physics
did you know an octave is a doubling you can hear?
440 vibrations a second and 880 belong to the same musical family.
440 Hz → 880 Hz
one octave higher
the frequency doubles; equal-tempered semitones divide that ratio into twelve steps.
- one cycle
- twice as many cycles
- an octave higher
the idea
two pitches an octave apart have a frequency ratio of two to one. a tone at 440 hertz and one at 880 hertz are both called A in the usual note-naming system, even though one is higher.
how it works
frequency counts cycles per second. doubling it fits two cycles into the time previously occupied by one. the higher tone aligns repeatedly with the lower tone's cycle, giving the pair a simple relationship.
go deeper
why ratios instead of differences?
two pitches an octave apart have a frequency ratio of two to one. a tone at 440 hertz and one at 880 hertz are both called A in the usual note-naming system, even though one is higher.
musical intervals repeat through frequency ratios. moving from 220 to 440 Hz and from 440 to 880 Hz produces the same octave interval, despite different numerical increases.
fit twice as many vibrations into a second
frequency counts cycles per second. doubling it fits two cycles into the time previously occupied by one. the higher tone aligns repeatedly with the lower tone's cycle, giving the pair a simple relationship.
real instruments contain overtones as well as a fundamental frequency. their mix helps distinguish a piano from a violin playing the same note, while the frequency ratio describes the interval.
why are piano semitones not equal numbers of hertz?
in twelve-tone equal temperament, twelve equal frequency ratios must multiply to two. each semitone therefore multiplies frequency by the twelfth root of two, about 1.05946.
semitone ratio = 2^(1/12); after 12 steps: [2^(1/12)]¹² = 2
is equal temperament the only tuning?
- no. different tuning systems arrange intervals differently. the two-to-one octave is distinct from the choice of how to divide it.
- pitch perception is richer than a frequency equation. timbre, loudness, context, and the listener all influence what we hear.
sources & further reading
concepts: frequency · octaves · logarithmic scales
2 minute read