Octave

A perfect octave between two Cs

In music, an octave (Latin : octavus: eighth) or perfect octave (sometimes called the diapason) ^[2] is a series of eight notes occupying the interval between (and including) two notes, one having twice the frequency of vibration of the other. 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] [4]} The interval between the first and second harmonics of the harmonic series is an octave. In Western music notation, notes separated by an octave (or multiple octaves) have the same name and are of the same pitch class.

Contents

• Explanation and definition
• Music theory
• Notation
• Octave of a pitch
• Ottava alta and bassa
• Equivalence
• See also
• References
• External links

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 8^a or 8^va (Italian : all'ottava), 8^va bassa (Italian : all'ottava bassa, sometimes also 8^vb), or simply 8 for the octave in the direction indicated by placing this mark above or below the staff.

Explanation and definition

An octave is the interval between one musical pitch and another with double or half its frequency. 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 +1⁄2 (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:

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

• 
  Oscillogram of middle C (262 Hz). (Scale: 1 square is equal to 1 millisecond)
• 
  C5, an octave above middle C. The frequency is twice that of middle C (523 Hz).
• 
  C3, an octave below middle C. The frequency is half that of middle C (131 Hz).

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 Cs 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 one or more octaves apart (see example under Equivalence, below).

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, 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.

Scientific	C−1	C0	C1	C2	C3	C4	C5	C6	C7	C8	C9
Helmholtz	C,,,	C,,	C,	C	c	c'	c''	c'''	c''''	c'''''	c''''''
Organ	64 Foot	32 Foot	16 Foot	8 Foot	4 Foot	2 Foot	1 Foot	3 Line	4 Line	5 Line	6 Line
Name	Dbl Contra	Sub Contra	Contra	Great	Small	1 Line	2 Line	3 Line	4 Line	5 Line	6 Line
MIDI Note	0	12	24	36	48	60	72	84	96	108	120

Ottava alta and bassa

Example of the same three notes expressed in three ways: (1) regularly, (2) in an 8^va bracket, and (3) in a 15^ma bracket

Similar example with 8^vb and 15^mb

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

The abbreviations col 8, coll' 8, and c. 8^va 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. ^[5]

Equivalence

Demonstration of octave equivalence. The melody to "Twinkle, Twinkle, Little Star" with parallel harmony. The melody is paralleled in three ways: (1) in octaves (consonant and equivalent); (2) in fifths (fairly consonant but not equivalent); and (3) in seconds (neither consonant nor equivalent).

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. ^[6]

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". ^[7] The conceptualization of pitch as having two dimensions, pitch height (absolute frequency) and pitch class (relative position within the octave), inherently include octave circularity. ^[7] Thus all C♯s (or all 1s, if C = 0), any number of octaves apart, are part of the same pitch class.

Octave equivalence is a part of most advanced ^{[ clarification needed ]} musical cultures, but is far from universal in "primitive" and early music. ^{[8] [ failed verification ] [9] [ 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. ^[10] 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". ^[11]

Monkeys experience octave equivalence, and its biological basis apparently is an octave mapping of neurons in the auditory thalamus of the mammalian brain. ^[12] Studies have also shown the perception of octave equivalence in rats, ^[13] human infants, ^[14] and musicians ^[15] but not starlings, ^[16] 4–9-year-old children, ^[17] or non-musicians. ^{[15] [7] [ clarification needed ]}

See also

• Blind octave  – Music composition and performance technique
• Decade  – Unit for measuring ratios on a logarithmic scale
• Eight-foot pitch  – Standard pitch designation
• Octave band  – Frequency band that spans one octave
• Octave species  – Classification of musical key or scale in ancient Greek music theory
• One-third octave
• Pitch circularity  – Fixed series of tones that appear to ascend or descend endlessly in pitch
• Pseudo-octave
• Pythagorean interval  – Musical interval
• Short octave  – Musical keyboard layout
• Solfège  – Music teaching method

Related Research Articles

In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.

In music, notes are distinct and isolatable sounds that act as the most basic building blocks for nearly all of music. This discretization facilitates performance, comprehension, and analysis. Notes may be visually communicated by writing them in musical notation.

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifths which are "pure" or perfect, with ratio . This is chosen because it is the next harmonic of a vibrating string, after the octave, and hence is the next most consonant "pure" interval, and the easiest to tune by ear. 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.

Pitch is a perceptual property that allows sounds to be ordered 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 is a major auditory attribute of musical tones, along with duration, loudness, and timbre.

In music, two written notes have enharmonic equivalence if they produce the same pitch but are notated differently. Similarly, written intervals, chords, or key signatures are considered enharmonic if they represent identical pitches that are notated differently. The term derives from Latin enharmonicus, in turn from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος, from ἐν ('in') and ἁρμονία ('harmony').

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 theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.

The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each. Typically, cents are used to express small intervals, to check intonation, or to compare the sizes of comparable intervals in different tuning systems. For humans, a single cent is too small to be perceived between successive notes.

In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, 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 minor second, 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, visually seen on a keyboard as the distance between two keys that are adjacent to each other. For example, C is adjacent to C♯; the interval between them is a semitone.

Piano tuning is the process of adjusting the tension of the strings of an acoustic piano so that the musical intervals between strings are in tune. The meaning of the term 'in tune', in the context of piano tuning, is not simply a particular fixed set of pitches. Fine piano tuning requires an assessment of the vibration interaction among notes, which is different for every piano, thus in practice requiring slightly different pitches from any theoretical standard. Pianos are usually tuned to a modified version of the system called equal temperament.

Scientific pitch notation (SPN), also known as American standard pitch notation (ASPN) and international pitch notation (IPN), 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, approximately equal to 1.0594631. 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. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". 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.

Quarter-comma meantone, or ⁠ 1 / 4 ⁠-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma ( 81 : 80 ), with respect to its just intonation used in Pythagorean tuning ; the result is ⁠ 3 / 2 ⁠ × [⁠ 80 / 81 ⁠]^{1 / 4} = ⁴√5 ≈ 1.49535, or a fifth of 696.578 cents. This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

In music, a fifteenth or double octave, abbreviated 15^ma, 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 analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory.

In music theory, an inversion is a rearrangement of the top-to-bottom elements in an interval, a chord, a melody, or a group of contrapuntal lines of music. In each of these cases, "inversion" has a distinct but related meaning. The concept of inversion also plays an important role in musical set theory.

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 Lorne Temes in 1970, appear to have also created this scale prior to Bohlen's discovery of it.

References

Sources

• Allen, David. 1967. "Octave Discriminability of Musical and Non-Musical Subjects". Psychonomic Science 7:421–22.
• Blackwell, H. R., & H. Schlosberg. 1943. "Octave Generalization, Pitch Discrimination, and Loudness Thresholds in the White Rat". Journal of Experimental Psychology 33:407–419.
• 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.
• Demany, Laurent, and Françoise Armand. 1984. "The Perceptual Reality of Tone Chroma in Early Infancy". Journal of the Acoustical Society of America 76:57–66.
• Sergeant, Desmond. 1983. "The Octave: Percept or Concept?" Psychology of Music 11, no. 1:3–18.

External links

Wikimedia Commons has media related to Octaves .

• Anatomy of an Octave by Kyle Gann