User:Inthar/Style guide

This page documents my xen math notation and its differences from conventional xen notation or conventional math notation.

Variables

• Capital italicized Latin letters may denote scales written cumulatively: i.e. with S(0) = 0 cents and S(i + p) = E + S(i) (p = length, E = equave) for every i.
  • S(n) = 100n cents
• Lowercase italicized Latin letters may denote (rotational equivalence classes of) scales written as steps, or abstract scale words. For example:
  • s(a, b, c) = abacaba
  • [math]\displaystyle{ \sum_{n=a}^{b-1}s(n) = S(b)-S(a) \ \text{if} \ s(n) := S(n+1)-S(n) }[/math]
• Bolded variables denote interval sizes (especially letters of scale words) and elements of lattices.
  • 5L 2s
• Sans serif function names are scale constructions, or more generally functions named more verbosely than is typical for conventional math notation.
  • [math]\displaystyle{ \mathsf{MOS}(5,2;6)(\mathbf{L}, \mathbf{s}) = \mathbf{LLLsLLs} }[/math]
  • Blackdye is [math]\displaystyle{ \mathsf{Flought}(\mathrm{Pyth}[5]; 10/9) }[/math]

Discrete sets

• For [math]\displaystyle{ k \in \mathbb{R} }[/math] and [math]\displaystyle{ n\in \mathbb{Z}_{\gt 0}, }[/math] [math]\displaystyle{ [n]_k }[/math] denotes [math]\displaystyle{ \{k, k+1, ..., k+n-1\}. }[/math] I may also use [math]\displaystyle{ [i:j] }[/math] for [math]\displaystyle{ [j-i]_i. }[/math] For n = 0, [0]_k is the empty set.

Words

• Zero-indexing is used for indices.
• A (linear) word is a function [math]\displaystyle{ w : [n]_0 \to \mathcal{A} }[/math] where [math]\displaystyle{ \mathcal{A} }[/math] is a set of letters and [math]\displaystyle{ n \in \mathbb{Z}_{\ge 0}. }[/math] n is called the length of w. The letter of w at index i is denoted w[i]. If 0 ≤ i < j ≤ |w| − 1, the slice notation w[i:j] denotes the (j − i)-letter word w[i]w[i+1]...w[j−1].
• A based circular word is a function [math]\displaystyle{ s: \mathbb{Z}/n \to \mathcal{A}, }[/math] where by abuse of notation, s[i] is used for s[i mod n]. The period of a based circular word s is the minimal [math]\displaystyle{ p, 1 \le p \le |s|, }[/math] such that for all i, [math]\displaystyle{ s[i+p]=s[i]. }[/math] If the period of s is equal to the length of s, then s is called primitive.
• A (free) circular word is a class of based circular words equivalent under rotation: [math]\displaystyle{ \{x\mapsto s[x], x\mapsto s[x+1], ..., x\mapsto s[x+|s|-1] \} }[/math] for s a based circular word. A based circular word may be called a mode of the corresponding free circular word or a rotation of the based circular word.
• The length of a linear, based circular, or free circular word s is denoted |s| or len(s).
• For circular words s, if i < j the slice notation s[i:j] denotes the (j − i)-letter word s[i]s[i+1]...s[j−1], where all indices are taken mod |s|.
• Substitution: If w is a linear or based circular word in X and possibly other letters, and u is a based circular word, then [math]\displaystyle{ \mathsf{subst}(w, \mathbf{X}, u) }[/math] denotes the word w but with the ith occurrence of X replaced with u[i] (for i ≥ 0).

Algebraic structures

• [math]\displaystyle{ \mathrm{JI}\langle p_1, ..., p_r \rangle }[/math] is the p₁.[...].p_r subgroup, the subgroup of [math]\displaystyle{ (\mathbb{Q}_{\gt 0}, \cdot) }[/math] generated by rationals [math]\displaystyle{ p_1, ..., p_r. }[/math] For not-necessarily-JI generators, [math]\displaystyle{ \mathrm{M}\langle p_1, ..., p_r \rangle }[/math] is used.
• If R is a commutative ring with 1, [math]\displaystyle{ R^r\langle a_1, ..., a_r\rangle }[/math] is the rank-r free R-module generated by basis elements [math]\displaystyle{ a_1, ..., a_r. }[/math] Ordered tuples in such modules are assumed to be in the given basis. Example: [math]\displaystyle{ \mathbf{m} + 3\mathbf{s} = (0,1,3) \in \mathbb{Z}^3\langle \mathbf{L}, \mathbf{m}, \mathbf{s}\rangle }[/math]

Miscellaneous

• [math]\displaystyle{ \log }[/math] with no subscript is base e.
• Temperament names are capitalized.