|
|
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.