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

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This is an appendix to our series "[[Dave Keenan & Douglas Blumeyer's guide to RTT]]", or "[[D&D's guide]]" for short. 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, or "[[D&D's guide]]" for short. 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.
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|[[tuning map|(tempered-prime) tuning map]]
|[[tuning map|(tempered-prime) tuning map]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒈 \\[-2pt] ¢ \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] · \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}  
Line 284: Line 284:
|[[interval span|(just) (interval) size]]
|[[interval span|(just) (interval) size]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒋 \\[-2pt] ¢ \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] · \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}  
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|[[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] ¢ \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] · \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}  
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|[[error|(interval) error]]
|[[error|(interval) error]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒓 \\[-2pt] ¢ \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] · \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}  
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|<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]]
|𝟙<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>(C)
|<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>
|<math>\small\mathsf{(C)}</math>
|<math>\small\mathsf{(C)}</math>
|complexity weight
|complexity weight
Line 428: Line 428:
|<math>s</math>
|<math>s</math>
|[[simplicity]]
|[[simplicity]]
|𝟙(S)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math>
|<math>\small\mathsf{(S)}</math>
|<math>\small\mathsf{(S)}</math>
|simplicity weight
|simplicity weight
Line 446: Line 446:
|<math>w</math>
|<math>w</math>
|[[weight]]
|[[weight]]
|𝟙(C) or 𝟙<math>\small\mathsf{(S)}</math>
|<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>
|<math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(S)}</math>
|complexity weight or simplicity weight
|complexity weight or simplicity weight
Line 465: Line 465:
|[[damage]]
|[[damage]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} |\mathrm{e}| \\[-2pt] ¢ \end{array}  
\begin{array} {c} |\mathrm{e}| \\[-2pt] {\small\mathsf{¢}} \end{array}  
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} w \\[-2pt] \text{(U, C, or S)} \end{array}  
\begin{array} {c} w \\[-2pt] \mathsf{(U, C, or\,S)} \end{array}  
</math>
</math>
| <math>\mathsf{¢}\small\mathsf{(U)}</math> or <math>\mathsf{¢}\small\mathsf{(C)}</math> or <math>\mathsf{¢}\small\mathsf{(S)}</math>
| <math>\mathsf{¢}\small\mathsf{(U)}</math> or <math>\mathsf{¢}\small\mathsf{(C)}</math> or <math>\mathsf{¢}\small\mathsf{(S)}</math>
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|[[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] ¢ \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] · \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}  
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|[[tempered target-interval size list]]
|[[tempered target-interval size list]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒕 \\[-2pt] ¢ \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] · \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}  
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|[[target-interval error list]]
|[[target-interval error list]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒓 \\[-2pt] ¢ \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] · \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}  
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|<math>W</math>
|<math>W</math>
|[[target-interval weight matrix]]
|[[target-interval weight matrix]]
|𝟙(C) or 𝟙(S) or 𝟙(U)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(U)}</math>
|<math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(S)}</math> or <math>\small\mathsf{(U)}</math>
|<math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(S)}</math> or <math>\small\mathsf{(U)}</math>
|complexity weight or simplicity weight
|complexity weight or simplicity weight
Line 646: Line 646:
|<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]]
|𝟙(C)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math>
|<math>\small\mathsf{(C)}</math>
|<math>\small\mathsf{(C)}</math>
|complexity weight
|complexity weight
Line 664: Line 664:
|<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]]
|𝟙(S)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math>
|<math>\small\mathsf{(S)}</math>
|<math>\small\mathsf{(S)}</math>
|simplicity weight
|simplicity weight
Line 681: Line 681:
|<math>|\textbf{e}|W</math>
|<math>|\textbf{e}|W</math>
|<math>\textbf{d}</math>
|<math>\textbf{d}</math>
|[[target-interval damage list]]<ref>Sometimes annotated units parentheses are dropped, but this is not compliant with SI standards, so we always keep them.</ref>
|[[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>
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} |\textbf{e}| \\[-2pt] ¢ \end{array}  
\begin{array} {c} |\textbf{e}| \\[-2pt] {\small\mathsf{¢}} \end{array}  
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} W \\[-2pt] (\text{U, C, or S}) \end{array}  
\begin{array} {c} W \\[-2pt] (\mathsf{U, C, or\,S}) \end{array}  
</math>
</math>
|<math>\mathsf{¢}\small\mathsf{(U)}</math>, <math>\mathsf{¢}\small\mathsf{(C)}</math>, or <math>\mathsf{¢}\small\mathsf{(S)}</math>
|<math>\mathsf{¢}\small\mathsf{(U)}</math>, <math>\mathsf{¢}\small\mathsf{(C)}</math>, or <math>\mathsf{¢}\small\mathsf{(S)}</math>
Line 1,103: Line 1,103:
|[[just tuning map|just(-prime) tuning map]]
|[[just tuning map|just(-prime) tuning map]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 1200 \\[-2pt] ¢ \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\text{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\text{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] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
