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

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{{breadcrumb}}{{texops}}{{texmap}}{{texzz}}
{{breadcrumb}}{{texops}}{{texmap}}{{texzz}}
This is an appendix to [[Dave Keenan]] & [[Douglas Blumeyer]]'s {{subpage|prev|text=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.  
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.  


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.
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.


See [[Extended_bra-ket_notation#Variant_including_curly_and_square_brackets|here]] for more information on our variation on extended bra-ket notation.
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 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:
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"
{| class="wikitable center-all"
Line 18: Line 18:
| &darr; Order
| &darr; Order
| &darr; Style &rarr;
| &darr; Style &rarr;
| Upright
| Roman (upright)
| ''Italic''
| ''Italic''
|-
|-
! scope="col" height="8px" ! colspan="2" | &nbsp;
! scope="col" height="8px" ! colspan="2" |
!  
!
! colspan="2" | &nbsp;
! colspan="2" |
|-
|-
| 0
| 0
| Plain
| lowercase
! rowspan="3" | &nbsp;
! rowspan="3" | &nbsp;
| Scalar with simple unit
| scalar (with simple unit)
| ''Scalar'' with no unit
| ''scalar'' (with no unit)
|-
|-
| 1
| 1
| '''Bold'''
| '''bold lowercase'''
| '''Vector'''
| '''vector'''
| '''''Map''''' (row vector)
| '''''map''''' (row vector)
|-
|-
| 2
| 2
| UPPERCASE
| UPPERCASE
| LIST or BASIS
| BASIS or LIST (of vectors)
| True ''MATRIX''
| ''MATRIX''
|}
|}


Line 47: Line 47:
=== Objects ===
=== Objects ===
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
|+ style="font-size: 105%;" |
|-
|-
! rowspan="2" | Equivalent<br />expressions
! rowspan="2" | Equivalent<br />expressions
Line 142: Line 143:
|  
|  
| <math>𝒎</math>
| <math>𝒎</math>
| [[map| (temperament) map]]
| [[map|(Temperament) map]]
|  
|  
| <math>\small 𝗴</math>/<math>\small 𝗽</math>
| <math>\small 𝗴</math>/<math>\small 𝗽</math>
Line 324: Line 325:
| Mnemonic: <math>\mathrm{o}</math>riginal size
| Mnemonic: <math>\mathrm{o}</math>riginal size
|-
|-
| <math>𝒈M\textbf{i} \\
| <math>𝒈M\textbf{i}</math><br />
𝒕\textbf{i}</math>
<math>𝒕\textbf{i}</math>
| <math>\mathrm{a}</math>
| <math>\mathrm{a}</math>
| {{subpage|tuning_fundamentals|uprev|s=Example 3|text=Tempered (interval) size}}
| {{subpage|tuning_fundamentals|uprev|s=Example 3|text=Tempered (interval) size}}
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| Mnemonic: <math>\mathrm{a}</math>ltered size
| Mnemonic: <math>\mathrm{a}</math>ltered size
|-
|-
| <math>𝒕\textbf{i} - 𝒋\textbf{i} \\
| <math>𝒕\textbf{i} - 𝒋\textbf{i}</math><br />
a - o \\
<math>a - o</math><br />
𝒓\textbf{i}</math>
<math>𝒓\textbf{i}</math>
| <math>\mathrm{e}</math>
| <math>\mathrm{e}</math>
| [[error|(Interval) error]]
| [[error|(Interval) error]]
Line 425: Line 426:
|  
|  
| <math>c</math>
| <math>c</math>
| {{subpage|tuning_fundamentals|uprev|s=complexity}}
| {{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{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>\small\mathsf{(C)}</math>
Line 581: Line 582:
| Mnemonic: <math>\textbf{o}</math>riginal size list
| Mnemonic: <math>\textbf{o}</math>riginal size list
|-
|-
| <math>𝒕\mathrm{T} \\
| <math>𝒕\mathrm{T}</math><br />
𝒈M\mathrm{T}</math>
<math>𝒈M\mathrm{T}</math>
| <math>\textbf{a}</math>
| <math>\textbf{a}</math>
| [[Tempered target-interval size list]]
| [[Tempered target-interval size list]]
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| Mnemonic: <math>\textbf{a}</math>ltered size list
| Mnemonic: <math>\textbf{a}</math>ltered size list
|-
|-
| <math>𝒕\mathrm{T} - 𝒋\mathrm{T}\\
| <math>𝒕\mathrm{T} - 𝒋\mathrm{T}</math><br />
\textbf{a} - \textbf{o} \\
<math>\textbf{a} - \textbf{o}</math><br />
𝒓\mathrm{T}
<math>𝒓\mathrm{T}</math>
</math>
| <math>\textbf{e}</math>
| <math>\textbf{e}</math>
| [[Target-interval error list]]
| [[Target-interval error list]]
Line 823: Line 823:


{| class="wikitable center-all mw-collapsible"
{| class="wikitable center-all mw-collapsible"
|+ style="font-size: 105%;" |
|-
|-
! Symbol
! Symbol
Line 869: Line 870:


{| class="wikitable center-all mw-collapsible"
{| class="wikitable center-all mw-collapsible"
|+ style="font-size: 105%;" |
|-
|-
! Damage weight
! Damage weight
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=== Damages ===
=== Damages ===
{| class="wikitable center-all mw-collapsible"
{| class="wikitable center-all mw-collapsible"
|+ style="font-size: 105%;" |
|-
|-
! colspan="2" | Quantity
! colspan="2" | Quantity
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=== Complexity and simplicity ===
=== Complexity and simplicity ===
{| class="wikitable center-all mw-collapsible"
{| class="wikitable center-all mw-collapsible"
|+ style="font-size: 105%;" |
|-
|-
! colspan="2" | Quantity
! colspan="2" | Quantity
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=== Objects ===
=== Objects ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" |
|-
|-
! rowspan="2" | Equivalent expressions
! rowspan="2" | Equivalent expressions
Line 1,196: Line 1,201:
|  
|  
|-
|-
| <math>𝒕 - 𝒋 \\
| <math>𝒕 - 𝒋</math><br />
1200×\slant{\mathbf{1}}L(P - I)</math>
<math>1200×\slant{\mathbf{1}}L(P - I)</math>
| <math>𝒓</math>
| <math>𝒓</math>
| [[retuning map|Retuning (or mistuning) map]]
| [[retuning map|Retuning (or mistuning) map]]
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| Mnemonic: <math>\mathrm{o}</math>riginal size
| Mnemonic: <math>\mathrm{o}</math>riginal size
|-
|-
| <math>𝒈M\textbf{i} \\
| <math>𝒈M\textbf{i}</math><br />
𝒕\textbf{i}</math>
<math>𝒕\textbf{i}</math>
| <math>\mathrm{a}</math>
| <math>\mathrm{a}</math>
| {{subpage|tuning fundamentals|uprev|s=Example 3|text=Tempered (interval) size}}
| {{subpage|tuning fundamentals|uprev|s=Example 3|text=Tempered (interval) size}}
Line 1,272: Line 1,277:
| Mnemonic: <math>\mathrm{a}</math>ltered size
| Mnemonic: <math>\mathrm{a}</math>ltered size
|-
|-
| <math>𝒕\textbf{i} - 𝒋\textbf{i} \\
| <math>𝒕\textbf{i} - 𝒋\textbf{i}</math><br />
a - o \\
<math>a - o</math><br />
𝒓\textbf{i}</math>
<math>𝒓\textbf{i}</math>
| <math>\mathrm{e}</math>
| <math>\mathrm{e}</math>
| [[error|(Interval) error]]
| [[error|(Interval) error]]
Line 1,510: Line 1,515:
| Mnemonic: <math>\textbf{a}</math>ltered size list
| Mnemonic: <math>\textbf{a}</math>ltered size list
|-
|-
| <math>𝒕\mathrm{T} - 𝒋\mathrm{T} \\
| <math>𝒕\mathrm{T} - 𝒋\mathrm{T}</math><br />
𝒓\mathrm{T} \\
<math>𝒓\mathrm{T}</math><br />
\textbf{a} - \textbf{o}</math>
<math>\textbf{a} - \textbf{o}</math>
| <math>\textbf{e}</math>
| <math>\textbf{e}</math>
| [[Target-interval error list]]
| [[Target-interval error list]]
Line 1,701: Line 1,706:
|-
|-
|  
|  
| <math>\llzigzag·\,\rrzigzag\! _p</math>
| {{llzigzag}}<math>\,·\,</math>{{rrzigzag}}<math>_p</math>
| [[Power sum]] (<math>p</math>-sum)
| [[Power sum]] (<math>p</math>-sum)
|  
|  
Line 1,793: Line 1,798:
|-
|-
|  
|  
| <math>\norm{\cdot}_q</math>
| <math>\norm{·}_q</math>
| [[Power norm]] (<math>p</math>-norm)
| [[Power norm]] (<math>p</math>-norm)
|  
|  
Line 1,870: Line 1,875:
=== Tuning schemes ===
=== Tuning schemes ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" |
|-
|-
! colspan="3" rowspan="3" | Retuning (or mistuning) magnitude
! colspan="3" rowspan="3" | Retuning (or mistuning) magnitude
Line 2,078: Line 2,084:
=== Damages ===
=== Damages ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" |
|-
|-
! colspan="2" | Quantity
! colspan="2" | Quantity
Line 2,115: Line 2,122:
=== Complexity and simplicity ===
=== Complexity and simplicity ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" |
|-
|-
! colspan="2" | quantity
! colspan="2" | Quantity
! colspan="2" | unit
! colspan="2" | Unit
|-
|-
! abbreviation
! Abbreviation
! name
! Name
! symbol
! Symbol
! name
! Name
|-
|-
| C
| C
| complexity
| Complexity
| <math>\small\mathsf{(C)}</math>
| <math>\small\mathsf{(C)}</math>
| complexity weight
| Complexity weight
|-
|-
| EC
| EC
Line 2,135: Line 2,143:
|-
|-
| S
| S
| simplicity
| Simplicity
| <math>\small\mathsf{(S)}</math>
| <math>\small\mathsf{(S)}</math>
| simplicity weight
| Simplicity weight
|-
|-
| ES
| ES
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=== Objects ===
=== Objects ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" |
|-
|-
! rowspan="2" | equivalent expressions
! rowspan="2" | Equivalent expressions
! rowspan="2" | variable
! rowspan="2" | Variable
! rowspan="2" | name
! rowspan="2" | Name
! colspan="3" | units
! colspan="3" | Units
! colspan="2" | shape
! colspan="2" | Shape
! colspan="2" | type
! colspan="2" | Type
! colspan="2" | EBK notation
! colspan="2" | EBK notation
! colspan="4" | subobjects
! colspan="4" | Subobjects
! rowspan="2" | notes
! rowspan="2" | Notes
|-
|-
! unreduced
! Unreduced
! reduced
! Reduced
! read as
! Read as
! unreduced
! Unreduced
! reduced
! Reduced
! numeric
! Numeric
! structural
! Structural
! row-first
! Row-first
! col-first
! Col-first
! row
! Row
! col
! Column
! diag
! Diagonal
! entry
! Entry
|-
|-
! colspan="17" | mapping
! colspan="17" | Mapping
|-
|-
|  
|  
| <math>\textbf{i}</math>
| <math>\textbf{i}</math>
| [[interval| (just) interval]]
| [[interval|(Just) interval]]
|  
|  
| <math>\small 𝗽</math>
| <math>\small 𝗽</math>
| primes
| Primes
|  
|  
| <math>\scriptsize (d, 1)</math>
| <math>\scriptsize (d, 1)</math>
| integer
| Integer
| vector
| Vector
|  
|  
| [...⟩
| [...⟩
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|  
|  
| <math>\mathrm{i}_i</math>
| <math>\mathrm{i}_i</math>
| specific type: vector ([[prime-count vector]] or PC-vector)
| Specific type: vector ([[prime-count vector]] or PC-vector)
jargon name: monzo
Jargon name: monzo
|-
|-
|  
|  
| <math>M</math>
| <math>M</math>
| [[Mapping| (temperament) mapping (matrix)]]
| [[Mapping|(Temperament) mapping (matrix)]]
|  
|  
| <math>\small 𝗴</math>/<math>\small 𝗽</math>
| <math>\small 𝗴</math>/<math>\small 𝗽</math>
| generators per prime
| Generators per prime
|  
|  
| <math>\scriptsize (r, d)</math>
| <math>\scriptsize (r, d)</math>
| integer
| Integer
| matrix
| Matrix
| [⟨...] ...}
| [⟨...] ...}
| ⟨[...} ...]
| ⟨[...} ...]
