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 In music, consonance and dissonance are categorizations of simultaneous or successive Sound, sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness ...

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 a 12-note approximately equally divided scale, any interval can be defined in terms of an appropriate number of semitones (e.g. a

whole tone In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more deta ...

or major second is 2 semitones wide, a

major third In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...

4 semitones, and a

perfect fifth In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval fro ...

7 semitones. In

music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...

, a distinction is made between a diatonic semitone, or minor second (an interval encompassing two different staff positions, e.g. from C to D) and a chromatic semitone or augmented unison (an interval between two notes at the same staff position, e.g. from C to C). These are

enharmonically equivalent 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. The enharmonic spelling of a written no ...

when

twelve-tone equal temperament Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That resultin ...

is used, but are not the same thing in

meantone temperament Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. Me ...

, where the diatonic semitone is distinguished from and smaller than the chromatic semitone (augmented unison). See for more details about this terminology. In

all semitones are equal in size (100 cents). In other tuning systems, "semitone" refers to a family of intervals that may vary both in size and name. In

Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...

, seven semitones out of twelve are diatonic, with ratio 256:243 or 90.2 cents (

Pythagorean limma 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 no ...

), and the other five are chromatic, with ratio 2187:2048 or 113.7 cents (

Pythagorean apotome 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 no ...

); they differ by the

Pythagorean comma In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C ...

of ratio 531441:524288 or 23.5 cents. In

quarter-comma meantone Quarter-comma meantone, or -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 ...

, seven of them are diatonic, and 117.1 cents wide, while the other five are chromatic, and 76.0 cents wide; they differ by the lesser

diesis In classical music from Western culture, a diesis ( , plural dieses ( , "difference"; Greek: δίεσις "leak" or "escape"Benson, Dave (2006). ''Music: A Mathematical Offering'', p.171. . Based on the technique of playing the aulos, where p ...

of ratio 128:125 or 41.1 cents. 12-tone scales tuned in

just intonation In music, just intonation or pure intonation is the tuning of musical intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to ...

typically define three or four kinds of semitones. For instance, Asymmetric

five-limit tuning 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 ...

yields chromatic semitones with ratios 25:24 (70.7 cents) and 135:128 (92.2 cents), and diatonic semitones with ratios 16:15 (111.7 cents) and 27:25 (133.2 cents). For further details, see

below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) Bottom may refer to: Anatomy and sex * Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...

. The condition of having semitones is called hemitonia; that of having no semitones is

anhemitonia Musicology commonly classifies scales as either hemitonic or anhemitonic. Hemitonic scales contain one or more semitones, while anhemitonic scales do not contain semitones. For example, in traditional Japanese music, the anhemitonic ''yo'' sc ...

. A

musical scale In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Often, especially in the ...

or chord containing semitones is called hemitonic; one without semitones is anhemitonic.

Minor second

The ''minor second'' occurs in the

major scale The major scale (or Ionian mode) is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at double i ...

, between the third and fourth degree, (''mi'' (E) and ''fa'' (F) in C major), and between the seventh and eighth degree (''ti'' (B) and ''do'' (C) in C major). It is also called the ''diatonic semitone'' because it occurs between steps in the

diatonic scale In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...

. The minor second is abbreviated m2 (or −2). Its inversion is the ''

major seventh In music from Western culture, a seventh is a musical interval encompassing seven staff positions (see Interval number for more details), and the major seventh is one of two commonly occurring sevenths. It is qualified as ''major'' because it i ...

'' (''M7'' or ''Ma7''). . Here,

middle C C or Do is the first note and semitone of the C major scale, the third note of the A minor scale (the relative minor of C major), and the fourth note (G, A, B, C) of the Guidonian hand, commonly pitched around 261.63 Hz. The actual frequen ...

is followed by D, which is a tone 100 cents sharper than C, and then by both tones together.

Melodically A melody (from Greek μελῳδία, ''melōidía'', "singing, chanting"), also tune, voice or line, is a linear succession of musical tones that the listener perceives as a single entity. In its most literal sense, a melody is a combinat ...

, this interval is very frequently used, and is of particular importance in

cadences In Western musical theory, a cadence (Latin ''cadentia'', "a falling") is the end of a phrase in which the melody or harmony creates a sense of full or partial resolution, especially in music of the 16th century onwards.Don Michael Randel (1999) ...

