# Dave Keenan & Douglas Blumeyer's guide to RTT: conventions for names, variables, units, and notations

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This is an appendix to Dave Keenan & Douglas Blumeyer's guide to RTT, or "D&D's guide" for short. The tables in this article present our recommendations for communicating about regular temperament theory (RTT), in particular the names and notations for temperament matrices, tuning schemes, interval complexities, and measurement units.

Our recommendations are designed to make this topic easy to learn for musicians who do not have technical backgrounds, though we have generally deferred to established mathematical, scientific, and engineering conventions for the benefit of those who do.

For more information on our variation on extended bra-ket notation, please see Extended bra-ket notation: Variant including curly and square brackets.

We've followed a symbol formatting pattern, explained by the table below, which we hope serves as an aid to quickly identifying objects and remembering their properties and purposes, but at the least we hope our choices are unobtrusive. In short, the objects with simple units of primes, generators or cents, i.e. the things which are actually audible in our application, are distinguished by upright formatting, while other variables are italic as is conventional. This is crossed with the mathematical convention that objects of order-1 like vectors are bolded and order-2 like matrices are uppercased:

 units → simple units compound or no units ↓ order ↓ style → upright italic 0 plain scalar with simple unit scalar with no unit 1 bold vector map (row vector) 2 UPPERCASE LIST or BASIS true MATRIX

We present our conventions here in three separate sections, one for each level of this article series: basic, intermediate, and advanced. The basic section contains only information covered in the basic part of the series, the intermediate section contains both basic and intermediate, and the advanced section contains it all (that is to say, the sections are cumulative)[1]. We expect that for most readers, the basic tier will be the best reference (this is the reference designed primarily for musicians interested in RTT, as opposed to scientists, engineers, mathematicians, or theoreticians), and so we've left the other two sections initially collapsed.

## Basic

### Objects

equivalent expressions variable name units shape type EBK notation subobjects notes
unreduced reduced read as unreduced reduced numeric structural row-first col-first row col diag entry
mapping
$\textbf{i}$ (just) interval $\small ๐ฝ$ primes $\scriptsize (d, 1)$ integer vector [...โฉ $\mathrm{i}_i$ specific type: vector (prime-count vector or PC-vector)

jargon name: monzo

$M$ (temperament) mapping (matrix) $\small ๐ด$/$\small ๐ฝ$ generators per prime $\scriptsize (r, d)$ integer matrix [โจ...] ...} โจ[...} ...] $๐_i$ $m_{ij}$ jargon name: val list
$M\textbf{i}$ $\textbf{y}$ mapped interval $\scriptsize \begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\small ๐ด$ generators $\scriptsize \! \! \begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (r, 1)$ integer vector [...} specific type: generator-count vector (GC-vector)

jargon name: tmonzo; mnemonic: $\textbf{y}$nterval

$๐$ (temperament) map $\small ๐ด$/$\small ๐ฝ$ generators per prime $\scriptsize (1, d)$ integer vector โจ...] $m_i$ jargon name: val
$d$ dimensionality $\scriptsize (1, 1)$ integer scalar
$r$ rank $\scriptsize (1, 1)$ integer scalar
tuning
${\large\textbf{๐}}\hspace{2mu}$ log-prime map $\small\mathsf{oct}$/$\small ๐ฝ$ octaves per prime $\scriptsize (1, d)$ real vector โจ...] ${\large ๐}\hspace{2mu}_i$
$1200ร{\large\textbf{๐}}\hspace{2mu}$ $๐$ just(-prime) tuning map $\mathsf{ยข}$/$\small ๐ฝ$ cents per prime $\scriptsize (1, d)$ real vector โจ...] $j_i$
$๐$ generator tuning map $\mathsf{ยข}$/$\small ๐ด$ cents per generator $\scriptsize (1, r)$ real vector {...] $g_i$
$๐M$ $๐$ (tempered-prime) tuning map $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ด} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} ๐ \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} ๐ฝ \end{array}$ $\mathsf{ยข}$/$\small ๐ฝ$ cents per prime $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{r}) \end{array} \! \! \begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array} \! \!$ $\scriptsize (1, d)$ real vector โจ...] $t_i$
$๐ - ๐$ $๐$ retuning (or mistuning) map $\mathsf{ยข}$/$\small ๐ฝ$ cents per prime $\scriptsize (1, d)$ real vector โจ...] $r_i$ previous name: prime error map
$๐\textbf{i}$ $\mathrm{o}$ (just) (interval) size $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (1, 1)$ real scalar mnemonic: $\mathrm{o}$riginal size
$๐M\textbf{i} \\ ๐\textbf{i}$ $\mathrm{a}$ tempered (interval) size $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (1, 1)$ real scalar mnemonic: $\mathrm{a}$ltered size
$๐\textbf{i} - ๐\textbf{i} \\ a - o \\ ๐\textbf{i}$ $\mathrm{e}$ (interval) error $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (1, 1)$ real scalar
optimization
$p$ optimization power $\scriptsize (1, 1)$ real scalar
$โช\,ยท\,โซ_p$ power mean ($p$-mean) $\scriptsize (1, 1)$ real scalar
damage
$c$ complexity $\small\mathsf{๐}\scriptsize\mathsf{(C)}$[2] $\small\mathsf{(C)}$ complexity weight $\scriptsize (1, 1)$ real scalar
$\dfrac1c$ $s$ simplicity $\small\mathsf{๐}\scriptsize\mathsf{(S)}$ $\small\mathsf{(S)}$ simplicity weight $\scriptsize (1, 1)$ real scalar
$c$ or $s$ $w$ weight $\small\mathsf{๐}\scriptsize\mathsf{(C)}$ or ๐$\small\mathsf{(S)}$ $\small\mathsf{(C)}$ or $\small\mathsf{(S)}$ complexity weight or simplicity weight $\scriptsize (1, 1)$ real scalar
$| \mathrm{e}| w$ $\mathrm{d}$ damage $\scriptsize \begin{array} {c} | \mathrm{e}| \\[-2pt] {\small\mathsf{ยข}} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} w \\[-2pt] \mathsf{(U, C, or\,S)} \end{array}$ $\mathsf{ยข}\small\mathsf{(U)}$ or $\mathsf{ยข}\small\mathsf{(C)}$ or $\mathsf{ยข}\small\mathsf{(S)}$ (see damages table) $\scriptsize \! \! \begin{array} {c} | \mathrm{e}| \\[-3pt] (1, \cancel{1}) \end{array} \! \! \begin{array} {c} w \\[-3pt] (\cancel{1}, 1) \end{array} \! \!$ $\scriptsize (1, 1)$ real scalar
target-intervals
$\mathrm{T}$ target-interval list $\small ๐ฝ$ primes $\scriptsize (d, k)$ integer matrix [[...โฉ ...] $\textbf{t}_i$ $\mathrm{t}_{ij}$
$M\mathrm{T}$ $\mathrm{Y}$ mapped target-interval list $\scriptsize \begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\small ๐ด$ generators $\scriptsize \! \! \begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} \! \!$ $\scriptsize (r, k)$ integer matrix [[...} ...] $\textbf{y}_i$ $\mathrm{y}_{ij}$ mnemonic: looks like bent-up 'T', or cross between 'M' and 'T'
$๐\mathrm{T}$ $\textbf{o}$ target-interval (just) size list $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} \! \!$ $\scriptsize (1, k)$ real list [...] $\mathrm{o}_i$ mnemonic: $\textbf{o}$riginal size list
$๐\mathrm{T} \\ ๐M\mathrm{T}$ $\textbf{a}$ tempered target-interval size list $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} \! \!$ $\scriptsize (1, k)$ real list [...] $\mathrm{a}_i$ mnemonic: $\textbf{a}$ltered size list
$๐\mathrm{T} - ๐\mathrm{T}\\ \textbf{a} - \textbf{o} \\ ๐\mathrm{T}$ $\textbf{e}$ target-interval error list $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} \! \!$ $\scriptsize (1, k)$ real list [...] $\mathrm{e}_i$
$C$ or $S$ $W$ target-interval weight matrix $\small\mathsf{๐}\scriptsize\mathsf{(C)}$ or $\small\mathsf{๐}\scriptsize\mathsf{(S)}$ or $\small\mathsf{๐}\scriptsize\mathsf{(U)}$ $\small\mathsf{(C)}$ or $\small\mathsf{(S)}$ or $\small\mathsf{(U)}$ complexity weight or simplicity weight $\scriptsize (k, k)$ real matrix [[...] ...] $๐$ $w_i$
$C$ target-interval complexity weight matrix $\small\mathsf{๐}\scriptsize\mathsf{(C)}$ $\small\mathsf{(C)}$ complexity weight $\scriptsize (k, k)$ real matrix [[...] ...] $๐$ $c_i$
$\dfrac1C$ $S$ target-interval simplicity weight matrix $\small\mathsf{๐}\scriptsize\mathsf{(S)}$ $\small\mathsf{(S)}$ simplicity weight $\scriptsize (k, k)$ real matrix [[...] ...] $๐$ $s_i$ entry-wise reciprocal of $C$
$| \textbf{e}| W$ $\textbf{d}$ target-interval damage list[3] $\scriptsize \begin{array} {c} | \textbf{e}| \\[-2pt] {\small\mathsf{ยข}} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} W \\[-2pt] (\mathsf{U, C, or\,S}) \end{array}$ $\mathsf{ยข}\small\mathsf{(U)}$, $\mathsf{ยข}\small\mathsf{(C)}$, or $\mathsf{ยข}\small\mathsf{(S)}$ weighted cents $\scriptsize \! \! \begin{array} {c} | \textbf{e}| \\[-3pt] (1, \cancel{k}) \end{array} \! \! \begin{array} {c} W \\[-3pt] (\cancel{k}, k) \end{array} \! \!$ $\scriptsize (1, k)$ real list [...] $\mathrm{d}_i$
$k$ target-interval count $\scriptsize (1, 1)$ integer scalar mnemonic: $k$ount
held-intervals
$\mathrm{H}$ held-interval basis $\small ๐ฝ$ primes $\scriptsize (d, h)$ matrix [[...โฉ ...] $\textbf{h}_i$ $\mathrm{h}_{ij}$
$h$ held-interval count $\scriptsize (1, 1)$ integer scalar
exploring temperaments
$\mathrm{C}$ comma basis $\small ๐ฝ$ primes $\scriptsize (d, n)$ integer matrix [[...โฉ ...] $\textbf{c}_i$ $\mathrm{c}_{ij}$ jargon name: monzo list
$\textbf{c}$ comma $\small ๐ฝ$ primes $\scriptsize (d, 1)$ integer vector [...โฉ $\mathrm{c}_i$ specific type: vector (prime-count vector or PC-vector)

