Monzos and interval space
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This is an expert page. It is written to allow experienced readers to learn more about the advanced elements of the topic. The corresponding beginner page for this topic is Monzo. |
This page gives the formal mathematical definition of a monzo and shows its relation to interval space.
Definition
A p-limit rational number q can by definition be factored into primes of size less than or equal to p, giving
[math]q = 2^{e_2} \, 3^{e_3} \, 5^{e_5} \dotso p^{e_p}[/math]
where the exponents are integers (positive, negative, or zero.) This is often written in ket vector (→ Wikipedia: Bra-ket notation) notation as
[math]|e_2 \, e_3 \, e_5 \dotso e_p\rangle[/math]
in which case it is called a monzo, where the name refers to the enthusiastic advocacy of Joe Monzo.
The Tenney height of this monzo is given by
[math]\| |e_2 \, e_3 \dotso e_p \rangle \| = |e_2| + |e_3| \log_2 3 + \dotsb + |e_p| \log_2 p[/math]
which is a vector space norm; hence we may embed the p-limit monzos into a normed vector I space of dimension n = π (p) via a map M:monzos ⟶ I. The monzos under this embedding now define a lattice, which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2 (k), then the norm becomes the standard L1 norm. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while [1 0⟩ represents 2, so does [0 log3 (2)⟩.
Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have Tenney-Euclidean interval space instead of Tenney interval space. Explicitly, if we take the monzo [e2 e3 … ep⟩ then the Tenney-Euclidean norm, or TE norm, of it is
[math]\sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2}[/math]
and if the coordinates are the weighted interval space coordinates, then the TE norm is the standard Euclidean, or L2, norm.
Alternate definition
Given a rational number q, we can rewrite it in monzo form by the following definition:
[math]q = |v_2 (q) \,v_3 (q) \, v_5 (q) \dotso v_p (q)\rangle[/math]
The Tenney height of this monzo is given by
[math]\| |v_2 (q) \, v_3 (q) \dotso v_p (q) \rangle \| = |v_2 (q)| + |v_3 (q)| \log_2 3 + \dotsb + |v_p (q)| \log_2 p[/math]
Where vp (q) is the p-adic valuation of q.
Example
The 5-limit interval 16/15 factors as 24 3-1 5-1, so it has a monzo representation of [4 -1 -1⟩. In weighted coordinates, that becomes [4 -log2 (3) -log2 (5)⟩, approximately [4 -1.585 -2.322⟩.
The TE norm is therefore
[math]\sqrt{(4^2 + \log_2(3)^2 + \log_2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889. [/math]