At its simplest, a harmonic series is just a sequence of consecutive integers, beginning with one – 1, 2, 3, 4, and so forth. In the context of sound, those integers are multiplication factors, applied to some fundamental frequency. If we use 'x' to represent that fundamental frequency, then its harmonic series is 1x (the fundamental itself), 2x, 3x, 4x, and so forth. These are also called the first harmonic (again, the fundamental itself), the second harmonic, the third harmonic, the fourth harmonic, ..., and are collectively called the members of the harmonic series defined by a particular fundamental. Different fundamentals, different harmonic series. The position of a harmonic in such a series is its harmonic number.
If someone were to ask you how you would determine the difference between two members of a harmonic series, your first instinct might be to subtract the smaller from the larger. Again using 'x' to represent the frequency of the fundamental, you might be tempted to say that the difference between the fifth harmonic (5x) and the third harmonic (3x) is 2x, and while this isn't wrong exactly (the frequency of the interference pattern between them, or 'beat frequency' would in fact be 2x), it's also not the best answer for how these two harmonics sound together, simultaneously or sequentially, as musical notes.
A better answer is that the perceived relationship or 'interval' between these two harmonics is the ratio between their harmonic numbers, in this example 5:3. The interval between any two members of a harmonic series can be expressed in this way, as a ratio of the integers which mark the positions of the specific harmonics in the series, and, in this, all harmonic series are the same, because the interval between a pair of harmonics in one series will sound very much like the interval between the same pair of harmonics in another series with a different fundamental. A 5:3 interval is a 5:3 interval, whether it's 50:30 Hz or 500:300 Hz; in each case the size of the step from one to the other sounds much the same.
I have fallen into the habit of representing these ratios in terms of the powers of prime numbers, for example representing 9:8 as (3^2):(2^3), where '^' is an exponentiation operator. For most purposes this is a matter of hunch on my part. Perhaps others can make an argument for the musical relevance of prime factors; I cannot, except in one particular use, which I'll get to in a later installment.
Next up, how harmonic series can be related and combined into structures.
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