Dave Keenan & Douglas Blumeyer's guide to RTT/Conventions for names, variables, units, and notations: Difference between revisions

This is an appendix to [[Dave Keenan]] & [[Douglas Blumeyer]]'s guide to RTT. 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.  

See [[Extended bra-ket notation#Variant including curly and square brackets|here]] for more information on our variation on extended bra-ket notation.

We've followed a variable styling convention, explained in 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 linear-algebra convention that objects of order-1 like vectors are bolded and order-2 like matrices are uppercased:

{| class="wikitable center-all"

!  

| Roman (upright)

! scope="col" height="8px" ! colspan="2" |

! colspan="2" |

| lowercase

| scalar (with simple unit)

| ''scalar'' (with no unit)

| '''bold lowercase'''

| '''vector'''

| '''''map''''' (row vector)

| BASIS or LIST (of vectors)

| ''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)<ref group="note">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.</ref>. 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.

{| class="wikitable mw-collapsible"

|+ style="font-size: 105%;" |

! rowspan="2" | Equivalent<br />expressions

! Column

! Diagonal

| [[interval|(Just) interval]]

Jargon name: Monzo

| [[Mapping|(Temperament) mapping (matrix)]]

| Jargon name: Val list

\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}  

| [[map|(Temperament) map]]

|  

| [[dimensionality]]

|  

| [[just tuning map|Just(-prime) tuning map]]

| <math>𝒈M</math>

| [[tuning map|(Tempered-prime) tuning map]]

| <math>\scriptsize

| <math>\scriptsize

| <math>𝒕 - 𝒋</math>

| [[retuning map|Retuning (or mistuning) map]]

| [[interval span|(Just) (interval) size]]

| Cents

\begin{array} {c} 𝒋 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  

\begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array}

| <math>𝒈M\textbf{i}</math><br />

<math>𝒕\textbf{i}</math>

| {{subpage|tuning_fundamentals|uprev|s=Example 3|text=Tempered (interval) size}}

| Cents

\begin{array} {c} 𝒕 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  

\begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array}

| Mnemonic: <math>\mathrm{a}</math>ltered size

| <math>𝒕\textbf{i} - 𝒋\textbf{i}</math><br />

<math>a - o</math><br />

<math>𝒓\textbf{i}</math>

| [[error|(Interval) error]]

| Cents

\begin{array} {c} 𝒓 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  

\begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array}

| <math>\llangle\,·\,\rrangle_p</math>

| {{subpage|tuning_fundamentals|uprev|s=Complexity}}

| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math><ref group="note">For educational purposes, we use the 𝟙 symbol here to represent the implicit [[Wikipedia:Dimensionless_quantity| 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 {{nowrap|''w'' {{=}} ''d''/{{!}}''e''{{!}}}} whose units are {{nowrap|¢(W) / ¢}} and the cents cancel.</ref>

| <math>\small\mathsf{(C)}</math>

| <math>\abs{\mathrm{e}} w</math>

| <math>\scriptsize

| <math>\mathsf{¢}\small\mathsf{(U)}</math> or <math>\mathsf{¢}\small\mathsf{(C)}</math> or <math>\mathsf{¢}\small\mathsf{(S)}</math>

| (See damages table)

| <math>\scriptsize

\begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array}  

\begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array}  

| {{subpage|tuning_fundamentals|uprev|s=primes|text=Target-interval (just) size list}}

\begin{array} {c} 𝒋 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  

\begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array}

| <math>𝒕\mathrm{T}</math><br />

<math>𝒈M\mathrm{T}</math>

\begin{array} {c} 𝒕 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  

\begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array}

| <math>𝒕\mathrm{T} - 𝒋\mathrm{T}</math><br />

<math>\textbf{a} - \textbf{o}</math><br />

<math>𝒓\mathrm{T}</math>

\begin{array} {c} 𝒓 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  

\begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array}

| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(U)}</math>

| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(S)}</math> or <math>\small\mathsf{(U)}</math>

| {{subpage|tuning_fundamentals|uprev|s=complexity-weight_damage|text=Target-interval complexity weight matrix}}

| {{subpage|tuning fundamentals|uprev|s=complexity-weight_damage|text=Target-interval simplicity weight matrix}}

| Entry-wise reciprocal of <math>C</math>

| <math>\abs{\textbf{e}} W</math>

| [[Target-interval damage list]]<ref group="note">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.</ref>

| <math>\scriptsize

\begin{array} {c} W \\[-2pt] (\mathsf{U, C, \text{or}\,S}) \end{array}

| <math>\mathsf{¢}\small\mathsf{(U)}</math>, <math>\mathsf{¢}\small\mathsf{(C)}</math>, or <math>\mathsf{¢}\small\mathsf{(S)}</math>

| <math>\scriptsize (1, k)</math>

| Specific type: vector ([[prime-count vector]] or PC-vector)

