Just intonation

Just intonation is any musical tuning in which the frequencies of notes are related by whole number ratios. Any interval tuned in this way is called a just interval; in other words, the two notes are members of the same harmonic series. Although in theory two notes tuned in the frequency ratio 1024:927 might be said to be justly tuned, in practice only ratios using quite small numbers tend to be called just. Intervals used are then capable of being more consonant.

The diatonic scale in just intonation

It is possible to tune the familiar diatonic scale or chromatic scale in just intonation, in many ways, all of which make certain chords purely tuned and as consonant as possible, and others considerably more dissonant and indeed seeming out-of-tune to modern ears (see below for more on this).

Related Topics:
Diatonic scale - Chromatic scale - Below

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The prominent notes of a given scale are tuned so that the ratios of their frequencies are comprised of relatively small integers. For example, in the key of G major, the ratio of the frequencies of the notes G:D (perfect fifth) is 2:3, while that of G:C (perfect fourth) is 3:4.

Related Topics:
Major - Perfect fifth - Perfect fourth

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All ratios that involve the prime numbers of 2, 3 and 5

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can be built out of the following 3 basic intervals:

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• s=16:15 Semitone
• t=10:9 Minor tone
• T=9:8 Major tone

from which we get:

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• 6:5 = Ts (minor third)
• 5:4 = Tt (major third)
• 4:3 = Tts (perfect fourth)
• 3:2 = TTts (perfect fifth)
• 2:1 = TTTttss (octave)

It gives rise to the following scale in the key of G (this is only one possibility):

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G A B C D E F# G

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T t s T t T s

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with ratios w.r.t. G of

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A 9/8, B 5/4, C 4/3, D 3/2

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E 5/3, F# 15/8 and G 2/1

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