What is "mediant"

Mediant \Me"di*ant\, n. [L. medians, p. p. of mediare to halve: cf. It. mediante, F. m['e]diante.] (Mus.) The third above the keynote; -- so called because it divides the interval between the tonic and dominant into two thirds.

"third note of the diatonic scale," 1753, from Italian mediante, from Late Latin mediantem (nominative medians) "dividing in the middle," present participle of mediare "to be in the middle" (see medial (adj.)). So called from being midway between the tonic and the dominant.

n. 1 (context music English) The third degree of the diatonic scale. 2 (context mathematics English) A rational number whose numerator is the sum of the numerators of two other given rational numbers and whose denominator is the sum of the denominators of those same two other rational numbers.

n. (music) the third note of a diatonic scale; midway between the tonic and the dominant

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. It is sung as mi in solfege. Similarly, the submediant is halfway between the tonic and subdominant. The fifth note is almost always a perfect fifth, while the third note can equally be a minor or major third.

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.

In Roman numeral analysis, the mediant chord can take several forms. In major scales, the mediant chord is minor and is noted with the Roman numeral iii. In a natural minor scale, the mediant occurs as a major chord, noted 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 a subtonic to a leading tone, creating an augmented mediant chord, noted with the Roman numeral III.

For example, in the C major scale (white keys on a piano, starting on C), the mediant is the note E; and the mediant chord is E-minor consisting of the notes E, G, and B. Therefore, Em is the iii chord in the C major scale. Also, in the A natural minor scale (same white keys, but now starting on A), the mediant is the note C; and the mediant chord is C (or C-major) consisting of the notes C, E, and G. Therefore, C is the III chord in the A (natural) minor scale. However, if the harmonic minor scale is used, G would be raised to G, changing the C chord to Caug, consisting of the notes C, E, and G. Therefore, Caug is the III chord in the A harmonic minor scale.

"Mediant" also refers to a relationship of musical keys. For example, relative to the key of A (natural) minor, the key of C major is the mediant, and 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 forms, since it is very easy to tonicize III in minor (no need to alter notes). Tonicization of III in major is quite rare in classical harmony, compared with, say, modulation to the V in major, but mediant tonicization in major is an important feature of late romantic music.

In mathematics, the mediant of two fractions

$$\frac {a} {c} \text{ and } \frac {b} {d}$$

is

$$\frac {a + b} {c + d}.$$

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 usual addition of fractions.

In general, this is an operation on fractions rather than on rational numbers. That is to say, for two rational numbers q, q, the value of the mediant depends on how the rational numbers are expressed using integer pairs. For example, the mediant of 1/1 and 1/2 is 2/3, but the mediant of 2/2 and 1/2 is 3/4.

A way around this, where required, is to specify that both rationals are to be represented as fractions in their lowest terms (with c > 0, d > 0). With such a restriction, mediant becomes a well-defined binary operation on rationals.

The Stern-Brocot tree provides an enumeration of all positive rational numbers, in lowest terms, obtained purely by iterative computation of the mediant according to a simple algorithm.