. In the perfect and deceptive cadences it appears as a resolution of the

leading-tone In music theory, a leading-tone (also called a subsemitone, and a leading-note in the UK) is a note or pitch which resolves or "leads" to a note one semitone higher or lower, being a lower and upper leading-tone, respectively. Typically, ''the ...

to the tonic. In the

plagal cadence In Western musical theory, a cadence (Latin ''cadentia'', "a falling") is the end of a phrase in which the melody or harmony creates a sense of full or partial resolution, especially in music of the 16th century onwards.Don Michael Randel (1999 ...

, it appears as the falling of the

subdominant In music, the subdominant is the fourth tonal degree () of the diatonic scale. It is so called because it is the same distance ''below'' the tonic as the dominant is ''above'' the tonicin other words, the tonic is the dominant of the subdomina ...

to the

mediant In music, the mediant (''Latin'': to be in the middle) is the third scale degree () of a diatonic scale, being the note halfway between the tonic and the dominant.Benward & Saker (2003), p.32. In the movable do solfège system, the mediant note i ...

. It also occurs in many forms of the

imperfect cadence In Western musical theory, a cadence (Latin ''cadentia'', "a falling") is the end of a phrase in which the melody or harmony creates a sense of full or partial resolution, especially in music of the 16th century onwards.Don Michael Randel (19 ...

, wherever the tonic falls to the leading-tone.

Harmonically In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. However, ...

, the interval usually occurs as some form of dissonance or a

nonchord tone A nonchord tone (NCT), nonharmonic tone, or embellishing tone is a note in a piece of music or song that is not part of the implied or expressed chord set out by the harmonic framework. In contrast, a chord tone is a note that is a part of the f ...

that is not part of the

functional harmony In music, function (also referred to as harmonic function) is a term used to denote the relationship of a chord"Function", unsigned article, ''Grove Music Online'', . or a scale degree to a tonal centre. Two main theories of tonal functions ex ...

. It may also appear in inversions of a

major seventh chord In music, a major seventh chord is a seventh chord in which the third is a major third above the root and the seventh is a major seventh above the root. The major seventh chord, sometimes also called a ''Delta chord'', can be written as maj7, M7, , ...

, and in many

added tone chord An added tone chord, or added note chord, is a non- tertian chord composed of a triad and an extra "added" note. Any tone that is not a seventh factor is commonly categorized as an added tone. It can be outside the tertian sequence of ascendin ...

In unusual situations, the minor second can add a great deal of character to the music. For instance,

Frédéric Chopin Frédéric François Chopin (born Fryderyk Franciszek Chopin; 1 March 181017 October 1849) was a Polish composer and virtuoso pianist of the Romantic period, who wrote primarily for solo piano. He has maintained worldwide renown as a leadin ...

's Étude Op. 25, No. 5 opens with a melody accompanied by a line that plays fleeting minor seconds. These are used to humorous and whimsical effect, which contrasts with its more lyrical middle section. This eccentric dissonance has earned the piece its nickname: the "wrong note" étude. This kind of usage of the minor second appears in many other works of the Romantic period, such as

Modest Mussorgsky Modest Petrovich Mussorgsky ( rus, link=no, Модест Петрович Мусоргский, Modest Petrovich Musorgsky , mɐˈdɛst pʲɪˈtrovʲɪtɕ ˈmusərkskʲɪj, Ru-Modest Petrovich Mussorgsky version.ogg; – ) was a Russian compo ...

's '' Ballet of the Unhatched Chicks''. More recently, the music to the movie ''

Jaws Jaws or Jaw may refer to: Anatomy * Jaw, an opposable articulated structure at the entrance of the mouth ** Mandible, the lower jaw Arts, entertainment, and media * Jaws (James Bond), a character in ''The Spy Who Loved Me'' and ''Moonraker'' * ...

'' exemplifies the minor second.

In other temperaments

a 16:15 minor second arises in the C

between B & C and E & F, and is, "the sharpest dissonance found in the ajor scale."Paul, Oscar (1885).
A manual of harmony for use in music-schools and seminaries and for self-instruction
', p. 165.

