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Sound waves in the real world of musical instruments often do have simple ratios like these. (See Standing Waves and Musical Instruments for more about this.) In fact, a vibrating string or a tube of vibrating air will generate a whole series of waves, called a harmonic series , that have fairly simple ratios. Musicians describe sounds in terms of pitch rather than frequency and call the distance between two pitches (how far apart their frequencies are) the interval between the pitches. The simple-ratio intervals between the harmonic-series notes are called pure intervals . (The specific names of the intervals, such as "perfect fifth" are based on music notation and traditions rather than physics. If you need to understand interval names, please see Interval .)
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.
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 musicians in Western musical genres typically do not use pure intervals; instead they use a tuning system called equal temperament . (If you would like to know more about how and why this choice was made, please read Tuning Systems .) In equal temperament, the ratios for notes in equal temperament are based on the twelfth root of two. (For more discussion and practice with roots and equal temperament, please see Powers, Roots, and Equal Temperament .) This evens out the intervals between the notes so that scales are more uniform, but it makes the math less simple.
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. )
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