Octave

The word octave describes two things. Most importantly, an octave is the same note 8 notes above or below a given note; in terms of a major or minor scale. In terms of the smallest distance between two notes – the semitone – an octave is twelve semitones above or below a given note. (See scale examples below). “Octave” may also be used to measure the range in a group or series of notes, as in “the melody’s range is an octave”; meaning the distance between the lowest and highest notes is an octave, 8 notes apart.

Notes an octave apart are called “octaves”. Because octaves are the same note they have the same letter name. Octaves are the same note in a different register and should be heard as “higher” or “lower”, but not “different”. (Individual octaves may be identified by adding a number to the name. For example, middle C is C4, C3 is an octave below and C5 an octave above). There are reasons, based on naturally occurring phenomenon, we hear octaves as the same note. I go into that at the bottom of the page.

C in four octaves. (Middle C is the third note):

In terms of scales:
Using “octave” in its measuring sense, in a single octave major or minor scale the first and last notes are an octave apart and have the same name. A one octave ascending C major scale:

The smallest distance between notes is a semitone. There are 12 notes in Western music and the chromatic scale uses all of them, each separated by a semitone. An octave is exactly 12 semitones above or below a given note. Below is an ascending and descending C chromatic scale. The Cs are 12 semitones apart, an octave in range from bottom to top.

 
The Science of Pitch, In a Nutshell
The reason octaves sound the same is rooted in the harmonic series and sound wave frequencies. Each pitch we hear is a composite of many others, in an ascending order known as the harmonic series. The harmonic series is a naturally occurring phenomenon; it exists in nature. Scientists call the pitches harmonic partials; the fundamental is the first one. Musicians call the pitches overtones; the first overtone is the second harmonic partial.

The first three overtones above low C (C2). (The first four harmonic partials):

The harmonic series is an arithmetic one. Meaning, in this case, each successive pitch is an ascending integer factor times the original – the fundamental (f). So: 1f is the fundamental; 2f is the next pitch, an octave up; 3f is an octave plus a perfect 5th, a G; 4f is two octaves above, etc…

Notice the first overtone above a pitch is exactly double its sound wave frequency (2f): it is an octave above. And the fundamental note (C) occurs three times in two octaves in the first four harmonic partials (1f, 2f, 4f).

In terms of sound waves, for any given note, we know an octave above vibrates at exactly double its frequency. (An octave below vibrates at exactly half its frequency). A note 2 octaves above a given note vibrates at 4 times its frequency; 3 octaves above at 8 times; 4 octaves above at 16 times, etc. Notice the frequency compounds exponentially. We hear pitches vibrating at this frequency relationship to the fundamental as the same note.

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