### Units

We recommend using a narrow no-break space (U+202F) between quantities and their units.[4] For how to type this, see the WinCompose section below.

symbol name vectorized
$\small ๐ด$ generators yes
$\small ๐ฝ$ primes yes
$\mathsf{ยข}$[5] cents
$\mathsf{ยข}\small\mathsf{(U)}$ unity-weighted cents
$\mathsf{ยข}\small\mathsf{(C)}$ complexity-weighted cents
$\mathsf{ยข}\small\mathsf{(S)}$ simplicity-weighted cents
$\small\mathsf{oct}$ octaves
$\small\mathsf{(C)}$ complexity weight
$\small\mathsf{(S)}$ simplicity weight

### Tuning schemes

 damage weight optimization power systematic name โ minimax-U complexity minimax-C 1/complexity minimax-S 2 miniRMS-U complexity miniRMS-C 1/complexity miniRMS-S 1 miniaverage-U complexity miniaverage-C 1/complexity miniaverage-S

### Damages

quantity unit
abbreviation name symbol name
U-damage unity-weight damage $\mathsf{ยข}\small\mathsf{(U)}$ unity-weighted cents
C-damage complexity-weight damage $\mathsf{ยข}\small\mathsf{(C)}$ complexity-weighted cents
S-damage simplicity-weight damage $\mathsf{ยข}\small\mathsf{(S)}$ simplicity-weighted cents

### Complexity and simplicity

quantity unit
abbreviation name symbol name
C complexity $\small\mathsf{(C)}$ complexity weight
S simplicity $\small\mathsf{(S)}$ simplicity weight

$% \slant{} command approximates italics to allow slanted bold characters, including digits, in MathJax. \def\slant#1{\style{display:inline-block;margin:-.05em;transform:skew(-14deg)translateX(.03em)}{#1}} % Latex equivalents of the wiki templates llzigzag and rrzigzag for double zigzag brackets. \def\llzigzag{\hspace{-1.6mu}\style{display:inline-block;transform:scale(.62,1.24)translateY(.07em);font-family:sans-serif}{๊จ\hspace{-3mu}๊จ}\hspace{-1.6mu}} \def\rrzigzag{\hspace{-1.6mu}\style{display:inline-block;transform:scale(-.62,1.24)translateY(.07em);font-family:sans-serif}{๊จ\hspace{-3mu}๊จ}\hspace{-1.6mu}}$

## Intermediate

### Objects

equivalent expressions variable name units shape type EBK notation subobjects notes
unreduced reduced read as unreduced reduced numeric structural row-first col-first row col diag entry
mapping
$\textbf{i}$ (just) interval $\small ๐ฝ$ primes $\scriptsize (d, 1)$ integer vector [...โฉ $\mathrm{i}_i$ specific type: vector (prime-count vector or PC-vector)

jargon name: monzo

$M$ (temperament) mapping (matrix) $\small ๐ด$/$\small ๐ฝ$ generators per prime $\scriptsize (r, d)$ integer matrix [โจ...] ...} โจ[...} ...] $๐_i$ $m_{ij}$ jargon name: val list
$M\textbf{i}$ $\textbf{y}$ mapped interval $\scriptsize \begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\small ๐ด$ generators $\scriptsize \! \! \begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (r, 1)$ integer vector [...} specific type: generator-count vector (GC-vector)

