Semitone

This article is about the musical interval. For the printing method, seeHalftone.

Asemitone,also called aminor second,half step,or ahalf tone,is the smallestmusical intervalcommonly used in Western tonal music,and it is considered the mostdissonantwhen sounded harmonically. It is defined as the interval between two adjacent notes in a12-tone scale(or half of awhole 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♯;the interval between them is a semitone.

semitone

Inverse	major seventh(for minor second);diminished octave(for augmented unison);augmented octave(for diminished unison)
Name
Other names	minor second, diatonic semitone, augmented unison, diminished unison, chromatic semitone
Abbreviation	m2; A1
Size
Semitones	1
Interval class	1
Just interval	16:15,17:16,27:25, 135:128,25:24,256:243
Cents
12-Tone equal temperament	100
Just intonation	112,105, 133, 92,71,90

Minor second

In a 12-note approximately equally divided scale, any interval can be defined in terms of an appropriate number of semitones (e.g. awhole toneor major second is 2 semitones wide, amajor third4 semitones, and aperfect fifth7 semitones).

Inmusic theory,a distinction is madebetween adiatonic semitone,orminor second(an interval encompassing two differentstaff positions,e.g. from C to D♭) and achromatic semitoneoraugmented unison(an interval between two notes at the same staff position, e.g. from C to C♯). These areenharmonically equivalentif and only iftwelve-tone equal temperamentis used; for example, they are not the same thing inmeantone temperament,where the diatonic semitone is distinguished from and larger than the chromatic semitone (augmented unison), or inPythagorean tuning,where the diatonic semitone is smaller instead. SeeInterval (music) § Numberfor more details about this terminology.

Intwelve-tone equal temperamentall 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. InPythagorean tuning,seven semitones out of twelve are diatonic, with ratio 256:243 or 90.2 cents (Pythagorean limma), and the other five are chromatic, with ratio 2187:2048 or 113.7 cents (Pythagorean apotome); they differ by thePythagorean commaof ratio 531441:524288 or 23.5 cents. Inquarter-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 lesserdiesisof ratio 128:125 or 41.1 cents. 12-tone scales tuned injust intonationtypically define three or four kinds of semitones. For instance,Asymmetricfive-limit tuningyields 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, seebelow.

The condition of having semitones is called hemitonia; that of having no semitones isanhemitonia.Amusical scaleorchordcontaining semitones is called hemitonic; one without semitones is anhemitonic.

Minor second

The melodic minor second is an integral part of most cadences of theCommon practice period.

minor second

Inverse	major seventh
Name
Other names	semitone, half step
Abbreviation	m2
Size
Semitones	1
Interval class	1
Cents
12-Tone equal temperament	100.0

Theminor secondoccurs in themajor scale,between the third and fourth degree, (mi(E) andfa(F) in C major), and between the seventh and eighth degree (ti(B) anddo(C) in C major). It is also called thediatonic semitonebecause it occurs betweenstepsin thediatonic scale.The minor second is abbreviatedm2(or−2). Its inversion is themajor seventh(M7orMa7).

Listen to a minor second in equal temperament^ⓘ.Here,middle Cis followed by D♭,which is a tone 100centssharper than C, and then by both tones together.

Melodically,this interval is very frequently used, and is of particular importance incadences.In theperfectanddeceptive cadencesit appears as a resolution of theleading-toneto thetonic.In theplagal cadence,it appears as the falling of thesubdominantto themediant.It also occurs in many forms of theimperfect cadence,wherever the tonic falls to the leading-tone.

Harmonically,the interval usually occurs as some form ofdissonanceor anonchord tonethat is not part of thefunctional harmony.It may also appear in inversions of amajor seventh chord,and in manyadded tone chords.

Harmonic minor second inJ. S. Bach's Prelude in C major from theWTC,book 1, mm. 7–9.The minor second may be viewed as asuspensionof theBresolving into the followingA minor seventhchord.

In unusual situations, the minor second can add a great deal of character to the music. For instance,Frédéric Chopin'sÉtude Op. 25, No. 5opens 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 theRomanticperiod, such asModest Mussorgsky'sBallet of the Unhatched Chicks.More recently, the music to the movieJawsexemplifies the minor second.

