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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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== Basic ==
== Basic ==
=== Objects ===
=== Objects ===
 
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible"
! rowspan="2" | Equivalent expressions
! rowspan="2" | Equivalent expressions
! rowspan="2" | Variable
! rowspan="2" | Variable
Line 74: Line 73:
|  
|  
| <math>\textbf{i}</math>
| <math>\textbf{i}</math>
| [[interval|(Just) interval]]
| [[interval| (Just) interval]]
|  
|  
| <math>\small 𝗽</math>
| <math>\small 𝗽</math>
Line 88: Line 87:
|  
|  
| <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>
Line 120: Line 119:
| Generators
| Generators
| <math>\scriptsize  
| <math>\scriptsize  
\!\!
\! \!  
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array}  
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array}  
\!\!  
\! \!  
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}  
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\!\!  
\! \!  
</math>
</math>
| <math>\scriptsize (r, 1)</math>
| <math>\scriptsize (r, 1)</math>
Line 140: Line 139:
|  
|  
| <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 156: Line 155:
| Jargon name: val
| Jargon name: val
|-
|-
|  
| <math>n + r</math>
| <math>d</math>
| <math>d</math>
| [[Dimensionality]]
| [[Dimensionality]]
Line 174: Line 173:
|  
|  
|-
|-
|  
| <math>d - n</math>
| <math>r</math>
| <math>r</math>
| [[Rank]]
| [[Rank]]
|
|
|
|
| <math>\scriptsize (1, 1)</math>
| Integer
| Scalar
|
|
|
|
|
|
|
|-
| <math>d - r</math>
| <math>n</math>
| [[Nullity]]
|  
|  
|  
|  
Line 197: Line 214:
| <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
Line 214: Line 231:
| <math>1200×{\large\textbf{𝓁}}\hspace{2mu}</math>
| <math>1200×{\large\textbf{𝓁}}\hspace{2mu}</math>
| <math>𝒋</math>
| <math>𝒋</math>
| [[just tuning map|Just(-prime) tuning map]]
| [[just tuning map| Just(-prime) tuning 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
Line 248: Line 265:
|  
|  
|-
|-
| <math>𝒈M</math>
|  
| <math>𝒕</math>
| <math>𝒕</math>
| [[tuning map|(Tempered-prime) tuning map]]
| [[tuning map| (Tempered-prime) tuning map]]
| <math>\scriptsize
|  
\begin{array} {c} 𝒈 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} 𝑀 \\[-2pt] \cancel{𝗴} \hspace{-2mu} / \hspace{-2mu} 𝗽 \end{array}
</math>
| <math>\mathsf{¢}</math>/<math>\small 𝗽</math>
| <math>\mathsf{¢}</math>/<math>\small 𝗽</math>
| Cents per prime
| Cents per prime
| <math>\scriptsize
|  
\!\!
\begin{array} {c} 𝒈 \\[-3pt] (1, \cancel{r}) \end{array}
\!\!
\begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array}
\!\!
</math>
| <math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
| Real
| Real
Line 274: Line 281:
|  
|  
| <math>t_i</math>
| <math>t_i</math>
|
|  
|-
|-
| <math>𝒕 - 𝒋</math>
| <math>𝒕 - 𝒋 \\
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
Line 292: Line 300:
|  
|  
| <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 303: Line 311:
</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] (1, \cancel{d}) \end{array}  
\!\!  
\! \!  
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\begin{array} {c} \mathbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\!\!
\! \!  
</math>
</math>
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
Line 325: Line 333:
𝒕\textbf{i}</math>
𝒕\textbf{i}</math>
| <math>\mathrm{a}</math>
| <math>\mathrm{a}</math>
| [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Example_3|Tempered (interval) size]]
| [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Example_3| 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 332: Line 340:
</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] (1, \cancel{d}) \end{array}  
\!\!  
\! \!  
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\!\!
\! \!  
</math>
</math>
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
Line 349: Line 357:
|  
|  
|  
|  
| Mnemonic: <math>\mathrm{a}</math>ltered size
| Mnemonic: <math>\mathrm{o}</math>ltered size
|-
|-
| <math>𝒕\textbf{i} - 𝒋\textbf{i} \\
| <math>𝒕\textbf{i} - 𝒋\textbf{i} \\
Line 355: Line 363:
𝒓\textbf{i}</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 362: Line 370:
</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] (1, \cancel{d}) \end{array}  
\!\!  
\! \!  
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\!\!
\! \!  
</math>
</math>
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
Line 381: Line 389:
|  
|  
|-
|-
! colspan="17" |Optimization
! colspan="17" | Optimization
|-
|-
|  
|  
Line 403: Line 411:
|  
|  
| <math>⟪\,·\,⟫_p</math>
| <math>⟪\,·\,⟫_p</math>
| P[[power mean|ower mean]] (<math>p</math>-mean)
| [[Power mean]] (<math>p</math>-mean)
|  
|  
|  
|  
Line 423: Line 431:
|  
|  
| <math>c</math>
| <math>c</math>
| [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity|Complexity]]
| [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity| Complexity]]
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math><ref>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 ''w'' = ''d'' / |''e''| whose units are ¢(W) / ¢ and the cents cancel.</ref>
| colspan="3" rowspan="3" | (see complexities section of complexities and simplicities table)
| <math>\small\mathsf{(C)}</math>
| Complexity weight
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
Line 442: Line 448:
| <math>s</math>
| <math>s</math>
| [[Simplicity]]
| [[Simplicity]]
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math>
| <math>\small\mathsf{(S)}</math>
| Simplicity weight
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
Line 460: Line 463:
| <math>w</math>
| <math>w</math>
| [[Weight]]
| [[Weight]]
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> or 𝟙<math>\small\mathsf{(S)}</math>
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(S)}</math>
| Complexity weight or simplicity weight
|  
|  
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
Line 475: Line 475:
|  
|  
|-
|-
| <math>|\mathrm{e}|w</math>
| <math>| \mathrm{e}| w</math>
| <math>\mathrm{d}</math>
| <math>\mathrm{d}</math>
| [[Damage]]
| [[Damage]]
| <math>\scriptsize
| colspan="3" | (see damages table)
\begin{array} {c} |\mathrm{e}| \\[-2pt] {\small\mathsf{¢}} \end{array}
|  
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} w \\[-2pt] \mathsf{(U, C, or\,S)} \end{array}
</math>
| <math>\mathsf{¢}\small\mathsf{(U)}</math> or <math>\mathsf{¢}\small\mathsf{(C)}</math> or <math>\mathsf{¢}\small\mathsf{(S)}</math>
| (See damages table)
| <math>\scriptsize
\!\!
