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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 116: Line 117:
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
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 255: Line 256:
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} 𝒈 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array}  
\begin{array} {c} 𝒈 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} 𝑀 \\[-2pt] \cancel{𝗴} \hspace{-2mu} / \hspace{-2mu} 𝗽 \end{array}  
\begin{array} {c} 𝑀 \\[-2pt] \cancel{𝗴} \hspace{-2mu} / \hspace{-2mu} 𝗽 \end{array}  
</math>
</math>
Line 301: Line 302:
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</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}}
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
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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]]
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
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|-
|-
|  
|  
| <math>\llangle\,\cdot\,\rrangle_p</math>
| <math>\llangle\,·\,\rrangle_p</math>
| [[Power mean]] (<math>p</math>-mean)
| [[Power mean]] (<math>p</math>-mean)
|  
|  
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|  
|  
| <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 482: Line 483:
| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} \abs{\mathrm{e}} \\[-2pt] {\small\mathsf{¢}} \end{array}  
\begin{array} {c} \abs{\mathrm{e}} \\[-2pt] {\small\mathsf{¢}} \end{array}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} w \\[-2pt] \mathsf{(U, C, \text{or}\,S)} \end{array}  
\begin{array} {c} w \\[-2pt] \mathsf{(U, C, \text{or}\,S)} \end{array}  
</math>
</math>
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| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
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| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</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]]
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
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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]]
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
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| <math>\scriptsize  
| <math>\scriptsize  
\begin{array} {c} \abs{\textbf{e}} \\[-2pt] {\small\mathsf{¢}} \end{array}  
\begin{array} {c} \abs{\textbf{e}} \\[-2pt] {\small\mathsf{¢}} \end{array}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} W \\[-2pt] (\mathsf{U, C, \text{or}\,S}) \end{array}  
\begin{array} {c} W \\[-2pt] (\mathsf{U, C, \text{or}\,S}) \end{array}  
</math>
</math>
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
Line 934: Line 937:
=== 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 1,026: Line 1,031:
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
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]]
Line 1,220: Line 1,225:
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
Line 1,243: Line 1,248:
| 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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
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]]
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
Line 1,323: Line 1,328:
|-
|-
|  
|  
| <math>\llangle\,\cdot\,\rrangle_p</math>
| <math>\llangle\,·\,\rrangle_p</math>
| [[Power mean]] (<math>p</math>-mean)
| [[Power mean]] (<math>p</math>-mean)
|  
|  
Line 1,431: Line 1,436:
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
Line 1,459: Line 1,464:
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
Line 1,487: Line 1,492:
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
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]]
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
Line 1,701: Line 1,706:
|-
|-
|  
|  
| <math>\llzigzag\cdot\,\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
Line 2,148: Line 2,156:
=== 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 2,217: Line 2,226:
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
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}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
\begin{array} {c} G_{\text{j}} \\[-2pt] \cancel{𝗽} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array}
\begin{array} {c} G_{\text{j}} \\[-2pt] \cancel{𝗽} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} M_{\text{j}} \\[-2pt] \cancel{𝗴} \hspace{-2mu} / \hspace{-2mu} 𝗽 \end{array}
\begin{array} {c} M_{\text{j}} \\[-2pt] \cancel{𝗴} \hspace{-2mu} / \hspace{-2mu} 𝗽 \end{array}
</math>
</math>
Line 2,358: Line 2,367:
\begin{array} {c} \slant{\mathbf{1}} \\[-3pt] \left(\cancel{1}, \cancel{d}\right) \end{array}
\begin{array} {c} \slant{\mathbf{1}} \\[-3pt] \left(\cancel{1}, \cancel{d}\right) \end{array}
\! \!  
\! \!  
\begin{array} {c} L \\[-3pt] \left(\cancel{d}, \cancel{d}) \end{array}
\begin{array} {c} L \\[-3pt] \left(\cancel{d}, \cancel{d}\right) \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
\! \!  
\! \!  
