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This page documents my xen math notation and its differences from conventional xen notation or conventional math notation.
This document is a style guide. Pages are not required to follow it, but any page that does should link to this page using <code>{{<nowiki />User:Inthar/Template:Notation}}</code>.
== Variables ==
 
* Capital italicized Latin letters may denote scales written cumulatively.
== Math notation ==
This section documents notation that may differ from conventional xen notation or conventional math notation.
 
=== Variables ===
<!--* Capital italicized Latin letters may denote 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'') = 100''n'' cents
** ''S''(''n'') = 100''n'' cents
* Lowercase italicized Latin letters may denote (rotational equivalence classes of) scales written as steps, or abstract scale [[word]]s. For example:
* Lowercase italicized Latin letters may denote (rotational equivalence classes of) scales written as steps, or abstract scale [[word]]s. For example:
** ''s''('''a''', '''b''', '''c''') = '''abacaba'''
** ''s''('''a''', '''b''', '''c''') = '''abacaba'''
** <math>\sum_{n=a}^{b-1}s(n) = S(b)-S(a) \ \text{if} \ s(n) := S(n+1)-S(n)</math>
** <math>\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.
* Bolded variables denote interval sizes (especially letters of scale words) and elements of lattices. This is optional, but may be used for visual clarity, particularly in pages with more mathematical notation. '''0''' is the unison.
** 5'''L''' 2'''s'''
** 5'''L'''&nbsp;2'''s'''
* Sans serif function names are scale constructions, or more generally functions named more verbosely than in conventional math notation.
* Sans serif function names are scale constructions, or more generally functions named more verbosely than is typical for conventional math notation. The page should define any sans-serif functions clearly, as it should any notation not specifically stated on this page.
** <math>\mathsf{MOS}(5,2;6)(\mathbf{L}, \mathbf{s}) = \mathbf{LLLsLLs}</math>
** Blackdye is <math>\mathsf{Flought}(\mathrm{Pyth}[5]; 10/9)</math>
** Blackdye is <math>\mathsf{Fl}(\mathrm{Pyth}[5]; 10/9)</math>
 
== Discrete sets ==
=== Sets ===
* For <math>k \in \mathbb{R}</math> and <math>n\in \mathbb{Z}_{>0},</math> <math>[n]_k</math> denotes <math>\{k, k+1, ..., k+n-1\}.</math> I may also use <math>[i:j]</math> for <math>[j-i]_i.</math> For ''n'' = 0, [0]<sub>k</sub> is the empty set.
* For conciseness the following notation is provided for ranges. For <math>x \in \mathbb{R}</math> and <math>n\in \mathbb{Z}_{>0},</math> <math>[n]_x</math> denotes <math>\{x, x+1, ..., x+n-1\}.</math> [0]<sub>x</sub> is the empty set, and [ω]<sub>''x''</sub> is the set <math>\{x + n : n \in \mathbb{Z}_{\geq 0}\}.</math>
== Words ==
** You may also use <math>[i:j]</math> for <math>[j-i]_i</math> and <math>[i:]</math> for <math>[\omega]_i.</math>
* Zero-indexing is used for indices.
 