\begin{array} {c} |\mathrm{e}| \\[-3pt] (1, \cancel{1}) \end{array}
\!\!
\begin{array} {c} w \\[-3pt] (\cancel{1}, 1) \end{array}
\!\!
</math>
| <math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
| Real
| Real
Line 503: Line 491:
|  
|  
|-
|-
! colspan="17" | Target intervals
! colspan="17" | Target-intervals
|-
|-
|  
|  
Line 534: Line 522:
| Generators
| Generators
| <math>\scriptsize  
| <math>\scriptsize  
\!\!
\! \!  
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array}  
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array}  
\!\!  
\! \!  
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}  
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}  
\!\!  
\! \!  
</math>
</math>
| <math>\scriptsize (r, k)</math>
| <math>\scriptsize (r, k)</math>
Line 549: Line 537:
|  
|  
| <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>
| [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Primes|Target-interval (just) size list]]
| [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Primes| 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 562: Line 550:
| Cents
| Cents
| <math>\scriptsize  
| <math>\scriptsize  
\!\!  
\! \!  
\begin{array} {c} 𝒋 \\[-3pt] (1, \cancel{d}) \end{array}  
\begin{array} {c} 𝒋 \\[-3pt] (1, \cancel{d}) \end{array}  
\!\!  
\! \!  
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\!\!
\! \!  
</math>
</math>
| <math>\scriptsize (1, k)</math>
| <math>\scriptsize (1, k)</math>
Line 577: Line 565:
|  
|  
| <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>𝒕\mathrm{T}</math>
𝒈M\mathrm{T}</math>
| <math>\textbf{a}</math>
| <math>\textbf{a}</math>
| [[Tempered target-interval size list]]
| [[Tempered target-interval size list]]
Line 591: Line 578:
| Cents
| Cents
| <math>\scriptsize  
| <math>\scriptsize  
\!\!  
\! \!  
\begin{array} {c} 𝒕 \\[-3pt] (1, \cancel{d}) \end{array}  
\begin{array} {c} 𝒕 \\[-3pt] (1, \cancel{d}) \end{array}  
\!\!  
\! \!  
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\!\!
\! \!  
</math>
</math>
| <math>\scriptsize (1, k)</math>
| <math>\scriptsize (1, k)</math>
Line 606: Line 593:
|  
|  
| <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} \\
\textbf{a} - \textbf{o} \\
𝒓\mathrm{T} \\
𝒓\mathrm{T}
\textbf{a} - \textbf{o}</math>
</math>
| <math>\textbf{e}</math>
| <math>\textbf{e}</math>
| [[Target-interval error list]]
| [[Target-interval error list]]
Line 622: Line 608:
| Cents
| Cents
| <math>\scriptsize  
| <math>\scriptsize  
\!\!  
\! \!  
\begin{array} {c} 𝒓 \\[-3pt] (1, \cancel{d}) \end{array}  
\begin{array} {c} 𝒓 \\[-3pt] (1, \cancel{d}) \end{array}  
\!\!  
\! \!  
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\!\!
\! \!  
</math>
</math>
| <math>\scriptsize (1, k)</math>
| <math>\scriptsize (1, k)</math>
Line 641: Line 627:
| <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]]
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(U)}</math>
| colspan="3" rowspan="3" | (See complexities and simplicities table)
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(S)}</math> or <math>\small\mathsf{(U)}</math>
| Complexity weight or simplicity weight
|  
|  
| <math>\scriptsize (k, k)</math>
| <math>\scriptsize (k, k)</math>
Line 659: Line 643:
|  
|  
| <math>C</math>
| <math>C</math>
| [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity-weight_damage|Target-interval complexity weight Matrix]]
| [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity-weight_damage| Target-interval complexity weight matrix]]
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math>
| <math>\small\mathsf{(C)}</math>
| Complexity weight
|  
|  
| <math>\scriptsize (k, k)</math>
| <math>\scriptsize (k, k)</math>
Line 677: Line 658:
| <math>\dfrac1C</math>
| <math>\dfrac1C</math>
| <math>S</math>
| <math>S</math>
| [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity-weight_damage|Target-interval simplicity weight Matrix]]
| [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity-weight_damage| Target-interval simplicity weight matrix]]
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math>
| <math>\small\mathsf{(S)}</math>
| Simplicity weight
|  
|  
| <math>\scriptsize (k, k)</math>
| <math>\scriptsize (k, k)</math>
Line 691: Line 669:
| <math>𝒔</math>
| <math>𝒔</math>
| <math>s_i</math>
| <math>s_i</math>
| entrywise reciprocal of <math>C</math>
| Entrywise reciprocal of <math>C</math>
|-
|-
| <math>|\textbf{e}|W</math>
| <math>| \textbf{e}| W</math>
| <math>\textbf{d}</math>
| <math>\textbf{d}</math>
| [[Target-interval damage list]]<ref>You may sometimes see annotated units without parentheses, such as "dBA", but this is not compliant with SI standards, so we always keep the parentheses.</ref>
| [[Target-interval damage list]]
| <math>\scriptsize
| colspan="3" | (See damages table)
\begin{array} {c} |\textbf{e}| \\[-2pt] {\small\mathsf{¢}} \end{array}
|  
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} W \\[-2pt] (\mathsf{U, C, or\,S}) \end{array}
</math>
| <math>\mathsf{¢}\small\mathsf{(U)}</math>, <math>\mathsf{¢}\small\mathsf{(C)}</math>, or <math>\mathsf{¢}\small\mathsf{(S)}</math>
| Weighted cents
| <math>\scriptsize
\!\!
\begin{array} {c} |\textbf{e}| \\[-3pt] (1, \cancel{k}) \end{array}
\!\!
\begin{array} {c} W \\[-3pt] (\cancel{k}, k) \end{array}
\!\!
