An example of an octave, from G4 to G5
For example, if one note has a frequency of 440 Hz, the note one octave above is at 880 Hz, and the note one octave below is at 220 Hz. The ratio of frequencies of two notes an octave apart is therefore 2:1. Further octaves of a note occur at 2n times the frequency of that note (where n is an integer), such as 2, 4, 8, 16, etc. and the reciprocal of that series. For example, 55 Hz and 440 Hz are one and two octaves away from 110 Hz because they are 1⁄2 (or 2−1) and 4 (or 22) times the frequency, respectively.
For Calculating the Octave Range between two given frequencies one has to simply enter :
Octave Range = log ( base 2 ) Higher Frequency / Lower Frequency
After the unison, the octave is the simplest interval in music. The human ear tends to hear both notes as being essentially "the same", due to closely related harmonics. Notes separated by an octave "ring" together, adding a pleasing sound to music. For this reason, notes an octave apart are given the same note name in the Western system of music notation—the name of a note an octave above A is also A. This is called octave equivalency, the assumption that pitches one or more octaves apart are musically equivalent in many ways, leading to the convention "that scales are uniquely defined by specifying the intervals within an octave". The conceptualization of pitch as having two dimensions, pitch height (absolute frequency) and pitch class (relative position within the octave), inherently include octave circularity. Thus all C♯s, or all 1s (if C = 0), in any octave are part of the same pitch class.
Octave equivalency is a part of most "advanced musical cultures", but is far from universal in "primitive" and early music.
The languages in which the oldest extant written documents on tuning are written, Sumerian and Akkadian, have no known word for "octave". However, it is believed that a set of cuneiform tablets that collectively describe the tuning of a nine-stringed instrument, believed to be a Babylonian lyre, describe tunings for seven of the strings, with indications to tune the remaining two strings an octave from two of the seven tuned strings.
Leon Crickmore recently proposed that "The octave may not have been thought of as a unit in its own right, but rather by analogy like the first day of a new seven-day week".
Monkeys experience octave equivalency, and its biological basis apparently is an octave mapping of neurons in the auditory thalamus of the mammalian brain. Studies have also shown the perception of octave equivalence in rats (Blackwell & Schlosberg 1943), human infants (Demany & Armand 1984), and musicians (Allen 1967) but not starlings (Cynx 1993), 4–9 year old children (Sergeant 1983), or nonmusicians (Allen 1967).
While octaves commonly refer to the perfect octave (P8), the interval of an octave in music theory encompasses chromatic alterations within the pitch class, meaning that G♮ to G♯ (13 semitones higher) is an Augmented octave (A8), and G♮ to G♭ (11 semitones higher) is a diminished octave (d8). The use of such intervals is rare, as there is frequently a preferable enharmonically-equivalent notation available (minor ninth and major seventh respectively), but these categories of octaves must be acknowledged in any full understanding of the role and meaning of octaves more generally in music.