Flute acoustics: measurement, modelling and design - School of ...

331
Appendix C
Quantifying music
The musician has words to describe the perceptual qualities of sound—words such as pitch,
volume and timbre. The physicist or mathematician has her own set of words to describe
sound—such as amplitude, frequency, speed, wavelength, spectral envelope and transient. To
an extent, this is perfectly natural, since the musician and the physicist are concerned with
different things. However, no two fields of inquiry are ever truly separate, and the union of music,
physics and mathematics has been a particularly fruitful one. The premise of this thesis is
that we can analyse a flute in order to give useful information to instrument makers, who are
thus enabled to make instruments better suited to the particular requirements of musicians.
Therefore, a brief discussion of the relationship between music and physics is in order.
The musician’s word pitch is related to the physicist’s word frequency, the number of times
a periodic wave repeats itself in a fixed time interval. Frequency is measured in hertz (Hz). Low
pitched notes have low frequencies, and high pitched notes high frequencies. The note A4 (the
A above middle C on a piano) usually has a nominal frequency of 440 Hz, and the frequencies of
all other notes are calculated based on this reference value. (Often, the frequency of other notes
is calculated according to the equal-tempered scale. Woodwind makers usually design their
instruments to play in many different keys, and the equal-tempered scale is a compromise.)
An octave in music is an interval between two notes that differ in frequency by a factor of two.
In the equal-tempered scale a semitone corresponds to a frequency ratio of 2 1/12 . The semitone
is further divided into 100 cents (2 1/1200 ).
The volume of a musical sound is related to the amplitude of the physical wave, but because
human perception of sound is frequency-dependent it depends also on the spectral envelope
and on some details of the temporal envelope, such as attack transients and vibrato. The frequency
range of hearing is from 20 Hz to 20 kHz with a peak at around 1–3 kHz. Filters with
weighting factors exist to approximate the frequency response of the human ear (called A, B
and C weighting factors). Ears (like microphones) respond to changes in air pressure and so
sound amplitude is measured in pascals (Pa). Sound levels are commonly given in decibels
(dB), which is a logarithm of a pressure ratio.
Perhaps the most difficult musical term to relate to physics is the timbre of a sound. Timbre
is defined negatively: it is that quality which differs between two sounds of equal pitch and
loudness. Timbre has some relation to the spectral components of the wave. A musical note
with a frequency of 440 Hz is usually a mixture of harmonics, that is, the wave will contain
contributions at frequencies of 880 Hz, 1320 Hz etc. The relative strength of these harmonics
differ among different instruments, different volumes and among different notes on the same
instrument. A sound with many strong harmonics is generally considered bright while a sound
with a strong fundamental frequency and few harmonics is considered more mellow or soft. But