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Just Intonation, Part 5: The Arithmetic

Before we actually start constructing a scale, a quick review of fractions would be in order. If you're already a whiz at fractions, freel free to skip this section.

While I sometimes write tuning ratios with colons rather than fractions -- 9:8, for instance, rather than 9/8 -- for our purposes the two types of notation are interchangeable. Another technical point: When discussing just intonation, we keep the entire ratio in fractional form. You'll never see 3/2 written as 1-1/2, for instance.

Now let's look at how to handle fractions. Rule 1 is, to add two intervals, you need to multiply their ratios (fractions). To multiply two fractions, you multiply both the numerators (the upper numbers) and the denominators (the lower numbers), like this:

9/8 x 3/2 = 27/16 (because 9 x 3 = 27 and 8 x 2 = 16)

Rule 2 is, to subtract one interval from another, you divide the fractions. To divide one fraction by another, you flip the divisor on its head and multiply, like this:

9/8 divided by 3/2 is the same as 9/8 x 2/3, or 18/24

Rule 3 is, when the numerator and denominator of a fraction have a common factor, you can divide both of them by this factor to reduce the fraction to its simplest terms. In the case of 18/24, for instance, both the numerator and denominator are divisible by 6. So this fraction can be written more simply as 3/4.

Rule 4 is, if you want to compare two fractions, you need to give them the same denominator. For instance, let's suppose we want to know whether 7/4 or 5/3 is larger. It's not always easy to answer such a question just by looking at the raw numbers, but a little arithmetic saves the day. Here's how to get the answer:

7/4 x 3/3 = 21/12
5/3 x 4/4 = 20/12

In each case, we've multiplied by 1 (3/3 = 1, and so does 4/4), so the value of the fraction hasn't changed. And now, because we're comparing two fractions with the same denominator, it's easy to see that 7/4 is larger than 5/3. How much larger? The difference, which we get by subtracting, is 1/12, but that's misleading. What we're interested in, for tuning purposes, is the ratio between the two, which is obtained by dividing. 21/12 divided by 20/12 is the same as 21/12 x 12/20, so the ratio of 7/4 to 5/3 is 21/20.

As noted elsewhere in this tutorial, the ratio of the interval we call the octave is 2:1. This fact allows us to manipulate tuning ratios easily: We can shift a note up an octave by multiplying it by 2, or shift it down an octave by dividing by 2.

For instance, let's assume the frequency ratio between two notes (perhaps a B-flat and the C below it) is 7/4. If we shift the B-flat down an octave, they'll have a ratio of 7/8 (because 7/4 x 1/2 = 7/8). If we shift the C down an octave or the B-flat up an octave, the interval will be 7/2 (because 7/4 x 2/1 = 14/4, and 14/4 reduces to 7/2 when we reduce it to its simplest form).

For convenience, interval ratios are usually stated in a form greater than 1/1 and less than 2/1. The 7/8 ratio just mentioned, for example, is the ratio of a just B-flat to the C above it. To compare this rather large whole-step with other whole-steps that we may create, we need to find the absolute size of the interval. This is done by inverting the fraction: The interval from the 7/8 B-flat to the C above it is 8/7.