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A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale (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; 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 In music theory, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four Semitone, half steps or two ...
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 f ...
7 semitones). In
music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first is the "Elements of music, ...
, 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 if and only if 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 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 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. The condition of having semitones is called hemitonia; that of having no semitones is anhemitonia. A musical scale or chord containing semitones is called hemitonic; one without semitones is anhemitonic.


Minor second

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 steps in the
diatonic scale In music theory a diatonic scale is a heptatonic scale, heptatonic (seven-note) 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 eith ...
. 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 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 Piano solo, solo piano. He has maintained worldwide renown ...
'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 (; ; ; – ) was a Russian composer, one of the group known as "The Five (composers), The Five." He was an innovator of Music of Russia, Russian music in the Romantic music, Romantic period and strove to achieve a ...
'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 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, 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 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 ...
, 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 (; ) is a series of four notes separated by three interval (music), intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion (approx. 498 cent (m ...
, 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 music, sacred and secular music of Western Europe during the Middle Ages, from approximately the 6th to 15th centuries. It is the Dates of classical music eras, first and longest major era of Western class ...
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 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 In music theory, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four Semitone, half steps or two ...
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 \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. . 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, in the 16th century the repeated melodic semitone became associated with weeping, see: passus duriusculus, 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 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 and American composer, music theorist, teacher and writer. He was among the first Modernism (music), modernists who transformed the practice of harmony in 20th-centu ...
,
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 Hunga ...
, and
Igor Stravinsky Igor Fyodorovich Stravinsky ( – 6 April 1971) was a Russian composer and conductor with French citizenship (from 1934) and American citizenship (from 1945). He is widely considered one of the most important and influential 20th-century c ...
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 An equal temperament is a musical temperament or Musical tuning#Tuning systems, tuning system that approximates Just intonation, just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequency, frequencie ...
, 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 An equal temperament is a musical temperament or Musical tuning#Tuning systems, tuning system that approximates Just intonation, just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequency, frequencie ...
, 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 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 major sixth equals nine semitones. There are many approximations, rational or otherwise, to the equal-tempered semitone. To cite a few: :*18 / 17 \approx 99.0 \text
suggested by Vincenzo Galilei and used by luthiers of the
Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
, :*\sqrt \approx 100.4 \text
suggested by Marin Mersenne as a constructible and more accurate alternative, :*(139 / 138 )^8 \approx 99.9995 \text
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 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. 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. :\frac = \frac \approx 90.2 \text It can be thought of as the difference between three
octaves In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
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 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''.) :\frac = \frac \approx 113.7\text It can be thought of as the difference between four perfect
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
s 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 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  cents), called the just diatonic semitone. This is a practical just semitone, since it is the interval that occurs twice within the diatonic scale between a: :
major third In music theory, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four Semitone, half steps or two ...
(5:4) and perfect fourth (4:3) \ \left(\ \tfrac \div \tfrac = \tfrac\ \right)\ , and a : major seventh (15:8) and the perfect octave (2:1) \ \left(\ \tfrac \div \tfrac = \tfrac\ \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 () or 1.0416... (approximately 70.7 cents). It is the interval between a
major third In music theory, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four Semitone, half steps or two ...
(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 used 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. A chromatic scale defines 12 semitones as the 12 intervals between the 13 adjacent notes, spanning a full octave (e.g. from C to C). The 12 semitones produced by a commonly used version of 5 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 and E (6:5 and 5:4): : S_1 = \tfrac \div \tfrac = \tfrac \approx 70.7 \ \hbox ; Larger chromatic semitone : or ''major chromatic semitone'', or ''larger limma'', or ''major chroma'', e.g. between C and an accute C (C raised by a syntonic comma) (1:1 and 135:128): : S_2 = \tfrac \times \tfrac = \tfrac \approx 92.2 \ \hbox ; Just diatonic semitone: or ''smaller'', or ''minor diatonic semitone'', e.g. between E and F (5:4 to 4:3): : S_3 = \tfrac \div \tfrac = \tfrac \approx 111.7 \ \hbox ; Larger diatonic semitone: or ''greater'' or ''major 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 \div \tfrac = \tfrac \approx 133.2 \ \hbox The most frequently occurring semitones are the just ones (, 16:15, and , 25:24): S occurs at 6 short intervals out of 12, 3 times, twice, and at 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 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 The diaschisma (or diacisma) is a small interval (music), musical interval defined as the difference between three octaves and four perfect fifths plus two just major third, major thirds (in just intonation). It can be represented by the ratio 2 ...
(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 tuning there is the septimal diatonic semitone of 15:14 () available in between the 5 limit major seventh (15:8) and the 7 limit minor seventh / harmonic 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 Cent (music), cents. The septimal chromatic semitone may be derived ...
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 Cowell, while Partch 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 tone (13:12 or 138.57 cents) and tridecimal 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, 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 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-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 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

* 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


References


Further reading

* Grout, Donald Jay, and Claude V. Palisca. ''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) Units of level