Theodore Baker Theodore Baker (June 3, 1851"Passed Away," ''Musical America'' (Nov. 10, 1934), p. 32."Dr. Theodore Baker," ''Musical Courier'' (Nov. 3, 1934), p. 20. – October 12, 1934) Mendelssohn dominants

The augmented unison, the interval produced by the augmentation, or widening by one half step, of the perfect unison, does not occur between diatonic scale steps, but instead between a scale step and a

chromatic Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, ...

alteration of the same step. It is also called a ''chromatic semitone''. The augmented unison is abbreviated A1, or aug 1. Its inversion is the ''

diminished octave In music from Western culture, a diminished octave () is an interval produced by narrowing a perfect octave by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . Specific example of an d8 not given ...

'' (''d8'', or ''dim 8''). The augmented unison is also the inversion of the

augmented octave In Western tonal music theory, an augmented octave is the sum of a perfect octave and an augmented unison or chromatic semitone. It is the interval between two notes, with the same note letter on staff positions an octave apart, whose altera ...

, because the interval of the diminished unison does not exist. This is because a unison is always made larger when one note of the interval is changed with an accidental.

, an augmented unison very frequently occurs when proceeding to a chromatic chord, such as a

secondary dominant A secondary chord is an analytical label for a specific harmonic device that is prevalent in the tonal idiom of Western music beginning in the common practice period: the use of diatonic functions for tonicization. Secondary chords are a typ ...

, a

diminished seventh chord The diminished seventh chord is a four-note chord (a seventh chord) composed of a root note, together with a minor third, a diminished fifth, and a diminished seventh above the root: (1, 3, 5, 7). For example, the diminished seventh ...

, or an

augmented sixth chord In music theory, an augmented sixth chord contains the interval of an augmented sixth, usually above its bass tone. This chord has its origins in the Renaissance, was further developed in the Baroque, and became a distinctive part of the musi ...

. Its use is also often the consequence of a melody proceeding in semitones, regardless of harmonic underpinning, e.g. D, D, E, F, F. (Restricting the notation to only minor seconds is impractical, as the same example would have a rapidly increasing number of accidentals, written enharmonically as D, E, F, G, A). Liszt augmented unison

, augmented unisons are quite rare in tonal repertoire. In the example to the right,

Liszt Franz Liszt, in modern usage ''Liszt Ferenc'' . Liszt's Hungarian passport spelled his given name as "Ferencz". An orthographic reform of the Hungarian language in 1922 (which was 36 years after Liszt's death) changed the letter "cz" to simpl ...

had written an E against an E in the bass. Here E was preferred to a D to make the tone's function clear as part of an F

dominant seventh In music theory, a dominant seventh chord, or major minor seventh chord, is a seventh chord, usually built on the fifth degree of the major scale, and composed of a root, major third, perfect fifth, and minor seventh. Thus it is a major triad tog ...

chord, and the augmented unison is the result of superimposing this harmony upon an E

pedal point In music, a pedal point (also pedal note, organ point, pedal tone, or pedal) is a sustained tone, typically in the bass, during which at least one foreign (i.e. dissonant) harmony is sounded in the other parts. A pedal point sometimes function ...

. In addition to this kind of usage, harmonic augmented unisons are frequently written in modern works involving tone clusters, such as

Iannis Xenakis Giannis Klearchou Xenakis (also spelled for professional purposes as Yannis or Iannis Xenakis; el, Γιάννης "Ιωάννης" Κλέαρχου Ξενάκης, ; 29 May 1922 – 4 February 2001) was a Romanian-born Greek-French avant-garde ...

' ''Evryali'' for piano solo.

History

The semitone appeared in the music theory of Greek antiquity as part of a diatonic or chromatic

tetrachord In music theory, a tetrachord ( el, τετράχορδoν; lat, tetrachordum) is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency propo ...

, and it has always had a place in the diatonic scales of Western music since. The various modal scales of

medieval music Medieval music encompasses the sacred and secular music of Western Europe during the Middle Ages, from approximately the 6th to 15th centuries. It is the first and longest major era of Western classical music and followed by the Renaissance ...

theory were all based upon this diatonic pattern of tones and semitones. Though it would later become an integral part of the musical

cadence In Western musical theory, a cadence (Latin ''cadentia'', "a falling") is the end of a phrase in which the melody or harmony creates a sense of full or partial resolution, especially in music of the 16th century onwards.Don Michael Randel (1999) ...

, in the early polyphony of the 11th century this was not the case.

