of Microprocessors

14 MUSICAL ApPUCATIONS OF MICROPROCESSORS
Listening tests have shown that the relation between frequency and
pitch is an approximately exponential one. Thus, the increase from 100 Hz to
200 Hz represents a doubling in frequency so an equivalent pitch increase
starting from 5 kHz would require doubling again to 10 kHz.
Musical Pitch
Musical pitch has its own system of measurement. Unlike frequency,
the units are relative rather than absolute. The most fundamental unit is the
octave. If tone B is one octave higher than tone A, then its frequency is exactly
twice as high and the sensation of pitch would be twice as high. (In tests with
musically inexperienced laymen, tone B would typically be judged to be less
than twice as high in pitch as tone A, but such a tendency is usually
eliminated by musical training.) Other units are the half-Jtep, which is 1/12
ofan octave or a frequency ratio of 1.05946, and the cent} which is 1/100 ofa
half-step or a ratio of about 1.00059, which is roughly 0.06%. A half-step is
also the difference in pitch between two directly adjacent keys on a conventionally
tuned piano. For moderately loud sounds around 1 kHz, the smallest
change in frequency that can be perceived is around 5 cents.
Since these pitch units are purely relative, a basis point is needed if an
absolute pitch scale is to be defined. One such basis point is the international
pitch standard, which defines the note, A above middle-C, as being 440.0
Hz. The corresponding frequencies of all other musical notes can be obtained
by applying the proper ratios to the 440-Hz standard.
Table 1-1 gives the frequencies of some musical notes. Note that two
systems of tuning are represented, although there are others. The most
popular tuning system is equal temperment, which is based solely on the
frequency ratio of the half-step being the twelfth root of 2.0 or approximately
1.05946. The term equal temperment means that all half-steps are exactly
the same size. The other system represented is the just system of tuning,
which is based on rational fraction ratios with small numbers for numerator
and denominator. The table shows these ratios in both fractional- form and
decimal form for comparison with the equally tempered scale frequencies.
Note that the octave ratio is exact in both scales, but there are small differences
in all of the other ratios.
Of the two scales, the just-tuned one is more musically accurate and
pleasing to the ear particularly when chords are played. Musical accuracy
here means accuracy of the important musical intervals such as the fifth,
which is ideally a ratio of 3:2, and the third, which should be 5:4. Its
disadvantage is that not all half-steps are the same size; thus, transposition
froin one key to another is not easily achieved. For example, the just scale
shown is for the key ofA major, meaning that the basis frequency chosen for
application of the rational fraction ratios was 440 Hz. If another just scale