Radical interval

A radical interval is an interval whose ratio can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to just intervals which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in equal tunings such as edos, and also occur in eigenmonzo tunings of regular temperaments. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as 2^1/2 × 3^-1/13). Because of this, radical intervals can be expressed as monzos, like just intervals.

Fractional monzos

For the sake of clarity, monzos representing radical intervals are called fractional monzos. Mathematically, fractional monzos behave the same as ordinary monzos, except that elements have been extended to allow them to be rational numbers. If [e₂ e₃ … e_p⟩ is a fractional monzo, then it represents 2^e₂ 3^e₃ … p^e_p just as with an ordinary monzo. Hence, for instance, [1/13 -1/13 7/26⟩ represents the interval 2^1/13 3^-1/13 5^7/26. By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an n-th root of a positive rational number; for instance from our example, (312500/9)^1/26, which may also be written as 1\26ed312500/9.

By multiplying each monzo entry by the cent value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the just tuning map in cents 1200⋅⟨log₂(2) log₂(3) … log₂(p)]) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example (1/13)⋅1200 - (1/13)⋅1200⋅log₂(3) + (7/26)⋅1200⋅log₂(5) = 696.1648 cents.

Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one n-th root of a positive rational number which corresponds to it.

Tunings in terms of radical intervals

Any number that can be expressed as a root corresponds to a radical interval, so radical intervals can be used to express the degrees of equal tunings. For example, 12edo's fifth can be expressed as [7/12⟩ or 2^7/12, and the Bohlen–Pierce supermajor third may be expressed as [0 3/13⟩ or 3^3/13.

What this additionally unlocks is the ability to stack intervals from multiple edo systems. For example, one could define intervals in the 12edo.13edt subgroup by specifying their monzo, for example a subminor third of about 261 cents can be generated by [7/12 -3/13⟩. This also introduces the potential for dividing intervals outside of pure edo systems: one method of building scales can be to divide just intervals into portions. This is similar to temperaments like slendric, and is identical to defining an eigenmonzo or rational comma-fraction tuning of these temperaments, except that while those temperaments are sometimes understood through a 2-step process of (1) equally dividing a just interval and (2) assigning the divisions to another just interval, radical intervals provide a framework for skipping the second step (if you deem it unnecessary). In fact, the structure of slendric can be described as equating [3 0 0 -1⟩ and [-1/3 1/3⟩.

Radical subgroups

A radical subgroup may be notated in the same manner as a normal subgroup, except where the elements are names of equal tunings. For example, quarter-comma meantone intervals can be considered to be radical intervals in the 2.4ed5 subgroup.

Fractional monzos in projection matrices

Main article: Projection matrices