Fractional monzo

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A fractional monzo or fmonzo is like an ordinary monzo except that coefficients 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. By taking a dot product with ⟨cents (2) cents (3) … cents (p)] (or in layman's terms, multiplying each monzo entry by the cent value of the corresponding prime) the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (1/13)×1200.0 - (1/13)×cents (3) + (7/26)×cents (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 nth root of a positive rational number which corresponds to it.

Tunings in terms of fractional monzos

Fractional monzos can be used to notate any number that can be expressed as a root, so they can be used to express the degrees of equal tunings. For example, 12edo's fifth can be expressed as [7/12⟩, and the Bohlen-Pierce supermajor third may be expressed as [0 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 follow a 2-step process of 1) equally dividing a just interval and 2) assigning the divisions to another just interval, fractional monzos provide a framework for skipping the second step (if you deem it unnecessary). In fact, slendric can be described as equating [3 0 0 -1⟩ and [-1/3 1/3⟩.

Use in projection matrices

Main article: Projection matrices