Line 2,210: Line 2,219:
|  
|  
| <math>m_{ij}</math>
| <math>m_{ij}</math>
| jargon name: val list
| Jargon name: val list
|-
|-
| <math>M\textbf{i}</math>
| <math>M\textbf{i}</math>
| <math>\textbf{y}</math>
| <math>\textbf{y}</math>
| [[mapped interval]]
| [[Mapped interval]]
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} M \\[-2pt] 𝗴 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
\begin{array} {c} M \\[-2pt] 𝗴 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
Line 2,224: Line 2,233:
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array}  
\begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array}  
\! \!  
\! \!  
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array}
\! \!  
\! \!  
</math>
</math>
| <math>\scriptsize (r, 1)</math>
| <math>\scriptsize (r, 1)</math>
| integer
| Integer
| vector
| Vector
|  
|  
| [...}
| [...}
Line 2,238: Line 2,247:
|  
|  
|  
|  
| specific type: [[generator-count vector]] (GC-vector)
| Specific type: [[generator-count vector]] (GC-vector)
jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval
Jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval
|-
|-
|  
|  
| <math>𝒎</math>
| <math>𝒎</math>
| [[map| (temperament) map]]
| [[map|(Temperament) map]]
|  
|  
| <math>\small 𝗴</math>/<math>\small 𝗽</math>
| <math>\small 𝗴</math>/<math>\small 𝗽</math>
| generators per prime
| Generators per prime
|  
|  
| <math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
| integer
| Integer
| vector
| Vector
| ⟨...]
| ⟨...]
|  
|  
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|  
|  
| <math>m_i</math>
| <math>m_i</math>
| jargon name: val
| Jargon name: val
|-
|-
| <math>n + r</math>
| <math>n + r</math>
| <math>d</math>
| <math>d</math>
| [[dimensionality]]
| [[Dimensionality]]
|  
|  
|  
|  
Line 2,267: Line 2,276:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| integer
| Integer
| scalar
| Scalar
|  
|  
|  
|  
Line 2,279: Line 2,288:
| <math>d - n</math>
| <math>d - n</math>
| <math>r</math>
| <math>r</math>
| [[rank]]
| [[Rank]]
|  
|  
|  
|  
Line 2,285: Line 2,294:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| integer
| Integer
| scalar
| Scalar
|  
|  
|  
|  
Line 2,297: Line 2,306:
| <math>d - r</math>
| <math>d - r</math>
| <math>n</math>
| <math>n</math>
| [[nullity]]
| [[Nullity]]
|  
|  
|  
|  
Line 2,303: Line 2,312:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| integer
| Integer
| scalar
| Scalar
|  
|  
|  
|  
Line 2,313: Line 2,322:
|  
|  
|-
|-
! colspan="17" | tuning
! colspan="17" | Tuning
|-
|-
| <math>\slant{\mathbf{1}}L</math>
| <math>\slant{\mathbf{1}}L</math>
| <math>{\large\textbf{𝓁}}\hspace{2mu}</math>
| <math>{\large\textbf{𝓁}}\hspace{2mu}</math>
| [[log-prime map]]
| [[Log-prime map]]
|  
|  
| <math>\small\mathsf{oct}</math>/<math>\small 𝗽</math>
| <math>\small\mathsf{oct}</math>/<math>\small 𝗽</math>
| octaves per prime
| Octaves per prime
|  
|  
| <math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
| real
| Real
| vector
| Vector
| ⟨...]
| ⟨...]
|  
|  
Line 2,333: Line 2,342:
|  
|  
|-
|-
| <math>1200×\slant{\mathbf{1}}LG_{\text{j}}M_{\text{j}} \\
| <math>1200×\slant{\mathbf{1}}LG_{\text{j}}M_{\text{j}}</math><br />
1200×\slant{\mathbf{1}}L \\
<math>1200×\slant{\mathbf{1}}L</math><br />
𝒈_{\text{j}}M_{\text{j}}</math>
<math>𝒈_{\text{j}}M_{\text{j}}</math>
| <math>𝒋</math>
| <math>𝒋</math>
| [[just tuning map| just(-prime) tuning map]]
| [[just tuning map|Just(-prime) tuning map]]
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
Line 2,351: Line 2,360:
</math>
</math>
| <math>\mathsf{¢}</math>/<math>\small 𝗽</math>
| <math>\mathsf{¢}</math>/<math>\small 𝗽</math>
| cents per prime
| Cents per prime
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
\begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array}
\begin{array} {c} 1200 \\[-3pt] \left(1, \cancel{1}\right) \end{array}
\! \!  
\! \!  
\begin{array} {c} \slant{\mathbf{1}} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array}
\begin{array} {c} \slant{\mathbf{1}} \\[-3pt] \left(\cancel{1}, \cancel{d}\right) \end{array}
\! \!  
\! \!  
\begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array}
\begin{array} {c} L \\[-3pt] \left(\cancel{d}, \cancel{d}\right) \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
\! \!  
\! \!  
\begin{array} {c} G_{\text{j}} \\[-3pt] (\cancel{d}, \cancel{r}) \end{array}
\begin{array} {c} G_{\text{j}} \\[-3pt] \left(\cancel{d}, \cancel{r}\right) \end{array}
\! \!  
\! \!  
\begin{array} {c} M_{\text{j}} \\[-3pt] (\cancel{r}, d) \end{array}
\begin{array} {c} M_{\text{j}} \\[-3pt] \left(\cancel{r}, d\right) \end{array}
\! \!  
\! \!  
</math>
</math>
| <math>\scriptsize (1, d_{\text{p}})</math>
| <math>\scriptsize \left(1, d_{\text{p}}\right)</math>
| real
| Real
| vector
| Vector
| ⟨...]
| ⟨...]
|  
|  
Line 2,379: Line 2,388:
| <math>1200×\slant{\mathbf{1}}LG</math>
| <math>1200×\slant{\mathbf{1}}LG</math>
| <math>𝒈</math>
| <math>𝒈</math>
| [[generator tuning map]]
| [[Generator tuning map]]
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
Line 2,391: Line 2,400:
</math>
</math>
| <math>\mathsf{¢}</math>/<math>\small 𝗴</math>
| <math>\mathsf{¢}</math>/<math>\small 𝗴</math>
| cents per generator
| Cents per generator
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
Line 2,405: Line 2,414:
</math>
</math>
| <math>\scriptsize (1, r)</math>
| <math>\scriptsize (1, r)</math>
| real
| Real
| vector
| Vector
| {...]
| {...]
|  
|  
Line 2,415: Line 2,424:
|  
|  
|-
|-
| <math>1200×\slant{\mathbf{1}}LGM \\
| <math>1200×\slant{\mathbf{1}}LGM</math><br />
1200×\slant{\mathbf{1}}LP \\
<math>1200×\slant{\mathbf{1}}LP</math><br />
𝒈M</math>
<math>𝒈M</math>
| <math>𝒕</math>
| <math>𝒕</math>
| [[tuning map| (tempered-prime) tuning map]]
| [[tuning map|(Tempered-prime) tuning map]]
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
Line 2,433: Line 2,442:
</math>
</math>
| <math>\mathsf{¢}</math>/<math>\small 𝗽</math>
| <math>\mathsf{¢}</math>/<math>\small 𝗽</math>
| cents per prime
| Cents per prime
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
\begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array}
\begin{array} {c} 1200 \\[-3pt] \left(1, \cancel{1}\right) \end{array}
\! \!  
\! \!  
\begin{array} {c} \slant{\mathbf{1}} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array}
\begin{array} {c} \slant{\mathbf{1}} \\[-3pt] \left(\cancel{1}, \cancel{d}\right) \end{array}
\! \!  
\! \!  
\begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array}
\begin{array} {c} L \\[-3pt] \left(\cancel{d}, \cancel{d}\right) \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
\! \!  
\! \!  
\begin{array} {c} G \\[-3pt] (\cancel{d}, \cancel{r}) \end{array}
\begin{array} {c} G \\[-3pt] \left(\cancel{d}, \cancel{r}\right) \end{array}
\! \!  
\! \!  
\begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array}
\begin{array} {c} M \\[-3pt] \left(\cancel{r}, d\right) \end{array}
\! \!  
\! \!  
</math>
</math>
| <math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
| real
| Real
| vector
| Vector
| ⟨...]
| ⟨...]
|  
|  
Line 2,459: Line 2,468:
|  
|  
|-
|-
| <math>𝒕 - 𝒋 \\
| <math>𝒕 - 𝒋</math><br />
1200×\slant{\mathbf{1}}L(P - I)</math>
<math>1200×\slant{\mathbf{1}}L(P - I)</math>
| <math>𝒓</math>
| <math>𝒓</math>
| [[retuning map| retuning (or mistuning) map]]
| [[retuning map|Retuning (or mistuning) map]]
|  
|  
| <math>\mathsf{¢}</math>/<math>\small 𝗽</math>
| <math>\mathsf{¢}</math>/<math>\small 𝗽</math>
| cents per prime
| Cents per prime
|  
|  
| <math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
| real
| Real
| vector
| Vector
| ⟨...]
| ⟨...]
|  
|  
Line 2,476: Line 2,485:
|  
|  
| <math>r_i</math>
| <math>r_i</math>
| previous name: prime error map
| Previous name: prime error map
|-
|-
| <math>𝒋\textbf{i}</math>
| <math>𝒋\textbf{i}</math>
| <math>\mathrm{o}</math>
| <math>\mathrm{o}</math>
| [[interval span| (just) (interval) size]]
| [[interval span|(Just) (interval) size]]
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
\begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
Line 2,487: Line 2,496:
</math>
</math>
| <math>\mathsf{¢}</math>
| <math>\mathsf{¢}</math>
| cents
| Cents
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
\begin{array} {c} 𝒋 \\[-3pt] (1, \cancel{d}) \end{array}  
\begin{array} {c} 𝒋 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  
\! \!  
\! \!  
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array}
\! \!  
\! \!  
</math>
</math>
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 2,504: Line 2,513:
|  
|  
|  
|  
| mnemonic: <math>\mathrm{o}</math>riginal size
| Mnemonic: <math>\mathrm{o}</math>riginal size
|-
|-
| <math>𝒈M\textbf{i} \\
| <math>𝒈M\textbf{i}</math><br />
𝒕\textbf{i}</math>
<math>𝒕\textbf{i}</math>
| <math>\mathrm{a}</math>
| <math>\mathrm{a}</math>
| {{subpage|tuning fundamentals|uprev|s=Example 3|text=tempered (interval) size}}
| {{subpage|tuning fundamentals|uprev|s=Example 3|text=Tempered (interval) size}}
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
\begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
Line 2,519: Line 2,528:
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
\begin{array} {c} 𝒕 \\[-3pt] (1, \cancel{d}) \end{array}  
\begin{array} {c} 𝒕 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  
\! \!  
