# Octave

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A perfect octave between two C's
 Inverse unison - P8 12 0 2:1 [1] 1200 [1] 1200 [1]

In music, an octave (Latin : octavus: eighth) or perfect octave (sometimes called the diapason) [2] is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music," the use of which is "common in most musical systems." [3] The interval between the first and second harmonics of the harmonic series is an octave.

## Contents

In Western music notation, notes separated by an octave (or multiple octaves) have the same letter name and are of the same pitch class.

To emphasize that it is one of the perfect intervals (including unison, perfect fourth, and perfect fifth), the octave is designated P8. Other interval qualities are also possible, though rare. The octave above or below an indicated note is sometimes abbreviated 8a or 8va (Italian : all'ottava), 8va bassa (Italian : all'ottava bassa, sometimes also 8vb), or simply 8 for the octave in the direction indicated by placing this mark above or below the staff.

## Explanation and definition

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 ${\displaystyle 2^{n}}$ 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 +12 (or ${\displaystyle 2^{-1}}$) and 4 (or ${\displaystyle 2^{2}}$) times the frequency, respectively.

The number of octaves between two frequencies is given by the formula:

${\displaystyle {\text{Number of octaves}}=\log _{2}\left({\frac {f_{2}}{f_{1}}}\right)}$

## Music theory

Most musical scales are written so that they begin and end on notes that are an octave apart. For example, the C major scale is typically written C D E F G A B C (shown below), the initial and final C's being an octave apart. Because of octave equivalence, notes in a chord that are one or more octaves apart are said to be doubled (even if there are more than two notes in different octaves) in the chord. The word is also used to describe melodies played in parallel in more than multiple octaves.

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.

## Notation

### Octave of a pitch

Octaves are identified with various naming systems. Among the most common are the scientific, Helmholtz, organ pipe, MIDI[ citation needed ], and MIDI note systems. In scientific pitch notation, a specific octave is indicated by a numerical subscript number after note name. In this notation, middle C is C4, because of the note's position as the fourth C key on a standard 88-key piano keyboard, while the C an octave higher is C5.

ScientificC−1C0C1C2C3C4C5C6C7C8C9
HelmholtzC,,,C,,C,Ccc'c''c'''c''''c'''''c''''''
Organ64 Foot32 Foot16 Foot8 Foot4 Foot2 Foot1 Foot3 Line4 Line5 Line6 Line
NameDbl ContraSub ContraContraGreatSmall1 Line2 Line3 Line4 Line5 Line6 Line
MIDI−5−4−3−2−1012345
MIDI Note01224364860728496108120

### Ottava alta and bassa

An example of the same three notes expressed in three ways: (1) regularly, (2) in an 8va bracket, and (3) in a 15ma bracket
A similar example with 8vb and 15mb

The notation 8a or 8va is sometimes seen in sheet music, meaning "play this an octave higher than written" (all' ottava: "at the octave" or all' 8va). 8a or 8va stands for ottava, the Italian word for octave (or "eighth"); the octave above may be specified as ottava alta or ottava sopra). Sometimes 8va is used to tell the musician to play a passage an octave lower (when placed under rather than over the staff), though the similar notation 8vb (ottava bassa or ottava sotto) is also used. Similarly, 15ma (quindicesima) means "play two octaves higher than written" and 15mb (quindicesima bassa) means "play two octaves lower than written."

The abbreviations col 8, coll' 8, and c. 8va stand for coll'ottava, meaning "with the octave", i.e. to play the notes in the passage together with the notes in the notated octaves. Any of these directions can be cancelled with the word loco, but often a dashed line or bracket indicates the extent of the music affected. [4] [ verification needed ]

## Equivalence

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. The interval is so natural to humans that when men and women are asked to sing in unison, they typically sing in octave. [5]

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 equivalence, 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". [6] The conceptualization of pitch as having two dimensions, pitch height (absolute frequency) and pitch class (relative position within the octave), inherently include octave circularity. [6] Thus all Cs, or all 1s (if C = 0), in any octave are part of the same pitch class.

Octave equivalence is a part of most advanced musical cultures, but is far from universal in "primitive" and early music. [7] [ failed verification ] [8] [ clarification needed ] 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. [9] 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". [10]

Monkeys experience octave equivalence, and its biological basis apparently is an octave mapping of neurons in the auditory thalamus of the mammalian brain. [11] Studies have also shown the perception of octave equivalence in rats ( Blackwell & Schlosberg 1943 ), human infants ( Demany & Armand 1984 ), [12] and musicians ( Allen 1967 ) but not starlings ( Cynx 1993 ), 4–9 year old children ( Sergeant 1983 ), or nonmusicians ( Allen 1967 ). [6]

## Related Research Articles

An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, which gives an equal perceived step size as pitch is perceived roughly as the logarithm of frequency.