jargon name: tmonzo; mnemonic: $\textbf{y}$nterval

$๐$ (temperament) map $\small ๐ด$/$\small ๐ฝ$ generators per prime $\scriptsize (1, d)$ integer vector โจ...] $m_i$ jargon name: val
$n + r$ $d$ dimensionality $\scriptsize (1, 1)$ integer scalar
$d - n$ $r$ rank $\scriptsize (1, 1)$ integer scalar
$d - r$ $n$ nullity $\scriptsize (1, 1)$ integer scalar
tuning
${\large\textbf{๐}}\hspace{2mu}$ log-prime map $\small\mathsf{oct}$/$\small ๐ฝ$ octaves per prime $\scriptsize (1, d)$ real vector โจ...] ${\large ๐}\hspace{2mu}_i$
$1200ร{\large\textbf{๐}}\hspace{2mu}$ $๐$ just(-prime) tuning map $\mathsf{ยข}$/$\small ๐ฝ$ cents per prime $\scriptsize (1, d)$ real vector โจ...] $j_i$
$๐$ generator tuning map $\mathsf{ยข}$/$\small ๐ด$ cents per generator $\scriptsize (1, r)$ real vector {...] $g_i$
$๐$ (tempered-prime) tuning map $\mathsf{ยข}$/$\small ๐ฝ$ cents per prime $\scriptsize (1, d)$ real vector โจ...] $t_i$
$๐ - ๐ \\ 1200ร\slant{\mathbf{1}}L(P - I)$ $๐$ retuning (or mistuning) map $\mathsf{ยข}$/$\small ๐ฝ$ cents per prime $\scriptsize (1, d)$ real vector โจ...] $r_i$ previous name: prime error map
$๐\textbf{i}$ $\mathrm{o}$ (just) (interval) size $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (1, 1)$ real scalar mnemonic: $\mathrm{o}$riginal size
$๐M\textbf{i} \\ ๐\textbf{i}$ $\mathrm{a}$ tempered (interval) size $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (1, 1)$ real scalar mnemonic: $\mathrm{a}$ltered size
$๐\textbf{i} - ๐\textbf{i} \\ a - o \\ ๐\textbf{i}$ $\mathrm{e}$ (interval) error $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (1, 1)$ real scalar
optimization
$p$ optimization power $\scriptsize (1, 1)$ real scalar
$โช\,ยท\,โซ_p$ power mean ($p$-mean) $\scriptsize (1, 1)$ real scalar
damage
$c$ complexity (see complexities section of complexities and simplicities table) $\scriptsize (1, 1)$ real scalar
$\dfrac1c$ $s$ simplicity (see simplicities section of complexities and simplicities table) $\scriptsize (1, 1)$ real scalar
$c$ or $s$ $w$ weight (see complexities and simplicities table) $\scriptsize (1, 1)$ real scalar
$| \mathrm{e}| w$ $\mathrm{d}$ damage (see damages table) $\scriptsize (1, 1)$ real scalar
target-intervals
$\mathrm{T}$ target-interval list $\small ๐ฝ$ primes $\scriptsize (d, k)$ integer matrix [[...โฉ ...] $\textbf{t}_i$ $\mathrm{t}_{ij}$
$M\mathrm{T}$ $\mathrm{Y}$ mapped target-interval list $\scriptsize \begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\small ๐ด$ generators $\scriptsize \! \! \begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} \! \!$ $\scriptsize (r, k)$ integer matrix [[...} ...] $\textbf{y}_i$ $\mathrm{y}_{ij}$ mnemonic: looks like bent-up 'T', or cross between 'M' and 'T'
$๐\mathrm{T}$ $\textbf{o}$ target-interval (just) size list $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} \! \!$ $\scriptsize (1, k)$ real list [...] $\mathrm{o}_i$ mnemonic: $\textbf{o}$riginal size list
$๐\mathrm{T}$ $\textbf{a}$ tempered target-interval size list $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} \! \!$ $\scriptsize (1, k)$ real list [...] $\mathrm{a}_i$ mnemonic: $\textbf{a}$ltered size list
$๐\mathrm{T} - ๐\mathrm{T} \\ ๐\mathrm{T} \\ \textbf{a} - \textbf{o}$ $\textbf{e}$ target-interval error list $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} \! \!$ $\scriptsize (1, k)$ real list [...] $\mathrm{e}_i$
$C$ or $S$ $W$ target-interval weight matrix (see complexities and simplicities table) $\scriptsize (k, k)$ real matrix [[...] ...] $๐$ $w_i$
$C$ target-interval complexity weight matrix (see complexities section of complexities and simplicities table) $\scriptsize (k, k)$ real matrix [[...] ...] $๐$ $c_i$
$\dfrac1C$ $S$ target-interval simplicity weight matrix (see simplicities section of complexities and simplicities table) $\scriptsize (k, k)$ real matrix [[...] ...] $๐$ $s_i$ entry-wise reciprocal of $C$
$| \textbf{e}| W$ $\textbf{d}$ target-interval damage list (see damages table) $\scriptsize (1, k)$ real list [...] $\mathrm{d}_i$
$k$ target-interval count $\scriptsize (1, 1)$ integer scalar mnemonic: $k$ount
held-intervals
$\mathrm{H}$ held-interval basis $\small ๐ฝ$ primes $\scriptsize (d, h)$ matrix [[...โฉ ...] $\textbf{h}_i$ $\mathrm{h}_{ij}$
$h$ held-interval count $\scriptsize (1, 1)$ integer scalar
exploring temperaments
$\mathrm{C}$ comma basis $\small ๐ฝ$ primes $\scriptsize (d, n)$ integer matrix [[...โฉ ...] $\textbf{c}_i$ $\mathrm{c}_{ij}$ jargon name: monzo list
$\textbf{c}$ comma $\small ๐ฝ$ primes $\scriptsize (d, 1)$ integer vector [...โฉ $\mathrm{c}_i$ specific type: vector (prime-count vector or PC-vector)
computation
$\llzigzagยท\,\rrzigzag\! _p$ power sum ($p$-sum) $\scriptsize (1, 1)$ real scalar
all-interval tuning schemes
$\mathrm{I}$ $\mathrm{T}_{\text{p}}$ prime proxy target-interval list $\small ๐ฝ$ primes $\scriptsize (d, d)$ integer matrix โจ[...โฉ ...] $\mathbf{1}$
$X$ complexity prescaler $\small\mathsf{๐}\scriptsize\mathsf{(C)}$ $\small\mathsf{(C)}$ complexity weight $\scriptsize (d, d)$ real matrix [โจ...] ...โฉ $๐$ $x_i$
$\text{diag}({\large\textbf{๐}}\hspace{2mu})$ $L$ log-prime matrix $\small\mathsf{oct}$/$\small ๐ฝ$ octaves per prime $\scriptsize (d, d)$ real matrix [โจ...] ...โฉ โจ[...โฉ ...] ${\large\textbf{๐}}\hspace{2mu}_i$ ${\large\textbf{๐}}\hspace{2mu}$ ${\large ๐}\hspace{2mu}_{ij}$
$q$ interval complexity norm power $\scriptsize (1, 1)$ real scalar
$โ ยท โ_q$ power norm ($p$-norm) $\scriptsize (1, 1)$ real scalar
$\dfrac1{1-\frac1q}$ $\text{dual}(q)$ dual norm power $\scriptsize (1, 1)$ real scalar
$โX\mathbf{i}โ_q$ interval complexity $\small\mathsf{(C)}$ $\scriptsize (1, 1)$ real scalar
$โ๐X^{-1}โ_{\text{dual}(q)}$ retuning magnitude $\mathsf{ยข}\small\mathsf{(C^{-1})}$ $\scriptsize (1, 1)$ real scalar

### Units

Same as the basic level.