</math>
| <math>\scriptsize (1, k)</math>
| <math>\scriptsize (1, k)</math>
| Real
| Real
Line 737: Line 703:
|  
|  
|  
|  
| mnemonic: <math>k</math>ount
| Mnemonic: <math>k</math>ount
|-
|-
! colspan="17" | Held-intervals
! colspan="17" | Held-intervals
Line 795: Line 761:
|  
|  
| <math>\mathrm{c}_{ij}</math>
| <math>\mathrm{c}_{ij}</math>
| Jargon name: Monzo List
| Jargon name: monzo list
|-
|-
|  
|  
Line 814: Line 780:
| <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)
|}
===Units===
We recommend using a narrow no-break space (U+202F) between quantities and their units.<ref>Per https://physics.nist.gov/cuu/Units/checklist.html and https://academia.stackexchange.com/questions/54885/should-there-be-a-space-between-a-value-and-the-units-used
.</ref> For how to type this, see the [[#WinCompose|WinCompose]] section below.
{| class="wikitable center-all mw-collapsible"
!Symbol
!Name
!Vectorized
|-
|-
|<math>\small 𝗴</math>
! colspan="17" | Computation
|Generators
|Yes
|-
|-
|<math>\small 𝗽</math>
|
|Primes
| <math>\llzigzag·\,\rrzigzag\! _p</math>
|Yes
| [[Power sum]] (<math>p</math>-sum)
|
|
|
|
| <math>\scriptsize (1, 1)</math>
| Real
| Scalar
|
|
|
|
|
|
|  
|-
|-
|<math>\mathsf{¢}</math><ref>It seems there is no standard symbol for a musical cent, except the word spelled in full (see https://en.wikipedia.org/wiki/Cent_(music)). But it seems unlikely anyone will interpret the cent currency symbol "¢" following a number in a musical context as anything other than musical cents.</ref>
! colspan="17" | All-interval tuning schemes
|Cents
|
|-
|-
|<math>\mathsf{¢}\small\mathsf{(U)}</math>
| <math>\mathrm{I}</math>
|Unity-weighted cents
| <math>\mathrm{T}_{\text{p}}</math>
|
| [[Prime proxy target-interval list]]
|
| <math>\small 𝗽</math>
| Primes
|
| <math>\scriptsize (d, d)</math>
| Integer
| Matrix
|
| ⟨[...⟩ ...]
|
|
| <math>\mathbf{1}</math>
|  
|  
|-
|-
|<math>\mathsf{¢}\small\mathsf{(C)}</math>
|
|Complexity-weighted cents
| <math>X</math>
|
| [[Complexity prescaler]]
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math>
| <math>\small\mathsf{(C)}</math>
| complexity weight
|
| <math>\scriptsize (d, d)</math>
| Real
| Matrix
| [⟨...] ...⟩
|
|
|
| <math>𝒙</math>
| <math>x_i</math>
|  
|-
|-
|<math>\mathsf{¢}\small\mathsf{(S)}</math>
| <math>\text{diag}({\large\textbf{𝓁}}\hspace{2mu})</math>
|Simplicity-weighted cents
| <math>L</math>
|
| [[Log-prime matrix]]
|
| <math>\small\mathsf{oct}</math>/<math>\small 𝗽</math>
| Octaves per prime
|
| <math>\scriptsize (d, d)</math>
| Real
| Matrix
| [⟨...] ...⟩
| ⟨[...⟩ ...]
| <math>{\large\textbf{𝓁}}\hspace{2mu}_i</math>
|
| <math>{\large\textbf{𝓁}}\hspace{2mu}</math>
| <math>{\large 𝓁}\hspace{2mu}_{ij}</math>
|  
|-
|-
|<math>\small\mathsf{oct}</math>
|
|Octaves
| <math>q</math>
|
| [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_all-interval_tuning_schemes#Dual_norms| Interval complexity norm power]]
|
|
|
|
| <math>\scriptsize (1, 1)</math>
| Real
| Scalar
|
|
|
|
|
|
|  
|-
|-
|<math>\small\mathsf{(C)}</math>
|
|Complexity weight
| <math>‖ · ‖_q</math>
|
| [[Power norm]] (<math>p</math>-norm)
|
|
|
|
| <math>\scriptsize (1, 1)</math>
| Real
| Scalar
|
|
|
|
|
|
|  
|-
|-
|<math>\small\mathsf{(S)}</math>
| <math>\dfrac1{1-\frac1q}</math>
|Simplicity weight
| <math>\text{dual}(q)</math>
|
| [[Dave_Keenan_%26_Douglas_Blumeyer's_guide_to_RTT:_all-interval_tuning_schemes#Dual_norms| Dual norm power]]
|}
|
 
|
===Tuning schemes===
|
Copied from [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning fundamentals#Systematic tuning scheme names]].
