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{{About|the musical interval|the printing method|Halftone}}
[[File:Minor second on C.png|thumb|Minor second[[File:Minor second on C.mid]]]]

A '''semitone''' (also called a '''half step''' or '''half tone''')<ref>''Semitone'', ''half step'', ''half tone'', ''halftone'', and ''half-tone'' are all variously used in sources.[http://www.merriam-webster.com/dictionary/half+step][http://www.merriam-webster.com/dictionary/half%20tone][http://dictionary.reference.com/browse/half%20tone][https://books.google.com/books?id=sTMbuSQdqPMC&q=a+half+step+is+called+a+semitone&pg=PA19][https://books.google.com/books?id=iYgSJSxWW2sC]
<br />[[Aaron Copland]], [[Leonard Bernstein]], and others use "half tone".[https://books.google.com/books?id=dsyPycO3GfgC&q=the+twelve+chromatic+ones,+arranged+in+the+following+order:+two+whole+tones+followed+by+a+half+tone,+plus+three+whole+tones+followed+by+a+half+tone.&pg=PA41]
[https://books.google.com/books?id=d-zwOLoDIcEC&q=Now,+if+you+remember+that+the+step+from+any+note+on+the+piano+to+the+note+just+next+to+it,+whether+it%27s+black+or+white,+is+a+step+of+a+half+tone,+you+can+see+that+the+entire+piano+keyboard+is+made+up+of+only+half+tones,+one+after+another.&pg=PA185][https://books.google.com/books?id=PoM6AAAAMAAJ&q=almost+a+half-tone+sharp][https://books.google.com/books?id=1nSqLzjZBKwC&q=This+is+easy+on+the+guitar,+since+from+one+fret+to+the+next+is+a+half+tone+(also+called+half+step+or+semitone).&pg=PA116]
<br />One source says that ''step'' is "chiefly US",[https://web.archive.org/web/20060301170305/http://www.askoxford.com/concise_oed/step] and that ''half-tone'' is "chiefly N. Amer."[https://archive.today/20130117121046/http://www.askoxford.com/concise_oed/halftone]
</ref> is the smallest [[interval (music)|musical interval]] commonly used in Western tonal music,<ref>Miller, Michael. [https://books.google.com/books?id=sTMbuSQdqPMC ''The Complete Idiot's Guide to Music Theory, 2nd ed'']. [Indianapolis, Indiana]: Alpha, 2005. {{ISBN|1-59257-437-8}}. p. 19.</ref> and it is considered the most [[Consonance and dissonance#Dissonance|dissonant]]<ref>{{cite book | title = Sound: An Elementary Text-book for Schools and Colleges | first = John Walton |last = Capstick | publisher = Cambridge University Press | year = 1913 | url = https://books.google.com/books?id=bwNJAAAAIAAJ&q=most+dissonant-interval+semitone+intitle:sound&pg=PA227 }}</ref> when sounded harmonically.
It is defined as the interval between two adjacent notes in a [[chromatic scale|12-tone scale]] (or half of a [[whole step]]), visually seen on a keyboard as the distance between two keys that are adjacent to each other. For example, C is adjacent to C{{music|sharp}}; the interval between them is a semitone.<ref>{{Cite web |title=musictheory.net |url=https://www.musictheory.net/lessons/20 |access-date=2024-01-04 |website=www.musictheory.net}}</ref>

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]] or major second is 2 semitones wide, a [[major third]] 4 semitones, and a [[perfect fifth]] 7 semitones).

In [[music theory]], a distinction is made<ref name="Wharram">{{cite book | title=Elementary Rudiments of Music | edition=2nd| last=Wharram| first=Barbara| year=2010| page=17| publisher=Frederick Harris Music| location=Mississauga, Ontario| isbn=978-1-55440-283-0}}</ref> between a '''diatonic semitone''', or '''minor second''' (an interval encompassing two different [[staff position]]s, e.g. from C to D{{music|flat}}) and a '''chromatic semitone''' or '''augmented unison''' (an interval between two notes at the same staff position, e.g. from C to C{{music|sharp}}). These are [[Enharmonic|enharmonically equivalent]] if and only if [[Equal temperament|twelve-tone equal temperament]] is used; for example, they are not the same thing in [[meantone temperament]], where the diatonic semitone is distinguished from and larger than the chromatic semitone (augmented unison), or in [[Pythagorean tuning]], where the diatonic semitone is smaller instead. See {{slink|Interval (music)|Number}} for more details about this terminology.