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|[[generator tuning map]]
|[[generator tuning map]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 1200 \\[-2pt] ¢ \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\text{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\text{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] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
Line 1,185: Line 1,185:
|[[tuning map|(tempered-prime) tuning map]]
|[[tuning map|(tempered-prime) tuning map]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 1200 \\[-2pt] ¢ \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\text{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\text{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] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
Line 1,246: Line 1,246:
|[[interval span|(just) (interval) size]]
|[[interval span|(just) (interval) size]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒋 \\[-2pt] ¢ \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] · \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}  
Line 1,275: Line 1,275:
|[[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] ¢ \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] · \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}  
Line 1,305: Line 1,305:
|[[error|(interval) error]]
|[[error|(interval) error]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒓 \\[-2pt] ¢ \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] · \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}  
Line 1,485: Line 1,485:
|[[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] ¢ \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] · \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}  
Line 1,513: Line 1,513:
|[[tempered target-interval size list]]
|[[tempered target-interval size list]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒕 \\[-2pt] ¢ \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] · \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}  
Line 1,543: Line 1,543:
|[[target-interval error list]]
|[[target-interval error list]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒓 \\[-2pt] ¢ \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] · \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}  
Line 1,926: Line 1,926:
|[[JI generator tuning map]]
|[[JI generator tuning map]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 1200 \\[-2pt] ¢ \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\text{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\text{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] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
Line 2,001: Line 2,001:
|<math>C_{\text{p}}</math>
|<math>C_{\text{p}}</math>
|[[complexity prescaler]]
|[[complexity prescaler]]
|𝟙(C)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math>
|<math>\small\mathsf{(C)}</math>
|<math>\small\mathsf{(C)}</math>
|complexity weight
|complexity weight
Line 2,019: Line 2,019:
|<math>S_{\text{p}}</math>
|<math>S_{\text{p}}</math>
|[[simplicity prescaler]]
|[[simplicity prescaler]]
|𝟙(S)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math>
|<math>\small\mathsf{(S)}</math>
|<math>\small\mathsf{(S)}</math>
|simplicity weight
|simplicity weight
Line 2,548: Line 2,548:
|[[just tuning map|just(-prime) tuning map]]
|[[just tuning map|just(-prime) tuning map]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 1200 \\[-2pt] ¢ \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\text{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\text{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] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
Line 2,590: Line 2,590:
|[[generator tuning map]]
|[[generator tuning map]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 1200 \\[-2pt] ¢ \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\text{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\text{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] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
Line 2,630: Line 2,630:
|[[tuning map|(tempered-prime) tuning map]]
|[[tuning map|(tempered-prime) tuning map]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 1200 \\[-2pt] ¢ \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\text{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\text{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] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
Line 2,691: Line 2,691:
|[[interval span|(just) (interval) size]]
|[[interval span|(just) (interval) size]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒋 \\[-2pt] ¢ \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] · \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}  
Line 2,720: Line 2,720:
|[[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] ¢ \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] · \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}  
Line 2,750: Line 2,750:
|[[error|(interval) error]]
|[[error|(interval) error]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒓 \\[-2pt] ¢ \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] · \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}  
Line 2,930: Line 2,930:
|[[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] ¢ \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] · \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}  
Line 2,958: Line 2,958:
|[[tempered target-interval size list]]
|[[tempered target-interval size list]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒕 \\[-2pt] ¢ \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] · \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}  
Line 2,988: Line 2,988:
|[[target-interval error list]]
|[[target-interval error list]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 𝒓 \\[-2pt] ¢ \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] · \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}  
Line 3,422: Line 3,422:
|[[JI generator tuning map]]
|[[JI generator tuning map]]
|<math>\scriptsize  