|
 
| <math>\scriptsize (1, 1)</math>
{| class="wikitable center-all mw-collapsible"
| Real
| Scalar
|  
|  
|
|
|
|
|  
|-
|-
!Damage weight
|
!Optimization power
| <math>‖X\mathbf{i}‖_q</math>
!Systematic name
| [[Interval complexity]]
|
| <math>\small\mathsf{(C)}</math>
|
|
| <math>\scriptsize (1, 1)</math>
| Real
| Scalar
|
|
|
|
|
|
|
|-
|-
|<none>
|  
| rowspan="3" |∞
| <math>‖𝒓X^{-1}‖_{\text{dual}(q)}</math>
|Minimax-U
| [[Retuning magnitude]]
|-
|  
|Complexity
| <math>\mathsf{¢}\small\mathsf{(C^{-1})}</math>
|Minimax-C
|  
|-
|  
|1/complexity
| <math>\scriptsize (1, 1)</math>
|Minimax-S
| Real
|-
| Scalar
|<none>
|  
| rowspan="3" |2
|  
|MiniRMS-U
|  
|-
|  
|Complexity
|  
|MiniRMS-C
|  
|-
|  
|1/complexity
|MiniRMS-S
|-
|&lt;none&gt;
| rowspan="3" |1
|Miniaverage-U
|-
|Complexity
|Miniaverage-C
|-
|1/complexity
|Miniaverage-S
|}
 
===Damages===
{| class="wikitable center-all mw-collapsible"
! colspan="2" |Quantity
! colspan="2" |Unit
|-
!Abbreviation
!Name
!Symbol
!Name
|-
|U-damage
|Unity-weight damage
|<math>\mathsf{¢}\small\mathsf{(U)}</math>
|Unity-weighted cents
|-
|C-damage
|Complexity-weight damage
|<math>\mathsf{¢}\small\mathsf{(C)}</math>
|Complexity-weighted cents
|-
|S-damage
|Simplicity-weight damage
|<math>\mathsf{¢}\small\mathsf{(S)}</math>
|Simplicity-weighted cents
|}
 
===Complexity and simplicity===
{| class="wikitable center-all mw-collapsible"
! colspan="2" |Quantity
! colspan="2" |Unit
|-
!Abbreviation
!Name
!Symbol
!Name
|-
|C
|Complexity
|<math>\small\mathsf{(C)}</math>
|Complexity weight
|-
|S
|Simplicity
|<math>\small\mathsf{(S)}</math>
|Simplicity weight
|}
 
<math>
% \slant{} command approximates italics to allow slanted bold characters, including digits, in MathJax.
\def\slant#1{\style{display:inline-block;margin:-.05em;transform:skew(-14deg)translateX(.03em)}{#1}}
% Latex equivalents of the wiki templates llzigzag and rrzigzag for double zigzag brackets.
\def\llzigzag{\hspace{-1.6mu}\style{display:inline-block;transform:scale(.62,1.24)translateY(.07em);font-family:sans-serif}{ꗨ\hspace{-3mu}ꗨ}\hspace{-1.6mu}}
\def\rrzigzag{\hspace{-1.6mu}\style{display:inline-block;transform:scale(-.62,1.24)translateY(.07em);font-family:sans-serif}{ꗨ\hspace{-3mu}ꗨ}\hspace{-1.6mu}}
</math>
 
==Intermediate==
===Objects===
{| class="wikitable mw-collapsible mw-collapsed"
! rowspan="2" |Equivalent expressions
! rowspan="2" |Variable
! rowspan="2" |Name
! colspan="3" |Units
! colspan="2" |Shape
! colspan="2" |Type
! colspan="2" |EBK notation
! colspan="4" |Subobjects
! rowspan="2" |Notes
|-
!Unreduced
!Reduced
!Read as
!Unreduced
!Reduced
!Numeric
!Structural
!Row-first
!Col-first
!Row
!Col
!Diag
!Entry
|-
! colspan="17" |Mapping
|-
|
|<math>\textbf{i}</math>
|[[interval|(Just) interval]]
|
|<math>\small 𝗽</math>
|Primes
|
|<math>\scriptsize (d, 1)</math>
|Integer
|Vector
|
|[...⟩
|
|
|
|<math>\mathrm{i}_i</math>
|Specific type: vector ([[prime-count vector]] or PC-vector)
Jargon name: monzo
|-
|
|<math>M</math>
|[[Mapping|(Temperament) mapping (matrix)]]
|
|<math>\small 𝗴</math>/<math>\small 𝗽</math>
|Generators per prime
|
|<math>\scriptsize (r, d)</math>
|Integer
|Matrix
|[⟨...] ...}
|⟨[...} ...]
|<math>𝒎_i</math>
|
|
|<math>m_{ij}</math>
|Jargon name: val list
|-
|<math>M\textbf{i}</math>
|<math>\textbf{y}</math>
|[[Mapped interval]]
|<math>\scriptsize
\begin{array} {c} M \\[-2pt] 𝗴 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}
</math>
|<math>\small 𝗴</math>
|Generators
|<math>\scriptsize
\!\!
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array}
\!\!
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\!\!
</math>
|<math>\scriptsize (r, 1)</math>
|Integer
|Vector
|
|[...}
|
|
|
|
|Specific type: [[generator-count vector]] (GC-vector)
Jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval
|-
|
|<math>𝒎</math>
|[[map|(Temperament) map]]
|
|<math>\small 𝗴</math>/<math>\small 𝗽</math>
|Generators per prime
|
|<math>\scriptsize (1, d)</math>
|Integer
|Vector
|⟨...]
|
|
|
|
|<math>m_i</math>
|Jargon name: val
|-
|<math>n + r</math>
|<math>d</math>
|[[Dimensionality]]
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|Integer
|Scalar
|
|
|
|
|
|
|
|-
|<math>d - n</math>
|<math>r</math>
|[[Rank]]
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|Integer
|Scalar
|
|
|
|
|
|
|
|-
|<math>d - r</math>
|<math>n</math>
|[[Nullity]]
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|Integer
|Scalar
|
|
|
|
|
|
|
|-
! colspan="17" |Tuning
|-
|
|<math>{\large\textbf{𝓁}}\hspace{2mu}</math>
|[[Log-prime map]]
|
|<math>\small\mathsf{oct}</math>/<math>\small 𝗽</math>
|Octaves per prime
|
|<math>\scriptsize (1, d)</math>
|Real
|Vector
|⟨...]