\! \!  
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array}
\! \!  
\! \!  
</math>
</math>
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 2,533: Line 2,542:
|  
|  
|  
|  
| mnemonic: <math>\mathrm{a}</math>ltered size
| Mnemonic: <math>\mathrm{a}</math>ltered size
|-
|-
| <math>𝒕\textbf{i} - 𝒋\textbf{i} \\
| <math>𝒕\textbf{i} - 𝒋\textbf{i}</math><br />
a - o \\
<math>a - o</math><br />
𝒓\textbf{i}</math>
<math>𝒓\textbf{i}</math>
| <math>\mathrm{e}</math>
| <math>\mathrm{e}</math>
| [[error| (interval) error]]
| [[error|(Interval) error]]
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} 𝒓 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
\begin{array} {c} 𝒓 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
Line 2,546: Line 2,555:
</math>
</math>
| <math>\mathsf{¢}</math>
| <math>\mathsf{¢}</math>
| cents
| Cents
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
\begin{array} {c} 𝒓 \\[-3pt] (1, \cancel{d}) \end{array}  
\begin{array} {c} 𝒓 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  
\! \!  
\! \!  
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array}
\! \!  
\! \!  
</math>
</math>
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 2,565: Line 2,574:
|  
|  
|-
|-
! colspan="17" | optimization
! colspan="17" | Optimization
|-
|-
|  
|  
| <math>p</math>
| <math>p</math>
| [[optimization power]]
| [[Optimization power]]
|  
|  
|  
|  
Line 2,575: Line 2,584:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 2,586: Line 2,595:
|-
|-
|  
|  
| <math>⟪\,·\,⟫_p</math>
| <math>\llangle\,·\,\rrangle_p</math>
| [[power mean]] (<math>p</math>-mean)
| [[Power mean]] (<math>p</math>-mean)
|  
|  
|  
|  
Line 2,593: Line 2,602:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 2,603: Line 2,612:
|  
|  
|-
|-
! colspan="17" | damage
! colspan="17" | Damage
|-
|-
|  
|  
| <math>c</math>
| <math>c</math>
| {{subpage|tuning_fundamentals|uprev|s=complexity}}
| {{subpage|Tuning_fundamentals|prev|s=complexity}}
| colspan="3" | (see complexities section of complexities and simplicities table)
| colspan="3" | (See complexities section of complexities and simplicities table)
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 2,623: Line 2,632:
| <math>\dfrac1c</math>
| <math>\dfrac1c</math>
| <math>s</math>
| <math>s</math>
| [[simplicity]]
| [[Simplicity]]
| colspan="3" | (see simplicities section of complexities and simplicities table)
| colspan="3" | (See simplicities section of complexities and simplicities table)
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 2,639: Line 2,648:
| <math>c</math> or <math>s</math>
| <math>c</math> or <math>s</math>
| <math>w</math>
| <math>w</math>
| [[weight]]
| [[Weight]]
| colspan="3" | (see complexities and simplicities table)
| colspan="3" | (See complexities and simplicities table)
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 2,653: Line 2,662:
|  
|  
|-
|-
| <math>| \mathrm{e}| w</math>
| <math>\abs{\mathrm{e}} w</math>
| <math>\mathrm{d}</math>
| <math>\mathrm{d}</math>
| [[damage]]
| [[Damage]]
| colspan="3" | (see damages table)
| colspan="3" | (See damages table)
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 2,669: Line 2,678:
|  
|  
|-
|-
! colspan="17" | target-intervals
! colspan="17" | Target-intervals
|-
|-
|  
|  
| <math>\mathrm{T}</math>
| <math>\mathrm{T}</math>
| [[target-interval list]]
| [[Target-interval list]]
|  
|  
| <math>\small 𝗽</math>
| <math>\small 𝗽</math>
| primes
| Primes
|  
|  
| <math>\scriptsize (d, k)</math>
| <math>\scriptsize (d, k)</math>
| integer
| Integer
| matrix
| Matrix
|  
|  
| [[...⟩ ...]
| [[...⟩ ...]
Line 2,691: Line 2,700:
| <math>M\mathrm{T}</math>
| <math>M\mathrm{T}</math>
| <math>\mathrm{Y}</math>
| <math>\mathrm{Y}</math>
| [[mapped target-interval list]]
| [[Mapped target-interval list]]
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} M \\[-2pt] 𝗴 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
\begin{array} {c} M \\[-2pt] 𝗴 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
Line 2,698: Line 2,707:
</math>
</math>
| <math>\small 𝗴</math>
| <math>\small 𝗴</math>
| generators
| Generators
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array}  
\begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array}  
\! \!  
\! \!  
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}  
\begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array}  
\! \!  
\! \!  
</math>
</math>
| <math>\scriptsize (r, k)</math>
| <math>\scriptsize (r, k)</math>
| integer
| Integer
| matrix
| Matrix
|  
|  
| [[...} ...]
| [[...} ...]
Line 2,715: Line 2,724:
|  
|  
| <math>\mathrm{y}_{ij}</math>
| <math>\mathrm{y}_{ij}</math>
| mnemonic: looks like bent-up 'T', or cross between 'M' and 'T'
| Mnemonic: looks like bent-up 'T', or cross between 'M' and 'T'
|-
|-
| <math>𝒋\mathrm{T}</math>
| <math>𝒋\mathrm{T}</math>
| <math>\textbf{o}</math>
| <math>\textbf{o}</math>
| {{subpage|tuning fundamentals|uprev|s=primes|text=target-interval (just) size list}}
| {{subpage|tuning fundamentals|uprev|s=primes|text=Target-interval (just) size list}}
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
\begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
Line 2,726: Line 2,735:
</math>
</math>
| <math>\mathsf{¢}</math>
| <math>\mathsf{¢}</math>
| cents
| Cents
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
\begin{array} {c} 𝒋 \\[-3pt] (1, \cancel{d}) \end{array}  
\begin{array} {c} 𝒋 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  
\! \!  
\! \!  
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array}
\! \!  
\! \!  
</math>
</math>
| <math>\scriptsize (1, k)</math>
| <math>\scriptsize (1, k)</math>
| real
| Real
| list
| List
| [...]
| [...]
|  
|  
Line 2,743: Line 2,752:
|  
|  
| <math>\mathrm{o}_i</math>
| <math>\mathrm{o}_i</math>
| mnemonic: <math>\textbf{o}</math>riginal size list
| Mnemonic: <math>\textbf{o}</math>riginal size list
|-
|-
| <math>𝒕\mathrm{T}</math>
| <math>𝒕\mathrm{T}</math>
| <math>\textbf{a}</math>
| <math>\textbf{a}</math>
| [[tempered target-interval size list]]
| [[Tempered target-interval size list]]
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
\begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}  
Line 2,754: Line 2,763:
</math>
</math>
| <math>\mathsf{¢}</math>
| <math>\mathsf{¢}</math>
| cents
| Cents
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
\begin{array} {c} 𝒕 \\[-3pt] (1, \cancel{d}) \end{array}  
\begin{array} {c} 𝒕 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  
\! \!  
\! \!  
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array}
\! \!  
\! \!  
</math>
</math>
| <math>\scriptsize (1, k)</math>
| <math>\scriptsize (1, k)</math>
| real
| Real
| list
| List
| [...]
| [...]
|  
|  
Line 2,771: Line 2,780:
|  
|  
| <math>\mathrm{a}_i</math>
| <math>\mathrm{a}_i</math>
| mnemonic: <math>\textbf{a}</math>ltered size list
| Mnemonic: <math>\textbf{a}</math>ltered size list
|-
|-
| <math>𝒕\mathrm{T} - 𝒋\mathrm{T} \\
| <math>𝒕\mathrm{T} - 𝒋\mathrm{T}</math><br />
𝒓\mathrm{T} \\
<math>𝒓\mathrm{T}</math><br />
\textbf{a} - \textbf{o}</math>
<math>\textbf{a} - \textbf{o}</math>
| <math>\textbf{e}</math>
| <math>\textbf{e}</math>
| [[target-interval error list]]
| [[target-interval error list]]
Line 2,784: Line 2,793:
</math>
</math>
| <math>\mathsf{¢}</math>
| <math>\mathsf{¢}</math>
| cents
| Cents
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
\begin{array} {c} 𝒓 \\[-3pt] (1, \cancel{d}) \end{array}  
\begin{array} {c} 𝒕 \\[-3pt] \left(1, \cancel{d}\right) \end{array}  
\! \!  
\! \!  
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array}
\! \!  
\! \!  
</math>
</math>
| <math>\scriptsize (1, k)</math>
| <math>\scriptsize (1, k)</math>
| real
| Real
| list
| List
| [...]
| [...]
|  
|  
Line 2,805: Line 2,814:
| <math>C</math> or <math>S</math>
| <math>C</math> or <math>S</math>
| <math>W</math>
| <math>W</math>
| [[target-interval weight matrix]]
| [[Target-interval weight matrix]]
| colspan="3" | (see complexities and simplicities table)
| colspan="3" | (See complexities and simplicities table)
|  
|  
| <math>\scriptsize (k, k)</math>
| <math>\scriptsize (k, k)</math>
| real
| Real
| matrix
| Matrix
|  
|  
| [[...] ...]
| [[...] ...]
Line 2,821: Line 2,830:
|  
|  
| <math>C</math>
| <math>C</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 complexity weight matrix}}
| colspan="3" | (see complexities section of complexities and simplicities table)
| colspan="3" | (See complexities section of complexities and simplicities table)
|  
|  
| <math>\scriptsize (k, k)</math>
| <math>\scriptsize (k, k)</math>
| real
| Real
| matrix
| Matrix
|  
|  
| [[...] ...]
| [[...] ...]
Line 2,837: Line 2,846:
| <math>\dfrac1C</math>
| <math>\dfrac1C</math>
| <math>S</math>
| <math>S</math>
| {{subpage|tuning fundamentals|uprev|s=complexity-weight_damage|text=target-interval simplicity weight matrix}}
| {{subpage|tuning fundamentals|uprev|s=complexity-weight_damage|text=Target-interval simplicity weight matrix}}
| colspan="3" | (see simplicities section of complexities and simplicities table)
| colspan="3" | (See simplicities section of complexities and simplicities table)
|  
|  
| <math>\scriptsize (k, k)</math>
| <math>\scriptsize (k, k)</math>
| real
| Real
| matrix
| Matrix
|  
|  
| [[...] ...]
| [[...] ...]