Guido of Arezzo Guido of Arezzo ( it, Guido d'Arezzo; – after 1033) was an Italian music theorist and pedagogue of High medieval music. A Benedictine monk, he is regarded as the inventor—or by some, developer—of the modern staff notation that had a ma ...

suggested instead in his ''

Micrologus The ''Micrologus'' is a treatise on Medieval music written by Guido of Arezzo, dating to approximately 1026. It was dedicated to Tedald, Bishop of Arezzo. This treatise outlines singing and teaching practice for Gregorian chant, and has considera ...

'' other alternatives: either proceeding by whole tone from a

major second In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more deta ...

to a unison, or an ''occursus'' having two notes at a

move by contrary motion toward a unison, each having moved a whole tone. "As late as the 13th century the half step was experienced as a problematic interval not easily understood, as the irrational remainder between the perfect fourth and the

ditone In music, a ditone (, from , "of two tones") is the interval of a major third. The size of a ditone varies according to the sizes of the two tones of which it is compounded. The largest is the Pythagorean ditone, with a ratio of 81:64, also cal ...

\left(\begin \frac \end /  = \begin \frac \end\right)

." In a melodic half step, no "tendency was perceived of the lower tone toward the upper, or of the upper toward the lower. The second tone was not taken to be the ‘goal’ of the first. Instead, the half step was avoided in clausulae because it lacked clarity as an interval." Dahlhaus, Carl, trans. Gjerdingen, Robert O. ''Studies in the Origin of Harmonic Tonality''. Princeton University Press: Princeton, 1990. . Marenzio solo e pensoso chromatic

However, beginning in the 13th century

begin to require motion in one voice by half step and the other a whole step in contrary motion. These cadences would become a fundamental part of the musical language, even to the point where the usual accidental accompanying the minor second in a cadence was often omitted from the written score (a practice known as

musica ficta ''Musica ficta'' (from Latin, "false", "feigned", or "fictitious" music) was a term used in European music theory from the late 12th century to about 1600 to describe pitches, whether notated or added at the time of performance, that lie outside ...

). By the 16th century, the semitone had become a more versatile interval, sometimes even appearing as an augmented unison in very

passages.

Semantically Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...

, in the 16th century the repeated melodic semitone became associated with weeping, see: passus duriusculus,

lament bass In music, the lament bass is a ground bass, built from a descending perfect fourth from tonic to dominant, with each step harmonized.Brover-Lubovsky, Bella (2008). ''Tonal Space in the Music of Antonio Vivaldi'', p.151-52. . The diatonic versi ...

, and

pianto In music, the ''pianto'' (en:crying) is the motif of a descending minor second, has represented laments and been associated textually with weeping, sighing (called the Mannheim sigh by Hugo Riemann); or pain, grief, etc.; since the 16th centur ...

. By the

Baroque era The Baroque (, ; ) is a style of architecture, music, dance, painting, sculpture, poetry, and other arts that flourished in Europe from the early 17th century until the 1750s. In the territories of the Spanish and Portuguese empires including th ...

(1600 to 1750), the tonal harmonic framework was fully formed, and the various musical functions of the semitone were rigorously understood. Later in this period the adoption of

well temperament Well temperament (also good temperament, circular or circulating temperament) is a type of tempered tuning described in 20th-century music theory. The term is modeled on the German word ''wohltemperiert''. This word also appears in the title of ...

s for instrumental tuning and the more frequent use of

enharmonic 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. The enharmonic spelling of a written n ...

equivalences increased the ease with which a semitone could be applied. Its function remained similar through the Classical period, and though it was used more frequently as the language of tonality became more chromatic in the Romantic period, the musical function of the semitone did not change. In the 20th century, however, composers such as

Arnold Schoenberg Arnold Schoenberg or Schönberg (, ; ; 13 September 187413 July 1951) was an Austrian-American composer, music theorist, teacher, writer, and painter. He is widely considered one of the most influential composers of the 20th century. He was as ...

Béla Bartók Béla Viktor János Bartók (; ; 25 March 1881 – 26 September 1945) was a Hungarian composer, pianist, and ethnomusicologist. He is considered one of the most important composers of the 20th century; he and Franz Liszt are regarded as H ...