### Tuning schemes

retuning (or mistuning) magnitude damage target

intervals

systematic name previously named tuning schemes that are specific types of this tuning scheme of interest?
weight optimization
interval complexity slope initial name power
initial name power initial name power initial name multiplier abbreviated read ("____ tuning scheme")
<n/a> maximum โ (t) taxicab 1 S simplicity-weight 1/complexity <n/a> minimax โ all minimax-S minimax simplicity-weight damage "TOP"/"T1"/"TIPTOP"*, "CTOP", "POTOP"/"POTT"*, " BOP", " Weil", "Kees" yes
<n/a> Euclidean 2 E Euclidean 2 minimax-ES minimax Euclideanized-simplicity-weight damage " TE"/"T2"/"TOP-RMS", " CTE", " POTE", "Frobenius", "BE", "WE", "KE"
<n/a> <n/a> U unity-weight <none> <set> <set> minimax-U <set> minimax unity-weight-damage " minimax" yes
(t) taxicab 1 S simplicity-weight 1/complexity <set> minimax-S <set> minimax simplicity-weight damage yes
E Euclidean 2 <set> minimax-ES <set> minimax Euclideanized-simplicity-weight damage
(t) taxicab 1 C complexity-weight complexity <set> minimax-C <set> minimax complexity-weight damage yes
E Euclidean 2 <set> minimax-EC <set> minimax Euclideanized-complexity-weight damage
<n/a> U unity-weight <none> miniRMS 2 <set> miniRMS-U <set> miniRMS unity-weight damage "least squares" yes
(t) taxicab 1 S simplicity-weight 1/complexity <set> miniRMS-S <set> miniRMS simplicity-weight damage yes
E Euclidean 2 <set> miniRMS-ES <set> miniRMS Euclideanized-simplicity-weight damage
(t) taxicab 1 C complexity-weight complexity <set> miniRMS-C <set> miniRMS complexity-weight damage yes
E Euclidean 2 <set> miniRMS-EC <set> miniRMS Euclideanized-complexity-weight damage
<n/a> U unity-weight <none> miniaverage 1 <set> miniaverage-U <set> miniaverage unity-weight damage yes
(t) taxicab 1 S simplicity-weight 1/complexity <set> miniaverage-S <set> miniaverage simplicity-weight damage yes
E Euclidean 2 <set> miniaverage-ES <set> miniaverage Euclideanized-simplicity-weight damage
(t) taxicab 1 C complexity-weight complexity <set> miniaverage-C <set> miniaverage complexity-weight damage yes
E Euclidean 2 <set> miniaverage-EC <set> miniaverage Euclideanized-complexity-weight damage

### Damages

quantity unit
abbreviation name symbol name
U-damage unity-weight damage $\mathsf{ยข}\small\mathsf{(U)}$ unity-weighted cents
C-damage complexity-weight damage $\mathsf{ยข}\small\mathsf{(C)}$ complexity-weighted cents
EC-damage Euclideanized-complexity-weight damage $\mathsf{ยข}$$\small\mathsf{(EC)}$ Euclideanized-complexity-weighted cents
S-damage simplicity-weight damage $\mathsf{ยข}\small\mathsf{(S)}$ simplicity-weighted cents
ES-damage Euclideanized-simplicity-weight damage $\mathsf{ยข}$$\small\mathsf{(ES)}$ Euclideanized-simplicity-weighted cents

### Complexity and simplicity

quantity unit
abbreviation name symbol name
C complexity $\small\mathsf{(C)}$ complexity weight
EC Euclideanized complexity $\small\mathsf{(EC)}$ Euclideanized-complexity weight
S simplicity $\small\mathsf{(S)}$ simplicity weight
ES Euclideanized simplicity $\small\mathsf{(ES)}$ Euclideanized-simplicity weight

## Advanced

### Objects

equivalent expressions variable name units shape type EBK notation subobjects notes
unreduced reduced read as unreduced reduced numeric structural row-first col-first row col diag entry
mapping
$\textbf{i}$ (just) interval $\small ๐ฝ$ primes $\scriptsize (d, 1)$ integer vector [...โฉ $\mathrm{i}_i$ specific type: vector (prime-count vector or PC-vector)

jargon name: monzo

$M$ (temperament) mapping (matrix) $\small ๐ด$/$\small ๐ฝ$ generators per prime $\scriptsize (r, d)$ integer matrix [โจ...] ...} โจ[...} ...] $๐_i$ $m_{ij}$ jargon name: val list
$M\textbf{i}$ $\textbf{y}$ mapped interval $\scriptsize \begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\small ๐ด$ generators $\scriptsize \! \! \begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (r, 1)$ integer vector [...} specific type: generator-count vector (GC-vector)