In other temperaments

Injust intonationa 16:15 minor second arises in the Cmajor scalebetween B & C and E & F, and is "the sharpestdissonancefound in the [major]scale."Play B & C^ⓘ

Augmented unison

Main article:Augmented unison

Augmented unison on C

Augmented unisons often appear as a consequence ofsecondary dominants,such as those in the soprano voice of thissequencefromFelix Mendelssohn'sSong Without WordsOp. 102 No. 3, mm. 47–49.

Theaugmented unison,the interval produced by theaugmentation,or widening by one half step, of the perfect unison,does not occur between diatonic scale steps, but instead between a scale step and achromaticalteration of the same step. It is also called achromatic semitone.The augmented unison is abbreviatedA1,oraug 1.Its inversion is thediminished octave(d8,ordim 8). The augmented unison is also the inversion of theaugmented octave,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.

Melodically,an augmented unison very frequently occurs when proceeding to a chromatic chord, such as asecondary dominant,adiminished seventh chord,or anaugmented sixth chord.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 asD,E♭,F♭,G,A).

Franz Liszt's secondTranscendental Étude,measure 63

Harmonically,augmented unisons are quite rare in tonal repertoire. In the example to the right,Liszthad written anE♭against anE♮in the bass. HereE♭was preferred to aD♯to make the tone's function clear as part of anFdominant seventhchord, and the augmented unison is the result of superimposing this harmony upon anEpedal point.

In addition to this kind of usage, harmonic augmented unisons are frequently written in modern works involvingtone clusters,such asIannis Xenakis'Evryalifor piano solo.

History

The semitone appeared in the music theory of Greek antiquity as part of a diatonic or chromatictetrachord,and it has always had a place in the diatonic scales of Western music since. The variousmodalscales ofmedieval musictheory were all based upon this diatonic pattern oftonesand semitones.

Though it would later become an integral part of the musicalcadence,in the early polyphony of the 11th century this was not the case.Guido of Arezzosuggested instead in hisMicrologusother alternatives: either proceeding by whole tone from amajor secondto a unison, or anoccursushaving two notes at amajor thirdmove 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 theditone$\left({\begin{matrix}{\frac {4}{3}}\end{matrix}}/{{\begin{matrix}({\frac {9}{8}})\end{matrix}}^{2}}={\begin{matrix}{\frac {256}{243}}\end{matrix}}\right)$."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 inclausulaebecause it lacked clarity as an interval. "

Dramatic chromatic scale in the opening measures ofLuca Marenzio'sSolo e pensoso,ca. 1580.

However, beginning in the 13th centurycadencesbegin 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 asmusica ficta). By the 16th century, the semitone had become a more versatile interval, sometimes even appearing as an augmented unison in verychromaticpassages.Semantically,in the 16th century the repeated melodic semitone became associated with weeping, see:passus duriusculus,lament bass,andpianto.

By theBaroque era(1600 to 1750), thetonalharmonic framework was fully formed, and the various musical functions of the semitone were rigorously understood. Later in this period the adoption ofwell temperamentsfor instrumental tuning and the more frequent use ofenharmonicequivalences increased the ease with which a semitone could be applied. Its function remained similar through theClassicalperiod, and though it was used more frequently as the language of tonality became more chromatic in theRomanticperiod, the musical function of the semitone did not change.

In the 20th century, however, composers such asArnold Schoenberg,Béla Bartók,andIgor Stravinskysought 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 ofHenry Cowell). By now, enharmonic equivalence was a commonplace property ofequal 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 thetuningsystem used.Meantone temperamentshave two distinct types of semitones, but in the exceptional case ofequal temperament,there is only one. The unevenly distributedwell temperamentscontain 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

Inmeantonesystems, there are two different semitones. This results because of the break in thecircle of fifthsthat 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 commonquarter-comma meantone,tuned as a cycle oftemperedfifthsfrom E♭to G♯,the chromatic and diatonic semitones are 76.0 and 117.1 cents wide respectively.

