A semitone, also called a half step or a half tone, is the smallest
musical interval
In music theory, an interval is a difference in pitch between two sounds.
An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or ha ...
commonly used in Western tonal music, and it is considered the most
dissonant when sounded harmonically.
It is defined as the interval between two adjacent notes in a
12-tone scale. For example, C is adjacent to C; the interval between them is a semitone.
In 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
between a diatonic semitone, or minor second (an interval encompassing two different
s, 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 when
twelve-tone equal temperament is used, but are not the same thing in
meantone temperament, where the diatonic semitone is distinguished from and smaller than the chromatic semitone (augmented unison). See 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 limma), and the other five are chromatic, with ratio 2187:2048 or 113.7 cents (
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,
Asymmetric
Asymmetric may refer to:
*Asymmetry in geometry, chemistry, and physics
Computing
* Asymmetric cryptography, in public-key cryptography
*Asymmetric digital subscriber line, Internet connectivity
* Asymmetric multiprocessing, in computer architect ...
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
below
Below may refer to:
*Earth
* Ground (disambiguation)
* Soil
* Floor
* Bottom (disambiguation)
* Less than
*Temperatures below freezing
* Hell or underworld
People with the surname
* Ernst von Below (1863–1955), German World War I general
* Fr ...
.
The condition of having semitones is called hemitonia; that of having no semitones is
anhemitonia. A
musical scale or
chord
Chord may refer to:
* Chord (music), an aggregate of musical pitches sounded simultaneously
** Guitar chord a chord played on a guitar, which has a particular tuning
* Chord (geometry), a line segment joining two points on a curve
* 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 doub ...
, 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
Step(s) or STEP may refer to:
Common meanings
* Steps, making a staircase
* Walking
* Dance move
* Military step, or march
** Marching
Arts Films and television
* ''Steps'' (TV series), Hong Kong
* ''Step'' (film), US, 2017
Literature
* ...
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 st ...
. The minor second is abbreviated m2 (or −2). Its inversion is the ''
major seventh'' (''M7'' or ''Ma7'').
. Here,
middle C is followed by D, which is a tone 100
cents sharper than C, and then by both tones together.
Melodically, this interval is very frequently used, and is of particular importance in
cadences. In the
perfect
Perfect commonly refers to:
* Perfection, completeness, excellence
* Perfect (grammar), a grammatical category in some languages
Perfect may also refer to:
Film
* Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama
* Perfect (2018 f ...
and
deceptive cadences it appears as a resolution of the
leading-tone to the
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.
Harmonically, the interval usually occurs as some form of
dissonance or a
nonchord tone that is not part of the
functional harmony. It may also appear in inversions of a
major seventh chord, and in many
added tone chords.
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
Romantic may refer to:
Genres and eras
* The Romantic era, an artistic, literary, musical and intellectual movement of the 18th and 19th centuries
** Romantic music, of that era
** Romantic poetry, of that era
** Romanticism in science, of that e ...
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'' exemplifies the minor second.
In other temperaments
In
just intonation a 16:15 minor second arises in the C
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 doub ...
between B & C and E & F, and is, "the sharpest
dissonance found in the
ajorscale
Scale or scales may refer to:
Mathematics
* Scale (descriptive set theory), an object defined on a set of points
* Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original
* Scale factor, a number ...
."
[Paul, Oscar (1885). ]
A manual of harmony for use in music-schools and seminaries and for self-instruction
', p. 165. Theodore Baker, trans. G. Schirmer.
Augmented unison
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 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. 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 a
secondary dominant
A secondary chord is an music analysis, analytical label for a specific harmony, harmonic device that is prevalent in the tonality, tonal idiom of Western culture#Music, Western music beginning in the common practice period: the use of Diatonic a ...
, 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, 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).
Harmonically, augmented unisons are quite rare in tonal repertoire. In the example to the right,
Liszt 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 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
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 pr ...
, and it has always had a place in the diatonic scales of Western music since. The various
modal scales of
medieval music 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 ( ...
, 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 remainder between the perfect fourth and the
ditone ." 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. .]
However, beginning in the 13th century
cadences 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). By the 16th century, the semitone had become a more versatile interval, sometimes even appearing as an augmented unison in very
chromatic 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 compu ...
, in the 16th century the repeated melodic semitone became associated with weeping, see:
passus duriusculus
In music theory, a chromatic fourth, or ''passus duriusculus'',Monelle, Raymond (2000). ''The Sense of Music: Semiotic Essays'', p.73. . is a melody or melodic fragment spanning a perfect fourth with all or almost all chromatic Interval (music), ...
