Harmonic
A harmonic is a whole-number multiple of the fundamental frequency of a sound. It is an element of the harmonic series.
The timbre of a periodic sound, such as a bowed violin or the human voice, contains a nearly infinite amount of harmonic partials, starting with 1f, 2f, 3f, 4f, … where f is the fundamental frequency. Each of these harmonics has a distinct amplitude, generally decreasing as the 'height' of the harmonic increases. The span between any two of these harmonics is a just interval. If the harmonics are numbered such that the fundamental is number 1, the octave is 2, etc., then the interval's ratio is given by the two numbers. For example the interval between the 3rd and 4th harmonics is 4/3.
The ancient Greeks called these harmonics "multiples", and considered them to be a unique interval class separate from superparticular and superpartient intervals.
A subharmonic is a unit fraction of the fundamental frequency of a sound: f/1, f/2, f/3, f/4, …. It is an element of the subharmonic series.
For individual articles on each harmonic, see Category: Harmonics.
Odd harmonic
An odd harmonic is a harmonic where the frequency ratio is an odd number. The first few odd harmonics are 1/1, 3/1, 5/1, 7/1, 9/1, 11/1, etc. Odd harmonics are significant in that they create distinct pitch classes, since any even harmonic is a whole number of octaves above an odd harmonic.
An odd limit is the set of all just intervals where the largest odd factor in the numerator and denominator both do not exceed a specified bound. For example, the 5-odd-limit consists of all ratios where the only allowable odd factors are 1, 3, and 5; those being 1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, and any whole number of octaves above those intervals.
We can also restrict the set of usable intervals to only allow odd harmonics. Such intervals include 3/1, 5/3, 9/7, 27/25, etc. Here, using 3/1 as the interval of equivalence (a "tritave") is natural, since the 3rd harmonic is the smallest odd harmonic after the fundamental. Edts divide the tritave into a whole number of equal steps, analogous to how edos divide the octave.
Prime harmonic
A prime interval or prime harmonic is a harmonic which as a ratio of frequencies is a prime number; that is, a number such as 2, 3, 5, 7, 11, … which is divisible only by itself and 1. It is an element of the prime harmonic series.
Any interval of just intonation (JI) can be expressed in terms of a product of prime intervals, allowing us to decompose a complex JI interval into simpler parts. A prime interval itself cannot be expressed by other prime intervals, so no prime intervals are redundant for reconstructing the entirety of JI. For those reasons and for the fact that prime intervals occur in harmonic series, they form a very important basis (literally and mathematically) for JI.
For example, the octave is a prime interval whereas the intervals 5/3 or even 1/1 are not. In traditional ratio notation, the prime intervals are 2/1, 3/1, 5/1, 7/1, 11/1 etc.
The monzo notation of each prime interval consists of all-zeros except for a single entry equal to 1: (2: [1⟩, 3: [0 1⟩, 5: [0 0 1⟩, 7: [0 0 0 1⟩, 11: [0 0 0 0 1⟩, …)
Prime harmonics are often used to define harmonic limits, also known as prime limits.
Highly composite harmonic
A highly composite interval or highly composite harmonic is a harmonic which has a ratio of frequencies is a highly composite number; that is, a number such as 1, 2, 4, 6, 12, … each of which has more divisors than the number before it.
Highly composite intervals are significant for having the largest number of ways to be approached through harmonic chords for its size.
Highly composite intervals can be thought of as the opposite of prime harmonics, since highly composite numbers are "antiprime", but they are not complementary sets, as they share a common element (2/1) and there are infinitely many harmonics which are neither prime nor highly composite.