Chromatic semitone		76.0						76.0				76.0				76.0						76.0
Pitch	C		C♯		D		E♭		E		F		F♯		G		G♯		A		B♭		B		C
Cents	0.0		76.0		193.2		310.3		386.3		503.4		579.5		696.6		772.6		889.7		1006.8		1082.9		1200.0
Diatonic semitone				117.1		117.1				117.1				117.1				117.1		117.1				117.1

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 temperamentis 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 temperamentis 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 of2^1/12(approximately 1.05946), or 100 cents, and is 11.7 cents narrower than the 16:15 ratio (its most common form injust intonation,discussed below).

All diatonic intervals can be expressed as an equivalent number of semitones. For instance amajor sixthequals nine semitones.

There are many approximations,rationalor otherwise, to the equal-tempered semitone. To cite a few:

• $18/17\approx 99.0{\text{ cents,}}$
  suggested byVincenzo Galileiand used byluthiersof theRenaissance,

• ${\sqrt[{4}]{\frac {2}{3-{\sqrt {2}}}}}\approx 100.4{\text{ cents,}}$
  suggested byMarin Mersenneas aconstructibleand more accurate alternative,

• $(139/138)^{8}\approx 99.9995{\text{ cents,}}$
  used byJulián Carrilloas part of a sixteenth-tone system.

For more examples, see Pythagorean and Just systems of tuning below.

Well temperament

There are many forms ofwell 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 adiatonicandchromaticsemitone in the tuning. Well temperament was constructed so thatenharmonicequivalence 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, eachkeyhad a slightly different sonic color or character, beyond the limitations of conventional notation.

Pythagorean tuning

Pythagorean limma on C

Pythagorean apotome on C

Pythagorean limma as five descending just perfect fifths from C (the inverse is B+)

Pythagorean apotome as seven just perfect fifths

Like meantone temperament,Pythagorean tuningis a brokencircle of fifths.This creates two distinct semitones, but because Pythagorean tuning is also a form of 3-limitjust intonation,these semitones are rational. Also, unlike most meantone temperaments, the chromatic semitone is larger than the diatonic.

ThePythagorean diatonic semitonehas a ratio of 256/243 (play^ⓘ), and is often called thePythagorean limma.It is also sometimes called thePythagorean minor semitone.It is about 90.2 cents.

${\frac {256}{243}}={\frac {2^{8}}{3^{5}}}\approx 90.2{\text{ cents}}$

It can be thought of as the difference between threeoctavesand fivejust fifths,and functions as adiatonic semitonein aPythagorean tuning.

ThePythagorean chromatic semitonehas a ratio of 2187/2048 (play^ⓘ). It is about 113.7cents.It may also be called thePythagorean apotomeor thePythagorean major semitone.(SeePythagorean interval.)

${\frac {2187}{2048}}={\frac {3^{7}}{2^{11}}}\approx 113.7{\text{ cents}}$

It can be thought of as the difference between four perfectoctavesand sevenjust fifths,and functions as achromatic semitonein aPythagorean tuning.

The Pythagorean limma and Pythagorean apotome areenharmonicequivalents (chromatic semitones) and only aPythagorean commaapart, in contrast to diatonic and chromatic semitones inmeantone temperamentand 5-limitjust intonation.

Just 5-limit intonation

16:15diatonic semitone

16:15 diatonic semitone

'Larger' or major limma on C

Relationship between the 4 common 5 limit semitones

A minor second injust intonationtypically corresponds to a pitchratioof 16:15 (play^ⓘ) or 1.0666... (approximately 111.7cents), called thejust diatonic semitone.This is a practical just semitone, since it is the interval that occurs twice within the diatonic scale between a:

major third(5:4) andperfect fourth(4:3)$\ \left(\ {\tfrac {4}{3}}\div {\tfrac {5}{4}}={\tfrac {16}{15}}\ \right)\,$and a

major seventh(15:8) and theperfect octave(2:1)$\ \left(\ {\tfrac {2}{1}}\div {\tfrac {15}{8}}={\tfrac {16}{15}}\ \right)~.$

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" (sharp) in just intonation is a different, smaller semitone, with frequency ratio 25:24 (play^ⓘ) or 1.0416... (approximately 70.7 cents). It is the interval between amajor 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). ComposerBen Johnstonused a sharp (♯) to indicate a note is raised 70.7 cents, or a flat (♭) to indicate a note is lowered 70.7 cents.(This is the standard practice for just intonation, but not for all other microtunings.)