|<math>\scriptsize  
\begin{array} {c} 1200 \\[-2pt] ¢ \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\text{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\text{oct}} \end{array}
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\text{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] · \end{array}
\begin{array} {c} \\[-2pt] · \end{array}
\\ \scriptsize \quad  
\\ \scriptsize \quad  
Line 3,497: Line 3,497:
|<math>C_{\text{p}}</math>
|<math>C_{\text{p}}</math>
|[[complexity pretransformer]]
|[[complexity pretransformer]]
|𝟙(C) or 𝟙(<alt<ref>In these tables, "alternative" means any complexity other than the default of log-product complexity, and "alt" stands for its abbreviation.</ref>>-C)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math><ref>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
Line 3,515: Line 3,515:
|<math>S_{\text{p}}</math>
|<math>S_{\text{p}}</math>
|[[simplicity pretransformer]]
|[[simplicity pretransformer]]
|𝟙(S) or 𝟙<math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math>
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{S)}</math>
|<math>\small\mathsf{(S)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math>
|<math>\small\mathsf{(S)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math>
|simplicity weight or <alternative>-simplicity weight
|simplicity weight or <alternative>-simplicity weight
Line 3,896: Line 3,896:
|
|
|-
|-
|<math>\mathsf{¢}</math>(<weight>)
|<math>\mathsf{¢}\small{(}</math><weight><math>\small\mathsf{)}</math>
|weighted cents
|weighted cents
|
|
Line 4,262: Line 4,262:
|E-<alt>-C-damage
|E-<alt>-C-damage
|Euclideanized-<alternative>-complexity-weight damage
|Euclideanized-<alternative>-complexity-weight damage
|<math>\mathsf{¢}</math>(E-<alt>-C)
|<math>\mathsf{¢}</math><math>\small\mathsf{(E}</math>-<alt>-<math>\small\mathsf{C)}</math>
|Euclideanized-<alternative>-complexity-weighted cents
|Euclideanized-<alternative>-complexity-weighted cents
|-
|-
Line 4,282: Line 4,282:
|E-<alt>-S-damage
|E-<alt>-S-damage
|Euclideanized-<alternative>-simplicity-weight damage
|Euclideanized-<alternative>-simplicity-weight damage
|<math>\mathsf{¢}</math>(E-<alt>-S)
|<math>\mathsf{¢}</math><math>\small\mathsf{(E}</math>-<alt>-<math>\small\mathsf{S)}</math>
|Euclideanized-<alternative>-simplicity-weighted cents
|Euclideanized-<alternative>-simplicity-weighted cents
|}
|}
Line 4,300: Line 4,300:
|C
|C
|complexity
|complexity
|𝟙(C) = (C)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> = <math>\small\mathsf{(C)}</math>
|complexity weight
|complexity weight
|-
|-
|<alt>-C
|<alt>-C
|<alternative> complexity
|<alternative> complexity
|𝟙<math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math> = (<alternative>-C)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math> = <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math>
|<alternative>-complexity weight
|<alternative>-complexity weight
|-
|-
|EC
|EC
|Euclideanized complexity
|Euclideanized complexity
|𝟙<math>\small\mathsf{(EC)}</math> = <math>\small\mathsf{(EC)}</math>
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(EC)}</math> = <math>\small\mathsf{(EC)}</math>
|Euclideanized-complexity weight
|Euclideanized-complexity weight
|-
|-
|E-<alt>-C
|E-<alt>-C
|Euclideanized-<alternative> complexity
|Euclideanized-<alternative> complexity
|𝟙(E-<alt>-C) = (E-<alternative>-C)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(E}</math>-<alt>-<math>\scriptsize\mathsf{C)}</math> = <math>\small\mathsf{(E}</math>-<alt>-<math>\small\mathsf{C)}</math>
|Euclideanized-<alternative>-complexity weight
|Euclideanized-<alternative>-complexity weight
|-
|-
|S
|S
|simplicity
|simplicity
|𝟙(S) = (S)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math> = <math>\small\mathsf{(S)}</math>
|simplicity weight
|simplicity weight
|-
|-
|<alt>-S
|<alt>-S
|<alternative> simplicity
|<alternative> simplicity
|𝟙<math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math> = (<alternative>-S)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{S)}</math> = <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math>
|<alternative>-simplicity weight
|<alternative>-simplicity weight
|-
|-
|ES
|ES
|Euclideanized simplicity
|Euclideanized simplicity
|𝟙<math>\small\mathsf{(ES)}</math> = <math>\small\mathsf{(ES)}</math>
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(ES)}</math> = <math>\small\mathsf{(ES)}</math>
|Euclideanized-simplicity weight
|Euclideanized-simplicity weight
|-
|-
|E-<alt>-S
|E-<alt>-S
|Euclideanized-<alternative> simplicity
|Euclideanized-<alternative> simplicity
|𝟙(E-<alt>-S) = (E-<alternative>-S)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(E}</math>-<alt>-<math>\scriptsize\mathsf{S)}</math> = <math>\small\mathsf{(E}</math>-<alt>-<math>\small\mathsf{S)}</math>
|Euclideanized-<alternative>-simplicity weight
|Euclideanized-<alternative>-simplicity weight
|}
|}
Line 4,351: Line 4,351:
|S
|S
|simplicity
|simplicity
|𝟙(S) = (S)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math> = <math>\small\mathsf{(S)}</math>
|simplicity weight
|simplicity weight
|-
|-
|<alt>-S
|<alt>-S
|<alternative> simplicitity
|<alternative> simplicitity
|𝟙<math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math> = (<alternative>-S)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{S)}</math> = <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math>
|<alternative>-simplicity
|<alternative>-simplicity
|-
|-
|ES
|ES
|Euclideanized simplicity
|Euclideanized simplicity
|𝟙<math>\small\mathsf{(ES)}</math> = <math>\small\mathsf{(ES)}</math>
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(ES)}</math> = <math>\small\mathsf{(ES)}</math>
|Euclideanized-simplicicty weight
|Euclideanized-simplicicty weight
|-
|-
|E-<alt>-S
|E-<alt>-S
|Euclideanized-<alternative> simplicity
|Euclideanized-<alternative> simplicity
|𝟙(E-<alt>-S) = (E-<alternative>-S)
|<math>\small\mathsf{𝟙}\scriptsize\mathsf{(E}</math>-<alt>-<math>\scriptsize\mathsf{S)}</math> = <math>\small\mathsf{(E}</math>-<alt>-<math>\small\mathsf{S)}</math>
|Euclideanized-<alternative>-simplicity
|Euclideanized-<alternative>-simplicity
|}
|}
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