In music, just intonation or pure intonation is the attempt to tune all musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and may be called a just interval; when it is sounded, no beating is heard. Just intervals consist of members of a single harmonic series of an implied fundamental. For example, in the diagram, the notes G3 and C4 may be tuned as members of the harmonic series of the lowest C, in which case their frequencies will be 3 and 4 times, respectively, the fundamental frequency and their interval ratio equal to 4:3; they may also be tuned differently.

In music, a note is a symbol denoting a musical sound. In English usage a note is also the sound itself.

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈702 cents wide.

In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

Pitch is a perceptual property of sounds that allows their ordering on a frequency-related scale, or more commonly, pitch is the quality that makes it possible to judge sounds as "higher" and "lower" in the sense associated with musical melodies. Pitch can be determined only in sounds that have a frequency that is clear and stable enough to distinguish from noise. Pitch is a major auditory attribute of musical tones, along with duration, loudness, and timbre.

In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. Thus, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. The term is derived from Latin enharmonicus, from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος (enarmónios), from ἐν (en)+ἁρμονία (harmonía).

A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth is the fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, because the note F is the fifth semitone above C, and there are four staff positions between C and F. Diminished and augmented fourths span the same number of staff positions, but consist of a different number of semitones.

In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave position." Important to musical set theory, a pitch class is "all pitches related to each other by octave, enharmonic equivalence, or both." Thus, using scientific pitch notation, the pitch class "C" is the set

A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale. For example, C is adjacent to C; the interval between them is a semitone.

In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apart. Depending on the complexity of the relationships under consideration, the models may be multidimensional. Models of pitch space are often graphs, groups, lattices, or geometrical figures such as helixes. Pitch spaces distinguish octave-related pitches. When octave-related pitches are not distinguished, we have instead pitch class spaces, which represent relationships between pitch classes. Chordal spaces model relationships between chords.

Scientific pitch notation is a method of specifying musical pitch by combining a musical note name and a number identifying the pitch's octave.

The twelfth root of two or is an algebraic irrational number. It is most important in Western music theory, where it represents the frequency ratio of a semitone in twelve-tone equal temperament. This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals as consisting of different numbers of a single interval, the equal tempered semitone. A semitone itself is divided into 100 cents.

In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning system, and another F tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B and A are both approximated by the same interval although they are a septimal kleisma apart.

In acoustics, a beat is an interference pattern between two sounds of slightly different frequencies, perceived as a periodic variation in volume whose rate is the difference of the two frequencies.

In music, a fifteenth or double octave, abbreviated 15ma, is the interval between one musical note and another with one-quarter the wavelength or quadruple the frequency. It has also been referred to as the bisdiapason. The fourth harmonic, it is two octaves. It is referred to as a fifteenth because, in the diatonic scale, there are 15 notes between them if one counts both ends. Two octaves do not make a sixteenth, but a fifteenth. In other contexts, the term two octaves is likely to be used.

Music theory has no axiomatic foundation in modern mathematics, although some interesting work has recently been done in this direction, yet the basis of musical sound can be described mathematically and exhibits "a remarkable array of number properties". Elements of music such as its form, rhythm and metre, the pitches of its notes and the tempo of its pulse can be related to the measurement of time and frequency, offering ready analogies in geometry.

In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2, 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12. If the A above middle C is 440 Hz, the perfect fifth above it would be E, at (440*1.5=) 660 Hz, while the equal tempered E5 is 659.255 Hz.

Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.

The 833 cents scale is a musical tuning and scale proposed by Heinz Bohlen based on combination tones, an interval of 833.09 cents, and, coincidentally, the Fibonacci sequence. The golden ratio is , which as a musical interval is 833.09 cents. In the 833 cents scale this interval is taken as an alternative to the octave as the interval of repetition, however the golden ratio is not regarded as an equivalent interval. Other music theorists such as Walter O'Connell, in his 1993 "The Tonality of the Golden Section", and Loren Temes appear to have also created this scale prior to Bohlen's discovery of it.

## References

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5. "Music". Vox Explained. Event occurs at 12:50. Retrieved 2018-11-01. When you ask men and women to sing in unison, what typically happens is they actually sing an octave apart.
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8. e.g., Nettl, 1956; Sachs, C. and Kunst, J. (1962). Cited in Burns, Edward M. (1999), p. 217.
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• Cynx, Jeffrey. 1996. "Neuroethological Studies on How Birds Discriminate Song". In Neuroethology of Cognitive and Perceptual Processes, edited by C. F. Moss and S. J. Shuttleworth, 63. Boulder: Westview Press.
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