For high school and above, a discussion of the relationship of musical intervals and frequency ratios, with examples and exercises.
In order to really understand tuning, the harmonic series, intervals, and harmonic relationships, it is very useful to understand a little bit about the physics of sound and to be comfortable discussing ratios, fractions, and decimals. This lesson is a short review of some basic math concepts for students who want to understand some of the math and physics principles that underlie music theory.
Ratios, fractions, and decimals are basically three different ways of saying the same thing. (So are percents, but they don't have anything to do with music.)
If you have two apples and three oranges, that's five pieces of fruit altogether. you can say:
- The ratio of apples to oranges is 2:3, or the ratio of oranges to apples is 3:2.
- The ratio of apples to total fruit is 2:5, or the ratio of oranges to total fruit is 3:5.
- 2/5 of the fruit are apples, and 3/5 of the fruit are oranges.
- There are 2/3 as many apples as oranges, and 1 and 1/2 times (or 3/2) as many oranges as apples.
- There are 1.5 times as many oranges as apples, or there are only .67 times as many apples as oranges.
- 0.4 (Four tenths) of the fruit is apples, and 0.6 (six tenths) of the fruit is oranges.
You should be able to see where the numbers for the ratios and fractions are coming from. If you don't understand where the decimal numbers are coming from, remember that a fraction can be understood as a quick way of writing a division problem. To get the decimal that equals a fraction, divide the numerator by the denominator.
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An adult is walking with a child. for every step the adult takes, the child has to take two steps to keep up. this can be expressed as:
- The ratio of adult to child steps is 1:2, or the ratio of child to adult steps is 2:1.
- The adult takes half as many (1/2) steps as the child, or the child takes twice as many (2/1) steps as the adult.
- The adult takes 0.5 as many steps as the child, or the child takes 2.0 times as many steps as the adult.
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The factory sends shirts to the store in packages of 10. Each package has 3 small, 3 medium, and 4 large shirts. How many different ratios, fractions, and decimals can you write to describe this situation?
- Ratio of small to medium is 3:3. Like fractions, ratios can be reduced to lowest terms, so ratio of 1:1 is also correct.
- Ratio of small to large, or medium to large, is 3:4; ratio of large to either of the others is 4:3.
- Ratio of small or medium to total is 3:10; ratio of large to total is 4:10.
- 3/10, or 0.3, of the shirts, are small; 3/10, or 0.3 of the shirts are medium, and 4/10, or 0.4 of the shirts, are large.
- There are 3/4 as many small or medium shirts as there are large shirts, and there are 4/3 as many large shirts as small or medium shirts.
- If you made more ratios, fractions, and decimals by combining various groups (say ratio of small and medium to large is 6:4, and so on), give yourself extra credit.
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What has all this got to do with music? Quite a bit, as a matter of fact. For example, every note in standard music notation is a fraction of a beat, and every beat is a fraction of a measure. You can explore the relationship between fractions and rhythm in
Fractions, Multiples, Beats, and Measures ,
Duration and
Time Signature .