music

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.

Scale of just intonation in A, heard melodically.


Scale of just intonation in A, heard as intervals.