Glebe Terrace

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Filed under Music

“What’s an octave?”.  Apart from saying “the same note played higher”, I can’t really answer that question without stepping back and giving a bit more of the picture.

There’s a lot that can be, and has been, written about sounds – most of which needs some university physics to properly understand. How do I describe this so my children can understand it? For the younger ones, I use a keyboard and show them “the same note played higher”, when they get a bit older (say high school) and are able to understand more of science & mathematics and can pull them together we go a bit further …

Sound & Frequency

At a simple level, sound is air being shaken.  How fast it is shaken is called frequency; how hard it is shaken is called volume.  Because the shaking happens quite fast we measure it as the number of complete “shakes” that happen in one second.  If one shake takes one second we call it one hertz (or 1 Hz for short); if the air is shaken a bit faster at twenty shakes in one second we call it 20 Hz 1.

Human hearing is generally accepted to cover the frequency range from 20 Hz (low-ptiched) to 20,000 Hz (high-pitched).  Any one note is the air being vibrated at a particular frequency, if you increase the frequency the note sounds “higher”.

We hear differences in pitch by multiples, not additions.  If two low notes are played and they differ by a semi-tone, to hear the same semi-tone difference at a much higher pitch requires two notes to be played whose frequencies are in the same ratio as the lower notes, not that differ by the same number of hertz.

Example: if a low note plays at 20 Hz, the note one semi-tone above it is roughly 21.19 Hz.  This is a difference of about 1.19 Hz and a ratio of 1.0595.  If a high note plays at 2000 Hz the note one semi-tone above it is 2119 Hz, not 2001.19 Hz – the ratio stays the same, not the difference.

Octaves & Semi-Tones

An octave change is defined as a doubling or halving of frequency – that’s it.

Play any note; to go up an octave doubles the frequency of the note and to go down an octave we halve the frequency.  For example, Middle C has a frequency of approximately 261 Hz 2 so the next C upwards has a frequency of 522 Hz and the next C above that is at 1044 Hz and the C above that is 2088 Hz then 4166 Hz and so on.

Each octave is split evenly into 12 semi-tones.  Because each octave is a doubling of frequency and our hearing detects changes in multiples this splitting gets a little tricky to work out.  Each semi-tone is different from the previous note by a multiple of a magic number3 which is approximately 1.05946 3.  To raise a note by a semi-tone multiply its frequency by this number.


Footnotes

1. A demonstration with a skipping rope tied to a post usually helps here, although I never managed more than 3 Hz.

2. Some say it’s 256 Hz, some say it’s as high as 278 Hz, I learnt from children to say “whatever” and let the purists argue about it.

3. Mathematically this is calculated by:  10(log(2))/12.