jargon name: tmonzo; mnemonic: $\textbf{y}$nterval

$๐$ (temperament) map $\small ๐ด$/$\small ๐ฝ$ generators per prime $\scriptsize (1, d)$ integer vector โจ...] $m_i$ jargon name: val
$n + r$ $d$ dimensionality $\scriptsize (1, 1)$ integer scalar
$d - n$ $r$ rank $\scriptsize (1, 1)$ integer scalar
$d - r$ $n$ nullity $\scriptsize (1, 1)$ integer scalar
tuning
$\slant{\mathbf{1}}L$ ${\large\textbf{๐}}\hspace{2mu}$ log-prime map $\small\mathsf{oct}$/$\small ๐ฝ$ octaves per prime $\scriptsize (1, d)$ real vector โจ...] ${\large ๐}\hspace{2mu}_i$
$1200ร\slant{\mathbf{1}}LG_{\text{j}}M_{\text{j}} \\ 1200ร\slant{\mathbf{1}}L \\ ๐_{\text{j}}M_{\text{j}}$ $๐$ just(-prime) tuning map $\scriptsize \begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \\ \scriptsize \quad \begin{array} {c} G_{\text{j}} \\[-2pt] \cancel{๐ฝ} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ด} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} M_{\text{j}} \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} ๐ฝ \end{array}$ $\mathsf{ยข}$/$\small ๐ฝ$ cents per prime $\scriptsize \! \! \begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array} \! \! \begin{array} {c} \slant{\mathbf{1}} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array} \! \! \begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array} \\ \scriptsize \quad \! \! \begin{array} {c} G_{\text{j}} \\[-3pt] (\cancel{d}, \cancel{r}) \end{array} \! \! \begin{array} {c} M_{\text{j}} \\[-3pt] (\cancel{r}, d) \end{array} \! \!$ $\scriptsize (1, d_{\text{p}})$ real vector โจ...] $j_i$
$1200ร\slant{\mathbf{1}}LG$ $๐$ generator tuning map $\scriptsize \begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \\ \scriptsize \quad \begin{array} {c} G \\[-2pt] \cancel{๐ฝ} \hspace{-2mu} / \hspace{-2mu} ๐ด \end{array}$ $\mathsf{ยข}$/$\small ๐ด$ cents per generator $\scriptsize \! \! \begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array} \! \! \begin{array} {c} \slant{\mathbf{1}} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array} \! \! \begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array} \\ \scriptsize \quad \! \! \begin{array} {c} G \\[-3pt] (\cancel{d}, r) \end{array} \! \!$ $\scriptsize (1, r)$ real vector {...] $g_i$
$1200ร\slant{\mathbf{1}}LGM \\ 1200ร\slant{\mathbf{1}}LP \\ ๐M$ $๐$ (tempered-prime) tuning map $\scriptsize \begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \\ \scriptsize \quad \begin{array} {c} G \\[-2pt] \cancel{๐ฝ} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ด} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} M \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} ๐ฝ \end{array}$ $\mathsf{ยข}$/$\small ๐ฝ$ cents per prime $\scriptsize \! \! \begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array} \! \! \begin{array} {c} \slant{\mathbf{1}} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array} \! \! \begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array} \\ \scriptsize \quad \! \! \begin{array} {c} G \\[-3pt] (\cancel{d}, \cancel{r}) \end{array} \! \! \begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array} \! \!$ $\scriptsize (1, d)$ real vector โจ...] $t_i$
$๐ - ๐ \\ 1200ร\slant{\mathbf{1}}L(P - I)$ $๐$ retuning (or mistuning) map $\mathsf{ยข}$/$\small ๐ฝ$ cents per prime $\scriptsize (1, d)$ real vector โจ...] $r_i$ previous name: prime error map
$๐\textbf{i}$ $\mathrm{o}$ (just) (interval) size $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (1, 1)$ real scalar mnemonic: $\mathrm{o}$riginal size
$๐M\textbf{i} \\ ๐\textbf{i}$ $\mathrm{a}$ tempered (interval) size $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (1, 1)$ real scalar mnemonic: $\mathrm{a}$ltered size
$๐\textbf{i} - ๐\textbf{i} \\ a - o \\ ๐\textbf{i}$ $\mathrm{e}$ (interval) error $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (1, 1)$ real scalar
optimization
$p$ optimization power $\scriptsize (1, 1)$ real scalar
$โช\,ยท\,โซ_p$ power mean ($p$-mean) $\scriptsize (1, 1)$ real scalar
damage
$c$ complexity (see complexities section of complexities and simplicities table) $\scriptsize (1, 1)$ real scalar
$\dfrac1c$ $s$ simplicity (see simplicities section of complexities and simplicities table) $\scriptsize (1, 1)$ real scalar
$c$ or $s$ $w$ weight (see complexities and simplicities table) $\scriptsize (1, 1)$ real scalar
$| \mathrm{e}| w$ $\mathrm{d}$ damage (see damages table) $\scriptsize (1, 1)$ real scalar
target-intervals
$\mathrm{T}$ target-interval list $\small ๐ฝ$ primes $\scriptsize (d, k)$ integer matrix [[...โฉ ...] $\textbf{t}_i$ $\mathrm{t}_{ij}$
$M\mathrm{T}$ $\mathrm{Y}$ mapped target-interval list $\scriptsize \begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\small ๐ด$ generators $\scriptsize \! \! \begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} \! \!$ $\scriptsize (r, k)$ integer matrix [[...} ...] $\textbf{y}_i$ $\mathrm{y}_{ij}$ mnemonic: looks like bent-up 'T', or cross between 'M' and 'T'
$๐\mathrm{T}$ $\textbf{o}$ target-interval (just) size list $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} \! \!$ $\scriptsize (1, k)$ real list [...] $\mathrm{o}_i$ mnemonic: $\textbf{o}$riginal size list
$๐\mathrm{T}$ $\textbf{a}$ tempered target-interval size list $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} \! \!$ $\scriptsize (1, k)$ real list [...] $\mathrm{a}_i$ mnemonic: $\textbf{a}$ltered size list
$๐\mathrm{T} - ๐\mathrm{T} \\ ๐\mathrm{T} \\ \textbf{a} - \textbf{o}$ $\textbf{e}$ target-interval error list $\scriptsize \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\mathsf{ยข}$ cents $\scriptsize \! \! \begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} \! \!$ $\scriptsize (1, k)$ real list [...] $\mathrm{e}_i$
$C$ or $S$ $W$ target-interval weight matrix (see complexities and simplicities table) $\scriptsize (k, k)$ real matrix [[...] ...] $๐$ $w_i$ or $w_{ij}$
$C$ target-interval complexity weight matrix (see complexities section of complexities and simplicities table) $\scriptsize (k, k)$ real matrix [[...] ...] $๐$ $c_i$
$\dfrac1C$ $S$ target-interval simplicity weight matrix (see simplicities section of complexities and simplicities table) $\scriptsize (k, k)$ real matrix [[...] ...] $๐$ $s_i$ entry-wise reciprocal of $C$
$| \textbf{e}| W \\ 1200ร\slant{\mathbf{1}}L| P - I| \mathrm{T}W$ $\textbf{d}$ target-interval damage list (see damages table) $\scriptsize (1, k)$ real list [...] $\mathrm{d}_i$
$k$ target-interval count $\scriptsize (1, 1)$ integer scalar mnemonic: $k$ount
held-intervals
$\mathrm{H}$ held-interval basis $\small ๐ฝ$ primes $\scriptsize (d, h)$ matrix [[...โฉ ...] $\textbf{h}_i$ $\mathrm{h}_{ij}$
$h$ held-interval count $\scriptsize (1, 1)$ integer scalar
exploring temperaments
$\mathrm{C}$ comma basis $\small ๐ฝ$ primes $\scriptsize (d, n)$ integer matrix [[...โฉ ...] $\textbf{c}_i$ $\mathrm{c}_{ij}$ jargon name: monzo list
$\textbf{c}$ comma $\small ๐ฝ$ primes $\scriptsize (d, 1)$ integer vector [...โฉ $\mathrm{c}_i$ specific type: vector (prime-count vector or PC-vector)