|
|
|
|
|<math>{\large 𝓁}\hspace{2mu}_i</math>
|
|-
|<math>1200×{\large\textbf{𝓁}}\hspace{2mu}</math>
|<math>𝒋</math>
|[[just tuning map|Just(-prime) tuning map]]
|
|<math>\mathsf{¢}</math>/<math>\small 𝗽</math>
|Cents per prime
|
|<math>\scriptsize (1, d)</math>
|Real
|Vector
|⟨...]
|
|
|
|
|<math>j_i</math>
|
|-
|
|<math>𝒈</math>
|[[Generator tuning map]]
|
|<math>\mathsf{¢}</math>/<math>\small 𝗴</math>
|Cents per generator
|
|<math>\scriptsize (1, r)</math>
|Real
|Vector
|{...]
|
|
|
|
|<math>g_i</math>
|
|-
|
|<math>𝒕</math>
|[[tuning map|(Tempered-prime) tuning map]]
|
|<math>\mathsf{¢}</math>/<math>\small 𝗽</math>
|Cents per prime
|
|<math>\scriptsize (1, d)</math>
|Real
|Vector
|⟨...]
|
|
|
|
|<math>t_i</math>
|
|-
|<math>𝒕 - 𝒋 \\
1200×\slant{\mathbf{1}}L(P - I)</math>
|<math>𝒓</math>
|[[retuning map|Retuning (or mistuning) map]]
|
|<math>\mathsf{¢}</math>/<math>\small 𝗽</math>
|Cents per prime
|
|<math>\scriptsize (1, d)</math>
|Real
|Vector
|⟨...]
|
|
|
|
|<math>r_i</math>
|Previous name: prime error map
|-
|<math>𝒋\textbf{i}</math>
|<math>\mathrm{o}</math>
|[[interval span|(Just) (interval) size]]
|<math>\scriptsize
\begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}
</math>
|<math>\mathsf{¢}</math>
|cents
|<math>\scriptsize
\!\!
\begin{array} {c} 𝒋 \\[-3pt] (1, \cancel{d}) \end{array}
\!\!
\begin{array} {c} \mathbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\!\!
</math>
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|Mnemonic: <math>\mathrm{O}</math>riginal size
|-
|<math>𝒈M\textbf{i} \\
𝒕\textbf{i}</math>
|<math>\mathrm{a}</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Example_3|Tempered (interval) size]]
|<math>\scriptsize
\begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}
</math>
|<math>\mathsf{¢}</math>
|cents
|<math>\scriptsize
\!\!
\begin{array} {c} 𝒕 \\[-3pt] (1, \cancel{d}) \end{array}
\!\!
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\!\!
</math>
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|Mnemonic: <math>\mathrm{O}</math>ltered size
|-
|<math>𝒕\textbf{i} - 𝒋\textbf{i} \\
a - o \\
𝒓\textbf{i}</math>
|<math>\mathrm{e}</math>
|[[error|(Interval) error]]
|<math>\scriptsize
\begin{array} {c} 𝒓 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{𝗽} \end{array}
</math>
|<math>\mathsf{¢}</math>
|cents
|<math>\scriptsize
\!\!
\begin{array} {c} 𝒓 \\[-3pt] (1, \cancel{d}) \end{array}
\!\!
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\!\!
</math>
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|-
! colspan="17" |Optimization
|-
|
|<math>p</math>
|[[Optimization power]]
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|-
|
|<math>⟪\,·\,⟫_p</math>
|[[Power mean]] (<math>p</math>-mean)
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|-
! colspan="17" |Damage
|-
|
|<math>c</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity|Complexity]]
| colspan="3" rowspan="3" |(see complexities section of complexities and simplicities table)
|
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|-
|<math>\dfrac1c</math>
|<math>s</math>
|[[Simplicity]]
|
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|-
|<math>c</math> or <math>s</math>
|<math>w</math>
|[[Weight]]
|
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|-
|<math>|\mathrm{e}|w</math>
|<math>\mathrm{d}</math>
|[[Damage]]
| colspan="3" |(see damages table)
|
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|-
! colspan="17" |Target-intervals
|-
|
|<math>\mathrm{T}</math>
|[[Target-interval list]]
|
|<math>\small 𝗽</math>
|Primes
|
|<math>\scriptsize (d, k)</math>
|Integer
|Matrix
|
|[[...⟩ ...]
|
|<math>\textbf{t}_i</math>
|
|<math>\mathrm{t}_{ij}</math>
|
|-
|<math>M\mathrm{T}</math>
|<math>\mathrm{Y}</math>
|[[Mapped target-interval list]]
|<math>\scriptsize
\begin{array} {c} M \\[-2pt] 𝗴 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}
</math>
|<math>\small 𝗴</math>
|Generators
|<math>\scriptsize
\!\!
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array}
\!\!
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\!\!
</math>
|<math>\scriptsize (r, k)</math>
|Integer
|Matrix
|
|[[...} ...]
|
|<math>\textbf{y}_i</math>
|
|<math>\mathrm{y}_{ij}</math>
|Mnemonic: looks like bent-up 'T', or cross between 'M' and 'T'
|-
|<math>𝒋\mathrm{T}</math>
|<math>\textbf{o}</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Primes|Target-interval (just) size list]]
|<math>\scriptsize
\begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}
</math>
|<math>\mathsf{¢}</math>
|Cents
|<math>\scriptsize
\!\!
\begin{array} {c} 𝒋 \\[-3pt] (1, \cancel{d}) \end{array}
\!\!
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\!\!
</math>
|<math>\scriptsize (1, k)</math>
|Real
|List
|[...]
|
|
|
|
|<math>\mathrm{o}_i</math>
|Mnemonic: <math>\textbf{o}</math>riginal size list
|-
|<math>𝒕\mathrm{T}</math>
|<math>\textbf{a}</math>
|[[Tempered target-interval size list]]
|<math>\scriptsize
\begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}
</math>
|<math>\mathsf{¢}</math>
|Cents
|<math>\scriptsize
\!\!
\begin{array} {c} 𝒕 \\[-3pt] (1, \cancel{d}) \end{array}
\!\!
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\!\!
