Summary: For high school and above, a discussion of the relationship of musical intervals and frequency ratios, with examples and exercises.
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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.)
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?
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.
The discussion here will focus on the relationship between ratio, frequency, and musical intervals. The interval between two pitches depends on the ratio of their frequencies. There are simple, ideal ratios as expressed in a harmonic series, and then there is the more complex reality of equal temperament, in which the frequency ratios are not so simple and are best written as roots or decimals. Here is one more exercise before we go on to discussions of music.
The kind of sound waves that music is made of are a lot like the adult and child walking along steadily in the example above. Low notes have long wavelengths, like the long stride of an adult. Their frequencies, like the frequency of the adult's steps, are low. High notes have shorter wavelengths, like the small stride of the child. Their frequencies, like the frequency of a child's steps, are higher. (See Sound, Physics and Music for more on this.)
It is easy to spot simple frequency relationships, like 2:1, but what about more complicated ratios? Remember that you are saying the ratio of one frequency to another IS (equals) another ratio(or fraction or decimal). This idea can be written as a simple mathematical expression. With enough information and a little bit of algebra, you can solve this equation for any number that you don't have.
If you remember enough algebra, you'll notice that the units for frequency in this equation must be the same: if frequency #1 is in hertz, frequency #2 must be in hertz also. In all the examples and problems below, I am going to assume all frequencies are in hertz (waves per second), but you can use any frequency unit as long as they are both the same. Most musicians don't talk about frequency much, and when they do, they rarely mention units, but just say, for example, "A 440".
Say you would like to compare the frequencies of two sounds. Sound #1 is 630 and sound #2 is 840. If you use the expression given above and do the division on a calculator, the answer will be a decimal. If you simply reduce the fraction to lowest terms, or if you know the fraction that these decimals represent, you can see that you have a simple ratio of 3:4. Notice that if you switch the frequencies in the expression, the ratio also switches from 3:4 to 4:3. So it doesn't really matter which frequency you put on top; you will get the right answer as long as you keep track of which frequency is which.
Perhaps you would like to find the frequency of a note that is a perfect fifth higher or lower than another note. A quick look at the harmonic series here shows you that the ratio of frequencies of a perfect fifth is 3:2. The higher number in the ratio will be the higher-sounding note. So if you want the frequency of the note that is a perfect fifth higher than A 440, you use the ratio 3:2 (that is, the fraction 3/2). If you want the note that is a perfect fifth lower than A 440, you use the ratio 2:3 (the fraction 2/3).
In this example, I have done the algebra for you to show that you are really using the same equation as in example 1, just rearranged a bit. If you are uncomfortable using algebra, use the red expression if you know the interval but don't know one of the frequencies.
Pure intervals that are found in the physical world (such as on strings or in brass tubes)are nice simple ratios like 2:3. But the ratios for notes in equal temperament are based on the twelfth root of two.
Say you would like to compare a pure major third from the harmonic series to a equal temperament major third.
By comparing the ratios as decimal numbers, you can see that a pure major third is quite a bit smaller than an equal temperament major third.
A note has frequency 220. Using the pure intervals of the harmonic series, what is the frequency of the note that is a perfect fourth higher? What is the frequency of the note that is a major third lower?
The frequency of one note is 1333. The frequency of another note is 1121. What equal temperament interval will these two notes sound like? (Hint: compare the frequencies, and then compare your answer to the frequencies in the equal temperament figure above.)