computation
$\llzigzagยท\,\rrzigzag\! _p$ power sum ($p$-sum) $\scriptsize (1, 1)$ real scalar
all-interval tuning schemes
$\mathrm{I}$ $\mathrm{T}_{\text{p}}$ prime proxy target-interval list $\small ๐ฝ$ primes $\scriptsize (d, d)$ integer matrix โจ[...โฉ ...] $\slant{\mathbf{1}}$
$X$ complexity pretransformer $\small\mathsf{๐}\scriptsize\mathsf{(C)}$ or $\small\mathsf{๐}\scriptsize\mathsf{(}$<alt>-$\scriptsize\mathsf{C)}$[6] $\small\mathsf{(C)}$ or $\small\mathsf{(}$<alt>-$\small\mathsf{C)}$ complexity weight or <alternative>-complexity weight $\scriptsize (d, d)$ or $\scriptsize (d+1, d+1)$ real matrix [โจ...] ...โฉ $๐_i$ $๐$ $x_i$ or $x_{ij}$
$\text{diag}({\large\textbf{๐}}\hspace{2mu})$ $L$ log-prime matrix $\small\mathsf{oct}$/$\small ๐ฝ$ octaves per prime $\scriptsize (d, d)$ real matrix [โจ...] ...โฉ โจ[...โฉ ...] ${\large\textbf{๐}}\hspace{2mu}_i$ ${\large\textbf{๐}}\hspace{2mu}$ ${\large ๐}\hspace{2mu}_{ij}$
$q$ interval complexity norm power $\scriptsize (1, 1)$ real scalar
$โ ยท โ_q$ power norm ($p$-norm) $\scriptsize (1, 1)$ real scalar
$\dfrac1{1-\frac1q}$ $\text{dual}(q)$ dual norm power $\scriptsize (1, 1)$ real scalar
$โX\mathbf{i}โ_q$ interval complexity $\small\mathsf{(C)}$ or $\small\mathsf{(}$<alt>-$\small\mathsf{C)}$ $\scriptsize (1, 1)$ real scalar
$โ๐X^{-1}โ_{\text{dual}(q)}$ retuning magnitude $\mathsf{ยข}\small\mathsf{(C^{-1})}$ or $\mathsf{ยข}\small\mathsf{(}$<alt>-$\small\mathsf{C^{-1})}$ $\scriptsize (1, 1)$ real scalar
alternative complexities
$๐$ prime list[7] $\scriptsize (1, d)$ integer list [...] $p_i$
$\slant{\mathbf{1}}$ summation map $\scriptsize (1, d)$ integer vector โจ...] $1$
$1200$ octaves-to-cents conversion ยข/oct cents per octave $\scriptsize (1, 1)$ integer scalar
$Z$ size-sensitizing matrix $\scriptsize (d+1, d)$ real matrix [โจโฆ]...] $๐_i$ $z_{ij}$
non-standard domain bases
$B_s$ (domain) basis (change) matrix $\small ๐ฝ$/$\small ๐ฏ$ primes per nonprime basis elements $\scriptsize (d_p, d_b)$ integer matrix [[...] ...] [[...] ...] $b_i$ $b_{ij}$
$B_{Ls}$ $\small ๐$/$\small ๐ฏ$ superspace basis elements per (subspace) basis elements $\scriptsize (d_L, d_s)$
embedding and projection
$G$ generator embedding (matrix) $\small ๐ฝ$/$\small ๐ด$ primes per generator $\scriptsize (d, r)$ real matrix [{...] ...โฉ {[...โฉ ...] $๐_i$ $g_{ij}$
$G_cF^{-1}FM_c \\ \mathrm{V}\textit{ฮ}\mathrm{V}^{-1}$ $P$ projection (matrix) $\scriptsize \begin{array} {c} G \\[-2pt] ๐ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐ด} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} M \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} ๐ฝ \end{array}$ $\small ๐ฝ$/$\small ๐ฝ$ primes per prime $\scriptsize \! \! \begin{array} {c} G \\[-3pt] (d, \cancel{r}) \end{array} \! \! \begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array} \! \!$ $\scriptsize (d, d)$ real matrix [โจ...] ...โฉ โจ[...โฉ ...] $๐_i$ $p_i$
$GM\textbf{i}$ $P\textbf{i}$ projected interval $\scriptsize \begin{array} {c} G \\[-2pt] ๐ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐ด} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} M \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array}$ $\small ๐ฝ$ primes $\scriptsize \! \! \begin{array} {c} G \\[-3pt] (d, \cancel{r}) \end{array} \! \! \begin{array} {c} M \\[-3pt] (\cancel{r}, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \!$ $\scriptsize (d, 1)$ real vector [...โฉ specific type: vector (prime-count vector or PC-vector)
$\mathrm{U}$ unchanged-interval basis $\small ๐ฝ$ primes $\scriptsize (d, r)$ matrix [[...โฉ ...] $\textbf{u}_i$ $\mathrm{u}_{ij}$ jargon name: eigenmonzo list
$\textit{ฮ}$ scaling factor (eigenvalue) matrix $\scriptsize (d, d)$ matrix [โจโฆ] โฆโฉ โจ[โฆโฉ โฆ] $๐$ $ฮป_i$ mnemonic: $\mathrm{V}$ is mirrored of $\textit{ฮ}$ which it combines with to create the projection matrix; previous name: eigenvalue matrix
$\mathrm{V}$ unrotated vector (eigenvector) list $\small ๐ฝ$ primes $\scriptsize (d, d)$ matrix โจ[...โฉ ...] $\textbf{v}_i$ $\mathrm{v}_{ij}$ mnemonic: $\mathrm{V}$ is mirrored of $\textit{ฮ}$ which it combines with to create the projection matrix; jargon name: eigenmonzo and comma list
$F$ generator form matrix $\scriptsize (r, r)$ matrix [{...] โฆ} $๐_i$ $f_{ij}$
$I$ $M_{\text{j}}$ JI mapping (matrix) $\small ๐ด$/$\small ๐ฝ$ generators per prime $\scriptsize (d, d)$ integer matrix [โจ...] ...} โจ[...} ...] $\slant{\mathbf{1}}$
$I$ $G_{\text{j}}$ JI generator embedding (matrix) $\small ๐ฝ$/$\small ๐ด$ primes per generator $\scriptsize (d, d)$ integer matrix [{...] ...โฉ {[...โฉ ...] $\slant{\mathbf{1}}$
$K$ constraint (matrix) $\scriptsize (k, r)$ $\scriptsize \{0, +1, -1\}$ matrix [[...] ...] $๐_i$ $k_{ij}$ mnemonic: $K$onstraint
$๐$ (generator tuning map) blend map $\scriptsize (1, ฯ-1)$ real vector [...] $b_i$
$B$ (generator tuning map) blend matrix $\scriptsize (d, ฯ-1)$ real matrix [[...โฉ...] $๐_{i}$ $b_{ij}$
$D$ (generator tuning map) deltas matrix $\mathsf{ยข}$/$\small ๐ด$ cents per generator $\scriptsize (ฯ-1,r)$ real matrix [{...] ...] $๐น_i$ $๐ฟ_{ij}$
$ฯ$ tied basic minimax tuning count integer scalar
exterior algebra
$๐$ multimap $\small ๐ด$/$\small ๐ฝ$ generators per prime $\scriptsize (1, d)$ integer multivector โจ...] or โจโจ...]] or โจโจโจ...]]] ... $๐_i$
$๐$ multicomma $\small ๐ฝ$ primes $\scriptsize (1, n)$ integer multivector [...โฉ or [[...โฉโฉ or [[[...โฉโฉโฉ ... $๐_i$
$๐ง$ (generic temperament multivector) $\scriptsize (1, {{d}\choose{r}})$ or $\scriptsize (1, {{d}\choose{n}})$ integer multivector โจ...] or โจโจ...]] or โจโจโจ...]]] ... [...โฉ or [[...โฉโฉ or [[[...โฉโฉโฉ ... $๐ง_i$
$A$ (generic temperament matrix) $\scriptsize (g, d)$ or $\scriptsize (d, g)$ integer matrix [โจ...] ...} โจ[...} ...] or [[...โฉ ...] $๐_i$ $๐_i$ $๐$ $a_{ij}$
$v$ variance
$g$ grade $\scriptsize (1, 1)$ integer scalar
temperament addition
$\min(r, n)$ $g_\text{min}$ min-grade $\scriptsize (1, 1)$ integer scalar
$\max(r, n)$ $g_\text{max}$ max-grade $\scriptsize (1, 1)$ integer scalar
$L_\text{dep}$ linear-dependence basis $\scriptsize (l_\text{dep}, d)$ or $\scriptsize (d, l_\text{dep})$ integer matrix [โจ...]] or [[...] ...โฉ โจ[...]] or [[...โฉ ...] ${\large\textbf{๐}}\hspace{2mu}_{\text{dep}i}$ ${\large\textbf{๐}}\hspace{2mu}_{\text{dep}i}$ ${\large\textbf{๐}}\hspace{2mu}_\text{dep}$ ${\large ๐}\hspace{2mu}_{\text{dep}ij}$
$L_\text{ind}$ linear-independence basis $\scriptsize (l_\text{ind}, d)$ or $\scriptsize (d, l_\text{ind})$ integer matrix [โจ...]] or [[...] ...โฉ โจ[...]] or [[...โฉ ...] ${\large\textbf{๐}}\hspace{2mu}_{\text{ind}i}$ ${\large\textbf{๐}}\hspace{2mu}_{\text{ind}i}$ ${\large\textbf{๐}}\hspace{2mu}_\text{ind}$ ${\large ๐}\hspace{2mu}_{\text{ind}ij}$
$\dim(L_\text{dep})$ $l_\text{dep}$ linear-dependence $\scriptsize (1, 1)$ integer scalar
$\dim(L_\text{ind})$ $l_\text{ind}$ linear-independence $\scriptsize (1, 1)$ integer scalar