, and

Igor Stravinsky Igor Fyodorovich Stravinsky (6 April 1971) was a Russian composer, pianist and conductor, later of French (from 1934) and American (from 1945) citizenship. He is widely considered one of the most important and influential composers of the ...

sought alternatives or extensions of tonal harmony, and found other uses for the semitone. Often the semitone was exploited harmonically as a caustic dissonance, having no resolution. Some composers would even use large collections of harmonic semitones ( tone clusters) as a source of cacophony in their music (e.g. the early piano works of

Henry Cowell Henry Dixon Cowell (; March 11, 1897 – December 10, 1965) was an American composer, writer, pianist, publisher and teacher. Marchioni, Tonimarie (2012)"Henry Cowell: A Life Stranger Than Fiction" ''The Juilliard Journal''. Retrieved 19 June 202 ...

). By now, enharmonic equivalence was a commonplace property of

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...

, and instrumental use of the semitone was not at all problematic for the performer. The composer was free to write semitones wherever he wished.

Semitones in different tunings

The exact size of a semitone depends on the

tuning Tuning can refer to: Common uses * Tuning, the process of tuning a tuned amplifier or other electronic component * Musical tuning, musical systems of tuning, and the act of tuning an instrument or voice ** Guitar tunings ** Piano tuning, adjusti ...

system used.

Meantone temperament Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. Me ...

s have two distinct types of semitones, but in the exceptional case of

, there is only one. The unevenly distributed

s contain many different semitones.

, similar to meantone tuning, has two, but in other systems of just intonation there are many more possibilities.

Meantone temperament

meantone Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. Me ...

systems, there are two different semitones. This results because of the break in the

circle of fifths In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval ...

that occurs in the tuning system: diatonic semitones derive from a chain of five fifths that does not cross the break, and chromatic semitones come from one that does. The chromatic semitone is usually smaller than the diatonic. In the common

, tuned as a cycle of tempered fifths from E to G, the chromatic and diatonic semitones are 76.0 and 117.1 cents wide respectively. Extended meantone temperaments with more than 12 notes still retain the same two semitone sizes, but there is more flexibility for the musician about whether to use an augmented unison or minor second. 31-tone equal temperament is the most flexible of these, which makes an unbroken circle of 31 fifths, allowing the choice of semitone to be made for any pitch.

Equal temperament

12-tone equal temperament Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That resultin ...

is a form of meantone tuning in which the diatonic and chromatic semitones are exactly the same, because its circle of fifths has no break. Each semitone is equal to one twelfth of an octave. This is a ratio of 2^1/12 (approximately 1.05946), or 100 cents, and is 11.7 cents narrower than the 16:15 ratio (its most common form in

, discussed below). All diatonic intervals can be expressed as an equivalent number of semitones. For instance a

equals two semitones. There are many approximations,

rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...

or otherwise, to the equal-tempered semitone. To cite a few: :*

18 / 17 \approx 99.0 \text

suggested by

Vincenzo Galilei Vincenzo Galilei (born 3 April 1520, Santa Maria a Monte, Italy died 2 July 1591, Florence, Italy) was an Italian lutenist, composer, and music theorist. His children included the astronomer and physicist Galileo Galilei and the lute virtuoso and ...

and used by

luthier A luthier ( ; AmE also ) is a craftsperson who builds or repairs string instruments that have a neck and a sound box. The word "luthier" is originally French and comes from the French word for lute. The term was originally used for makers o ...

s of the

Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...

, :*

\approx 100.4 \text

suggested by

Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...

as a constructible and more accurate alternative, :*

(139 / 138 )^8  \approx 99.9995 \text

used by

Julián Carrillo Julián Carrillo Trujillo (January 28, 1875 – September 9, 1965) was a Mexican composer,Camp, Roderic Ai (1995). "Carrillo (Flores), Nabor" on ''Mexican Political Biographies, 1935–1993: Third Edition'', p. 121. . conductor, violi ...

as part of a sixteenth-tone system. For more examples, see Pythagorean and Just systems of tuning below.

Well temperament

There are many forms of

, but the characteristic they all share is that their semitones are of an uneven size. Every semitone in a well temperament has its own interval (usually close to the equal-tempered version of 100 cents), and there is no clear distinction between a ''diatonic'' and ''chromatic'' semitone in the tuning. Well temperament was constructed so that

equivalence could be assumed between all of these semitones, and whether they were written as a minor second or augmented unison did not effect a different sound. Instead, in these systems, each

key Key or The Key may refer to: Common meanings * Key (cryptography), a piece of information that controls the operation of a cryptography algorithm * Key (lock), device used to control access to places or facilities restricted by a lock * Key (map ...

had a slightly different sonic color or character, beyond the limitations of conventional notation.