 

 

| <math>\small 𝗽</math>

|-

| <math>\mathsf{¢}\small\mathsf{(U)}</math>

| Unity-weighted cents

| rowspan="3" | &infin;

| Minimax-U

| Complexity

| Minimax-C

| 1/Complexity

| Minimax-S

|-

| <none>

| rowspan="3" | 2

| MiniRMS-U

|-

| Complexity

| MiniRMS-C

|-

| 1/Complexity

| MiniRMS-S

| &lt;none&gt;

| rowspan="3" | 1

| Miniaverage-U

| Complexity

| Miniaverage-C

|-

| 1/Complexity

| Miniaverage-S

|}

 

{| class="wikitable center-all mw-collapsible"

|+ style="font-size: 105%;" |  

! colspan="2" | Quantity

! colspan="2" | Unit

| U-damage

| <math>\mathsf{¢}\small\mathsf{(U)}</math>

| Unity-weighted cents

| C-damage

| <math>\mathsf{¢}\small\mathsf{(C)}</math>

| Complexity-weighted cents

| Simplicity-weight damage

| <math>\mathsf{¢}\small\mathsf{(S)}</math>

| Simplicity-weighted cents

=== Complexity and simplicity ===

{| class="wikitable center-all mw-collapsible"

|+ style="font-size: 105%;" |

! colspan="2" | Quantity

! colspan="2" | Unit

! Abbreviation

! Name

| C

| Complexity

| <math>\small\mathsf{(C)}</math>

| Complexity weight

 

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%;" |  

! rowspan="2" | Equivalent expressions

! rowspan="2" | Variable

! rowspan="2" | Name

! colspan="3" | Units

! colspan="2" | Shape

! colspan="2" | Type

! colspan="2" | EBK notation

! colspan="4" | Subobjects

! rowspan="2" | Notes

|-

! Col-first

|-

! colspan="17" | Mapping

| <math>\textbf{i}</math>

| [[interval|(Just) interval]]

|  

| <math>\small 𝗽</math>

| Primes

|  

| <math>\scriptsize (d, 1)</math>

| Integer

| Vector

|  

| [...⟩

| <math>M</math>

| [[Mapping|(Temperament) mapping (matrix)]]

|  

| <math>\small 𝗴</math>/<math>\small 𝗽</math>

| Generators per prime

| <math>m_{ij}</math>

| <math>M\textbf{i}</math>

| <math>\textbf{y}</math>

| [[Mapped interval]]

| <math>\scriptsize

\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}

| <math>\scriptsize

\begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array}

\begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array}

| <math>𝒎</math>

| [[map|(Temperament) map]]

| <math>\small 𝗴</math>/<math>\small 𝗽</math>

| Generators per prime

| <math>m_i</math>

| Jargon name: val

| <math>n + r</math>

|  

| <math>d - r</math>

|  

| <math>\scriptsize (1, 1)</math>

| Integer

! colspan="17" | Tuning

|  

| [[Log-prime map]]

|  

| <math>\small\mathsf{oct}</math>/<math>\small 𝗽</math>

| <math>\scriptsize (1, d)</math>

| Real

|  

|  

|  

| <math>{\large 𝓁}\hspace{2mu}_i</math>

|  

| <math>1200×{\large\textbf{𝓁}}\hspace{2mu}</math>

| [[just tuning map|Just(-prime) tuning map]]

|  

| <math>\mathsf{¢}</math>/<math>\small 𝗽</math>

| <math>\scriptsize (1, d)</math>

| Real

|  

|  

| <math>j_i</math>

|  

|  

| <math>𝒕</math>

| <math>\mathsf{¢}</math>/<math>\small 𝗽</math>

| <math>\scriptsize (1, d)</math>

| Real

| <math>𝒕 - 𝒋</math><br />

<math>1200×\slant{\mathbf{1}}L(P - I)</math>

| <math>𝒓</math>

| [[retuning map|Retuning (or mistuning) map]]

|  

| <math>\mathsf{¢}</math>/<math>\small 𝗽</math>

| Cents per prime

|  

| <math>\scriptsize (1, d)</math>

| Real

| Vector

| ⟨...]

|  

|  

|  

|  

| <math>r_i</math>

| Previous name: prime error map

| Mnemonic: <math>\mathrm{o}</math>riginal size

| <math>𝒈M\textbf{i}</math><br />

<math>𝒕\textbf{i}</math>

| <math>\mathrm{a}</math>

| {{subpage|tuning fundamentals|uprev|s=Example 3|text=Tempered (interval) size}}

\begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  

| <math>\mathsf{¢}</math>

| Cents

\begin{array} {c} 𝒕 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  

\begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array}

| <math>\scriptsize (1, 1)</math>

| Real

|  

| Mnemonic: <math>\mathrm{a}</math>ltered size