Line 2,849: Line 2,858:
| <math>𝒔</math>
| <math>𝒔</math>
| <math>s_i</math>
| <math>s_i</math>
| entry-wise reciprocal of <math>C</math>
| Entry-wise reciprocal of <math>C</math>
|-
|-
| <math>| \textbf{e}| W \\
| <math>\abs{\textbf{e}} W</math><br />
1200×\slant{\mathbf{1}}L| P - I| \mathrm{T}W</math>
<math>1200×\slant{\mathbf{1}}L\abs{P - I} \mathrm{T}W</math>
| <math>\textbf{d}</math>
| <math>\textbf{d}</math>
| [[target-interval damage list]]
| [[Target-interval damage list]]
| colspan="3" | (see damages table)
| colspan="3" | (See damages table)
|  
|  
| <math>\scriptsize (1, k)</math>
| <math>\scriptsize (1, k)</math>
| real
| Real
| list
| List
| [...]
| [...]
|  
|  
Line 2,870: Line 2,879:
|  
|  
| <math>k</math>
| <math>k</math>
| [[target-interval count]]
| [[Target-interval count]]
|  
|  
|  
|  
Line 2,876: Line 2,885:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| integer
| Integer
| scalar
| Scalar
|  
|  
|  
|  
Line 2,884: Line 2,893:
|  
|  
|  
|  
| mnemonic: <math>k</math>ount
| Mnemonic: <math>k</math>ount
|-
|-
! colspan="17" | held-intervals
! colspan="17" | Held-intervals
|-
|-
|  
|  
| <math>\mathrm{H}</math>
| <math>\mathrm{H}</math>
| [[held-interval basis]]
| [[Held-interval basis]]
|  
|  
| <math>\small 𝗽</math>
| <math>\small 𝗽</math>
| primes
| Primes
|  
|  
| <math>\scriptsize (d, h)</math>
| <math>\scriptsize (d, h)</math>
|  
|  
| matrix
| Matrix
|  
|  
| [[...⟩ ...]
| [[...⟩ ...]
Line 2,908: Line 2,917:
|  
|  
| <math>h</math>
| <math>h</math>
| [[held-interval count]]
| [[Held-interval count]]
|  
|  
|  
|  
Line 2,914: Line 2,923:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| integer
| Integer
| scalar
| Scalar
|  
|  
|  
|  
Line 2,924: Line 2,933:
|  
|  
|-
|-
! colspan="17" | exploring temperaments
! colspan="17" | Exploring temperaments
|-
|-
|  
|  
| <math>\mathrm{C}</math>
| <math>\mathrm{C}</math>
| [[comma basis]]
| [[Comma basis]]
|  
|  
| <math>\small 𝗽</math>
| <math>\small 𝗽</math>
| primes
| Primes
|  
|  
| <math>\scriptsize (d, n)</math>
| <math>\scriptsize (d, n)</math>
| integer
| Integer
| matrix
| Matrix
|  
|  
| [[...⟩ ...]
| [[...⟩ ...]
Line 2,942: Line 2,951:
|  
|  
| <math>\mathrm{c}_{ij}</math>
| <math>\mathrm{c}_{ij}</math>
| jargon name: monzo list
| Jargon name: monzo list
|-
|-
|  
|  
| <math>\textbf{c}</math>
| <math>\textbf{c}</math>
| [[comma]]
| [[Comma]]
|  
|  
| <math>\small 𝗽</math>
| <math>\small 𝗽</math>
| primes
| Primes
|  
|  
| <math>\scriptsize (d, 1)</math>
| <math>\scriptsize (d, 1)</math>
| integer
| Integer
| vector
| Vector
|  
|  
| [...⟩
| [...⟩
Line 2,960: Line 2,969:
|  
|  
| <math>\mathrm{c}_i</math>
| <math>\mathrm{c}_i</math>
| specific type: vector ([[prime-count vector]] or PC-vector)
| Specific type: vector ([[prime-count vector]] or PC-vector)
|-
|-
! colspan="17" | computation
! colspan="17" | Computation
|-
|-
|  
|  
| <math>\llzigzag·\,\rrzigzag\! _p</math>
| {{llzigzag}}<math>\,·\,</math>{{rrzigzag}}<math>_p</math>
| [[power sum]] (<math>p</math>-sum)
| [[Power sum]] (<math>p</math>-sum)
|  
|  
|  
|  
Line 2,972: Line 2,981:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 2,982: Line 2,991:
|  
|  
|-
|-
! colspan="17" | all-interval tuning schemes
! colspan="17" | All-interval tuning schemes
|-
|-
| <math>\mathrm{I}</math>
| <math>\mathrm{I}</math>
| <math>\mathrm{T}_{\text{p}}</math>
| <math>\mathrm{T}_{\text{p}}</math>
| [[prime proxy target-interval list]]
| [[Prime proxy target-interval list]]
|  
|  
| <math>\small 𝗽</math>
| <math>\small 𝗽</math>
| primes
| Primes
|  
|  
| <math>\scriptsize (d, d)</math>
| <math>\scriptsize (d, d)</math>
| integer
| Integer
| matrix
| Matrix
|  
|  
| ⟨[...⟩ ...]
| ⟨[...⟩ ...]
Line 3,004: Line 3,013:
|  
|  
| <math>X</math>
| <math>X</math>
| {{subpage|alternative complexities|uprev|s=Prescaling_vs._pretransforming|text=complexity pretransformer}}
| {{subpage|alternative complexities|uprev|s=Prescaling_vs._pretransforming|text=Complexity pretransformer}}
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math><ref group="note">In these tables, "alternative" means any complexity other than the default of log-product complexity, and "alt" stands for its abbreviation.</ref>
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math><ref group="note">In these tables, "alternative" means any complexity other than the default of log-product complexity, and "alt" stands for its abbreviation.</ref>
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math>
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math>
| complexity weight or <alternative>-complexity weight
| Complexity weight or <alternative>-complexity weight
|  
|  
| <math>\scriptsize (d, d)</math> or <math>\scriptsize (d+1, d+1)</math>
| <math>\scriptsize (d, d)</math> or <math>\scriptsize (d+1, d+1)</math>
| real
| Real
| matrix
| Matrix
| [⟨...] ...⟩
| [⟨...] ...⟩
|  
|  
Line 3,022: Line 3,031:
| <math>\text{diag}({\large\textbf{𝓁}}\hspace{2mu})</math>
| <math>\text{diag}({\large\textbf{𝓁}}\hspace{2mu})</math>
| <math>L</math>
| <math>L</math>
| [[log-prime matrix]]
| [[Log-prime matrix]]
|  
|  
| <math>\small\mathsf{oct}</math>/<math>\small 𝗽</math>
| <math>\small\mathsf{oct}</math>/<math>\small 𝗽</math>
| octaves per prime
| Octaves per prime
|  
|  
| <math>\scriptsize (d, d)</math>
| <math>\scriptsize (d, d)</math>
| real
| Real
| matrix
| Matrix
| [⟨...] ...⟩
| [⟨...] ...⟩
| ⟨[...⟩ ...]
| ⟨[...⟩ ...]
Line 3,040: Line 3,049:
|  
|  
| <math>q</math>
| <math>q</math>
| {{subpage|all-interval_tuning_schemes|uprev|s=dual_norms|text=interval complexity norm power}}
| {{subpage|all-interval_tuning_schemes|uprev|s=dual_norms|text=Interval complexity norm power}}
|  
|  
|  
|  
Line 3,046: Line 3,055:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 3,057: Line 3,066:
|-
|-
|  
|  
| <math>‖ · ‖_q</math>
| <math>\norm{·}_q</math>
| [[power norm]] (<math>p</math>-norm)
| [[Power norm]] (<math>p</math>-norm)
|  
|  
|  
|  
Line 3,064: Line 3,073:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 3,076: Line 3,085:
| <math>\dfrac1{1-\frac1q}</math>
| <math>\dfrac1{1-\frac1q}</math>
| <math>\text{dual}(q)</math>
| <math>\text{dual}(q)</math>
| {{subpage|all-interval tuning schemes|uprev|s=dual_norms|text=dual norm power}}
| {{subpage|all-interval tuning schemes|uprev|s=dual_norms|text=Dual norm power}}
|  
|  
|  
|  
Line 3,082: Line 3,091:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 3,093: Line 3,102:
|-
|-
|  
|  
| <math>‖X\mathbf{i}‖_q</math>
| <math>\norm{X\mathbf{i}}_q</math>
| [[interval complexity]]
| [[Interval complexity]]
|  
|  
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math>
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math>
Line 3,100: Line 3,109:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 3,111: Line 3,120:
|-
|-
|  
|  
| <math>‖𝒓X^{-1}‖_{\text{dual}(q)}</math>
| <math>\norm{𝒓X^{-1}}_{\text{dual}(q)}</math>
| [[retuning magnitude]]
| [[Retuning magnitude]]
|  
|  
| <math>\mathsf{¢}\small\mathsf{(C^{-1})}</math> or <math>\mathsf{¢}\small\mathsf{(}</math><alt>-<math>\small\mathsf{C^{-1})}</math>
| <math>\mathsf{¢}\small\mathsf{(C^{-1})}</math> or <math>\mathsf{¢}\small\mathsf{(}</math><alt>-<math>\small\mathsf{C^{-1})}</math>
Line 3,118: Line 3,127:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| real
| Real
| scalar
| Scalar
|  
|  
|  
|  
Line 3,128: Line 3,137:
|  
|  
|-
|-
! colspan="17" | alternative complexities
! colspan="17" | Alternative complexities
|-
|-
|  
|  
| <math>𝒑</math>
| <math>𝒑</math>
| {{subpage|alternative complexities|uprev|s=formulas|text=prime list}}<ref group="note">May be used for a prime-limit or for any prime-only list.</ref>
| {{subpage|alternative complexities|uprev|s=formulas|text=Prime list}}<ref group="note">May be used for a prime-limit or for any prime-only list.</ref>
|  
|  
|  
|  
Line 3,138: Line 3,147:
|  
|  
| <math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
| integer
| Integer
| list
| List
| [...]
| [...]
|  
|  
Line 3,150: Line 3,159:
|  
|  
| <math>\slant{\mathbf{1}}</math>
| <math>\slant{\mathbf{1}}</math>
| {{subpage|alternative complexities|uprev|s=proportionality to size|text=summation map}}
| {{subpage|alternative complexities|uprev|s=proportionality to size|text=Summation map}}
|  
|  
|  
|  
Line 3,156: Line 3,165:
|  
|  
| <math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
| integer
| Integer
| vector
| Vector
| ⟨...]
| ⟨...]
|  
|  
Line 3,168: Line 3,177:
|  
|  
| <math>1200</math>
| <math>1200</math>
| {{subpage|alternative complexities|uprev|s=Proportionality to size|text=octaves-to-cents conversion}}
| {{subpage|alternative complexities|uprev|s=Proportionality to size|text=Octaves-to-cents conversion}}
|  
|  
| ¢/oct
| ¢/oct
| cents per octave
| Cents per octave
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| integer
| Integer
| scalar
| Scalar
|  
|  
|  
|  
Line 3,186: Line 3,195:
|  
|  
| <math>Z</math>
| <math>Z</math>
| {{subpage|alternative complexities|uprev|s=Normifying: size-sensitizing matrix|text=size-sensitizing matrix}}
| {{subpage|alternative complexities|uprev|s=Normifying: size-sensitizing matrix|text=Size-sensitizing matrix}}
|  
|  
|  
|  
Line 3,192: Line 3,201:
|  
|  
| <math>\scriptsize (d+1, d)</math>
| <math>\scriptsize (d+1, d)</math>
| real
| Real
| matrix
| Matrix
| [⟨…]...]
| [⟨…]...]
|  
|  
Line 3,202: Line 3,211:
|  
|  
|-
|-
! colspan="17" | non-standard domain bases
! colspan="17" | Non-standard domain bases
|-
|-
| rowspan="2" |  
| rowspan="2" |  
| <math>B_s</math>
| <math>B_s</math>
| rowspan="2" | [[Domain_basis#Basis_matrix_conversion| (domain) basis (change) matrix]]
| rowspan="2" | [[Domain_basis#Basis_matrix_conversion|(Domain) basis (change) matrix]]
| rowspan="2" |  
| rowspan="2" |  
| <math>\small 𝗽</math>/<math>\small 𝗯</math>
| <math>\small 𝗽</math>/<math>\small 𝗯</math>
| primes per nonprime basis elements
| Primes per nonprime basis elements
| rowspan="2" |  
| rowspan="2" |  
| <math>\scriptsize (d_p, d_b)</math>
| <math>\scriptsize (d_p, d_b)</math>
| rowspan="2" | integer
| rowspan="2" | Integer
| rowspan="2" | matrix
| rowspan="2" | Matrix
| rowspan="2" | [[...] ...]
| rowspan="2" | [[...] ...]
| rowspan="2" | [[...] ...]
| rowspan="2" | [[...] ...]
