Semitone

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. The minor second is abbreviated m2 (or −2). Its inversion is the major seventh (M7 or Ma7). Melodically, this interval is very frequently used, and is of particular importance in cadences. In the authentic 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. {{Image frame \new PianoStaff > } \new Staff \relative c' { \clef bass > } >> }} 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 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's "Ballet of the Unhatched Chicks" from Pictures at an Exhibition. More recently, the music to the movie Jaws exemplifies the minor second. ==Augmented unison==

The augmented unison, augmented prime Historically, this interval, like the tritone, is described as being and was associated with the . s, such as those in the soprano voice of this sequence from Felix Mendelssohn's Song Without Words Op. 102 No. 3, mm. 47–49. 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, 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). 's second Transcendental Étude, measure 63 Harmonically, augmented unisons are quite rare in tonal repertoire. In the example to the right, Liszt wrote 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 solo piano. Diminished unison The term diminished unison or diminished prime is also found occasionally. It is found once in Rameau's writings, for example, Other sources reject the possibility or utility of the diminished unison on the grounds that any alteration to the unison increases its size, thus augmenting rather than diminishing it. The term is sometimes justified as a negative-numbered interval, and also in terms of violin double-stopping technique on analogy to parallel intervals found on other strings. Some theoreticians make a distinction for this diminished form of the unison, stating it is only valid as a melodic interval, not a harmonic one. ==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 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 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 \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 in clausulae because it lacked clarity as an interval." 's Solo e pensoso, ca. 1580 |450x450px 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, 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. 's Three Piano Pieces, Op. 11 (m. 40) demonstrates completely unrestrained use of the semitone and related intervals. ==Semitones in different tunings==

The exact size of a semitone depends on the tuning system used. Pythagorean tuning has two distinct types of semitones. Meantone temperaments have two distinct types of semitones, but in the exceptional case of 12-tone equal temperament, there is only one. The unevenly distributed well temperaments contain many different semitones. In systems of just intonation, several types of semitones are encountered. Pythagorean tuning Pythagorean tuning, or 3-limit just intonation, is generated by a sequence of perfect fifths, which creates two distinct semitones. 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{256}{243} = \frac{2^8}{3^5} = \frac{2^3}{(3/2)^5} \approx 90.2 \text{ 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 or the Pythagorean major semitone. (See Pythagorean interval.) :\frac{2187}{2048} = \frac{3^7}{2^{11}} = \frac{(3/2)^7}{2^4} \approx 113.7\text{ cents} 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 only a Pythagorean comma apart, and may be considered enharmonic equivalents, in contrast to the diatonic and chromatic semitones in meantone temperament and 5-limit just intonation, which are farther apart. Meantone temperament Meantone temperaments are generated by a sequence of tempered perfect fifths of the same size, similarly to Pythagorean tuning, where the perfect fifths are pure instead. Likewise, the diatonic and chromatic semitones depend on the size of the fifth, where the diatonic semitone is derived by descending 5 fifths, and the chromatic semitone is derived by ascending 7 fifths. These are typically of different sizes, with the exception of 12 equal temperament, discussed below. Unlike in Pythagorean tuning, the chromatic semitone is usually smaller than the diatonic. In the common quarter-comma meantone, the chromatic and diatonic semitones are 76.0 and 117.1 cents wide respectively. They differ by the lesser diesis of ratio 128:125 or 41.1 cents. Quarter-comma meantone is largely used to create a 12-tone tuning, usually via a sequence of fifths from E to G. There is a break in the circle of fifths in this tuning; the interval between G and E does not represent a perfect fifth, but rather a diminished sixth. Such intervals are referred to as wolf intervals. Ascending by a semitone from an arbirary pitch, the resulting interval depends on this break: diatonic semitones derive from a descending chain of 5 fifths that does not cross the break, and chromatic semitones come from one that does, where the note is instead derived by an ascending chain of 7 fifths. Extended tunings 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. 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). There are many approximations, rational or otherwise, to the equal-tempered semitone. To cite a few: :*18 / 17 \approx 99.0 \text{ cents,}suggested by Vincenzo Galilei and used by luthiers of the Renaissance, :*\sqrt[4]{\frac{2}{3-\sqrt{2}}} \approx 100.4 \text{ cents,}suggested by Marin Mersenne as a constructible and more accurate alternative, :*(139 / 138 )^8 \approx 99.9995 \text{ cents,}used by Julián Carrillo as part of a sixteenth-tone system. 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. 5-limit just intonation |175x175px |175x175px |175x175px 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 interval that occurs twice within the diatonic scale between a: : major third (5:4) and perfect fourth (4:3) \ \left(\ \tfrac{4}{3} \div \tfrac{5}{4} = \tfrac{16}{15}\ \right)\ , and a : major seventh (15:8) and the perfect octave (2:1) \ \left(\ \tfrac{2}{1} \div \tfrac{15}{8} = \tfrac{16}{15}\ \right) ~. The 16:15 just minor second arises in Ptolemy's intense diatonic scale. Given a major scale starting on C, it occurs 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 (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{5}{4} \div \tfrac{6}{5} = \tfrac{25}{24} \approx 70.7 \ \hbox{cents} ; Larger chromatic semitone : or major chromatic semitone, or larger limma, or major chroma, 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==