</math>
|<math>\scriptsize (1, k)</math>
|Real
|List
|[...]
|
|
|
|
|<math>\mathrm{a}_i</math>
|Mnemonic: <math>\textbf{a}</math>ltered size list
|-
|<math>𝒕\mathrm{T} - 𝒋\mathrm{T} \\
𝒓\mathrm{T} \\
\textbf{a} - \textbf{o}</math>
|<math>\textbf{e}</math>
|[[Target-interval error list]]
|<math>\scriptsize
\begin{array} {c} 𝒓 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{𝗽} \end{array}
</math>
|<math>\mathsf{¢}</math>
|Cents
|<math>\scriptsize
\!\!
\begin{array} {c} 𝒓 \\[-3pt] (1, \cancel{d}) \end{array}
\!\!
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array}
\!\!
</math>
|<math>\scriptsize (1, k)</math>
|Real
|List
|[...]
|
|
|
|
|<math>\mathrm{e}_i</math>
|
|-
|<math>C</math> or <math>S</math>
|<math>W</math>
|[[Target-interval weight matrix]]
| colspan="3" rowspan="3" |(See complexities and simplicities table)
|
|<math>\scriptsize (k, k)</math>
|Real
|Matrix
|
|[[...] ...]
|
|
|<math>𝒘</math>
|<math>w_i</math>
|
|-
|
|<math>C</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity-weight_damage|Target-interval complexity weight matrix]]
|
|<math>\scriptsize (k, k)</math>
|Real
|Matrix
|
|[[...] ...]
|
|
|<math>𝒄</math>
|<math>c_i</math>
|
|-
|<math>\dfrac1C</math>
|<math>S</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity-weight_damage|Target-interval simplicity weight matrix]]
|
|<math>\scriptsize (k, k)</math>
|Real
|Matrix
|
|[[...] ...]
|
|
|<math>𝒔</math>
|<math>s_i</math>
|Entrywise reciprocal of <math>C</math>
|-
|<math>|\textbf{e}|W</math>
|<math>\textbf{d}</math>
|[[Target-interval damage list]]
| colspan="3" |(See damages table)
|
|<math>\scriptsize (1, k)</math>
|Real
|List
|[...]
|
|
|
|
|<math>\mathrm{d}_i</math>
|
|-
|
|<math>k</math>
|[[Target-interval count]]
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|Integer
|Scalar
|
|
|
|
|
|
|Mnemonic: <math>k</math>ount
|-
! colspan="17" |Held-intervals
|-
|
|<math>\mathrm{H}</math>
|[[Held-interval basis]]
|
|<math>\small 𝗽</math>
|Primes
|
|<math>\scriptsize (d, h)</math>
|
|Matrix
|
|[[...⟩ ...]
|
|<math>\textbf{h}_i</math>
|
|<math>\mathrm{h}_{ij}</math>
|
|-
|
|<math>h</math>
|[[Held-interval count]]
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|Integer
|Scalar
|
|
|
|
|
|
|
|-
! colspan="17" |Exploring temperaments
|-
|
|<math>\mathrm{C}</math>
|[[Comma basis]]
|
|<math>\small 𝗽</math>
|Primes
|
|<math>\scriptsize (d, n)</math>
|Integer
|Matrix
|
|[[...⟩ ...]
|
|<math>\textbf{c}_i</math>
|
|<math>\mathrm{c}_{ij}</math>
|Jargon name: monzo list
|-
|
|<math>\textbf{c}</math>
|[[Comma]]
|
|<math>\small 𝗽</math>
|Primes
|
|<math>\scriptsize (d, 1)</math>
|Integer
|Vector
|
|[...⟩
|
|
|
|<math>\mathrm{c}_i</math>
|Specific type: vector ([[prime-count vector]] or PC-vector)
|-
! colspan="17" |Computation
|-
|
|<math>\llzigzag·\,\rrzigzag\!_p</math>
|[[Power sum]] (<math>p</math>-sum)
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|-
! colspan="17" |All-interval tuning schemes
|-
|<math>\mathrm{I}</math>
|<math>\mathrm{T}_{\text{p}}</math>
|[[Prime proxy target-interval list]]
|
|<math>\small 𝗽</math>
|Primes
|
|<math>\scriptsize (d, d)</math>
|Integer
|Matrix
|
|⟨[...⟩ ...]
|
|
|<math>\mathbf{1}</math>
|
|
|-
|
|<math>X</math>
|[[Complexity prescaler]]
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math>
|<math>\small\mathsf{(C)}</math>
|complexity weight
|
|<math>\scriptsize (d, d)</math>
|Real
|Matrix
|[⟨...] ...⟩
|
|
|
|<math>𝒙</math>
|<math>x_i</math>
|
|-
|<math>\text{diag}({\large\textbf{𝓁}}\hspace{2mu})</math>
|<math>L</math>
|[[Log-prime matrix]]
|
|<math>\small\mathsf{oct}</math>/<math>\small 𝗽</math>
|Octaves per prime
|
|<math>\scriptsize (d, d)</math>
|Real
|Matrix
|[⟨...] ...⟩
|⟨[...⟩ ...]