Line 2,382: Line 2,391:
| <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}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
\begin{array} {c} G \\[-2pt] \cancel{𝗽} \hspace{-2mu} / \hspace{-2mu} 𝗴 \end{array}
\begin{array} {c} G \\[-2pt] \cancel{𝗽} \hspace{-2mu} / \hspace{-2mu} 𝗴 \end{array}
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}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
\begin{array} {c} G \\[-2pt] \cancel{𝗽} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array}
\begin{array} {c} G \\[-2pt] \cancel{𝗽} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} M \\[-2pt] \cancel{𝗴} \hspace{-2mu} / \hspace{-2mu} 𝗽 \end{array}
\begin{array} {c} M \\[-2pt] \cancel{𝗴} \hspace{-2mu} / \hspace{-2mu} 𝗽 \end{array}
</math>
</math>
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]]
Line 2,483: Line 2,492:
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
Line 2,506: Line 2,515:
| 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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
Line 2,535: Line 2,544:
| 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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
Line 2,586: Line 2,595:
|-
|-
|  
|  
| <math>\llangle\,\cdot\,\rrangle_p</math>
| <math>\llangle\,·\,\rrangle_p</math>
| [[Power mean]] (<math>p</math>-mean)
| [[Power mean]] (<math>p</math>-mean)
|  
|  
Line 2,694: Line 2,703:
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
Line 2,722: Line 2,731:
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
Line 2,750: Line 2,759:
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
Line 2,773: Line 2,782:
| 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]]
| <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}  
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}  
</math>
</math>
Line 2,851: Line 2,860:
| Entry-wise reciprocal of <math>C</math>
| Entry-wise reciprocal of <math>C</math>
|-
|-
| <math>\abs{\textbf{e}} W \\
| <math>\abs{\textbf{e}} W</math><br />
1200×\slant{\mathbf{1}}L\abs{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]]
Line 2,965: Line 2,974:
|-
|-
|  
|  
| <math>\llzigzag\cdot\,\rrzigzag\! _p</math>
| {{llzigzag}}<math>\,·\,</math>{{rrzigzag}}<math>_p</math>
| [[Power sum]] (<math>p</math>-sum)
| [[Power sum]] (<math>p</math>-sum)
|  
|  
Line 3,057: Line 3,066:
|-
|-
|  
|  
| <math>\norm{\cdot}_q</math>
| <math>\norm{·}_q</math>
| [[Power norm]] (<math>p</math>-norm)
| [[Power norm]] (<math>p</math>-norm)
|  
|  
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}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} M \\[-2pt] \cancel{𝗴} \hspace{-2mu} / \hspace{-2mu} 𝗽 \end{array}
\begin{array} {c} M \\[-2pt] \cancel{𝗴} \hspace{-2mu} / \hspace{-2mu} 𝗽 \end{array}
</math>
</math>
Line 3,281: Line 3,290:
| <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}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} M \\[-2pt] \cancel{𝗴} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} M \\[-2pt] \cancel{𝗴} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} \\[-2pt] \cdot \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}
</math>
</math>
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
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
 
intervals
! colspan="2" rowspan="4" | Systematic name
! colspan="2" rowspan="4" | Systematic name
! rowspan="5" | Previously named tuning schemes that are specific types of this tuning scheme
! rowspan="5" | Previously named tuning schemes that are specific types of this tuning scheme
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| colspan="3" | <none>
| colspan="3" | <none>
| rowspan="2" | (t)
| rowspan="2" | (t)
| rowspan="2" | Raxicab
| rowspan="2" | Taxicab
| rowspan="2" | 1
| rowspan="2" | 1
| rowspan="4" | S
| rowspan="4" | S
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=== 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
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=== 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
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== 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 {{nowrap|♯ as ⎄##| ♭ as ⎄bb|¢ as ⎄c/|√ as ⎄v/|° as ⎄00|₂ as ⎄-2|× as ⎄xx|⁻¹ as ⎄11|⟩ as ⎄&gt;&gt;|∞ 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 &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 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.
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=== 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
|- style="white-space: nowrap;"
|- style="white-space: nowrap;"
! scope="col" style="width: 130px;" | Compose-key sequence
! scope="col" style="width: 130px;" | Compose-key sequence
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| 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" style="white-space: nowrap;" | Double key sequences
! colspan="3" style="white-space: nowrap;" | Double key sequences
|-
|-
| ⎄␣␣
| ⎄␣␣
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|-
|-
| ⎄..
| ⎄..
| \cdot
| ·
| Middle dot (used to multiply units when juxtaposition is ambiguous)
| Middle dot (used to multiply units when juxtaposition is ambiguous)
|-
|-
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| Black star
| Black star
|-
|-
| ⎄{{''}}
| ⎄&#39;&#39;
| ′
| ′
| prime mark
| prime mark
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| Right etc. arrows
| Right etc. arrows
|-
|-
! colspan="3" rowspan="1" style="white-space: nowrap;" | Multiplication operators
! colspan="3" style="white-space: nowrap;" | Multiplication operators
|-
|-
| ⎄xx
| ⎄xx
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| Bullet (infix: fat dot product, Dorst)
| Bullet (infix: fat dot product, Dorst)
|-
|-
! colspan="3" rowspan="1" style="white-space: nowrap;" | Other operators
! colspan="3" style="white-space: nowrap;" | Other operators
|-
|-
| ⎄v/
| ⎄v/
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| Double-struck italic, ⎄{{pipe}} 0e ⎄{{pipe}} i (only a few)
| Double-struck italic, ⎄{{pipe}} 0e ⎄{{pipe}} i (only a few)
|-
|-
! colspan="3" rowspan="1" style="white-space: nowrap;" | Power statistics brackets
! colspan="3" style="white-space: nowrap;" | Power statistics brackets
|-
|-
| ⎄{{pipe}} {{pipe}}  
| ⎄{{pipe}} {{pipe}}  
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| 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" style="white-space: nowrap;" | Combining marks
! colspan="3" style="white-space: nowrap;" | Combining marks
|-
|-
| ⎄\-
| ⎄\-
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