In [[twelve-tone equal temperament]] 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]], seven semitones out of twelve are diatonic, with ratio 256:243 or 90.2 cents ([[#Pythagorean tuning|Pythagorean limma]]), and the other five are chromatic, with ratio 2187:2048 or 113.7 cents ([[#Pythagorean tuning|Pythagorean apotome]]); they differ by the [[Pythagorean comma]] of ratio 531441:524288 or 23.5 cents. In [[quarter-comma meantone]], 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]] of ratio 128:125 or 41.1 cents. 12-tone scales tuned in [[just intonation]] typically define three or four kinds of semitones. For instance, [[Five-limit tuning#Size of intervals|Asymmetric]] [[five-limit tuning]] 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 [[Semitone#Just intonation|below]].

The condition of having semitones is called hemitonia; that of having no semitones is [[Anhemitonic scale|anhemitonia]]. A [[Scale (music)|musical scale]] or [[Chord (music)|chord]] containing semitones is called hemitonic; one without semitones is anhemitonic.

==Minor second==

[[File:Cadence minor second.png|thumb|The melodic minor second is an integral part of most cadences of the [[Common practice period]].[[File:Cadence minor second V65-I.mid]]]]

{{Infobox musical interval
| main_interval_name = minor second
| inverse = [[major seventh]]
| other_names = 
| abbreviation = m2
| semitones = 1
| interval_class = 1
| just_interval = 256:243, 16:15,<ref name="Duffin">{{cite book|last1=Duffin|first1=Ross W.|title=How equal temperament ruined harmony : (and why you should care)|date=2008|publisher=W. W. Norton|location=New York|isbn=978-0-393-33420-3|page=163|edition=First published as a Norton paperback.|url=https://books.google.com/books?id=i5LC7Csnw7UC&q=how+equal+temperament+ruined+harmony|access-date=28 June 2017}}</ref> 27:25
| cents_equal_temperament = 100.0
| cents_24T_equal_temperament = 
| cents_just_intonation = 90.2, 111.7, 133.2
}}

The ''minor second'' occurs in the [[major scale]], 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 [[step (music)|steps]] in the [[diatonic scale]]. The minor second is abbreviated '''m2''' (or '''&minus;2'''). Its inversion is the ''[[major seventh]]'' (''M7'' or ''Ma7'').

{{Audio|Minor_Second_ET.ogg|Listen to a minor second in equal temperament}}. Here, [[middle C]] is followed by D{{music|flat}}, which is a tone 100 [[Cent (music)|cents]] sharper than C, and then by both tones together.

[[Melody|Melodically]], this interval is very frequently used, and is of particular importance in [[Cadence (music)|cadences]]. In the [[Interval (music)|perfect]] and [[Cadence (music)#Deceptive cadence|deceptive cadences]] it appears as a resolution of the [[leading-tone]] to the [[Tonic (music)|tonic]]. In the [[plagal cadence]], it appears as the falling of the [[subdominant]] to the [[mediant]]. It also occurs in many forms of the [[imperfect cadence]], wherever the tonic falls to the leading-tone.

[[Harmony|Harmonically]], the interval usually occurs as some form of [[Consonance and dissonance|dissonance]] or a [[nonchord tone]] that is not part of the [[Diatonic function|functional harmony]]. It may also appear in inversions of a [[major seventh chord]], and in many [[added tone chord]]s.

[[File:Bach minor second smaller.png|thumb|center|400px|Harmonic minor second in [[Johann Sebastian Bach|J. S. Bach]]'s Prelude in C major from the [[The Well-Tempered Clavier|WTC]], book 1, mm. 7–9.[[File:Bach minor second smaller BWV 846.mid]] The minor second may be viewed as a [[Suspension (music)|suspension]] of the ''B'' resolving into the following ''A minor seventh'' chord.]]