|<math>{\large\textbf{𝓁}}\hspace{2mu}_i</math>
|
|<math>{\large\textbf{𝓁}}\hspace{2mu}</math>
|<math>{\large 𝓁}\hspace{2mu}_{ij}</math>
|
|-
|
|<math>q</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_all-interval_tuning_schemes#Dual_norms|Interval complexity norm power]]
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|-
|
|<math>‖ · ‖_q</math>
|[[Power norm]] (<math>p</math>-norm)
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|-
|<math>\dfrac1{1-\frac1q}</math>
|<math>\text{dual}(q)</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer's_guide_to_RTT:_all-interval_tuning_schemes#Dual_norms|Dual norm power]]
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|-
|
|<math>‖X\mathbf{i}‖_q</math>
|[[Interval complexity]]
|
|<math>\small\mathsf{(C)}</math>
|
|
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|-
|
|<math>‖𝒓X^{-1}‖_{\text{dual}(q)}</math>
|[[Retuning magnitude]]
|
|<math>\mathsf{¢}\small\mathsf{(C^{-1})}</math>
|
|
|<math>\scriptsize (1, 1)</math>
|Real
|Scalar
|
|
|
|
|
|
|
|}
|}


Line 1,871: Line 954:
{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|-
|-
! colspan="3" rowspan="3" |Retuning (or mistuning) magnitude
! colspan="3" rowspan="3" | Retuning (or mistuning) magnitude
! colspan="9" |Damage
! colspan="9" | Damage
! rowspan="4" |Target intervals
! rowspan="4" | Target intervals
! colspan="2" rowspan="3" |Systematic name
! colspan="2" rowspan="3" | Systematic name
! rowspan="4" |Previously named tuning schemes that are specific types of this tuning scheme
! rowspan="4" | Previously named tuning schemes that are specific types of this tuning scheme
! rowspan="4" |of interest?
! rowspan="4" | of interest?
|-
|-
! colspan="6" |Weight
! colspan="6" | Weight
! colspan="3" rowspan="1" |Optimization
! colspan="3" rowspan="1" | Optimization
|-
|-
! colspan="3" |Interval complexity
! colspan="3" | Interval complexity
! colspan="3" rowspan="1" |Slope
! colspan="3" rowspan="1" | Slope
! colspan="1" rowspan="2" |Initial
! colspan="1" rowspan="2" | Initial
! colspan="1" rowspan="2" |Name
! colspan="1" rowspan="2" | Name
! colspan="1" rowspan="2" |Power
! colspan="1" rowspan="2" | Power
|-
|-
!Initial
! Initial
!Name
! Name
!Power
! Power
!Initial
! Initial
!Name
! Name
!Power
! Power
!Initial
! Initial
!Name
! Name
!Multiplier
! Multiplier
! colspan="1" |Abbreviated
! colspan="1" | Abbreviated
! colspan="1" |Read ("____ tuning scheme")
! colspan="1" | Read ("____ tuning scheme")
|-
|-
|<n/a>
| <n/a>
|Maximum
| Maximum
|∞
| ∞
|(t)
| (t)
|Taxicab
| Taxicab
|1
| 1
| rowspan="2" |S
| rowspan="2" | S
| rowspan="2" |Simplicity-weight
| rowspan="2" | Simplicity-weight
| rowspan="2" |1/complexity
| rowspan="2" | 1/complexity
| rowspan="17" |<n/a>
| rowspan="17" | <n/a>
| rowspan="7" |Minimax
| rowspan="7" | Minimax
| rowspan="7" |∞
| rowspan="7" | ∞
| rowspan="2" |All
| rowspan="2" | All
|Minimax-S
| Minimax-S
|Minimax simplicity-weight damage
| Minimax simplicity-weight damage
|"[[TOP]]"/"[[T1]]"/"[[TIPTOP]]"*, "[[CTOP]]", "[[POTOP]]"/"[[POTT]]"*, "[[BOP tuning|BOP]]", "[[Weil Norms, Tenney-Weil Norms, and TWp Interval and Tuning Space|Weil]]", "[[Kees]]"
| "[[TOP]]"/"[[T1]]"/"[[TIPTOP]]"*, "[[CTOP]]", "[[POTOP]]"/"[[POTT]]"*, "[[BOP tuning| BOP]]", "[[Weil Norms, Tenney-Weil Norms, and TWp Interval and Tuning Space| Weil]]", "[[Kees]]"
|Yes
| Yes
|-
|-
|<n/a>
| <n/a>
|Euclidean
| Euclidean
|2
| 2
|E
| E
|Euclidean
| Euclidean
|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]]", "[[Frobenius]]", "[[BE]]", "[[WE]]", "[[KE]]"
| "[[Tenney-Euclidean tuning| TE]]"/"[[T2]]"/"[[TOP-RMS]]", "[[CTE tuning| CTE]]", "[[POTE tuning| POTE]]", "[[Frobenius]]", "[[BE]]", "[[WE]]", "[[KE]]"
|
|  
|-
|-
| colspan="3" rowspan="15" |<n/a>
| colspan="3" rowspan="15" | <n/a>
| colspan="3" |<n/a>
| colspan="3" | <n/a>
|U
| U
|Unity-weight
| Unity-weight
|<none>
| <none>
| rowspan="15" |<set>
| rowspan="15" | <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
|-
|-
|(t)
| (t)
|Taxicab
| Taxicab
|1
| 1
| rowspan="2" |S
| rowspan="2" | S
| rowspan="2" |Simplicity-weight
| rowspan="2" | Simplicity-weight
| rowspan="2" |1/complexity
| rowspan="2" | 1/complexity
|<set> Minimax-S
| <set> Minimax-S
|<set> Minimax simplicity-weight damage
| <set> Minimax simplicity-weight damage
|
|  
|Yes
| Yes
|-
|-
|E
| E
|Euclidean
| Euclidean
|2
| 2
|<set> Minimax-ES
| <set> Minimax-ES
|<set> Minimax Euclideanized-simplicity-weight damage
| <set> Minimax Euclideanized-simplicity-weight damage
|
|  
|
|  
|-
|-
|(t)
| (t)
|Taxicab
| Taxicab
|1
| 1
| rowspan="2" |C
| rowspan="2" | C
| rowspan="2" |Complexity-weight
| rowspan="2" | Complexity-weight
| rowspan="2" |Complexity
| rowspan="2" | Complexity
|<set> Minimax-C
| <set> Minimax-C
|<set> Minimax complexity-weight damage
