octave scale on the piano
keyboard. However, the resulting octave formed from
a succession of 12 upward fifths and six downward
Fig. 15.5 Wave envelope of a major triad chord based on
the Pythagorean scale followed by the same chord on the
equal-tempered scale, with pronounced beats in the amplitude
arising from the departures from harmonicity in the
frequencies of the major third and perfect fifth
Part E 15.1