{{Listen|type=music|header='''[[Frédéric Chopin]]'s "wrong note" [[Études (Chopin)|Étude]]'''|filename=Frederic Chopin - Opus 25 - Twelve Grand Etudes - e minor.ogg|title=Étude Op. 25, No. 5|description=[[Martha Goldstein]] playing on an [[Sébastien Érard|Érard]] (1851)<br /><small>Opening bars</small>[[File:Chopin minor second.png|250px]]|pos=left}}

In unusual situations, the minor second can add a great deal of character to the music. For instance, [[Frédéric Chopin]]'s [[Étude Op. 25, No. 5 (Chopin)|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 music|Romantic]] period, such as [[Modest Mussorgsky]]'s ''[[Pictures at an Exhibition|Ballet of the Unhatched Chicks]]''. More recently, the music to the movie ''[[Jaws (film)#Music|Jaws]]'' exemplifies the minor second.

===In other temperaments===
In [[just intonation]] a 16:15 minor second arises in the C [[major scale]] between B & C and E & F, and is "the sharpest [[consonance and dissonance|dissonance]] found in the [major] [[musical scale|scale]]."<ref name="books.google.com">Paul, Oscar (1885). ''[https://archive.org/details/bub_gb_4WEJAQAAMAAJ <!-- quote=musical interval "pythagorean major third". --> A manual of harmony for use in music-schools and seminaries and for self-instruction]'', p. 165. [[Theodore Baker]], trans. G. Schirmer.</ref> {{audio|Just minor second in scale.mid|Play B & C}}

{{Wide image|Comparison of minor seconds.png|500px|<!--caption-->|400px|<!--left-->|alt=}}{{clear}}

==Augmented unison==
{{main|Augmented unison}}
[[File:Augmented unison on C.png|thumb|right|Augmented unison on C]]
[[File:Mendelssohn dominants.png|thumb|left|Augmented unisons often appear as a consequence of [[secondary dominant]]s, such as those in the soprano voice of this [[Sequence (music)|sequence]] from [[Felix Mendelssohn]]'s ''[[Songs without Words|Song Without Words]]'' Op. 102 No. 3, mm. 47–49.[[File:Mendelssohn dominants.mid]]]]
The '''augmented unison''', the interval produced by the [[augmentation (music)|augmentation]], or widening by one half step, of the perfect unison,<ref>Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p. 54. {{ISBN|978-0-07-294262-0}}. Specific example of an A1 not given but general example of perfect intervals described.</ref> does not occur between diatonic scale steps, but instead between a scale step and a [[chromatic]] 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]]'' (''d8'', or ''dim 8''). The augmented unison is also the inversion of the [[augmented octave]], because the interval of the diminished unison does not exist.<ref>Kostka and Payne (2003). ''Tonal Harmony'', p. 21. {{ISBN|0-07-285260-7}}. "There is no such thing as a diminished unison."</ref> This is because a unison is always made larger when one note of the interval is changed with an accidental.<ref>Day and Pilhofer (2007). ''Music Theory for Dummies'', p. 113. {{ISBN|0-7645-7838-3}}. "There is no such thing as a diminished unison, because no matter how you change the unisons with accidentals, you are adding half steps to the total interval."</ref><ref>{{Cite book
  | last = Surmani
  | first = Andrew
  | author-link = Andrew Surmani
  |author2=Karen Farnum Surmani |author3=Morton Manus
  | title = Alfred's Essentials of Music Theory: A Complete Self-Study Course for All Musicians
  | publisher = Alfred Music Publishing
  | year = 2009
  | pages = 135
  | id = Since lowering either note of a perfect unison would actually increase its size, the perfect unison cannot be diminished, only augmented.
  | isbn = 978-0-7390-3635-8 }}</ref>

[[Melody|Melodically]], an augmented unison very frequently occurs when proceeding to a chromatic chord, such as a [[secondary dominant]], a [[diminished seventh chord]], or an [[augmented sixth chord]]. Its use is also often the consequence of a melody proceeding in semitones, regardless of harmonic underpinning, e.g. '''D''', '''D{{music|sharp}}''', '''E''', '''F''', '''F{{music|sharp}}'''. (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{{music|flat}}''', '''F{{music|flat}}''', '''G{{music|doubleflat}}''', '''A{{music|tripleflat}}''').