### Units

symbol name vectorized
$\small ๐ด$ generators yes
$\small ๐ฝ$ primes yes
$\small ๐ฏ$ (subspace) basis elements yes
$\small ๐$ superspace basis elements yes
$\mathsf{ยข}$ cents
$\mathsf{ยข}\small{(}$<weight>$\small\mathsf{)}$ weighted cents
$\small\mathsf{oct}$ octaves

### Tuning schemes

retuning (or mistuning) magnitude damage target

intervals

systematic name previously named tuning schemes that are specific types of this tuning scheme of interest?
weight optimization
interval complexity slope initial name power
norm pretransformer norm power norm pretransformer norm power initial name multiplier
initial name multiplier initial name power initial name multiplier initial name power abbreviated read ("____ tuning scheme")
<none> <n/a> maximum โ <none> (t) taxicab 1 S simplicity-weight 1/complexity <n/a> minimax โ all minimax-S minimax simplicity-weight damage "TOP"/"T1"/"TIPTOP"*, "CTOP", "POTOP"/"POTT"* yes
<various> <various> minimax-<alt>-S minimax <alternative>-simplicity-weight damage " BOP", " Weil", "Kees" yes
<none> Euclidean 2 <none> E Euclidean 2 minimax-ES minimax Euclideanized-simplicity-weight damage " TE"/"T2"/"TOP-RMS", " CTE", " POTE" yes
<various> <various> minimax-E-<alt>-S minimax Euclideanized-<alternative>-simplicity-weight damage "Frobenius", "BE", "WE", "KE" yes
<n/a> <n/a> U unity-weight <none> <set> <set> minimax-U <set> minimax unity-weight damage " minimax" yes
<none> (t) taxicab 1 S simplicity-weight 1/complexity <set> minimax-S <set> minimax simplicity-weight damage yes
<various> <set> minimax-<alt>-S <set> minimax <alternative>-simplicity-weight damage
<none> E Euclidean 2 <set> minimax-ES <set> minimax Euclideanized-simplicity-weight damage
<various> <set> minimax-E-<alt>-S <set> minimax Euclideanized-<alternative>-simplicity-weight damage
<none> (t) taxicab 1 C complexity-weight complexity <set> minimax-C <set> minimax complexity-weight damage yes
<various> <set> minimax-<alt>-C <set> minimax <alternative>-complexity-weight damage
<none> E Euclidean 2 <set> minimax-EC <set> minimax Euclideanized-complexity-weight damage
<various> <set> minimax-E-<alt>-C <set> minimax Euclideanized-<alternative>-complexity-weight damage
<n/a> U unity-weight <none> miniRMS 2 <set> miniRMS-U <set> miniRMS unity-weight damage "least squares" yes
<none> (t) taxicab 1 S simplicity-weight 1/complexity <set> miniRMS-S <set> miniRMS simplicity-weight damage yes
<various> <set> miniRMS-<alt>-S <set> miniRMS <alternative>-simplicity-weight damage
<none> E Euclidean 2 <set> miniRMS-ES <set> miniRMS Euclideanized-simplicity-weight damage
<various> <set> miniRMS-E-<alt>-S <set> miniRMS Euclideanized-<alternative>-simplicity-weight damage
<none> (t) taxicab 1 C complexity-weight complexity <set> miniRMS-C <set> miniRMS complexity-weight damage yes
<various> <set> miniRMS-<alt>-C <set> miniRMS <alternative>-complexity-weight damage
<none> E Euclidean 2 <set> miniRMS-EC <set> miniRMS Euclideanized-complexity-weight damage
<various> <set> miniRMS-E-<alt>-C <set> miniRMS Euclideanized-<alternative>-complexity-weight damage
<n/a> U unity-weight <none> miniaverage 1 <set> miniaverage-U <set> miniaverage unity-weight damage yes
<none> (t) taxicab 1 S simplicity-weight 1/complexity <set> miniaverage-S <set> miniaverage simplicity-weight damage yes
<various> <set> miniaverage-<alt>-S <set> miniaverage <alternative>-simplicity-weight damage
<none> E Euclidean 2 <set> miniaverage-ES <set> miniaverage Euclideanized-simplicity-weight damage
<various> <set> miniaverage-E-<alt>-S <set> miniaverage Euclideanized-<alternative>-simplicity-weight damage
<none> (t) taxicab 1 C complexity-weight complexity <set> miniaverage-C <set> miniaverage complexity-weight damage yes
<various> <set> miniaverage-<alt>-C <set> miniaverage <alternative>-complexity-weight damage
<none> E Euclidean 2 <set> miniaverage-EC <set> miniaverage Euclideanized-complexity-weight damage
<various> <set> miniaverage-E-<alt>-C <set> miniaverage Euclideanized-<alternative>-complexity-weight damage

### Damages

quantity unit
abbreviation name symbol name
U-damage unity-weight damage $\mathsf{ยข}\small\mathsf{(U)}$ unity-weighted cents
C-damage complexity-weight damage $\mathsf{ยข}\small\mathsf{(C)}$ complexity-weighted cents
<alt>-C-damage <alternative>-complexity-weight damage $\mathsf{ยข}$$\small\mathsf{(}$<alt>-$\small\mathsf{C)}$ <alternative>-complexity-weighted cents
EC-damage Euclideanized-complexity-weight damage $\mathsf{ยข}$$\small\mathsf{(EC)}$ Euclideanized-complexity-weighted cents
E-<alt>-C-damage Euclideanized-<alternative>-complexity-weight damage $\mathsf{ยข}$$\small\mathsf{(E}$-<alt>-$\small\mathsf{C)}$ Euclideanized-<alternative>-complexity-weighted cents
S-damage simplicity-weight damage $\mathsf{ยข}\small\mathsf{(S)}$ simplicity-weighted cents
<alt>-S-damage <alternative>-simplicity-weight damage $\mathsf{ยข}$$\small\mathsf{(}$<alt>-$\small\mathsf{S)}$ <alternative>-simplicity-weighted cents
ES-damage Euclideanized-simplicity-weight damage $\mathsf{ยข}$$\small\mathsf{(ES)}$ Euclideanized-simplicity-weighted cents
E-<alt>-S-damage Euclideanized-<alternative>-simplicity-weight damage $\mathsf{ยข}$$\small\mathsf{(E}$-<alt>-$\small\mathsf{S)}$ Euclideanized-<alternative>-simplicity-weighted cents

### Complexity and simplicity

quantity unit
abbreviation name unit name
C complexity $\small\mathsf{๐}\scriptsize\mathsf{(C)}$ = $\small\mathsf{(C)}$ complexity weight
<alt>-C <alternative> complexity $\small\mathsf{๐}\scriptsize\mathsf{(}$<alt>-$\scriptsize\mathsf{C)}$ = $\small\mathsf{(}$<alt>-$\small\mathsf{C)}$ <alternative>-complexity weight
EC Euclideanized complexity $\small\mathsf{๐}\scriptsize\mathsf{(EC)}$ = $\small\mathsf{(EC)}$ Euclideanized-complexity weight
E-<alt>-C Euclideanized-<alternative> complexity $\small\mathsf{๐}\scriptsize\mathsf{(E}$-<alt>-$\scriptsize\mathsf{C)}$ = $\small\mathsf{(E}$-<alt>-$\small\mathsf{C)}$ Euclideanized-<alternative>-complexity weight
S simplicity $\small\mathsf{๐}\scriptsize\mathsf{(S)}$ = $\small\mathsf{(S)}$ simplicity weight
<alt>-S <alternative> simplicity $\small\mathsf{๐}\scriptsize\mathsf{(}$<alt>-$\scriptsize\mathsf{S)}$ = $\small\mathsf{(}$<alt>-$\small\mathsf{S)}$ <alternative>-simplicity weight
ES Euclideanized simplicity $\small\mathsf{๐}\scriptsize\mathsf{(ES)}$ = $\small\mathsf{(ES)}$ Euclideanized-simplicity weight
E-<alt>-S Euclideanized-<alternative> simplicity $\small\mathsf{๐}\scriptsize\mathsf{(E}$-<alt>-$\scriptsize\mathsf{S)}$ = $\small\mathsf{(E}$-<alt>-$\small\mathsf{S)}$ Euclideanized-<alternative>-simplicity weight

## WinCompose

Are you tired of every time web-searching for and copy-pasting special characters that you use over and over in RTT discussions, or would like to use if only it were easy, such as โฏ, โญ, ยข, โ, ยฐ, โ, ร, โปยน, โฉ, โ, and ฯ? Well, try WinCompose! This tool lets you communicate about these ideas without disrupting your train of thought, by typing these characters with simple and memorable key sequences. These sequences always begin with your chosen Compose-key, which defaults to being your right Alt key. When describing these sequences we represent this key with the symbol โ. So for example, you type โฏ as โ##, โญ as โbb, ยข as โc/, โ as โv/, ยฐ as โ00, โ as โ-2, ร as โxx, โปยน as โ11, โฉ as โ>>, โ as โ88, and ฯ as โ8f.

For Windows users, install WinCompose then copy-paste the contents of this file: https://dkeenan.com/XCompose.txt into your user sequences (Show sequences โ User-defined sequences โ Edit). Then save and reload. You can always choose to override or add alternatives to our sequences if you find others to be more intuitive.