Two other kinds of semitones are produced by 5 limit tuning. Achromatic scaledefines 12 semitones as the 12 intervals between the 13 adjacent notes, spanning a full octave (e.g. from C₄to C₅). The 12 semitones produced by acommonly used versionof 5 limit tuning have four different sizes, and can be classified as follows:

Just chromatic semitone

chromatic semitone,orsmaller,orminor chromatic semitonebetween harmonically related flats and sharps e.g. between E♭and E (6:5 and 5:4):

$S_{1}={\tfrac {5}{4}}\div {\tfrac {6}{5}}={\tfrac {25}{24}}\approx 70.7\ {\hbox{cents}}$

Larger chromatic semitone

ormajor chromatic semitone,orlarger limma,ormajor chroma,e.g. between C and an accute C♯(C♯raised by asyntonic comma) (1:1 and 135:128):

$S_{2}={\tfrac {25}{24}}\times {\tfrac {81}{80}}={\tfrac {135}{128}}\approx 92.2\ {\hbox{cents}}$

Just diatonic semitone

orsmaller,orminor diatonic semitone,e.g. between E and F (5:4 to 4:3):

$S_{3}={\tfrac {4}{3}}\div {\tfrac {5}{4}}={\tfrac {16}{15}}\approx 111.7\ {\hbox{cents}}$

Larger diatonic semitone

orgreaterormajor diatonic semitone,e.g. between A and B♭(5:3 to 9:5), or C and chromatic D♭(27:25), or F♯and G (25:18 and 3:2):

$S_{4}={\tfrac {9}{5}}\div {\tfrac {5}{3}}={\tfrac {27}{25}}\approx 133.2\ {\hbox{cents}}$

The most frequently occurring semitones are the just ones (S₃,16:15, andS₁,25:24): S₃occurs at 6 short intervals out of 12,S₁3 times,S₂twice, andS₄at only one interval (if diatonic D♭replaces chromatic D♭and sharp notes are not used).

The smaller chromatic and diatonic semitones differ from the larger by thesyntonic comma(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 thediaschisma(2048:2025 or 19.6 cents), the outer differ by the greater diesis (648:625 or 62.6 cents).

Extended just intonations

In7 limit tuningthere is theseptimal diatonic semitoneof 15:14 (play^ⓘ) available in between the 5 limitmajor seventh(15:8) and the7 limit minor seventh/harmonic seventh(7:4). There is also a smallerseptimal chromatic semitoneof 21:20 (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 limit neighbours, although the former was often implemented by theoristCowell,whilePartchused the latter as part ofhis 43 tone scale.

Under 11 limit tuning, there is a fairly commonundecimalneutral second(12:11) (play^ⓘ), but it lies on the boundary between the minor andmajor second(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 tridecimal⁠2/3⁠tone (13:12 or 138.57 cents) and tridecimal⁠1/3⁠tone (27:26 or 65.34 cents).

In 17 limit just intonation, the major diatonic semitone is 15:14 or 119.4 cents (Play^ⓘ), and the minor diatonic semitone is 17:16 or 105.0 cents,and septendecimal limma is 18:17 or 98.95 cents.

Though the namesdiatonicandchromaticare 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 thechromaticcounterpart to adiatonic16:15. 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-tone equal temperamentdistinguishes between the chromatic and diatonic semitones; in this tuning, the chromatic semitone is one step of the scale (play 63.2 cents^ⓘ), and the diatonic semitone is two (play 126.3 cents^ⓘ).31-tone equal temperamentalso distinguishes between these two intervals, which become 2 and 3 steps of the scale, respectively.53-EThas an even closer match to the two semitones with 3 and 5 steps of its scale while72-ETuses 4 (play 66.7 cents^ⓘ) and 7 (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-tone equal temperament
• List of meantone intervals
• List of musical intervals
• List of pitch intervals
• Approach chord
• Major second
• Neutral second
• Pythagorean interval
• Regular temperament

Further reading

• Grout, Donald Jay,andClaude V. Palisca.A History of Western Music, 6th ed.New York: Norton, 2001.ISBN0-393-97527-4.
• Hoppin, Richard H.Medieval Music.New York: W. W. Norton, 1978.ISBN0-393-09090-6.