# Just intonation

In music, just intonation (sometimes abbreviated as JI) or pure intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a pure or just interval. The two notes in any just interval are members of the same harmonic series.[a] Frequency ratios involving large integers such as 1024:927 are not generally said to be justly tuned. "Just intonation is the tuning system of the later ancient Greek modes as codified by Ptolemy; it was the aesthetic ideal of the Renaissance theorists; and it is the tuning practice of a great many musical cultures worldwide, both ancient and modern."[1] Just intonation can be contrasted and compared with equal temperament, which dominates Western instruments of fixed pitch and default MIDI tuning. In equal temperament, all notes are defined as multiples of the same basic interval. Two notes separated by the same number of steps always have exactly the same frequency ratio. However, except for doubled frequencies (octaves), no other intervals are exact ratios of integers. Each just interval differs a different amount from its analogous, equally tempered interval. Justly tuned intervals can be written as either ratios, with a colon (for example, 3:2), or as fractions, with a solidus (3 ? 2). For example, two tones, one at 300 Hertz (cycles per second), and the other at 200 hertz are both multiples of 100 Hz and as such members of the harmonic series built on 100 Hz. Recorded history Pythagorean tuning, perhaps the first tuning system to be theorized in the West,[2] is a system in which all tones can be found using powers of the ratio 3:2, an interval known as a perfect fifth. It is easier to think of this system as a cycle of fifths. Because a series of 12 fifths with ratio 3:2 does not reach the same pitch class it began with, this system uses a wolf fifth at the end of the cycle, to obtain its closure. Quarter-comma meantone obtained a more consonant tuning of the major and minor thirds, but when limited to twelve keys (see split keys), the system does not close, leaving a very dissonant diminished sixth between the first and last tones of the cycle of fifths. In Pythagorean tuning, the only highly consonant intervals were the perfect fifth and its inversion, the perfect fourth. The Pythagorean major third (81:64) and minor third (32:27) were dissonant, and this prevented musicians from using triads and chords, forcing them for centuries to write m

sic with relatively simple texture. In late Middle Ages, musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made consonant. For instance, if you decrease by a syntonic comma (81:80) the frequency of E, C-E (a major third), and E-G (a minor third) become just. Namely, C-E is flattened to a justly intonated ratio of and at the same time E-G is sharpened to the just ratio of The drawback is that the fifths A-E and E-B, by flattening E, become almost as dissonant as the Pythagorean wolf fifth. But the fifth C-G stays consonant, since only E has been flattened (C-E * E-G = 5/4 * 6/5 = 3/2), and can be used together with C-E to produce a C-major triad (C-E-G). By generalizing this simple rationale, Gioseffo Zarlino, in the late sixteenth century, created the first justly intonated 7-tone (diatonic) scale, which contained pure perfect fifths (3:2), pure major thirds, and pure minor thirds: F > A > C > E > G > B > D This is a sequence of just major thirds (M3, ratio 5:4) and just minor thirds (m3, ratio 6:5), starting from F: F + M3 + m3 + M3 + m3 + M3 + m3 Since M3 + m3 = P5 (perfect fifth), i.e. 5/4 * 6/5 = 3/2, this is exactly equivalent to the diatonic scale obtained in 5-limit just intonation. The Guqin has a musical scale based on harmonic overtone positions. The dots on its soundboard indicate the harmonic positions: 1/8, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 7/8.[3] [edit]Modern practice Primary forms of the just tone row from Ben Johnston's String Quartet No. 7, mov. 2[4] Play (help·info) and Play hexachords (help·info). Each permutation contains a just chromatic scale, however, transformations (transposition and inversion) produce pitches outside of the primary row form, as already occurs in the inversion of P0. Today, despite the dominance of repertoire composed under equal-tempered systems and the prominence of the piano in musical training, musicians often approach just intonation either by accident or design because it is much easier to find (and hear) a point of stability than a point of calculated instability. A cappella groups that depend on close harmonies, such as barbershop quartets, usually use just intonation by design. Bagpipes, tuned correctly, also use just intonation. There are several conventionally used instruments which, while not associated specifically with just intonation, can handle it quite well, including the trombone and the violin family of instruments.