[[File:Liszt augmented unison.png|thumb|right|[[Franz Liszt]]'s second [[Transcendental Études|Transcendental Étude]], measure 63]]
[[Harmony|Harmonically]], augmented unisons are quite rare in tonal repertoire. In the example to the right, [[Franz Liszt|Liszt]] had written an '''E{{music|flat}}''' against an '''E{{music|natural}}''' in the bass. Here '''E{{music|flat}}''' was preferred to a '''D{{music|sharp}}''' to make the tone's function clear as part of an '''F''' [[dominant seventh]] chord, and the augmented unison is the result of superimposing this harmony upon an '''E''' [[pedal point]].

In addition to this kind of usage, harmonic augmented unisons are frequently written in modern works involving [[tone clusters]], such as [[Iannis Xenakis]]' ''Evryali'' for piano solo.

==History==

The semitone appeared in the music theory of Greek antiquity as part of a diatonic or chromatic [[tetrachord]], and it has always had a place in the diatonic scales of Western music since. The various [[Musical mode|modal]] scales of [[medieval music]] theory were all based upon this diatonic pattern of [[whole tone|tones]] and semitones.

Though it would later become an integral part of the musical [[Cadence (music)|cadence]], in the early polyphony of the 11th century this was not the case. [[Guido of Arezzo]] suggested instead in his ''[[Micrologus]]'' other alternatives: either proceeding by whole tone from a [[major second]] to a unison, or an ''occursus'' having two notes at a [[major third]] 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 {{sic}} remainder between the perfect fourth and the [[ditone]] <math>\left(\begin{matrix} \frac{4}{3} \end{matrix} / {{\begin{matrix} (\frac{9}{8}) \end{matrix}}^2} = \begin{matrix} \frac{256}{243} \end{matrix}\right)</math>." 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 [[Clausula (music)|clausulae]] because it lacked clarity as an interval."<ref name="Dahlhaus">[[Carl Dahlhaus|Dahlhaus, Carl]], trans. Gjerdingen, Robert O. ''Studies in the Origin of Harmonic Tonality''. Princeton University Press: Princeton, 1990. {{ISBN|0-691-09135-8}}.</ref>

[[File:Marenzio solo e pensoso chromatic.png|thumb|Dramatic chromatic scale in the opening measures of [[Luca Marenzio]]'s ''Solo e pensoso'', ca. 1580 [[File:Marenzio solo e pensoso opening.MID]]]]
However, beginning in the 13th century [[cadence (music)|cadences]] begin to require motion in one voice by half step and the other a whole step in contrary motion.<ref name="Dahlhaus"/> 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]]). By the 16th century, the semitone had become a more versatile interval, sometimes even appearing as an augmented unison in very [[chromatic]] passages. [[Music semiotics|Semantically]], in the 16th century the repeated melodic semitone became associated with weeping, see: [[chromatic fourth|passus duriusculus]], [[lament bass]], and [[pianto]].

By the [[Baroque music|Baroque era]] (1600 to 1750), the [[tonality|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]]s for instrumental tuning and the more frequent use of [[enharmonic]] equivalences increased the ease with which a semitone could be applied. Its function remained similar through the [[Classical music|Classical]] period, and though it was used more frequently as the language of tonality became more chromatic in the [[Romantic music|Romantic]] period, the musical function of the semitone did not change.

In the 20th century, however, composers such as [[Arnold Schoenberg]], [[Béla Bartók]], and [[Igor Stravinsky]] 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]]). By now, enharmonic equivalence was a commonplace property of [[equal temperament]], 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 [[Musical tuning|tuning]] system used. [[Meantone temperament]]s have two distinct types of semitones, but in the exceptional case of [[equal temperament]], there is only one. The unevenly distributed [[well temperament]]s contain many different semitones. [[Pythagorean tuning]], similar to meantone tuning, has two, but in other systems of just intonation there are many more possibilities.

===Meantone temperament===

In [[meantone temperament|meantone]] systems, there are two different semitones. This results because of the break in the [[circle of fifths]] 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 [[quarter-comma meantone]], tuned as a cycle of [[Musical temperament|tempered]] [[Perfect fifth|fifths]] from E{{music|flat}} to G{{music|sharp}}, the chromatic and diatonic semitones are 76.0 and 117.1 cents wide respectively.