,
lament bass, and
pianto.
By the
Baroque era (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 temperaments 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 period, and though it was used more frequently as the language of tonality became more chromatic in the
Romantic
Romantic may refer to:
Genres and eras
* The Romantic era, an artistic, literary, musical and intellectual movement of the 18th and 19th centuries
** Romantic music, of that era
** Romantic poetry, of that era
** Romanticism in science, of that e ...
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 Hun ...
, 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
tuning system used.
Meantone temperaments have two distinct types of semitones, but in the exceptional case of
equal temperament, there is only one. The unevenly distributed
well temperaments 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 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 of ...
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
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 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
21/12 (approximately 1.05946), or 100 cents, and is 11.7 cents narrower than the 16:15 ratio (its most common form in
just intonation,
discussed below).
All diatonic intervals can be expressed as an equivalent number of semitones. For instance a
whole tone equals two semitones.
There are many approximations,
rational or otherwise, to the equal-tempered semitone. To cite a few:
:*
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 used by
luthiers 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 id ...
,
:*
suggested by
Marin Mersenne as a
constructible and more accurate alternative,
:*
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, violin ...
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 had a slightly different sonic color or character, beyond the limitations of conventional notation.
Pythagorean tuning
Like meantone temperament,
Pythagorean tuning is a broken
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 of ...
. 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 (), and is often called the Pythagorean limma. It is also sometimes called the ''Pythagorean minor semitone''. It is about 90.2 cents.
:
It can be thought of as the difference between three
octaves and five
just fifths, and functions as a
diatonic semitone in a
Pythagorean tuning.
The Pythagorean chromatic semitone has a ratio of 2187/2048 (). It is about 113.7
cents. It may also be called the Pythagorean apotome
[Rashed, Roshdi (ed.) (1996). ''Encyclopedia of the History of Arabic Science, Volume 2'', pp. 588, 608. Routledge. .] or the ''Pythagorean major semitone''. (''See
Pythagorean interval''.)
:
It can be thought of as the difference between four perfect
octaves and seven
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
A minor second in
just intonation typically corresponds to a pitch
ratio of 16:15 () or 1.0666... (approximately 111.7
cents), called the just diatonic semitone. This is a practical just semitone, since it is the difference between a
perfect fourth and
major third (
).
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. " Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p. 109, '' Perspectives of New Music'', 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 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):
:
*Larger, or major, chromatic semitone, or larger limma, or major chroma,
e.g. between C and C (1/1 and 135/128):
:
*Just, or smaller, or minor, diatonic semitone, e.g. between C and D (1/1 and 16/15):
:
*Larger, or major, diatonic semitone, e.g. between A and B (5/3 and 16/9):
:
The most frequently occurring semitones are the just ones (S
3 and S
1): S
3 occurs six times out of 12, S
1 three times, S
2 twice, and S
4 only once.
The smaller chromatic and diatonic semitones differ from the larger by the
syntonic 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 the
diaschisma (2048:2025 or 19.6 cents), the outer differ by the greater diesis (648:625 or 62.6 cents).
Extended just intonations
In
7-limit there is the
septimal diatonic semitone of 15:14 () available between the 5-limit
major seventh (15:8) and the
7-limit minor seventh (7:4). There is also a smaller
septimal chromatic semitone 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
Henry Cowell, 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 co ...
used the latter as part of
his 43-tone scale.
Under 11-limit tuning, there is a fairly common ''undecimal
neutral second'' (12:11) (), but it lies on the boundary between the minor and
major 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 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
In music, 19 Tone Equal Temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), or 19 ET, is the tempered scale derived by dividing the octave into 19 equal steps (equal frequency ratios). Each step represent ...
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 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).
See also
*
List of meantone intervals
*
List of musical intervals
*
List of pitch intervals
*
Approach chord
*
Major second
*
Neutral second
*
Pythagorean interval
*
Regular temperament
References
Further reading
*
Grout, Donald Jay, and
Claude V. Palisca
Claude Victor Palisca (24 November 1921 – 11 January 2001) was an American musicologist. An internationally recognized authority on early music, especially opera of the Renaissance music, Renaissance and Baroque music, Baroque periods, he was ...
. ''A History of Western Music, 6th ed''. New York: Norton, 2001. .
*
Hoppin, Richard H. ''Medieval Music''. New York: W. W. Norton, 1978. .
{{Twelve-tone technique
Minor intervals
Seconds (music)