Pythagorean tuning

Like meantone temperament,

is a broken

. This creates two distinct semitones, but because Pythagorean tuning is also a form of 3-limit

, these semitones are rational. Also, unlike most meantone temperaments, the chromatic semitone is larger than the diatonic. The Pythagorean diatonic semitone has a ratio of 256/243 (), and is often called the Pythagorean limma. It is also sometimes called the ''Pythagorean minor semitone''. It is about 90.2 cents. :

\frac = \frac \approx 90.2 \text

It can be thought of as the difference between three

octaves In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

and five just fifths, and functions as a

diatonic semitone 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 no ...

in a

. The Pythagorean chromatic semitone has a ratio of 2187/2048 (). It is about 113.7 cents. It may also be called the Pythagorean apotomeRashed, Roshdi (ed.) (1996). ''Encyclopedia of the History of Arabic Science, Volume 2'', pp. 588, 608. Routledge. . or the ''Pythagorean major semitone''. (''See

Pythagorean interval In musical tuning theory, a Pythagorean interval is a musical interval with frequency ratio equal to a power of two divided by a power of three, or vice versa.Benson, Donald C. (2003). ''A Smoother Pebble: Mathematical Explorations'', p.56. . " ...

''.) :

\frac = \frac \approx 113.7\text

It can be thought of as the difference between four perfect

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

s and seven just fifths, and functions as a

chromatic semitone In modern Western tonal music theory an augmented unison or augmented prime is the interval between two notes on the same staff position, or denoted by the same note letter, whose alterations cause them, in ordinary equal temperament, to be one ...

in a

. The Pythagorean limma and Pythagorean apotome are

equivalents (chromatic semitones) and only a

apart, in contrast to diatonic and chromatic semitones in

and 5-limit

Just 5-limit intonation

A minor second in

typically corresponds to a pitch

ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

of 16:15 () or 1.0666... (approximately 111.7

cent Cent may refer to: Currency * Cent (currency), a one-hundredth subdivision of several units of currency * Penny (Canadian coin), a Canadian coin removed from circulation in 2013 * 1 cent (Dutch coin), a Dutch coin minted between 1941 and 1944 * ...

s), called the just diatonic semitone. This is a practical just semitone, since it is the difference between a

perfect fourth 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 (half steps, or half tones). For example, the ascending interval from C to ...

and

(

\tfrac \div \tfrac = \tfrac

). The 16:15 just minor second arises in the C major scale between B & C and E & F, and is, "the sharpest dissonance found in the scale." An augmented unison in just intonation is another semitone of 25:24 () or 1.0416... (approximately 70.7 cents). It is the difference between a 5:4 major third and a 6:5 minor third. Composer Ben Johnston uses a sharp () to indicate a note is raised 70.7 cents, or a flat () to indicate a note is lowered 70.7 cents.

John Fonville John Fonville is a flutist and composer. Fonville specializes in extended techniques on the flute, especially microtonality, and performs on instruments including a complete set of quarter tone ( Kingma system) flutes.Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p. 109, ''

Perspectives of New Music ''Perspectives of New Music'' (PNM) is a peer-reviewed academic journal specializing in music theory and analysis. It was established in 1962 by Arthur Berger and Benjamin Boretz (who were its initial editors-in-chief). ''Perspectives'' was first ...

'', vol. 29, no. 2 (Summer 1991), pp. 106–137. "...the 25/24 ratio is the sharp (#) ratio...this raises a note approximately 70.6 cents." Two other kinds of semitones are produced by 5-limit tuning. A

chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...

defines 12 semitones as the 12 intervals between the 13 adjacent notes forming a full octave (e.g. from C4 to C5). The 12 semitones produced by a commonly used version of 5-limit tuning have four different sizes, and can be classified as follows: *Just, or smaller, or minor, chromatic semitone, e.g. between E and E (6/5 and 5/4): :

S_1 =  \approx 70.7 \ \hbox

*Larger, or major, chromatic semitone, or larger limma, or major chroma, e.g. between C and C (1/1 and 135/128): :

S_2 =  \approx 92.2 \ \hbox

*Just, or smaller, or minor, diatonic semitone, e.g. between C and D (1/1 and 16/15): :

S_3 =  \approx 111.7 \ \hbox

*Larger, or major, diatonic semitone, e.g. between A and B (5/3 and 16/9): :

S_4 =  \approx 133.2 \ \hbox

The most frequently occurring semitones are the just ones (S₃ and S₁): S₃ occurs six times out of 12, S₁ three times, S₂ twice, and S₄ only once. The smaller chromatic and diatonic semitones differ from the larger by the

syntonic comma In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...