{| class="wikitable" style="text-align:center" align="center"
| bgcolor="#ffeeee" | '''Chromatic semitone'''
|
| colspan="2" bgcolor="#ffeeee" | <small>76.0</small>
| colspan="2" |
| colspan="2" |
| colspan="2" bgcolor="#ffeeee" | <small>76.0</small>
| colspan="2" |
| colspan="2" bgcolor="#ffeeee" | <small>76.0</small>
| colspan="2" |
| colspan="2" bgcolor="#ffeeee" | <small>76.0</small>
| colspan="2" |
| colspan="2" |
| colspan="2" bgcolor="#ffeeee" | <small>76.0</small>
| colspan="2" |
|-
| bgcolor="#fffbee" | '''Pitch'''
| colspan="2" bgcolor="#fffbee" | C
| colspan="2" bgcolor="#fffbee" | C{{music|sharp}}
| colspan="2" bgcolor="#fffbee" | D
| colspan="2" bgcolor="#fffbee" | E{{music|flat}}
| colspan="2" bgcolor="#fffbee" | E
| colspan="2" bgcolor="#fffbee" | F
| colspan="2" bgcolor="#fffbee" | F{{music|sharp}}
| colspan="2" bgcolor="#fffbee" | G
| colspan="2" bgcolor="#fffbee" | G{{music|sharp}}
| colspan="2" bgcolor="#fffbee" | A
| colspan="2" bgcolor="#fffbee" | B{{music|flat}}
| colspan="2" bgcolor="#fffbee" | B
| colspan="2" bgcolor="#fffbee" | C
|-
| bgcolor="#fffbee" | '''Cents'''
| colspan="2" bgcolor="#fffbee" | <small>0.0</small>
| colspan="2" bgcolor="#fffbee" | <small>76.0</small>
| colspan="2" bgcolor="#fffbee" | <small>193.2</small>
| colspan="2" bgcolor="#fffbee" | <small>310.3</small>
| colspan="2" bgcolor="#fffbee" | <small>386.3</small>
| colspan="2" bgcolor="#fffbee" | <small>503.4</small>
| colspan="2" bgcolor="#fffbee" | <small>579.5</small>
| colspan="2" bgcolor="#fffbee" | <small>696.6</small>
| colspan="2" bgcolor="#fffbee" | <small>772.6</small>
| colspan="2" bgcolor="#fffbee" | <small>889.7</small>
| colspan="2" bgcolor="#fffbee" | <small>1006.8</small>
| colspan="2" bgcolor="#fffbee" | <small>1082.9</small>
| colspan="2" bgcolor="#fffbee" | <small>1200.0</small>
|- <!--- this row inserted to avoid bug in Chrome causing misalignment of columns -->
| || || || || || || || || || || || || || || || || || || || || || || || || || ||
|-
| bgcolor="#eeeeff" | '''Diatonic semitone'''
| colspan="3" |
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
| colspan="2" |
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
| colspan="2" |
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
| colspan="2" |
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
| colspan="2" |
| colspan="2" bgcolor="#eeeeff" | <small>117.1</small>
|
|}

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-TET|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]] 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 [[Twelfth root of two|2<sup>1/12</sup>]] (approximately 1.05946), or 100 cents, and is 11.7 cents narrower than the 16:15 ratio (its most common form in [[just intonation]], [[#Just intonation|discussed below]]).

All diatonic intervals can be expressed as an equivalent number of semitones. For instance a [[major sixth]] equals nine semitones.

There are many approximations, [[Rational number|rational]] or otherwise, to the equal-tempered semitone. To cite a few:
:*<math>18 / 17 \approx 99.0 \text{ cents,}</math><br />suggested by [[Vincenzo Galilei]] and used by [[luthier]]s of the [[Renaissance music|Renaissance]],

:*<math>\sqrt[4]{\frac{2}{3-\sqrt{2}}} \approx 100.4 \text{ cents,}</math><br />suggested by [[Marin Mersenne]] as a [[Constructible number|constructible]] and more accurate alternative,

:*<math>(139 / 138 )^8  \approx 99.9995 \text{ cents,}</math><br />used by [[Julián Carrillo]] 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 [[well temperament]], 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 [[enharmonic]] 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 (music)|key]] had a slightly different sonic color or character, beyond the limitations of conventional notation.