Line 3,224: Line 3,233:
| <math>B_{Ls}</math>
| <math>B_{Ls}</math>
| <math>\small 𝗕</math>/<math>\small 𝗯</math>
| <math>\small 𝗕</math>/<math>\small 𝗯</math>
| superspace basis elements per (subspace) basis elements
| Superspace basis elements per (subspace) basis elements
| <math>\scriptsize (d_L, d_s)</math>
| <math>\scriptsize (d_L, d_s)</math>
|-
|-
! colspan="17" | embedding and projection
! colspan="17" | Embedding and projection
|-
|-
|  
|  
| <math>G</math>
| <math>G</math>
| [[generator embedding matrix| generator embedding (matrix)]]
| [[generator embedding matrix|Generator embedding (matrix)]]
|  
|  
| <math>\small 𝗽</math>/<math>\small 𝗴</math>
| <math>\small 𝗽</math>/<math>\small 𝗴</math>
| primes per generator
| Primes per generator
|  
|  
| <math>\scriptsize (d, r)</math>
| <math>\scriptsize (d, r)</math>
| real
| Real
| matrix
| Matrix
| [{...] ...⟩
| [{...] ...⟩
| {[...⟩ ...]
| {[...⟩ ...]
Line 3,247: Line 3,256:
|  
|  
|-
|-
| <math>G_cF^{-1}FM_c \\
| <math>G_cF^{-1}FM_c</math><br />
\mathrm{V}\textit{Λ}\mathrm{V}^{-1}</math>
<math>\mathrm{V}\textit{Λ}\mathrm{V}^{-1}</math>
| <math>P</math>
| <math>P</math>
| [[Projection matrix| projection (matrix)]]
| [[Projection matrix|Projection (matrix)]]
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} G \\[-2pt] 𝗽 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array}
\begin{array} {c} G \\[-2pt] 𝗽 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array}
Line 3,257: Line 3,266:
</math>
</math>
| <math>\small 𝗽</math>/<math>\small 𝗽</math>
| <math>\small 𝗽</math>/<math>\small 𝗽</math>
| primes per prime
| Primes per prime
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
\begin{array} {c} G \\[-3pt] (d, \cancel{r}) \end{array}
\begin{array} {c} G \\[-3pt] \left(d, \cancel{r}\right) \end{array}
\! \!  
\! \!  
\begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array}
\begin{array} {c} M \\[-3pt] \left(\cancel{r}, d\right) \end{array}
\! \!  
\! \!  
</math>
</math>
| <math>\scriptsize (d, d)</math>
| <math>\scriptsize (d, d)</math>
| real
| Real
| matrix
| Matrix
| [⟨...] ...⟩
| [⟨...] ...⟩
| ⟨[...⟩ ...]
| ⟨[...⟩ ...]
Line 3,278: Line 3,287:
| <math>GM\textbf{i}</math>
| <math>GM\textbf{i}</math>
| <math>P\textbf{i}</math>
| <math>P\textbf{i}</math>
| [[projected interval]]
| [[Projected interval]]
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} G \\[-2pt] 𝗽 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array}
\begin{array} {c} G \\[-2pt] 𝗽 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array}
Line 3,287: Line 3,296:
</math>
</math>
| <math>\small 𝗽</math>
| <math>\small 𝗽</math>
| primes
| Primes
| <math>\scriptsize  
| <math>\scriptsize  
\! \!  
\! \!  
\begin{array} {c} G \\[-3pt] (d, \cancel{r}) \end{array}
\begin{array} {c} G \\[-3pt] \left(d, \cancel{r}\right) \end{array}
\! \!  
\! \!  
\begin{array} {c} M \\[-3pt] (\cancel{r}, \cancel{d}) \end{array}
\begin{array} {c} M \\[-3pt] \left(\cancel{r}, \cancel{d}\right) \end{array}
\! \!  
\! \!  
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array}
\! \!  
\! \!  
</math>
</math>
| <math>\scriptsize (d, 1)</math>
| <math>\scriptsize (d, 1)</math>
| real
| Real
| vector
| Vector
|  
|  
| [...⟩
| [...⟩
Line 3,306: Line 3,315:
|  
|  
|  
|  
| specific type: vector ([[prime-count vector]] or PC-vector)
| Specific type: vector ([[prime-count vector]] or PC-vector)
|-
|-
|  
|  
| <math>\mathrm{U}</math>
| <math>\mathrm{U}</math>
| [[unchanged-interval basis]]
| [[Unchanged-interval basis]]
|  
|  
| <math>\small 𝗽</math>
| <math>\small 𝗽</math>
| primes
| Primes
|  
|  
| <math>\scriptsize (d, r)</math>
| <math>\scriptsize (d, r)</math>
|  
|  
| matrix
| Matrix
|  
|  
| [[...⟩ ...]
| [[...⟩ ...]
Line 3,324: Line 3,333:
|  
|  
| <math>\mathrm{u}_{ij}</math>
| <math>\mathrm{u}_{ij}</math>
| jargon name: eigenmonzo list
| Jargon name: eigenmonzo list
|-
|-
|  
|  
| <math>\textit{Λ}</math>
| <math>\textit{Λ}</math>
| [[scaling factor matrix| scaling factor (eigenvalue) matrix]]
| [[scaling factor matrix|Scaling factor (eigenvalue) matrix]]
|  
|  
|  
|  
Line 3,335: Line 3,344:
| <math>\scriptsize (d, d)</math>
| <math>\scriptsize (d, d)</math>
|  
|  
| matrix
| Matrix
| [⟨…] …⟩
| [⟨…] …⟩
| ⟨[…⟩ …]
| ⟨[…⟩ …]
Line 3,342: Line 3,351:
| <math>𝝀</math>
| <math>𝝀</math>
| <math>λ_i</math>
| <math>λ_i</math>
| mnemonic: <math>\mathrm{V}</math> is mirrored of <math>\textit{Λ}</math> which it combines with to create the projection matrix; previous name: eigenvalue matrix
| Mnemonic: <math>\mathrm{V}</math> is mirrored of <math>\textit{Λ}</math> which it combines with to create the projection matrix; previous name: eigenvalue matrix
|-
|-
|  
|  
| <math>\mathrm{V}</math>
| <math>\mathrm{V}</math>
| [[unrotated vector list| unrotated vector (eigenvector) list]]
| [[unrotated vector list|Unrotated vector (eigenvector) list]]
|  
|  
| <math>\small 𝗽</math>
| <math>\small 𝗽</math>
| primes
| Primes
|  
|  
| <math>\scriptsize (d, d)</math>
| <math>\scriptsize (d, d)</math>
|  
|  
| matrix
| Matrix
|  
|  
| ⟨[...⟩ ...]
| ⟨[...⟩ ...]
Line 3,360: Line 3,369:
|  
|  
| <math>\mathrm{v}_{ij}</math>
| <math>\mathrm{v}_{ij}</math>
| mnemonic: <math>\mathrm{V}</math> is mirrored of <math>\textit{Λ}</math> which it combines with to create the projection matrix; jargon name: eigenmonzo and comma list
| Mnemonic: <math>\mathrm{V}</math> is mirrored of <math>\textit{Λ}</math> which it combines with to create the projection matrix; jargon name: eigenmonzo and comma list
|-
|-
|  
|  
| <math>F</math>
| <math>F</math>
| [[generator form matrix]]
| [[Generator form matrix]]
|  
|  
|  
|  
Line 3,371: Line 3,380:
| <math>\scriptsize (r, r)</math>
| <math>\scriptsize (r, r)</math>
|  
|  
| matrix
| Matrix
| [{...] …}
| [{...] …}
|  
|  
Line 3,382: Line 3,391:
| <math>I</math>
| <math>I</math>
| <math>M_{\text{j}}</math>
| <math>M_{\text{j}}</math>
| [[Generator_embedding_optimization#Algebraic_setup| JI mapping (matrix)]]
| [[Generator_embedding_optimization#Algebraic_setup|JI mapping (matrix)]]
|  
|  
| <math>\small 𝗴</math>/<math>\small 𝗽</math>
| <math>\small 𝗴</math>/<math>\small 𝗽</math>
| generators per prime
| Generators per prime
|  
|  
| <math>\scriptsize (d, d)</math>
| <math>\scriptsize (d, d)</math>
| integer
| Integer
| matrix
| Matrix
| [⟨...] ...}
| [⟨...] ...}
| ⟨[...} ...]
| ⟨[...} ...]
Line 3,400: Line 3,409:
| <math>I</math>
| <math>I</math>
| <math>G_{\text{j}}</math>
| <math>G_{\text{j}}</math>
| [[Generator_embedding_optimization#Algebraic_setup| JI generator embedding (matrix)]]
| [[Generator_embedding_optimization#Algebraic_setup|JI generator embedding (matrix)]]
|  
|  
| <math>\small 𝗽</math>/<math>\small 𝗴</math>
| <math>\small 𝗽</math>/<math>\small 𝗴</math>
| primes per generator
| Primes per generator
|  
|  
| <math>\scriptsize (d, d)</math>
| <math>\scriptsize (d, d)</math>
| integer
| Integer
| matrix
| Matrix
| [{...] ...⟩
| [{...] ...⟩
| {[...⟩ ...]
| {[...⟩ ...]