* A ''(linear) word'' is a function <math>w : [n]_0 \to \mathcal{A}</math> where <math>\mathcal{A}</math> is a set of letters and <math>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''].  
=== Words ===
* A ''based circular word'' is a function <math> s: \mathbb{Z}/n \to \mathcal{A},</math> where by abuse of notation, ''s''[''i''] is used for ''s''[''i'' mod |''s''|]. A ''period'' of a circular word is the minimal <math>p, 1 \le p \le |s|,</math> such that for all ''i'', <math>s[i+p]=s[i].</math> If the period of ''s'' is equal to the length of ''s'', then ''s'' is called primitive.
* Zero-indexing is used for word indices.
* A ''(free) circular word'' is a class of based circular words equivalent under rotation: <math>\{x\mapsto s[x], x\mapsto s[x+1], ..., x\mapsto s[x+|s|-1] \}</math> for ''s'' a based circular word.
* A ''(linear) word'' is a function <math>w : [n]_0 \to \mathcal{A}</math> where <math>\mathcal{A}</math> is a set of letters and <math>n \in \mathbb{Z}_{\ge 0}</math> or {{nowrap|''n'' {{=}} &infin;}}. ''n'' is called the ''length'' of ''w''. The letter of ''w'' at index ''i'' is denoted ''w''[''i'']. If {{nowrap|0 &le; ''i'' &lt; ''j'' &le; {{!}}''w''{{!}} &minus; 1}}, the slice notation {{nowrap|''w''[''i'':''j'']}} denotes the {{nowrap|(''j'' &minus; ''i'')}}-letter word {{nowrap|''w''[''i'']''w''[''i'' + 1]...''w''[''j'' &minus; 1]}}.
* A ''based circular word'' is a function <math> s: \mathbb{Z}/n \to \mathcal{A},</math> where by abuse of notation, ''s''[''i''] is used for {{nowrap|''s''[''i'' mod ''n'']}}. The ''index period'' of a based circular word ''s'' is the minimal <math>p, 1 \le p \le |s|,</math> such that for all ''i'', <math>s[i+p]=s[i].</math> 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 <math>\{x\mapsto s[x], x\mapsto s[x+1], ..., x\mapsto s[x+|s|-1] \}</math> 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 rotation of the based circular word.
* The length of a linear, based circular, or free circular word ''s'' is denoted {{len|''s''}} or len(''s'').
* The length of a linear, based circular, or free circular word ''s'' is denoted {{len|''s''}} or len(''s'').
* Circular words may be treated as (based) infinite words. Thus for ''m'', ''n'' integers, ''m'' < ''n'', then ''s''[''m''] denotes ''s''[''m'' mod {{len|''s''}}], and the slice notation ''s''[''m'':''n''] denotes the (''n'' &minus; ''m'')-letter word ''s''[''m'']''s''[''m''+1]...''s''[''n''&minus;1], where all indices are taken mod {{len|''s''}}.
* For circular words ''s'', if {{nowrap|''i'' &lt; ''j''}} the slice notation {{nowrap|''s''[''i'':''j'']}} denotes the {{nowrap|(''j'' &minus; ''i'')}}-letter word {{nowrap|''s''[''i'']''s''[''i'' + 1]...''s''[''j'' &minus; 1]}}, where all indices are taken mod {{len|''s''}}.
* Substitution: If ''w'' is a linear or based circular word in '''X''' and possibly other letters, and ''u'' is a based circular word in '''b''' and '''c''', then <math>\mathsf{subst}(w, \mathbf{X}, u)</math> denotes the word ''w'' but with the ''i''th occurrence of '''X''' replaced with ''u''[''i''] (for ''i'' &ge; 0).
* Shifts: If ''s'' is a circular or infinite word, then for <math>k \in \mathbb{Z}, \ \sigma^k(s) = (x \mapsto s[x+k])</math> 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 <math>\mathsf{subst}(w, \mathbf{X}, u)</math> denotes the word ''w'' but with the ''i''th occurrence of '''X''' replaced with ''u''[''i''] (for {{nowrap|''i'' &ge; 0}}).


== Algebraic structures ==
=== Algebra ===
* <math>\mathrm{JI}\langle p_1, ..., p_r \rangle</math> is the ''p''<sub>1</sub>.[...].''p''<sub>''r''</sub> subgroup, the subgroup of <math>(\mathbb{Q}_{>0}, \cdot)</math> generated by rationals <math>p_1, ..., p_r.</math>
* <math>\mathrm{JI}( p_1, ..., p_r )</math> is the ''p''<sub>1</sub>.[...].''p''<sub>''r''</sub> subgroup, the subgroup of <math>(\mathbb{R}_{>0}, \cdot)</math> generated by rationals <math>p_1, ..., p_r.</math> For not-necessarily-JI generators, <math>\mathrm{Mul}(p_1, ..., p_r)</math> is used.
* If ''R'' is a commutative ring with 1, <math>R^r\langle a_1, ..., a_r\rangle</math> is the rank-''r'' free ''R''-module generated by basis elements <math>a_1, ..., a_r.</math> Ordered tuples in such modules are assumed to be in the given basis. Example: <math>\mathbf{m} + 3\mathbf{s} = (0,1,3) \in \mathbb{Z}^3\langle \mathbf{L}, \mathbf{m}, \mathbf{s}\rangle</math>
* If ''R'' is a commutative ring with 1, <math>R^r\langle a_1, ..., a_r\rangle</math> is the rank-''r'' free ''R''-module generated by basis elements <math>a_1, ..., a_r.</math> Ordered tuples in such modules are assumed to be in the given basis. Example: <math>\mathbf{m} + 3\mathbf{s} = (0,1,3) \in \mathbb{Z}^3\langle \mathbf{L}, \mathbf{m}, \mathbf{s}\rangle</math>
== Miscellaneous ==
 
