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"

|-

| Units →

| Simple units

| Compound or no units

| ↓ Order

| ↓ Style →

| Roman (upright)

| ''Italic''

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

!

! colspan="2" |

| lowercase

| scalar (with simple unit)

| ''scalar'' (with no unit)

| '''bold lowercase'''

| 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.

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

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

! rowspan="2" | Variable

! rowspan="2" | Name

! colspan="3" | Units

! colspan="2" | Shape

! colspan="2" | Type

! colspan="4" | Subobjects

! rowspan="2" | Notes

! Unreduced

! Reduced

! Read as

! Unreduced

! Reduced

! Numeric

! Structural

! Row-first

! Col-first

! Row

! Column

! Diagonal

! Entry

! colspan="17" | Mapping

| [[interval|(Just) interval]]

| Primes

| Integer

| Vector

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

Jargon name: Monzo

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

| Generators per prime

| Integer

| Matrix

| Jargon name: Val list

| [[Mapped interval]]

| Generators

| Integer

| Vector

| Specific type: [[Generator-count vector]] (GC-vector)

Jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval

| [[map|(Temperament) map]]

| Generators per prime

| Integer

| Vector

| Jargon name: val

| Integer

| Scalar

| [[Rank]]

| Integer

| Scalar

! colspan="17" | Tuning

| [[Log-prime map]]

| Octaves per prime

| Real

| Vector

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

| Cents per prime

| Real

| Vector

| [[Generator tuning map]]

| Cents per generator

| Real

| Vector

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

| Cents per prime

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

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

| Real

| Vector

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

| Cents per prime

| Real

| Vector

| Previous name: prime error 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}

| Real

| Scalar

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

| <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}

| Real

| Scalar

| 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}

| Real

| Scalar

! colspan="17" | Optimization

| [[Optimization power]]

| Real

| Scalar

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

| [[Power mean]] (<math>p</math>-mean)

| Real

| Scalar

! colspan="17" | Damage

| {{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>

| Complexity weight

| Real

| Scalar

| [[Simplicity]]

| Simplicity weight

| Real

| Scalar

| [[Weight]]

| Complexity weight or simplicity weight

| Real

| Scalar

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

| [[Damage]]

\begin{array} {c} \abs{\mathrm{e}} \\[-2pt] {\small\mathsf{¢}} \end{array}  

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

| (See damages table)

\begin{array} {c} \abs{\mathrm{e}} \\[-3pt] \left(1, \cancel{1}\right) \end{array}  

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

| Real

| Scalar

! colspan="17" | Target-intervals

| [[Target-interval list]]

| Primes

| Integer

| Matrix

| [[Mapped target-interval list]]

| Generators

\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}  

| Integer

| Matrix

| Mnemonic: looks like bent-up 'T', or cross between 'M' and 'T'

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

| Cents

\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}

| Real

| List

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

| <math>𝒕\mathrm{T}</math><br />

<math>𝒈M\mathrm{T}</math>

| [[Tempered target-interval size list]]

| Cents

\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}

| Real

| List

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

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

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

<math>𝒓\mathrm{T}</math>

| [[Target-interval error list]]

| Cents

\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}

| Real

| List

| [[Target-interval weight matrix]]

| Complexity weight or simplicity weight

| Real

| Matrix

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

| Complexity weight

| Real

| Matrix

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

| Simplicity weight

| Real

| 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>

\begin{array} {c} \abs{\textbf{e}} \\[-2pt] {\small\mathsf{¢}} \end{array}  

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

| Weighted cents

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

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

| Real

| List

| [[Target-interval count]]

| Integer

| Scalar

| Mnemonic: <math>k</math>ount

! colspan="17" | Held-intervals

| [[Held-interval basis]]

| Primes

| Matrix

| [[Held-interval count]]

| Integer

| Scalar

! colspan="17" | Exploring temperaments

| [[Comma basis]]

| Primes

| Integer

| Matrix

| Jargon name: monzo list

| [[Comma]]

| Primes

| Integer

| Vector

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

We recommend using a narrow no-break space (U+202F) between quantities and their units.<ref group="note">Per https://physics.nist.gov/cuu/Units/checklist.html and https://academia.stackexchange.com/questions/54885/should-there-be-a-space-between-a-value-and-the-units-used

.</ref> For how to type this, see the [[#WinCompose|WinCompose]] section below.

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

! Symbol

! Name

! Vectorized

| Generators

| Yes

| Primes

| Yes

| <math>\mathsf{¢}</math><ref group="note">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 "¢" following a number in a musical context as anything other than musical cents.</ref>

| Cents

| Unity-weighted cents

| Complexity-weighted cents

| Simplicity-weighted cents

| Octaves

| Complexity weight

| Simplicity weight

Copied from {{subpage|tuning fundamentals|uprev|s=Systematic tuning scheme names}}.

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

! Damage weight

! Optimization power

! Systematic name

| rowspan="3" | &infin;

| Minimax-U

| Complexity

| Minimax-C

| 1/Complexity

| Minimax-S

| MiniRMS-U

| Complexity

| MiniRMS-C

| 1/Complexity

| MiniRMS-S

| &lt;none&gt;

| Miniaverage-U

| Complexity

| Miniaverage-C

| 1/Complexity

| Miniaverage-S

|+ style="font-size: 105%;" |  

! colspan="2" | Quantity

! colspan="2" | Unit

! Symbol

! Name

| Unity-weight damage

| Unity-weighted cents

| Complexity-weight damage

| Complexity-weighted cents

| Simplicity-weight damage

| Simplicity-weighted cents

|+ style="font-size: 105%;" |  

! colspan="2" | Quantity

! colspan="2" | Unit

! Symbol

! Name

| Complexity

| Complexity weight

| Simplicity

| Simplicity weight

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

! rowspan="2" | Equivalent expressions

! rowspan="2" | Variable

! rowspan="2" | Name

! colspan="3" | Units

! colspan="2" | Shape

! colspan="2" | Type

! colspan="4" | Subobjects

! rowspan="2" | Notes

! Unreduced

! Reduced

! Read as

! Unreduced

! Reduced

! Numeric

! Structural

! Row-first

! Col-first

! Row

! Col

! Diag

! Entry

! colspan="17" | Mapping

| [[interval|(Just) interval]]

| Primes

| Integer

| Vector

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

Jargon name: monzo

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

| Generators per prime

| Integer

| Matrix

| Jargon name: val list

| [[Mapped interval]]

| Generators

\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}

| Integer

| Vector

| Specific type: [[generator-count vector]] (GC-vector)

Jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval

| [[map|(Temperament) map]]

| Generators per prime

| Integer

| Vector

| Jargon name: val

| [[Dimensionality]]

| Integer

| Scalar

| [[Rank]]

| Integer

| Scalar

| [[Nullity]]

| Integer

| Scalar

! colspan="17" | Tuning

| [[Log-prime map]]

| Octaves per prime

| Real

| Vector

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

| Cents per prime

| Real

| Vector

| [[Generator tuning map]]

| Cents per generator

| Real

| Vector

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

| Cents per prime

| Real

| Vector