User:Inthar/Math style guide

This document is a style guide. Pages are not required to follow it, but any page that does should link to this page using {{User:Inthar/Template:Notation}}. It documents notation that may differ from conventional xen notation or conventional math notation.

Basic musical objects

• A chord is a nonempty finite subset of pitch space (which may be represented as $\mathbb{ℝ}$), modulo pitch transposition (which may be represented as addition if $\mathbb{ℝ}$ is used).
• A scale is a nonempty discrete subset of pitch space invariant under transposition by a chosen equave $E\in {\mathbb{ℝ}}_{>0},$ modulo transposition.
• The period of a scale is the smallest positive $p$ such that the scale is invariant under transposition by $p.$
• A based scale is a scale with a chosen basepoint, called the tonic. By discreteness and the natural ordering on the reals, a based scale can be written as (an equivalence class of) a strictly increasing function $S:\mathbb{\mathbb{Z}}\to \mathbb{ℝ}$ such that $\cdots <S(-1)<S(0)<S(1)<\cdots$ where the image of $S$ is the scale and $S(0)$ is the tonic which we may assume to be 0 by transposition invariance.
• A mode is an equivalence class of based scales for a given scale that become equal after projecting pitch space onto pitch class space modulo the scale's period (not necessarily the equave).

Variables

• Capital italicized Latin letters may denote based scales written cumulatively: i.e. with S(0) = 0 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) based scales written as steps, or scale words written as functions of the given arguments. For example:
  • s(a, b, c) = abacaba
  • $\sum _{n=a}^{b-1}s(n)=S(b)-S(a)\text{if}s(n):=S(n+1)-S(n)$
• Bolded variables denote abstract letters of scale words (ideally not specific sizes) and elements of lattices. This is optional, but may be used for visual clarity, particularly in pages with more mathematical notation.
  • 5L 2s
• Sans serif text in typeset equations are scale constructions, or more generally variables and functions named more verbosely than is typical for conventional math notation. On XenBase, you should generally prefer verbose names for scale constructions; when using shorthand, define the shorthand clearly.

Sets

• For conciseness the following notation is provided for ranges. For $x\in \mathbb{ℝ}$ and $n\in {\mathbb{\mathbb{Z}}}_{>0},$ $[n{]}_x$ denotes the n-element set $\{x,x+1,...,x+n-1\}.$ [0]_x is the empty set, and [ω]_x is the set $\{x+n:n\in {\mathbb{\mathbb{Z}}}_{\ge 0}\}.$ You may also use:
  • $[i:j]$ for $[j-i{]}_i$ (i is included, j is excluded)
  • $[i:]$ for $[\omega {]}_i$
  • $[:j]$ for $\{j-1-n:n\in {\mathbb{\mathbb{Z}}}_{\ge 0}\}$
• Avoid $\mathbb{\mathbb{N}}.$ Use ${\mathbb{\mathbb{Z}}}_{>0}$ or ${\mathbb{\mathbb{Z}}}_{\ge 0}$ depending on which is meant.

Words

• Zero-indexing is used for word indices.
• A (linear) word is a function $w:[n{]}_0\to {𝒜}$ where ${𝒜}$ is a set (of elements usually called letters) and $n\in {\mathbb{\mathbb{Z}}}_{\ge 0}$ or n = ∞. 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].
• The length of a linear, based circular, or free circular word s is denoted |s| or len(s).
• A based circular word is a function $s:\mathbb{\mathbb{Z}}/n\to {𝒜},$ where by abuse of notation, s[i] is used for s[i mod n]. The index period of a based circular word s is the minimal $p,1\le p\le |s|,$ such that for all i, $s[i+p]=s[i].$ If the index period of s is equal to the length of s, then s is called primitive.
• A (free) circular word is an equivalence class of based circular words equivalent under rotation, i.e. a set of the form $\{x\mapsto s[x],x\mapsto s[x+1],...,x\mapsto s[x+|s|-1]\}$ for s a based circular word. Equivalently, a free circular word is an equivalence class of linear words of the same length under conjugacy. A based circular word may be called a mode of the corresponding free circular word or a mode/rotation of another based circular word.
• 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|.
• Shifts: If s is a circular or infinite word, then for $k\in \mathbb{\mathbb{Z}},{\sigma }^k(s)=(x\mapsto s[x+k])$ denotes s shifted to the left by k letters.
• Substitution: If w is a linear or based circular word in X and possibly other letters, and u is a based circular word, then $\mathsf{s}\mathsf{u}\mathsf{b}\mathsf{s}\mathsf{t}(w,\mathbf{𝐗},u)$ denotes the word w but with the ith occurrence of X replaced with u[i] (for i ≥ 0).

Algebra

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

Miscellaneous

• $\log$ with no subscript is base e.
• $s\mathsf{b}\mathsf{y}t$ denotes the cross-set of scales s and t. $s^{\mathsf{b}\mathsf{y}0}=\{\mathbf{𝟎}\},s^{\mathsf{b}\mathsf{y}n+1}=s\mathsf{b}\mathsf{y}{s}^{\mathsf{b}\mathsf{y}n}$ is the n-fold iterated cross-set.-
• "p, lest q" is shorthand for "p, for otherwise q, which is a contradiction".