- we might now rationalize the new 12-step scale by making
the frequency difference the same size (in log(2) throughout: 2
to the 1/12
| step |
frequency |
| 1 |
base_frequency * 2 to the 12/12th (2/1) |
| 12 |
base_frequency * 2 to the 11/12th |
| 11 |
base_frequency * 2 to the 10/12th |
| 10 |
base_frequency * 2 to the 9/12th |
| 9 |
base_frequency * 2 to the 8/12th |
| 8 |
base_frequency * 2 to the 7/12th |
| 7 |
base_frequency * 2 to the 6/12th |
| 6 |
base_frequency * 2 to the 5/12th |
| 5 |
base_frequency * 2 to the 4/12th |
| 4 |
base_frequency * 2 to the 3/12th |
| 3 |
base_frequency * 2 to the 2/12th |
| 2 |
base_frequency * 2 to the 1/12th |
| 1 |
base_frequency * 2 to the 0/12th (1/1) |
- here's the natural and rational scales side-by-side, using
100 Hz as the base frequency:
| natural scale |
rational scale |
| step |
frequency |
frequency |
step |
| 1 |
200.000 |
200.000 |
1 |
| 7 |
187.500 |
188.775 |
12 |
|
|
178.180 |
11 |
| 6 |
166.667 |
168.179 |
10 |
|
|
158.740 |
9 |
| 5 |
150.000 |
149.831 |
8 |
|
|
141.421 |
7 |
| 4 |
133.333 |
133.484 |
6 |
| 3 |
125.000 |
125.992 |
5 |
|
|
118.921 |
4 |
| 2 |
112.500 |
112.246 |
3 |
|
|
105.946 |
2 |
| 1 |
100.000 |
100.000 |
1 |
- the pitch of 12-step scale step 3 is no longer the same as
the pitch of 7-step scale 2, but it's very close
- the same is true for the other steps in the natural scale
and the steps in the rational scale they're close to
- we could create an instrument using the rational scale
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12...
If we now select steps from the 12-step scale, using the gap
pattern we used above (2, 2, 1, 2, 2, 2, 1), we can play music
that sounds a lot like the music we played on the instrument we
made that was tuned to the natural scale.
| 1 |
|
2 |
|
3 |
|
4 |
|
5 |
|
6 |
|
7 |
|
8 |
|
9 |
|
10 |
|
11 |
|
12 |
|
1 |
|
|
|
12-step rational scale |
| 1 |
|
|
|
2 |
|
|
|
3 |
|
4 |
|
|
|
5 |
|
|
|
6 |
|
|
|
7 |
|
1 |
|
|
|
7-step subset of the 12-step scale |
|
|
2 |
|
|
|
2 |
|
|
1 |
|
|
2 |
|
|
|
2 |
|
|
|
2 |
|
|
1 |
|
|
|
|
gaps |
This example begins the 7-step subset on step 1 of the 12-step
scale. But since the gaps in the rational scale are all the same,
we can now start the 7-step subset on any of the steps in the
12-step scale. For example, here's what starting it on 12-step
scale step 3 looks like:
| 1 |
|
2 |
|
3 |
|
4 |
|
5 |
|
6 |
|
7 |
|
8 |
|
9 |
|
10 |
|
11 |
|
12 |
|
1 |
|
2 |
|
3 |
|
|
|
12-step rational scale |
|
|
|
|
1 |
|
|
|
2 |
|
|
|
3 |
|
4 |
|
|
|
5 |
|
|
|
6 |
|
|
|
7 |
|
1 |
|
|
|
7-step subset of the 12-step scale |
|
|
|
|
|
|
2 |
|
|
|
2 |
|
|
1 |
|
|
2 |
|
|
|
2 |
|
|
|
2 |
|
|
1 |
|
|
|
|
gaps |