===Pythagorean tuning===
<!--[[Minor semitone]] and [[major semitone]] link directly here.-->
{{multiple image
| align = right
| image1 = Pythagorean limma on C.png
| width1 = 200
| alt1 = 
| caption1 = Pythagorean limma on C[[File:Pythagorean minor semitone on C.mid|90px]]
| image2 = Pythagorean apotome on C.png
| width2 = 200
| alt2 = 
| caption2 = Pythagorean apotome on C[[File:Pythagorean apotome on C.mid|90px]]
| footer = 
}}

{{multiple image
   | width1    = 200
   | image1    = Pythagorean limma.png
   | caption1  = Pythagorean limma as five descending just perfect fifths from C (the inverse is B+)
   | width2    = 200
   | image2    = Pythagorean apotome.png
   | caption2  = Pythagorean apotome as seven just perfect fifths
}}

Like meantone temperament, [[Pythagorean tuning]] is a broken [[circle of fifths]]. This creates two distinct semitones, but because Pythagorean tuning is also a form of 3-limit [[just intonation]], 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 ({{Audio|Pythagorean minor semitone on C.mid|play}}), and is often called the '''Pythagorean limma'''. It is also sometimes called the ''Pythagorean minor semitone''. It is about 90.2 cents.
:<math>\frac{256}{243} = \frac{2^8}{3^5} \approx 90.2 \text{ cents}</math>

It can be thought of as the difference between three [[octaves]] and five [[perfect fifth|just fifths]], and functions as a [[#Minor second|diatonic semitone]] in a [[Pythagorean tuning]].

The '''Pythagorean chromatic semitone''' has a ratio of 2187/2048 ({{Audio|Pythagorean apotome on C.mid|play}}). It is about 113.7 [[Cent (music)|cents]]. It may also be called the '''Pythagorean apotome'''<ref name="Rashed">Rashed, Roshdi (ed.) (1996). ''Encyclopedia of the History of Arabic Science, Volume 2'', pp. 588, 608. Routledge. {{ISBN|0-415-12411-5}}.</ref><ref>[[Hermann von Helmholtz]] (1885). ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'', p. 454.</ref><ref>Benson, Dave (2006). ''Music: A Mathematical Offering'', p. 369. {{ISBN|0-521-85387-7}}.</ref> or the ''Pythagorean major semitone''. (''See [[Pythagorean interval]]''.)
:<math>\frac{2187}{2048} = \frac{3^7}{2^{11}} \approx 113.7\text{ cents}</math>

It can be thought of as the difference between four perfect [[octave]]s and seven [[perfect fifth|just fifths]], and functions as a [[chromatic semitone]] in a [[Pythagorean tuning]].

The Pythagorean limma and Pythagorean apotome are [[enharmonic]] equivalents (chromatic semitones) and only a [[Pythagorean comma]] apart, in contrast to diatonic and chromatic semitones in [[meantone temperament]] and 5-limit [[just intonation]].

===Just 5-limit intonation {{anchor|Just intonation}}===
<!--[[Just diatonic semitone]], [[Just chromatic semitone]], and [[Semitone maximus]] redirect directly here.-->
[[File:Just diatonic semitone.png|thumb|right|16:15 [[#Minor second|diatonic semitone]]]]
[[File:Just diatonic semitone on C.png|thumb|right|16:15 diatonic semitone[[File:Just diatonic semitone on C.mid]]]]
[[File:Major limma on C.png|thumb|right|'Larger' or major limma on C[[File:Greater chromatic semitone on C.mid]]]]
[[File:Semitone 5-limit diamond.png|thumb|right|300px|Relationship between the 4&nbsp;common 5&nbsp;limit semitones]]

A minor second in [[just intonation]] typically corresponds to a pitch [[ratio]] of 16:15 ({{Audio|Just diatonic semitone on C.mid|play}}) or 1.0666... (approximately 111.7&nbsp;[[cent (music)|cent]]s), called the '''just diatonic semitone'''.<ref>{{cite journal |title={{grey|[no title cited]}} |publisher=Royal Society |place=Great Britain |year=1880 |quote=digitized 26 Feb 2008; Harvard University |journal=[[Proceedings of the Royal Society of London]] |volume=30 |page=531}}</ref> This is a practical just semitone, since it is the interval that occurs twice within the diatonic scale between a:
: [[major third]] (5:4) and [[perfect fourth]] (4:3) <math>\ \left(\ \tfrac{4}{3} \div \tfrac{5}{4} = \tfrac{16}{15}\ \right)\ ,</math> and a
: [[major seventh]] (15:8) and the [[perfect octave]] (2:1) <math>\ \left(\ \tfrac{2}{1} \div \tfrac{15}{8} = \tfrac{16}{15}\ \right) ~.</math>