Line 3,418: Line 3,427:
|  
|  
| <math>K</math>
| <math>K</math>
| [[Generator_embedding_optimization#How_to_build_constraint_matrices| constraint (matrix)]]
| [[Generator_embedding_optimization#How_to_build_constraint_matrices|Constraint (matrix)]]
|  
|  
|  
|  
Line 3,425: Line 3,434:
| <math>\scriptsize (k, r)</math>
| <math>\scriptsize (k, r)</math>
| <math>\scriptsize \{0, +1, -1\}</math>
| <math>\scriptsize \{0, +1, -1\}</math>
| matrix
| Matrix
| [[...] ...]
| [[...] ...]
|  
|  
Line 3,432: Line 3,441:
|  
|  
| <math>k_{ij}</math>
| <math>k_{ij}</math>
| mnemonic: <math>K</math>onstraint
| Mnemonic: <math>K</math>onstraint
|-
|-
|  
|  
| <math>𝒃</math>
| <math>𝒃</math>
| [[Generator embedding optimization#Generalizing to higher dimensions: the blend map| (generator tuning map) blend map]]
| [[Generator embedding optimization#Generalizing to higher dimensions: The blend map|(Generator tuning map) blend map]]
|  
|  
|  
|  
Line 3,442: Line 3,451:
|  
|  
| <math>\scriptsize (1, τ-1)</math>
| <math>\scriptsize (1, τ-1)</math>
| real
| Real
| vector
| Vector
| [...]
| [...]
|  
|  
Line 3,454: Line 3,463:
|  
|  
| <math>B</math>
| <math>B</math>
| [[Generator embedding optimization#How to identify tunings| (generator tuning map) blend matrix]]
| [[Generator embedding optimization#How to identify tunings|(Generator tuning map) blend matrix]]
|  
|  
|  
|  
Line 3,460: Line 3,469:
|  
|  
| <math>\scriptsize (d, τ-1)</math>
| <math>\scriptsize (d, τ-1)</math>
| real
| Real
| matrix
| Matrix
| [[...⟩...]
| [[...⟩...]
|  
|  
Line 3,472: Line 3,481:
|  
|  
| <math>D</math>
| <math>D</math>
| [[Generator embedding optimization#The deltas matrix| (generator tuning map) deltas matrix]]
| [[Generator embedding optimization#The deltas matrix|(Generator tuning map) deltas matrix]]
|  
|  
| <math>\mathsf{¢}</math>/<math>\small 𝗴</math>
| <math>\mathsf{¢}</math>/<math>\small 𝗴</math>
| cents per generator
| Cents per generator
|  
|  
| <math>\scriptsize (τ-1,r)</math>
| <math>\scriptsize (τ-1,r)</math>
| real
| Real
| matrix
| Matrix
| [{...] ...]
| [{...] ...]
|  
|  
Line 3,490: Line 3,499:
|  
|  
| <math>τ</math>
| <math>τ</math>
| [[Generator embedding optimization#The deltas matrix| tied basic minimax tuning count]]
| [[Generator embedding optimization#The deltas matrix|Tied basic minimax tuning count]]
|  
|  
|  
|  
Line 3,496: Line 3,505:
|  
|  
|  
|  
| integer
| Integer
| scalar
| Scalar
|  
|  
|  
|  
Line 3,506: Line 3,515:
|  
|  
|-
|-
! colspan="17" | exterior algebra
! colspan="17" | Exterior algebra
|-
|-
|  
|  
| <math>𝕞</math>
| <math>𝕞</math>
| [[multimap]]
| [[Multimap]]
|  
|  
| <math>\small 𝗴</math>/<math>\small 𝗽</math>
| <math>\small 𝗴</math>/<math>\small 𝗽</math>
| generators per prime
| Generators per prime
|  
|  
| <math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
| integer
| Integer
| multivector
| Multivector
| ⟨...] or ⟨⟨...]] or ⟨⟨⟨...]]] ...
| ⟨...] or ⟨⟨...]] or ⟨⟨⟨...]]] ...
|  
|  
Line 3,528: Line 3,537:
|  
|  
| <math>𝕔</math>
| <math>𝕔</math>
| [[multicomma]]
| [[Multicomma]]
|  
|  
| <math>\small 𝗽</math>
| <math>\small 𝗽</math>
| primes
| Primes
|  
|  
| <math>\scriptsize (1, n)</math>
| <math>\scriptsize (1, n)</math>
| integer
| Integer
| multivector
| Multivector
|  
|  
| [...⟩ or [[...⟩⟩ or [[[...⟩⟩⟩ ...
| [...⟩ or [[...⟩⟩ or [[[...⟩⟩⟩ ...
Line 3,546: Line 3,555:
|  
|  
| <math>𝕧</math>
| <math>𝕧</math>
| (generic temperament multivector)
| (Generic temperament multivector)
|  
|  
|  
|  
Line 3,552: Line 3,561:
|  
|  
| <math>\scriptsize (1, {{d}\choose{r}})</math> or <math>\scriptsize (1, {{d}\choose{n}})</math>
| <math>\scriptsize (1, {{d}\choose{r}})</math> or <math>\scriptsize (1, {{d}\choose{n}})</math>
| integer
| Integer
| multivector
| Multivector
| ⟨...] or ⟨⟨...]] or ⟨⟨⟨...]]] ...
| ⟨...] or ⟨⟨...]] or ⟨⟨⟨...]]] ...
| [...⟩ or [[...⟩⟩ or [[[...⟩⟩⟩ ...
| [...⟩ or [[...⟩⟩ or [[[...⟩⟩⟩ ...
Line 3,564: Line 3,573:
|  
|  
| <math>A</math>
| <math>A</math>
| (generic temperament matrix)
| (Generic temperament matrix)
|  
|  
|  
|  
Line 3,570: Line 3,579:
|  
|  
| <math>\scriptsize (g, d)</math> or <math>\scriptsize (d, g)</math>
| <math>\scriptsize (g, d)</math> or <math>\scriptsize (d, g)</math>
| integer
| Integer
| matrix
| Matrix
| [⟨...] ...}
| [⟨...] ...}
| ⟨[...} ...] or [[...⟩ ...]
| ⟨[...} ...] or [[...⟩ ...]
Line 3,582: Line 3,591:
|  
|  
| <math>v</math>
| <math>v</math>
| [[variance]]
| [[Variance]]
|  
|  
|  
|  
Line 3,600: Line 3,609:
|  
|  
| <math>g</math>
| <math>g</math>
| [[grade]]
| [[Grade]]
|  
|  
|  
|  
Line 3,606: Line 3,615:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| integer
| Integer
| scalar
| Scalar
|  
|  
|  
|  
Line 3,616: Line 3,625:
|  
|  
|-
|-
! colspan="17" | temperament addition
! colspan="17" | Temperament addition
|-
|-
| <math>\min(r, n)</math>
| <math>\min(r, n)</math>
| <math>g_\text{min}</math>
| <math>g_\text{min}</math>
| [[Temperament_addition#Introductory_examples| min-grade]]
| [[Temperament_addition#Introductory_examples|Min-grade]]
|  
|  
|  
|  
Line 3,626: Line 3,635:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| integer
| Integer
| scalar
| Scalar
|  
|  
|  
|  
Line 3,638: Line 3,647:
| <math>\max(r, n)</math>
| <math>\max(r, n)</math>
| <math>g_\text{max}</math>
| <math>g_\text{max}</math>
| [[Temperament_addition#Introductory_examples| max-grade]]
| [[Temperament_addition#Introductory_examples|Max-grade]]
|  
|  
|  
|  
Line 3,644: Line 3,653:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| integer
| Integer
| scalar
| Scalar
|  
|  
|  
|  
Line 3,656: Line 3,665:
|  
|  
| <math>L_\text{dep}</math>
| <math>L_\text{dep}</math>
| [[Temperament_addition#1._Find_the_.5Bmath.5DL_.7B.5Ctext.7Bdep.7D.7D.5B.2Fmath.5D| linear-dependence basis]]
| [[Temperament_addition#1._Find_the_.5Bmath.5DL_.7B.5Ctext.7Bdep.7D.7D.5B.2Fmath.5D|Linear-dependence basis]]
|  
|  
|  
|  
|  
|  
|  
|  
| <math>\scriptsize (l_\text{dep}, d)</math> or <math>\scriptsize (d, l_\text{dep})</math>
| <math>\scriptsize \left(l_\text{dep}, d\right)</math> or <math>\scriptsize \left(d, l_\text{dep}\right)</math>
| integer
| Integer
| matrix
| Matrix
| [⟨...]] or [[...] ...⟩
| [⟨...]] or [[...] ...⟩
| ⟨[...]] or [[...⟩ ...]
| ⟨[...]] or [[...⟩ ...]
Line 3,674: Line 3,683:
|  
|  
| <math>L_\text{ind}</math>
| <math>L_\text{ind}</math>
| [[Temperament_addition#Glossary| linear-independence basis]]
| [[Temperament_addition#Glossary|Linear-independence basis]]
|  
|  
|  
|  
|  
|  
|  
|  
| <math>\scriptsize (l_\text{ind}, d)</math> or <math>\scriptsize (d, l_\text{ind})</math>
| <math>\scriptsize \left(l_\text{ind}, d\right)</math> or <math>\scriptsize \left(d, l_\text{ind}\right)</math>
| integer
| Integer
| matrix
| Matrix
| [⟨...]] or [[...] ...⟩
| [⟨...]] or [[...] ...⟩
| ⟨[...]] or [[...⟩ ...]
| ⟨[...]] or [[...⟩ ...]
Line 3,692: Line 3,701:
| <math>\dim(L_\text{dep})</math>
| <math>\dim(L_\text{dep})</math>
| <math>l_\text{dep}</math>
| <math>l_\text{dep}</math>
| [[Temperament_addition#3._Linear_independence_between_temperaments| linear-dependence]]
| [[Temperament_addition#3._Linear_independence_between_temperaments|Linear-dependence]]
|  
|  
|  
|  
Line 3,698: Line 3,707:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| integer
| Integer
| scalar
| Scalar
|  
|  
|  
|  
Line 3,710: Line 3,719:
| <math>\dim(L_\text{ind})</math>
| <math>\dim(L_\text{ind})</math>
| <math>l_\text{ind}</math>
| <math>l_\text{ind}</math>
| [[Temperament_addition#3._Linear_independence_between_temperaments| linear-independence]]
| [[Temperament_addition#3._Linear_independence_between_temperaments|Linear-independence]]
|  
|  
|  
|  
Line 3,716: Line 3,725:
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| integer
| Integer
| scalar
| Scalar
|  
|  
|  
|  
Line 3,729: Line 3,738:
=== Units ===
=== Units ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" |
|-
|-
! symbol
! Symbol
! name
! Name
! vectorized
! Vectorized
|-
|-
| <math>\small 𝗴</math>
| <math>\small 𝗴</math>
| generators
| Generators
| yes
| Yes
|-
|-
| <math>\small 𝗽</math>
| <math>\small 𝗽</math>
| primes
| Primes
| yes
| Yes
|-
|-
| <math>\small 𝗯</math>
| <math>\small 𝗯</math>
| (subspace) basis elements
| (Subspace) basis elements
| yes
| Yes
|-
|-
| <math>\small 𝗕</math>
| <math>\small 𝗕</math>
| superspace basis elements
| Superspace basis elements
| yes
| Yes
|-
|-
| <math>\mathsf{¢}</math>
| <math>\mathsf{¢}</math>
| cents
| Cents
|  
|  
|-
|-
| <math>\mathsf{¢}\small{(}</math><weight><math>\small\mathsf{)}</math>
| <math>\mathsf{¢}\small{(}</math><weight><math>\small\mathsf{)}</math>
| weighted cents
| Weighted cents
|  
|  
|-
|-
| <math>\small\mathsf{oct}</math>
| <math>\small\mathsf{oct}</math>
| octaves
| Octaves
|  
|  
|}
|}
Line 3,765: Line 3,775:
=== Tuning schemes ===
=== Tuning schemes ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" |
|-
|-
! colspan="6" rowspan="3" | retuning (or mistuning) magnitude
! colspan="6" rowspan="3" | Retuning (or mistuning) magnitude
! colspan="12" rowspan="1" | damage
! colspan="12" rowspan="1" | Damage
! rowspan="5" | target
! rowspan="5" | Target<br />intervals
 
! colspan="2" rowspan="4" | Systematic name
intervals
! rowspan="5" | Previously named tuning schemes that are specific types of this tuning scheme
! colspan="2" rowspan="4" | systematic name
! rowspan="5" | Of interest?