* <math>\log</math> is base e.
=== Miscellaneous ===
* Temperament names are.capitalized.
* <math>\log</math> with no subscript is base ''e''.
* Avoid <math>\mathbb{N}.</math> Use <math>\mathbb{Z}_{>0}</math> or <math>\mathbb{Z}_{\ge 0}</math> depending on which is meant.
 
== To TeX or not to TeX? ==
{{todo|inline=1| expand}}


[[Category:Math]]
[[Category:Math]]

Latest revision as of 19:19, 7 January 2025

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}}.

Math notation

This section documents notation that may differ from conventional xen notation or conventional math notation.

Variables

  • Bolded variables denote interval sizes (especially letters of scale words) and elements of lattices. This is optional, but may be used for visual clarity, particularly in pages with more mathematical notation. 0 is the unison.
    • 5L 2s
  • Sans serif function names are scale constructions, or more generally functions named more verbosely than is typical for conventional math notation. The page should define any sans-serif functions clearly, as it should any notation not specifically stated on this page.
    • Blackdye is [math]\displaystyle{ \mathsf{Flought}(\mathrm{Pyth}[5]; 10/9) }[/math]

Sets

  • For conciseness the following notation is provided for ranges. For [math]\displaystyle{ x \in \mathbb{R} }[/math] and [math]\displaystyle{ n\in \mathbb{Z}_{\gt 0}, }[/math] [math]\displaystyle{ [n]_x }[/math] denotes [math]\displaystyle{ \{x, x+1, ..., x+n-1\}. }[/math] [0]x is the empty set, and [ω]x is the set [math]\displaystyle{ \{x + n : n \in \mathbb{Z}_{\geq 0}\}. }[/math]
    • You may also use [math]\displaystyle{ [i:j] }[/math] for [math]\displaystyle{ [j-i]_i }[/math] and [math]\displaystyle{ [i:] }[/math] for [math]\displaystyle{ [\omega]_i. }[/math]

Words

  • Zero-indexing is used for word 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] 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].
  • 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 index 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 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 [math]\displaystyle{ \{x\mapsto s[x], x\mapsto s[x+1], ..., x\mapsto s[x+|s|-1] \} }[/math] 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 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|.
  • Shifts: If s is a circular or infinite word, then for [math]\displaystyle{ k \in \mathbb{Z}, \ \sigma^k(s) = (x \mapsto s[x+k]) }[/math] 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 [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).

Algebra

  • [math]\displaystyle{ \mathrm{JI}( p_1, ..., p_r ) }[/math] is the p1.[...].pr subgroup, the subgroup of [math]\displaystyle{ (\mathbb{R}_{\gt 0}, \cdot) }[/math] generated by rationals [math]\displaystyle{ p_1, ..., p_r. }[/math] For not-necessarily-JI generators, [math]\displaystyle{ \mathrm{Mul}(p_1, ..., p_r) }[/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.
  • Avoid [math]\displaystyle{ \mathbb{N}. }[/math] Use [math]\displaystyle{ \mathbb{Z}_{\gt 0} }[/math] or [math]\displaystyle{ \mathbb{Z}_{\ge 0} }[/math] depending on which is meant.

To TeX or not to TeX?

Todo: expand
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