Sounds can be produced at a desired frequency by different methods. For example, a sound of 440 Hz can be created by actuating a loudspeaker with an oscillator tuned to this frequency. An air blast can be interrupted by a toothed wheel with 44 teeth, rotating at 10 revolutions/sec; this method is used in operating an ordinary siren. The sound of the speaker and that of the siren at the same frequency are very different in quality, but will correspond closely in pitch, equivalent to the A above middle C on a piano. The next higher A on the piano, the note one octave above, has a frequency of 880 Hz. Similarly, notes one or two octaves below have frequencies of 220 or 110 Hz, respectively. Thus, by definition, an octave is the interval between any two notes the frequencies of which are in a two-to-one ratio.
A fundamental law of harmony states that
two notes an octave apart, when sounded together, produce a euphonious
combination. A fifth and a major third produce successively less
euphonious combinations. Physically, an interval of a fifth consists of
two notes, the frequencies of which bear the arithmetical ratio three to
two, and a major third, the ratio five to four. Fundamentally, then, the
law of harmony states that two or more notes sound euphonious when played
together if their frequencies bear the ratio of small, whole numbers; if
the frequencies do not bear such ratios, a dissonance is produced. On a
fixed-pitch instrument, such as a piano, it is not possible to arrange the
notes so that all of these ratios hold exactly, and some compromise is
necessary in tuning, called the meantone system, or tempered scale.
Frequency (summary)
We perceive frequency as
"higher" or "lower" sounds. The frequency of a sound
is the number of cycles, or oscillations, a sound wave completes in a
given time. Frequency is measured in hertz, or cycles per second. Because
the amplitude (height) of the waves above remains constant, we are able to
hear the same note at different frequencies, from 110.00 Hz to 880.00 Hz.
Waves propagate at both higher and lower frequencies, but humans are
unable to hear them outside of a relatively narrow range.
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