! rowspan="5" | previously named tuning schemes that are specific types of this tuning scheme
! rowspan="5" | of interest?
|-
|-
! colspan="9" rowspan="1" | weight
! colspan="9" rowspan="1" | Weight
! colspan="3" rowspan="1" | optimization
! colspan="3" rowspan="1" | Optimization
|-
|-
! colspan="6" rowspan="1" | interval complexity
! colspan="6" rowspan="1" | Interval complexity
! colspan="3" rowspan="1" | slope
! colspan="3" rowspan="1" | Slope
! colspan="1" rowspan="3" | initial
! colspan="1" rowspan="3" | Initial
! colspan="1" rowspan="3" | name
! colspan="1" rowspan="3" | Name
! colspan="1" rowspan="3" | power
! colspan="1" rowspan="3" | Power
|-
|-
! colspan="3" rowspan="1" | norm pretransformer
! colspan="3" rowspan="1" | Norm pretransformer
! colspan="3" rowspan="1" | norm power
! colspan="3" rowspan="1" | Norm power
! colspan="3" rowspan="1" | norm pretransformer
! colspan="3" rowspan="1" | Norm pretransformer
! colspan="3" rowspan="1" | norm power
! colspan="3" rowspan="1" | Norm power
! colspan="1" rowspan="2" | initial
! colspan="1" rowspan="2" | Initial
! colspan="1" rowspan="2" | name
! colspan="1" rowspan="2" | Name
! colspan="1" rowspan="2" | multiplier
! colspan="1" rowspan="2" | Multiplier
|-
|-
! initial
! Initial
! name
! Name
! multiplier
! Multiplier
! initial
! Initial
! name
! Name
! power
! Power
! initial
! Initial
! name
! Name
! multiplier
! Multiplier
! initial
! Initial
! name
! Name
! power
! Power
! colspan="1" | abbreviated
! colspan="1" | Abbreviated
! colspan="1" | read ("____ tuning scheme")
! colspan="1" | Read ("____ tuning scheme")
|-
|-
| colspan="3" | <none>
| colspan="3" | <none>
| rowspan="4" | <n/a>
| rowspan="4" | <n/a>
| rowspan="2" | maximum
| rowspan="2" | Maximum
| rowspan="2" | ∞
| rowspan="2" | &infin;
| colspan="3" | <none>
| colspan="3" | <none>
| rowspan="2" | (t)
| rowspan="2" | (t)
| rowspan="2" | taxicab
| rowspan="2" | Taxicab
| rowspan="2" | 1
| rowspan="2" | 1
| rowspan="4" | S
| rowspan="4" | S
| rowspan="4" | simplicity-weight
| rowspan="4" | Simplicity-weight
| rowspan="4" | 1/complexity
| rowspan="4" | 1/Complexity
| rowspan="31" | <n/a>
| rowspan="31" | <n/a>
| rowspan="13" | minimax
| rowspan="13" | Minimax
| rowspan="13" | ∞
| rowspan="13" | &infin;
| rowspan="4" | all
| rowspan="4" | All
| minimax-S
| Minimax-S
| minimax simplicity-weight damage
| Minimax simplicity-weight damage
| "[[TOP]]"/"[[T1]]"/"[[TIPTOP]]"*, "[[CTOP]]", "[[POTOP]]"/"[[POTT]]"*
| "[[TOP]]"/"[[T1]]"/"[[TIPTOP]]"*, "[[CTOP]]", "[[POTOP]]"/"[[POTT]]"*
| yes
| yes
Line 3,829: Line 3,838:
| colspan="3" | <various>
| colspan="3" | <various>
| colspan="3" | <various>
| colspan="3" | <various>
| minimax-<alt>-S
| Minimax-<alt>-S
| minimax <alternative>-simplicity-weight damage
| Minimax <alternative>-simplicity-weight damage
| "[[BOP tuning| BOP]]", "[[Weil Norms, Tenney-Weil Norms, and TWp Interval and Tuning Space| Weil]]", "[[Kees]]"
| "[[BOP tuning|BOP]]", "[[Weil Norms, Tenney-Weil Norms, and TWp Interval and Tuning Space|Weil]]", "[[Kees]]"
| yes
| yes
|-
|-
Line 3,841: Line 3,850:
| rowspan="2" | Euclidean
| rowspan="2" | Euclidean
| rowspan="2" | 2
| rowspan="2" | 2
| minimax-ES
| Minimax-ES
| minimax Euclideanized-simplicity-weight damage
| Minimax Euclideanized-simplicity-weight damage
| "[[Tenney-Euclidean tuning| TE]]"/"[[T2]]"/"[[TOP-RMS]]", "[[CTE tuning| CTE]]", "[[POTE tuning| POTE]]"
| "[[Tenney-Euclidean tuning|TE]]"/"[[T2]]"/"[[TOP-RMS]]", "[[CTE tuning|CTE]]", "[[POTE tuning|POTE]]"
| yes
| yes
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| colspan="3" | <various>
| colspan="3" | <various>
| minimax-E-<alt>-S
| Minimax-E-<alt>-S
| minimax Euclideanized-<alternative>-simplicity-weight damage
| Minimax Euclideanized-<alternative>-simplicity-weight damage
| "[[Frobenius]]", "[[BE]]", "[[WE]]", "[[KE]]"
| "[[Frobenius]]", "[[BE]]", "[[WE]]", "[[KE]]"
| yes
| Yes
|-
|-
| colspan="6" rowspan="27" | <n/a>
| colspan="6" rowspan="27" | <n/a>
| colspan="6" | <n/a>
| colspan="6" | <n/a>
| U
| U
| unity-weight
| Unity-weight
| <none>
| <none>
| rowspan="27" | <set>
| rowspan="27" | <set>
| <set> minimax-U
| <set> Minimax-U
| <set> minimax unity-weight damage
| <set> Minimax unity-weight damage
| "[[Minimax tuning| minimax]]"
| "[[Minimax tuning|Minimax]]"
| yes
| yes
|-
|-
| colspan="3" | <none>
| colspan="3" | <none>
| rowspan="2" | (t)
| rowspan="2" | (t)
| rowspan="2" | taxicab
| rowspan="2" | Taxicab
| rowspan="2" | 1
| rowspan="2" | 1
| rowspan="4" | S
| rowspan="4" | S
| rowspan="4" | simplicity-weight
| rowspan="4" | Simplicity-weight
| rowspan="4" | 1/complexity
| rowspan="4" | 1/Complexity
| <set> minimax-S
| <set> Minimax-S
| <set> minimax simplicity-weight damage
| <set> Minimax simplicity-weight damage
|  
|  
| yes
| Yes
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| <set> minimax-<alt>-S
| <set> Minimax-<alt>-S
| <set> minimax <alternative>-simplicity-weight damage
| <set> Minimax <alternative>-simplicity-weight damage
|  
|  
|  
|  
Line 3,886: Line 3,895:
| rowspan="2" | Euclidean
| rowspan="2" | Euclidean
| rowspan="2" | 2
| rowspan="2" | 2
| <set> minimax-ES
| <set> Minimax-ES
| <set> minimax Euclideanized-simplicity-weight damage
| <set> Minimax Euclideanized-simplicity-weight damage
|  
|  
|  
|  
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| <set> minimax-E-<alt>-S
| <set> Minimax-E-<alt>-S
| <set> minimax Euclideanized-<alternative>-simplicity-weight damage
| <set> Minimax Euclideanized-<alternative>-simplicity-weight damage
|  
|  
|  
|  
Line 3,899: Line 3,908:
| colspan="3" | <none>
| colspan="3" | <none>
| rowspan="2" | (t)
| rowspan="2" | (t)
| rowspan="2" | taxicab
| rowspan="2" | Taxicab
| rowspan="2" | 1
| rowspan="2" | 1
| rowspan="4" | C
| rowspan="4" | C
| rowspan="4" | complexity-weight
| rowspan="4" | Complexity-weight
| rowspan="4" | complexity
| rowspan="4" | Complexity
| <set> minimax-C
| <set> Cinimax-C
| <set> minimax complexity-weight damage
| <set> Cinimax complexity-weight damage
|  
|  
| yes
| Yes
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| <set> minimax-<alt>-C
| <set> Minimax-<alt>-C
| <set> minimax <alternative>-complexity-weight damage
| <set> Minimax <alternative>-complexity-weight damage
|  
|  
|  
|  
Line 3,919: Line 3,928:
| rowspan="2" | Euclidean
| rowspan="2" | Euclidean
| rowspan="2" | 2
| rowspan="2" | 2
| <set> minimax-EC
| <set> Minimax-EC
| <set> minimax Euclideanized-complexity-weight damage
| <set> Minimax Euclideanized-complexity-weight damage
|  
|  
|  
|  
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| <set> minimax-E-<alt>-C
| <set> Minimax-E-<alt>-C
| <set> minimax Euclideanized-<alternative>-complexity-weight damage
| <set> Minimax Euclideanized-<alternative>-complexity-weight damage
|  
|  
|  
|  
Line 3,932: Line 3,941:
| colspan="6" | <n/a>
| colspan="6" | <n/a>
| U
| U
| unity-weight
| Unity-weight
| <none>
| <none>
| rowspan="9" | miniRMS
| rowspan="9" | MiniRMS
| rowspan="9" | 2
| rowspan="9" | 2
| <set> miniRMS-U
| <set> MiniRMS-U
| <set> miniRMS unity-weight damage
| <set> MiniRMS unity-weight damage
| "[[least squares]]"
| "[[Least squares]]"
| yes
| yes
|-
|-
| colspan="3" | <none>
| colspan="3" | <none>
| rowspan="2" | (t)
| rowspan="2" | (t)
| rowspan="2" | taxicab
| rowspan="2" | Taxicab
| rowspan="2" | 1
| rowspan="2" | 1
| rowspan="4" | S
| rowspan="4" | S
| rowspan="4" | simplicity-weight
| rowspan="4" | Simplicity-weight
| rowspan="4" | 1/complexity
| rowspan="4" | 1/Complexity
| <set> miniRMS-S
| <set> MiniRMS-S
| <set> miniRMS simplicity-weight damage
| <set> MiniRMS simplicity-weight damage
|  
|  
| yes
| Yes
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| <set> miniRMS-<alt>-S
| <set> MiniRMS-<alt>-S
| <set> miniRMS <alternative>-simplicity-weight damage
| <set> MiniRMS <alternative>-simplicity-weight damage
|  
|  
|  
|  
Line 3,963: Line 3,972:
| rowspan="2" | Euclidean
| rowspan="2" | Euclidean
| rowspan="2" | 2
| rowspan="2" | 2
| <set> miniRMS-ES
| <set> MiniRMS-ES
| <set> miniRMS Euclideanized-simplicity-weight damage
| <set> MiniRMS Euclideanized-simplicity-weight damage
|  
|  
|  
|  
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| <set> miniRMS-E-<alt>-S
| <set> MiniRMS-E-<alt>-S
| <set> miniRMS Euclideanized-<alternative>-simplicity-weight damage
| <set> MiniRMS Euclideanized-<alternative>-simplicity-weight damage
|  
|  
|  
|  
Line 3,976: Line 3,985:
| colspan="3" | <none>
| colspan="3" | <none>
| rowspan="2" | (t)
| rowspan="2" | (t)
| rowspan="2" | taxicab
| rowspan="2" | Taxicab
| rowspan="2" | 1
| rowspan="2" | 1
| rowspan="4" | C
| rowspan="4" | C
| rowspan="4" | complexity-weight
| rowspan="4" | Complexity-weight
| rowspan="4" | complexity
| rowspan="4" | Complexity
| <set> miniRMS-C
| <set> MiniRMS-C
| <set> miniRMS complexity-weight damage
| <set> MiniRMS complexity-weight damage
|  
|  
| yes
| yes
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| <set> miniRMS-<alt>-C
| <set> MiniRMS-<alt>-C
| <set> miniRMS <alternative>-complexity-weight damage
| <set> MiniRMS <alternative>-complexity-weight damage
|  
|  
|  
|  
Line 3,996: Line 4,005:
| rowspan="2" | Euclidean
| rowspan="2" | Euclidean
| rowspan="2" | 2
| rowspan="2" | 2
| <set> miniRMS-EC
| <set> MiniRMS-EC
