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JUST TUNING or JUST INTONATION

A TUNING of a SCALE in just intonation involves the usage of FREQUENCY ratios based on integer proportions as found in the HARMONIC SERIES, instead of, for instance, a division of the OCTAVE into exactly equal parts (as in the case of EQUAL TEMPERAMENT).

The two principal scales of just intonation are the major and minor, which have frequency ratios as follows:

major scale
1
9/8
5/4
4/3
3/2
5/3
15/8
2/1
minor scale
1
9/8
6/5
4/3
3/2
8/5
9/5
2/1

The tuning of this system results in an absence of BEATS, but differences in the size of the INTERVALs between adjacent notes in the scale, i.e. the whole tone and semintone. The result is a limited transposability of a scale (which is not a limitation with equal temperament). The following chart shows the various intervals produced between pairs of notes in either scale, as expressed in ratios and cents.

See also: Appendix C. Compare: INTONATION, PYTHAGOREAN SCALE, TEMPERED TUNING.

Interval
Frequency ratio from starting point
Cents from starting point

Unison
1:1
0

Semitone
16:15
111.731

Minor tone
10:9
182.404

Major tone
9:8
203.910

Minor third
6:5
315.641

Major third
5:4
386.314

Perfect fourth
4:3
498.045

Augmented fourth
45:32
590.224

Diminished fifth
64:45
609.777

Perfect fifth
3:2
701.955

Minor sixth
8:5
813.687

Major sixth
5:3
884.359

Harmonic minor seventh
7:4
968.826

Grave minor seventh
16:9
996.091

Minor seventh
9:5
1,017.597

Major seventh
15:8
1,088.269

Octave
2:1
1,200.000
Scale of Just Intonation.

Sound Example: Scale of just intonation in A, heard melodically.

Sound Example: Scale of just intonation in A, heard as intervals.
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