(81:80 or 21.5 cents). The smaller and larger chromatic semitones differ from the respective diatonic semitones by the same 128:125 diesis as the above meantone semitones. Finally, while the inner semitones differ by the

diaschisma The diaschisma (or diacisma) is a small musical interval defined as the difference between three octaves and four perfect fifths plus two major thirds (in just intonation). It can be represented by the ratio 2048:2025 and is about 19.5 cents. ...

(2048:2025 or 19.6 cents), the outer differ by the greater diesis (648:625 or 62.6 cents).

Extended just intonations

7-limit 7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14:11 ...

there is the

septimal diatonic semitone In music, a septimal diatonic semitone (or major diatonic semitoneHaluska, Jan (2003). ''The Mathematical Theory of Tone Systems'', p.xxiv & 25. .) is the interval 15:14 . It is about 119.44 cents. The septimal diatonic semitone may be derived f ...

of 15:14 () available between the 5-limit

(15:8) and the 7-limit minor seventh (7:4). There is also a smaller

septimal chromatic semitone In music, a septimal chromatic semitone or minor semitoneHaluska, Jan (2003). ''The Mathematical Theory of Tone Systems'', p.xxiv. . is the interval 21:20 (). It is about 84.47 cents. The septimal chromatic semitone may be derived from the har ...

of 21:20 () between a septimal minor seventh and a fifth (21:8) and an octave and a major third (5:2). Both are more rarely used than their 5-limit neighbours, although the former was often implemented by theorist

, while

Harry Partch Harry Partch (June 24, 1901 – September 3, 1974) was an American composer, music theorist, and creator of unique musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century com ...

used the latter as part of his 43-tone scale. Under 11-limit tuning, there is a fairly common ''undecimal

neutral second In music theory, a neutral interval is an interval that is neither a major nor minor, but instead in between. For example, in equal temperament, a major third is 400 cents, a minor third is 300 cents, and a neutral third is 350 cents. A neutral ...

'' (12:11) (), but it lies on the boundary between the minor and

(150.6 cents). In just intonation there are infinitely many possibilities for intervals that fall within the range of the semitone (e.g. the Pythagorean semitones mentioned above), but most of them are impractical. In 13-limit tuning, there is a tridecimal 2/3 tone (13:12 or 138.57 cents) and tridecimal 1/3 tone (27:26 or 65.34 cents). In 17-limit just intonation, the major diatonic semitone is 15:14 or 119.4 cents (), and the minor diatonic semitone is 17:16 or 105.0 cents, Prout, Ebenezer (2004). ''Harmony'', p. 325. . and septendecimal limma is 18:17 or 98.95 cents. Though the names ''diatonic'' and ''chromatic'' are often used for these intervals, their musical function is not the same as the two meantone semitones. For instance, 15:14 would usually be written as an augmented unison, functioning as the ''chromatic'' counterpart to a ''diatonic'' 16:15. These distinctions are highly dependent on the musical context, and just intonation is not particularly well suited to chromatic usage (diatonic semitone function is more prevalent).

Other equal temperaments

19-tone equal temperament distinguishes between the chromatic and diatonic semitones; in this tuning, the chromatic semitone is one step of the scale (), and the diatonic semitone is two (). 31-tone equal temperament also distinguishes between these two intervals, which become 2 and 3 steps of the scale, respectively. 53-ET has an even closer match to the two semitones with 3 and 5 steps of its scale while

72-ET In music, 72 equal temperament, called twelfth-tone, 72-TET, 72- EDO, or 72-ET, is the tempered scale derived by dividing the octave into twelfth-tones, or in other words 72 equal steps (equal frequency ratios). Each step represents a frequency ...

uses 4 () and 7 () steps of its scale. In general, because the smaller semitone can be viewed as the difference between a minor third and a major third, and the larger as the difference between a major third and a perfect fourth, tuning systems that closely match those just intervals (6/5, 5/4, and 4/3) will also distinguish between the two types of semitones and closely match their just intervals (25/24 and 16/15).