For Mac users, we refer you to this repo, which gives tools and instructions for setting up key bindings as compose rules in Mac OS, and even comes pre-packaged with our rules: https://github.com/cmloegcmluin/compose2keybindings

### Table of noteworthy sequences

Compose-key sequence resulting text description
Keyboard key symbols
โโโ โ compose key symbol (the right alt key by default)
โ\โฃ โฃ spacebar symbol
โ\โถ๏ธ etc. โถ๏ธ etc. right etc. arrow key symbols
โ\A or โ\O โฅ alt or option key symbol
โ\B โซ backspace key symbol
โ\C โฒ control key symbol
โ\D โฆ delete key symbol
โ\E โ escape key symbol
โ\L โช caps lock key symbol
โ\R or โ\.E โ return or enter key symbol
โ\S โง shift key symbol
โ\T โญพ tab key symbol
โ() โ dotted circle, represents any character (such as the character preceding a combining mark)
Double key sequences
โโฃโฃ โฏ narrow no-break space (used between quantities and their units)
โ.. ยท middle dot (used to multiply units when juxtaposition is ambiguous)
โ:: รท divide sign
โ;; โฬฒฬ combining overline and low line (undirected value)
โ| | โ power norm bracket
โ<< โจ left angle bracket
โ>> โฉ right angle bracket
โ~~ โ approximately equal
โ** โ black star
โ'' โฒ prime mark
โ11 โปยน power of -1 or inverse
โ22 through โ77 ยฒ ยณ โด โต โถ โท squared, cubed, fourth through seventh power
โ88 โ infinity
โ00 ยฐ degree sign
โnn โฟ superscript small n
โ-- โ subscript minus sign
โ__ โฬฒ combining low line (underline)
โ== โก modular congruence
โ// โ fraction slash (use with super and subscripts to create fractions)
โ## โฏ musical sharp
โbb โญ musical flat
โdd โ partial derivative
โff ฯ small phi symbol
โgg ษก single-storey (opentail) small g
โll โ script small L
โuu ยต micro sign
โxx ร multiplication sign
โDD โ delta (small difference) operator
โFF ฮฆ Greek capital phi
โQQ ฯ Greek capital letter archaic qoppa (small quotient operator)
โTT แต superscript capital T (matrix transpose)
โ++ โบ superscript plus sign (matrix pseudoinverse)
โโถ๏ธโถ๏ธ etc. โ etc. right etc. arrows
Multiplication operators
โxx ร multiplication sign
โXx or โxX โจฏ vector or cross product (barely distinguishable from multiplication sign)
โXX โ large multiplication sign (a better symbol for cross product)
โx* โ star operator (prefix: tensor complement, Hodge)
โX* โ asterisk operator (infix: scalar product, Dorst)
โx. โ dot (product) operator
โX. โข bullet (infix: fat dot product, Dorst)
Other operators
โv/ โ square root sign
โ3v/ โ cube root sign
โ4v/ โ fourth root sign
โ-+ โ subscript plus sign
โ-- โ subscript minus sign
โ-= โ subscript equals sign
โ++ โบ superscript plus sign (matrix pseudoinverse)
โ+- or โ+= ยฑ plus or minus sign
โ=+ โ minus or plus sign
โ=- โ minus sign
โ== โก modular congruence
โ/\ โง logical AND, wedge product, progressive product
โ\/ โจ logical OR, vee product, regressive product
โโ/\ โ larger logical AND, wedge product, progressive product
โโ\/ โ larger logical OR, vee product, regressive product
โ| _ โ left floor (infix: right contraction, Dorst)
โ_| โ right floor (infix: left contraction, Dorst)
โ| ^ โ left ceiling
โ^| โ right ceiling
โ'- โจฝ righthand interior product
โ-' โจผ (lefthand) interior product
โ-, ยฌ not sign (prefix: multivector complement)
โโ<> โ diamond operator (prefix: multivector dual)
โ(.) โจ entry-wise vector multiplication operator
โ(..) โ alternative entry-wise vector multiplication operator
โ(/) โ entry-wise vector division operator
Mathematical letter and digit prefixes
โ3โ ั cyrillic, โ3q is ya (example)
โ4โ โต hebrew, โ4a is aleph (example)
โ5โ ๐ fraktur, โ5a
โ6โ แต ยน โฏแชฒ โธ superscripts, โ6a โ61 โ688 โ68โฃ (not all letters, some only approximate) (same key as ^ but without shift)
โ68โ แต superscript greek, โ68b is superscript beta (only a few)
โ7โ ๐ถ script, โ7a
โ8โ ฮฑ greek, โ8a is alpha (by sound where possible otherwise letter-shape)
โ8.โ ฯ greek variants, โ8.s is final sigma
โ9โ ๐ ๐ ๐ ๐ ๐ ๐ bold, โ9a โ91 โ95โฃ โ97โฃ โ98โฃ โ90โฃ
โ95โ ๐ bold fraktur, โ95a
โ97โ ๐ช bold script, โ97a
โ98โ ๐ bold greek, โ98a is bold alpha
โ90โ ๐ bold italic, โ90a
โ908โ ๐ถ bold italic greek, โ908a is bold italic alpha
โ0โ ๐ italic, โ0a
โ08โ ๐ผ italic greek, โ08a is italic alpha
โ-โ โ แด โฏอ โ subscripts and small caps, โ-a โ-A โ-88 โ-8โฃ (not all letters, some only approximate) (same key as _ but without shift)
โ-8โ แตฆ subscript greek, โ-8b is subscript beta (only a few)
โ{โ ๐บ ๐ฃ ๐ซ sans-serif, โ{a โ{1 โ{9โฃ
โ{9โ ๐ฎ ๐ญ sans-serif bold, โ{9a โ{91
โ}โ ๐ ๐ท monospace, โ}a โ}1
โ| โ ๐ ๐ ๐  ๐ double-struck, โ| a โ| 1 โ| 8โฃ โ| 0โฃ
โ| 8โ โผ double-struck greek, โ| 8p (only a few)
โ| 0โ โ โ double-struck italic, โ| 0e โ| i (only a few)
Power statistics brackets
โ| | โ power-norm bracket
โ|-1 โโ 1-norm right bracket
โ|-2 โโ 2-norm right bracket
โ|-8 โโฏอ โ-norm right bracket
โโ<< โช left power-mean bracket
โโ>> โซ right power-mean bracket
โโ{{ โง left power-sum bracket (substitute for when HTML is not available)
โโ}} โง right power-sum bracket (substitute for when HTML is not available)
Combining marks
โ\- โฬถ combining strike-thru
โ^_ โฬ combining overline
โ__ โฬฒ combining low line
โ;; or โ-_ or โ_^ โฬฒฬ combining overline and low line (undirected value)

## Footnotes

1. โ The advanced section also contains conventions collected from other RTT-related articles Dave and Douglas have contributed to but are outside the main guide to RTT series.
2. โ For educational purposes, we use the ๐ symbol here to represent the implicit dimensionless unit that the weighting annotation "(C)" is attached to. But this symbol should not be shown in the reduced result. Another way to understand how we arrive at a bare annotation for the units of this quantity is to consider that w = d / | e| whose units are ยข(W) / ยข and the cents cancel.
3. โ You may sometimes see annotated units without parentheses, such as "dBA", but this is not compliant with SI standards, so we always keep the parentheses.
4. โ
5. โ It seems there is no standard symbol for a musical cent, except the word spelled in full (see https://en.wikipedia.org/wiki/Cent_(music)). But it seems unlikely anyone will interpret the cent currency symbol "ยข" following a number in a musical context as anything other than musical cents.
6. โ In these tables, "alternative" means any complexity other than the default of log-product complexity, and "alt" stands for its abbreviation.
7. โ May be used for a prime-limit or for any prime-only list.