The term mediant also refers to a relationship of musical keys. For example, relative to the key of A minor, the key of C major is the mediant, and it often serves as a mid-way point between I and V (hence the name). Tonicization or modulation to the mediant is quite common in pieces written in the minor mode and usually serves as the second theme group in sonata form since it is very easy to tonicize III in minor. Tonicization of III in major is quite rare in early classical harmony, compared with, say, modulation to the dominant in major. It becomes more common in late Haydn and Mozart and normal by middle-period Beethoven. Mediant tonicization in major is an important feature of Romantic music.
Schenkerian analysts consider this scale degree as expansion of the tonic since they have two common tones. On the other hand, in German theory derived from Hugo Riemann the mediant in major is considered the dominant parallel, Dp, and in minor the tonic parallel, tP.
Two points where are two representations of (possibly equivalent) rational numbers a/b and c/d. The line segments connecting the origin of coordinates to (b,a) and (d,c) form two adjacent sides in a parallelogram. The vertex of the parallelogram opposite to the origin of coordinates is the point (b+d,a+c), which is the mediant of a/b and c/d.
The nth Farey sequence F n is defined as the (ordered with respect to magnitude) sequence of reduced fractions a/b (with coprime a, b) such that b ≤ n. If two fractions a/c < b/d are adjacent (neighbouring) fractions in a segment of F n then the determinant relation bc-ad=1 mentioned above is generally valid and therefore the mediant is the simplest fraction in the interval (a/c, b/d), in the sense of being the fraction with the smallest denominator. Thus the mediant will then (first) appear in the (c + d)th Farey sequence and is the "next" fraction which is inserted in any Farey sequence between a/c and b/d. This gives the rule how the Farey sequences F n are successively built up with increasing n.
In Roman numeral analysis, the mediant chord can take several forms. In major scales, the mediant chord is a minor triad and is symbolized with the Roman numeral iii. In natural minor scales, the mediant is a major triad and is symbolized with the Roman numeral III. In harmonic minor scales and ascending melodic minor scales, the seventh scale degree is raised by a half step from the subtonic to the leading tone, creating an augmented triad that is symbolized with the Roman numeral III +.
In music, the mediant (Latin: to be in the middle ) is the third scale degree of a diatonic scale, being the note halfway between the tonic and the dominant. In the movable do solfège system, the mediant note is sung as mi. While the fifth scale degree is almost always a perfect fifth, the mediant can be a major or minor third.
That is to say, the numerator and denominator of the mediant are the sums of the numerators and denominators of the given fractions, respectively. It is sometimes called the freshman sum, as it is a common mistake in the early stages of learning about addition of fractions.
Technically, this is a binary operation on valid fractions (nonzero denominator), considered as ordered pairs of appropriate integers, a priori disregarding the perspective on rational numbers as equivalence classes of fractions. For example, the mediant of the fractions 1/1 and 1/2 is 2/3. However, if the fraction 1/1 is replaced by the fraction 2/2, which is an equivalent fraction denoting the same rational number 1, the mediant of the fractions 2/2 and 1/2 is 3/4. For a stronger connection to rational numbers the fractions may be required to be reduced to lowest terms, thereby selecting unique representatives from the respective equivalence classes.
There is not complete agreement on the definition of chromatic mediant relationships. Theorists such as Allen Forte define chromatic mediants conservatively, only allowing chromatic mediant chords of the same quality (major or minor) as described above. However, he describes an even more distant "doubly-chromatic mediant" relationship shared by two chords of the opposite mode, with roots a third apart and no common tones; for example C major and E minor or A minor, and A minor and C or F major,. Other less conservative theorists, such as Benward and Saker, include these additional chords of opposite quality and no shared tones in their default definition of chromatic mediants. Thus, by this more permissive definition, C major has six chromatic mediants: E major, A major, E major, A major, E minor and A minor.
In music, chromatic mediants are "altered mediant and submediant chords." A chromatic mediant relationship defined conservatively is a relationship between two sections and/or chords whose roots are related by a major third or minor third, and contain one common tone (thereby sharing the same quality, i.e. major or minor). For example, in the key of C major the diatonic mediant and submediant are E minor and A minor respectively. Their parallel majors are E major and A major. The mediants of the parallel minor of C major (C minor) are E major and A major. Thus, by this conservative definition, C major has four chromatic mediants: E major, A major, E major, and A major.
When a conservative chromatic mediant relationship involves seventh chords, "...the triad portions of the chords are both major or both minor." This pertains to the more permissive definition of chromatic mediant relationships as well.
* Assume that the pair of fractions a/c and b/d satisfies the determinant relation bc-ad=1. Then the mediant has the property that it is the simplest fraction in the interval (a/c, b/d), in the sense of being the fraction with the smallest denominator. More precisely, if the fraction a'/c' with positive denominator c' lies (strictly) between a/c and b/d, then its numerator and denominator can be written as and with two positive real (in fact rational) numbers. To see why the \lambda_i must be positive note that
The notion of mediant can be generalized to n fractions, and a generalized mediant inequality holds, a fact that seems to have been first noticed by Cauchy. More precisely, the weighted mediant m_w of n fractions is defined by (with w_i>0). It can be shown that m_w lies somewhere between the smallest and the largest fraction among the a_i/b_i.
Chromatic mediant chords were rarely used during the baroque and classical periods, though the chromatic mediant relationship was occasionally found between sections, but the chords and relationships became much more common during the romantic period and become even more prominent in post-romantic and impressionistic music. One author describes their use within phrases as, "surprising," even more so than the deceptive cadence, in part due to their rarity and in part due to their chromaticism (they come from 'outside' the key), while another says they are so rare that one should first eliminate the possibility that one is looking at a diatonic movement (presumably, borrowing), then make sure that it is not a secondary chord, and then, "finally," one may consider, "the likeliness of an actual chromatic mediant relationship."
In mathematics, the mediant of two fractions, generally made up of four positive integers
* The mediant inequality: An important property (also explaining its name) of the mediant is that it lies strictly between the two fractions of which it is the mediant: If a/c < b/d and, then
The Stern-Brocot tree provides an enumeration of all positive rational numbers via mediants in lowest terms, obtained purely by iterative computation of the mediant according to a simple algorithm.
After a short introduction on the tutti that begins with the mediant chord that transitions to the dominant seventh, the quiet theme, in E-flat major, first appears and then is subjected to a series of ten variations:
The third movement is a four-voice fugue in 6/8. It includes a countersubject typical of permutation fugues, which, unusually, engages in dialogue with the subject. Several features of the fugue suggest that it represented a considerable advance for Bach, especially considering that there are middle entries as far as the mediant and the dominant of the dominant. Somewhat unusually for Bach, the fugue includes very few episodes, the longest being the coda of the piece, which is based on various style brisé figures.