The Octave Divider utility makes this process easier. Simply enter the number of equal steps in the octave, and the utility will
return a table containing the ratios for each scale degree.
The most common tuning system in Western music, Equal Temperament, divides the octave into 12 equal steps. That is, the octave is divided into 12 intervals
with equal ratios of a frequency. For instance, the distance between C and G, a perfect 5th interval (P5), is the same ratio as the ratio between any other set of notes
that create a P5 such as A and E. In equal temperament, the ratio of the P5 is 1:1.498. Therefore, if one knows that the frequency of A is 440hz, simply
multiplying 440 x 1.1498 will yield the note a P5 above A (in this case E which has a frequency of 659.12 hz).
One of the joys of computer music is the ability to explore and create different tuning systems; however, many times, finding these ratios is somewhat cumbersome. Doing so generally requires, in the case of equal temperament, looking it up on a website or in a book. Csound is one application that allows users to control the frequency of a given sample, or to synthesize tuning systems. The example below transposes a sound file ("mid_f.aif") 4 times, one semitone at a time using traditional equal temperament.
sr = 44100
kr = 4410
ksmps = 10
inote = p4
asignal diskin "mid_f.aif", inote, 0
; ratios i1 0 5 1.059 -minor 2nd above base fq (scale degree1) i1 1 5 1.122 -major 2nd above base fq (scale degree2) i1 2 5 1.189 -minor 3rd above base fq (scale degree3) i1 3 5 1.260 -major 3rd above base fq (scale degree4)
But what if it was desired to divide the octave in more or less than 12 equal subdivisions? First, you must find the ratio for each interval, but since
traditional tuning is not being used, terms like "minor 2nd", "perfect 5th," and other traditional interval names become useless. Therefore, it may be best to
think in terms of the ratios between the base frequency and the scale degrees of the octave.
When dividing the octave into more or less than 12 equal subdivisions, a scientific calculator is required, and moreover each degree in the scale must be computed individually. For example, dividing the octave into 16 steps requires one to enter the equation: ratio = 2 X / 16 where X is the scale degree number. To find all 16 ratios, this must be done 16 times! Imagine if you wanted to divide the octave into 36, 47, or even 1500 equal steps!
Note that the implementation of ratios is exactly the same as that in equal temperament. For instance, if the octave is divided into 16 equal steps, and the desire is to find the frequency of the 13th scale degree above the base frequency, simply mulitplying that ratio by the base frequency will yield the frequency of the 13th scale degree (base fq * ratio = scale degree fq). For an octave divided into 16 equal subdivisions, the 13 scale degree above the base frequency of 440hz is 772.728 hz (440 * 1.7562 = 772.728).
Below is a very simple example of a scale with 16 equal subdivisions that uses a piano sample as the source sound (note there are 17 notes since the octave is included):