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".<ref name="books.google.com"/>

An "augmented unison" (sharp) in just intonation is a different, smaller semitone, with frequency ratio 25:24 ({{Audio|Just chromatic semitone on C.mid|play}}) or 1.0416... (approximately 70.7&nbsp;cents). It is the interval between a [[major third]] (5:4) and a minor third (6:5). In fact, it is the spacing between the minor and major thirds, sixths, and sevenths (but not necessarily the major and minor second). Composer [[Ben Johnston (composer)|Ben Johnston]] used a sharp ({{music|#}}) to indicate a note is raised 70.7&nbsp;cents, or a flat ({{Music|b}}) to indicate a note is lowered 70.7&nbsp;cents.<ref name=Fonville>{{cite journal |first=J. |last=Fonville |author-link=John Fonville |date=Summer 1991 |title=[[Ben Johnston (composer)|Ben Johnston]]'s extended just intonation – a guide for interpreters |journal=[[Perspectives of New Music]] |volume=29 |issue=2 |pages=106–137 |doi=10.2307/833435 |jstor=833435 |quote=...&nbsp;the {{sfrac|25|24}} ratio is the sharp ({{music|#}}) ratio ... this raises a note approximately 70.6&nbsp;cents.{{rp|style=ama|p=109}} }}</ref> (This is the standard practice for just intonation, but not for all other microtunings.)

Two other kinds of semitones are produced by 5&nbsp;limit tuning. A [[chromatic scale]] defines 12&nbsp;semitones as the 12&nbsp;intervals between the 13&nbsp;adjacent notes, spanning a full octave (e.g. from C{{sub|4}} to C{{sub|5}}). The 12&nbsp;semitones produced by a [[Five-limit tuning#Size of intervals|commonly used version]] of 5&nbsp;limit tuning have four different sizes, and can be classified as follows:

; Just chromatic semitone : ''chromatic semitone'', or ''smaller'', or ''minor chromatic semitone'' between harmonically related flats and sharps e.g. between E{{Music|b}} and E (6:5 and 5:4):
: <math> S_1 = \tfrac{5}{4} \div \tfrac{6}{5} = \tfrac{25}{24} \approx 70.7 \ \hbox{cents}</math>
; Larger chromatic semitone : or ''major chromatic semitone'', or ''larger limma'', or ''major chroma'',<ref name=Fonville/> e.g. between C and an accute&nbsp;C{{music|#}} (C{{music|#}} raised by a [[syntonic comma]]) (1:1 and 135:128):
: <math>S_2 = \tfrac{25}{24} \times \tfrac{81}{80} = \tfrac{135}{128} \approx 92.2 \ \hbox{cents}</math>
; Just diatonic semitone: or ''smaller'', or ''minor diatonic semitone'', e.g. between E and F (5:4 to 4:3):
: <math>S_3 = \tfrac{4}{3} \div \tfrac{5}{4} = \tfrac{16}{15} \approx 111.7 \ \hbox{cents}</math>
; Larger diatonic semitone: or ''greater'' or ''major diatonic semitone'', e.g. between A and B{{music|b}} (5:3 to 9:5), or C and chromatic D{{music|b}} (27:25), or F{{music|#}} and G (25:18 and 3:2):
: <math>S_4 = \tfrac{9}{5} \div \tfrac{5}{3} = \tfrac{27}{25} \approx 133.2 \ \hbox{cents}</math>

The most frequently occurring semitones are the just ones ({{mvar|S}}{{sub|3}}, 16:15, and {{mvar|S}}{{sub|1}}, 25:24): S{{sub|3}} occurs at 6&nbsp;short intervals out of 12, {{mvar|S}}{{sub|1}} 3&nbsp;times, {{mvar|S}}{{sub|2}} twice, and {{mvar|S}}{{sub|4}} at only one interval (if diatonic D{{music|b}} replaces chromatic D{{music|b}} and sharp notes are not used).

