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The minor intervals and the frequency ratios their correspond to are the second (9:8), minor third (6:5),
fourth (4:3), fifth (3:2), minor sixth (8:5), minor seventh (9:5), and octave (2:1).
The minor scale is sometimes also called the "natural" minor scale. The minor scale is constructed from
minor triads in an analogous manner to the construction of the major scale by major
triads. A minor triad is a group of three notes played together with the ratios of frequency
10:12:15. Within an octave, three such minor triads can be constructed as follows:
| C |
D |
E |
F |
G |
A |
B |
C' |
D' |
| 10 |
... |
12 |
... |
15 |
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| |
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|
10 |
... |
12 |
... |
15 |
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| |
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|
10 |
... |
12 |
... |
15 |
Therefore, the minor scale, like the major scale, also has just intonation. The minor scale was also used by the
Greeks, and was called the aeolian mode. Note that the same intervals are present in the minor scale as in the
major scale, although the order is different. However, the notes of the two do not correspond in frequency (E, A, and B
are different). That is, notes defined according to 1:1, 9:8, 5:4, 4:3, 3:2, 5:3, 15:8, and 2:1 (major scale) and 1:1,
9:8, 6:5, 4:3, 3:2, 8:5, 9:5, and 2:1 (minor scale) differ in the third, sixth, and seventh note. The problem becomes
more obvious with a numerical example. Defining C as 264 Hz, the major and minor scales can be seen to require different
frequencies for the three notes mentioned above:
| Key |
C |
D |
E |
F |
G |
A |
B |
C' |
D' |
| Major |
264 |
297 |
330 |
352 |
396 |
440 |
495 |
528 |
594 |
| Minor |
264 |
297 |
316.8 |
352 |
396 |
422.4 |
475.4 |
528 |
594 |
This difficulty can be ameliorated in a combined scale by adding additional notes for the minor scale frequencies which
do not correspond to major scale frequencies. Notes must be added between major notes D and E (D or E ), G and A (G or A ), and A and B (A or B ), for a total of 10 notes per octave: C (1:1), D (9:8), D (6:5), E (5:4), F (4:3), G (3:2), G (8:5), A (5:3), A (9:5), and B (15:8). However, the
only way to resolve these difficulties completely is to use equal temperament.
Harmonic Minor Scale, Melodic Minor Scale
© 1995-2005 Eric W. Weisstein
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