| <set> Minimax complexity-weight damage
|
|  
|Yes
| Yes
|-
|-
|E
| E
|Euclidean
| Euclidean
|2
| 2
|<set> Minimax-EC
| <set> Minimax-EC
|<set> Minimax Euclideanized-complexity-weight damage
| <set> Minimax Euclideanized-complexity-weight damage
|
|  
|
|  
|-
|-
| colspan="3" |<n/a>
| colspan="3" | <n/a>
|U
| U
|unity-weight
| unity-weight
|<none>
| <none>
| rowspan="5" |MiniRMS
| rowspan="5" | MiniRMS
| rowspan="5" |2
| rowspan="5" | 2
|<set> miniRMS-U
| <set> miniRMS-U
|<set> miniRMS unity-weight damage
| <set> miniRMS unity-weight damage
|"[[least squares]]"
| "[[least squares]]"
|Yes
| Yes
|-
|-
|(t)
| (t)
|Taxicab
| Taxicab
|1
| 1
| rowspan="2" |S
| rowspan="2" | S
| rowspan="2" |simplicity-weight
| rowspan="2" | simplicity-weight
| rowspan="2" |1/complexity
| rowspan="2" | 1/complexity
|<set> miniRMS-S
| <set> miniRMS-S
|<set> miniRMS simplicity-weight damage
| <set> miniRMS simplicity-weight damage
|
|  
|Yes
| Yes
|-
|-
|E
| E
|Euclidean
| Euclidean
|2
| 2
|<set> miniRMS-ES
| <set> miniRMS-ES
|<set> miniRMS Euclideanized-simplicity-weight damage
| <set> miniRMS Euclideanized-simplicity-weight damage
|
|  
|
|  
|-
|-
|(t)
| (t)
|Taxicab
| Taxicab
|1
| 1
| rowspan="2" |C
| rowspan="2" | C
| rowspan="2" |Complexity-weight
| rowspan="2" | Complexity-weight
| rowspan="2" |Complexity
| rowspan="2" | Complexity
|<set> miniRMS-C
| <set> miniRMS-C
|<set> miniRMS complexity-weight damage
| <set> miniRMS complexity-weight damage
|
|  
|Yes
| Yes
|-
|-
|E
| E
|Euclidean
| Euclidean
|2
| 2
|<set> miniRMS-EC
| <set> miniRMS-EC
|<set> miniRMS Euclideanized-complexity-weight damage
| <set> miniRMS Euclideanized-complexity-weight damage
|
|  
|
|  
|-
|-
| colspan="3" |<n/a>
| colspan="3" | <n/a>
|U
| U
|unity-weight
| unity-weight
|<none>
| <none>
| rowspan="5" |Miniaverage
| rowspan="5" | Miniaverage
| rowspan="5" |1
| rowspan="5" | 1
|<set> miniaverage-U
| <set> miniaverage-U
|<set> miniaverage unity-weight damage
| <set> miniaverage unity-weight damage
|
|  
|Yes
| Yes
|-
|-
|(t)
| (t)
|Taxicab
| Taxicab
|1
| 1
| rowspan="2" |S
| rowspan="2" | S
| rowspan="2" |Simplicity-weight
| rowspan="2" | Simplicity-weight
| rowspan="2" |1/complexity
| rowspan="2" | 1/complexity
|<set> miniaverage-S
| <set> miniaverage-S
|<set> miniaverage simplicity-weight damage
| <set> miniaverage simplicity-weight damage
|
|  
|Yes
| Yes
|-
|-
|E
| E
|Euclidean
| Euclidean
|2
| 2
|<set> miniaverage-ES
| <set> miniaverage-ES
|<set> miniaverage Euclideanized-simplicity-weight damage
| <set> miniaverage Euclideanized-simplicity-weight damage
|
|  
|
|  
|-
|-
|(t)
| (t)
|Taxicab
| Taxicab
|1
| 1
| rowspan="2" |C
| rowspan="2" | C
| rowspan="2" |Complexity-weight
| rowspan="2" | Complexity-weight
| rowspan="2" |Complexity
| rowspan="2" | Complexity
|<set> miniaverage-C
| <set> miniaverage-C
|<set> miniaverage complexity-weight damage
| <set> miniaverage complexity-weight damage
|
|  
|Yes
| Yes
|-
|-
|E
| E
|Euclidean
| Euclidean
|2
| 2
|<set> miniaverage-EC
| <set> miniaverage-EC
|<set> miniaverage Euclideanized-complexity-weight damage
| <set> miniaverage Euclideanized-complexity-weight damage
|
|  
|
|  
|}
|}


Line 2,079: Line 1,162:
{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|-
|-
! 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
|-
|-
|EC-damage
| EC-damage
|Euclideanized-complexity-weight damage
| Euclideanized-complexity-weight damage
|<math>\mathsf{¢}</math><math>\small\mathsf{(EC)}</math>
| <math>\mathsf{¢}</math><math>\small\mathsf{(EC)}</math>
|Euclideanized-complexity-weighted cents
| Euclideanized-complexity-weighted cents
|-
|-
|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
|-
|-
|ES-damage
| ES-damage
|Euclideanized-simplicity-weight damage
| Euclideanized-simplicity-weight damage
|<math>\mathsf{¢}</math><math>\small\mathsf{(ES)}</math>
| <math>\mathsf{¢}</math><math>\small\mathsf{(ES)}</math>
|Euclideanized-simplicity-weighted cents
| Euclideanized-simplicity-weighted cents
|}
|}


===Complexity and simplicity===
=== Complexity and simplicity ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
! 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
|Euclideanized complexity
| Euclideanized complexity
|<math>\small\mathsf{(EC)}</math>
| <math>\small\mathsf{(EC)}</math>
|Euclideanized-complexity weight
| Euclideanized-complexity weight
|-
|-
|S
| S
|Simplicity
| Simplicity
|<math>\small\mathsf{(S)}</math>
| <math>\small\mathsf{(S)}</math>
|Simplicity weight
| Simplicity weight
|-
|-
|ES
| ES
|Euclideanized simplicity
| Euclideanized simplicity
|<math>\small\mathsf{(ES)}</math>
| <math>\small\mathsf{(ES)}</math>
|Euclideanized-simplicity weight
| Euclideanized-simplicity weight
|}
|}


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