| <set> miniRMS Euclideanized-complexity-weight damage
| <set> MiniRMS Euclideanized-complexity-weight damage
|  
|  
|  
|  
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| <set> miniRMS-E-<alt>-C
| <set> MiniRMS-E-<alt>-C
| <set> miniRMS Euclideanized-<alternative>-complexity-weight damage
| <set> MiniRMS Euclideanized-<alternative>-complexity-weight damage
|  
|  
|  
|  
Line 4,009: Line 4,018:
| colspan="6" | <n/a>
| colspan="6" | <n/a>
| U
| U
| unity-weight
| Unity-weight
| <none>
| <none>
| rowspan="9" | miniaverage
| rowspan="9" | Miniaverage
| rowspan="9" | 1
| rowspan="9" | 1
| <set> miniaverage-U
| <set> Miniaverage-U
| <set> miniaverage unity-weight damage
| <set> Miniaverage unity-weight damage
|  
|  
| yes
| yes
Line 4,020: Line 4,029:
| colspan="3" | <none>
| colspan="3" | <none>
| rowspan="2" | (t)
| rowspan="2" | (t)
| rowspan="2" | taxicab
| rowspan="2" | Taxicab
| rowspan="2" | 1
| rowspan="2" | 1
| rowspan="4" | S
| rowspan="4" | S
| rowspan="4" | simplicity-weight
| rowspan="4" | Simplicity-weight
| rowspan="4" | 1/complexity
| rowspan="4" | 1/Complexity
| <set> miniaverage-S
| <set> Miniaverage-S
| <set> miniaverage simplicity-weight damage
| <set> Miniaverage simplicity-weight damage
|  
|  
| yes
| Yes
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| <set> miniaverage-<alt>-S
| <set> Miniaverage-<alt>-S
| <set> miniaverage <alternative>-simplicity-weight damage
| <set> Miniaverage <alternative>-simplicity-weight damage
|  
|  
|  
|  
Line 4,040: Line 4,049:
| rowspan="2" | Euclidean
| rowspan="2" | Euclidean
| rowspan="2" | 2
| rowspan="2" | 2
| <set> miniaverage-ES
| <set> Miniaverage-ES
| <set> miniaverage Euclideanized-simplicity-weight damage
| <set> Miniaverage Euclideanized-simplicity-weight damage
|  
|  
|  
|  
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| <set> miniaverage-E-<alt>-S
| <set> Miniaverage-E-<alt>-S
| <set> miniaverage Euclideanized-<alternative>-simplicity-weight damage
| <set> Miniaverage Euclideanized-<alternative>-simplicity-weight damage
|  
|  
|  
|  
Line 4,053: Line 4,062:
| colspan="3" | <none>
| colspan="3" | <none>
| rowspan="2" | (t)
| rowspan="2" | (t)
| rowspan="2" | taxicab
| rowspan="2" | Taxicab
| rowspan="2" | 1
| rowspan="2" | 1
| rowspan="4" | C
| rowspan="4" | C
| rowspan="4" | complexity-weight
| rowspan="4" | Complexity-weight
| rowspan="4" | complexity
| rowspan="4" | Complexity
| <set> miniaverage-C
| <set> Miniaverage-C
| <set> miniaverage complexity-weight damage
| <set> Miniaverage complexity-weight damage
|  
|  
| yes
| Yes
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| <set> miniaverage-<alt>-C
| <set> Miniaverage-<alt>-C
| <set> miniaverage <alternative>-complexity-weight damage
| <set> Miniaverage <alternative>-complexity-weight damage
|  
|  
|  
|  
Line 4,073: Line 4,082:
| rowspan="2" | Euclidean
| rowspan="2" | Euclidean
| rowspan="2" | 2
| rowspan="2" | 2
| <set> miniaverage-EC
| <set> Miniaverage-EC
| <set> miniaverage Euclideanized-complexity-weight damage
| <set> Miniaverage Euclideanized-complexity-weight damage
|  
|  
|  
|  
|-
|-
| colspan="3" | <various>
| colspan="3" | <various>
| <set> miniaverage-E-<alt>-C
| <set> Miniaverage-E-<alt>-C
| <set> miniaverage Euclideanized-<alternative>-complexity-weight damage
| <set> Miniaverage Euclideanized-<alternative>-complexity-weight damage
|  
|  
|  
|  
Line 4,087: Line 4,096:
=== Damages ===
=== Damages ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" |
|-
|-
! colspan="2" | quantity
! colspan="2" | Quantity
! colspan="2" | unit
! colspan="2" | Unit
|-
|-
! abbreviation
! Abbreviation
! name
! Name
! symbol
! Symbol
! name
! Name
|-
|-
| U-damage
| U-damage
| unity-weight damage
| Unity-weight damage
| <math>\mathsf{¢}\small\mathsf{(U)}</math>
| <math>\mathsf{¢}\small\mathsf{(U)}</math>
| unity-weighted cents
| Unity-weighted cents
|-
|-
| C-damage
| C-damage
| complexity-weight damage
| Complexity-weight damage
| <math>\mathsf{¢}\small\mathsf{(C)}</math>
| <math>\mathsf{¢}\small\mathsf{(C)}</math>
| complexity-weighted cents
| Complexity-weighted cents
|-
|-
| <alt>-C-damage
| <alt>-C-damage
Line 4,122: Line 4,132:
|-
|-
| S-damage
| S-damage
| simplicity-weight damage
| Simplicity-weight damage
| <math>\mathsf{¢}\small\mathsf{(S)}</math>
| <math>\mathsf{¢}\small\mathsf{(S)}</math>
| simplicity-weighted cents
| Simplicity-weighted cents
|-
|-
| <alt>-S-damage
| <alt>-S-damage
Line 4,144: Line 4,154:
=== Complexity and simplicity ===
=== Complexity and simplicity ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" |
|-
|-
! colspan="2" | quantity
! colspan="2" | Quantity
! colspan="2" | unit
! colspan="2" | Unit
|-
|-
! abbreviation
! Abbreviation
! name
! Name
! unit
! Unit
! name
! Name
|-
|-
| C
| C
| complexity
| Complexity
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> = <math>\small\mathsf{(C)}</math>
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> = <math>\small\mathsf{(C)}</math>
| complexity weight
| Complexity weight
|-
|-
| <alt>-C
| <alt>-C
Line 4,174: Line 4,185:
|-
|-
| S
| S
| simplicity
| Simplicity
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math> = <math>\small\mathsf{(S)}</math>
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math> = <math>\small\mathsf{(S)}</math>
| simplicity weight
| Simplicity weight
|-
|-
| <alt>-S
| <alt>-S
Line 4,195: Line 4,206:


== WinCompose ==
== 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 ♯, ♭, ¢, √, °, ₂, ×, ⁻¹, ⟩, ∞, and ϕ? Well, try [http://wincompose.info/ 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 ⎄. So for example, you type ♯ as ⎄##, ♭ as ⎄bb, ¢ as ⎄c/, √ as ⎄v/, ° as ⎄00, ₂ as ⎄-2, × as ⎄xx, ⁻¹ as ⎄11, ⟩ as ⎄>>, ∞ as ⎄88, and ϕ as ⎄8f.  
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 ♯, ♭, ¢, √, °, ₂, ×, {{inv}}, ⟩, ∞, and ϕ? Well, try [http://wincompose.info/ 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 ⎄. So for example, you type {{nowrap|♯ as <code>⎄##</code>|♭ as <code>⎄bb</code>|¢ as <code>⎄c/</code>|√ as <code>⎄v/</code>|° as <code>⎄00</code>|₂ as <code>⎄-2</code>|× as <code>⎄xx</code>|{{inv}} as <code>⎄11</code>|⟩ as <code>⎄&gt;&gt;</code>|∞ as <code>⎄88</code>|and ϕ as <code>⎄8f</code>}}.  


For Windows users, install WinCompose then copy-paste the contents of this file: https://dkeenan.com/XCompose.txt into your user sequences (Show sequences → User-defined sequences → 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 Windows users, install WinCompose then copy-paste the contents of this file: https://dkeenan.com/XCompose.txt into your user sequences (Show sequences &rarr; User-defined sequences &rarr; 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
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
Line 4,203: Line 4,214:
=== Table of noteworthy sequences ===
=== Table of noteworthy sequences ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|-
|+ style="font-size: 105%; white-space: nowrap;" | Dave Keenan & Douglas Blumeyer's compose-key sequences
! scope="col" width="130px" | Compose-key sequence
|- style="white-space: nowrap;"
! scope="col" width="75px" | Resulting text
! scope="col" style="width: 130px;" | Compose-key sequence
! scope="col" style="width: 75px;" | Resulting text
! Description
! Description
|-
|- style="white-space: nowrap;"
! colspan="3" rowspan="1" | Keyboard key symbols
! colspan="3" | Keyboard key symbols
|-
|-
| ⎄⎄⎄
| ⎄⎄⎄
Line 4,262: Line 4,274:
| Dotted circle, represents any character (such as the character preceding a combining mark)
| Dotted circle, represents any character (such as the character preceding a combining mark)
|-
|-
! colspan="3" rowspan="1" | Double key sequences
! colspan="3" style="white-space: nowrap;" | Double key sequences
|-
|-
| ⎄␣␣
| ⎄␣␣
Line 4,300: Line 4,312:
| Black star
| Black star
|-
|-
| ⎄{{''}}
| ⎄&#39;&#39;
| ′
| ′
| prime mark
| prime mark
Line 4,396: Line 4,408:
| Right etc. arrows
| Right etc. arrows
|-
|-
! colspan="3" rowspan="1" | Multiplication operators
! colspan="3" style="white-space: nowrap;" | Multiplication operators
|-
|-
| ⎄xx
| ⎄xx
Line 4,426: Line 4,438:
| Bullet (infix: fat dot product, Dorst)
| Bullet (infix: fat dot product, Dorst)
|-
|-
! colspan="3" rowspan="1" | Other operators
! colspan="3" style="white-space: nowrap;" | Other operators
|-
|-
| ⎄v/
| ⎄v/
Line 4,630: Line 4,642:
| Double-struck italic, ⎄{{pipe}} 0e ⎄{{pipe}} i (only a few)
| Double-struck italic, ⎄{{pipe}} 0e ⎄{{pipe}} i (only a few)
|-
|-
! colspan="3" rowspan="1" | Power statistics brackets
! colspan="3" style="white-space: nowrap;" | Power statistics brackets
|-
|-
| ⎄{{pipe}} {{pipe}}  
| ⎄{{pipe}} {{pipe}}  
Line 4,664: Line 4,676:
| Right power-sum bracket (substitute for {{rrzz}} when HTML is not available)
| Right power-sum bracket (substitute for {{rrzz}} when HTML is not available)
|-
|-
! colspan="3" rowspan="1" | Combining marks
! colspan="3" style="white-space: nowrap;" | Combining marks
|-
|-
| ⎄\-
| ⎄\-
Line 4,684: Line 4,696:


=== Keyboard map ===
=== Keyboard map ===
[[File:WinCompose keyboard map.png| 1000px]]
[[File:WinCompose keyboard map.png|1000px]]


== Footnotes ==
== Footnotes ==
Retrieved from "https://en.xen.wiki/w/Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT/Conventions_for_names,_variables,_units,_and_notations"