The smaller chromatic and diatonic semitones differ from the larger by the [[syntonic comma]] (81:80 or 21.5&nbsp;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]] (2048:2025 or 19.6&nbsp;cents), the outer differ by the greater diesis (648:625 or 62.6&nbsp;cents).

===Extended just intonations===
In [[7-limit|7&nbsp;limit tuning]] there is the [[septimal diatonic semitone]] of 15:14 ({{Audio|Septimal diatonic semitone on C.mid|play}}) available in between the 5&nbsp;limit [[major seventh]] (15:8) and the [[septimal minor seventh|7&nbsp;limit minor seventh]] / [[harmonic seventh]] (7:4). There is also a smaller [[septimal chromatic semitone]] of 21:20 ({{Audio|Septimal chromatic semitone on C.mid|play}}) 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&nbsp;limit neighbours, although the former was often implemented by theorist [[Henry Cowell|Cowell]], while [[Harry Partch|Partch]] used the latter as part of [[Harry Partch's 43-tone scale|his 43&nbsp;tone scale]].

Under 11&nbsp;limit tuning, there is a fairly common ''undecimal [[neutral second]]'' (12:11) ({{Audio|Neutral second on C.mid|play}}), but it lies on the boundary between the minor and [[major second]] (150.6&nbsp;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&nbsp;limit tuning, there is a tridecimal {{sfrac|2|3}} tone (13:12 or 138.57&nbsp;cents) and tridecimal {{sfrac|1|3}} tone (27:26 or 65.34&nbsp;cents).

In 17&nbsp;limit just intonation, the major diatonic semitone is 15:14 or 119.4&nbsp;cents ({{audio|Major diatonic semitone on C.mid|Play}}), and the minor diatonic semitone is 17:16 or 105.0&nbsp;cents,<ref name="Haluska">Haluska, Jan (2003). ''The Mathematical Theory of Tone Systems'', p. xxiv. {{ISBN|0-8247-4714-3}}. Overtone semitone.</ref><ref>{{cite book |author-link=Ebenezer Prout |last=Prout |first=E. |year=2004 |title=Harmony |page=325 |publisher=University Press of the Pacific |isbn=1-4102-1920-8}}</ref> and septendecimal limma is 18:17 or 98.95&nbsp;cents.

Though the names ''diatonic'' and ''chromatic'' are often used for these intervals, their musical function is not the same as the 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 use (diatonic semitone function is more prevalent).

===Other equal temperaments===

[[19 equal temperament|19-tone equal temperament]] distinguishes between the chromatic and diatonic semitones; in this tuning, the chromatic semitone is one step of the scale ({{Audio|1 step in 19-et on C.mid|play 63.2 cents}}), and the diatonic semitone is two ({{Audio|2 steps in 19-et on C.mid|play 126.3 cents}}). [[31 equal temperament|31-tone equal temperament]] also distinguishes between these two intervals, which become 2 and 3 steps of the scale, respectively. [[53 equal temperament|53-ET]] has an even closer match to the two semitones with 3 and 5 steps of its scale while [[72 equal temperament|72-ET]] uses 4 ({{Audio|4 steps in 72-et on C.mid|play 66.7 cents}}) and 7 ({{Audio|7 steps in 72-et on C.mid|play 116.7 cents}}) 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).

==See also==

*[[12 equal temperament|12-tone equal temperament]]
*[[List of meantone intervals]]
*[[Interval (music)|List of musical intervals]]
*[[List of pitch intervals]]
*[[Approach chord]]
*[[Major second]]
*[[Neutral second]]
*[[Pythagorean interval]]
*[[Regular temperament]]

==References==

{{reflist}}

==Further reading==

*[[Donald Jay Grout|Grout, Donald Jay]], and [[Claude V. Palisca]]. ''A History of Western Music, 6th ed''. New York: Norton, 2001. {{ISBN|0-393-97527-4}}.
*[[Richard Hoppin|Hoppin, Richard H.]] ''Medieval Music''. New York: W. W. Norton, 1978. {{ISBN|0-393-09090-6}}.

{{Intervals}}
{{Twelve-tone technique}}

[[Category:Minor intervals]]
[[Category:Seconds (music)]]
[[Category:Units of level]]