Dave Keenan & Douglas Blumeyer's guide to RTT/Conventions for names, variables, units, and notations: Difference between revisions
ArrowHead294 (talk | contribs) m →Complexity and simplicity: Capitalisation |
ArrowHead294 (talk | contribs) →Objects: Capitalisation and formatting |
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{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|- | |- | ||
! rowspan="2" | | ! rowspan="2" | Equivalent expressions | ||
! rowspan="2" | | ! rowspan="2" | Variable | ||
! rowspan="2" | | ! rowspan="2" | Name | ||
! colspan="3" | | ! colspan="3" | Units | ||
! colspan="2" | | ! colspan="2" | Shape | ||
! colspan="2" | | ! colspan="2" | Type | ||
! colspan="2" | EBK notation | ! colspan="2" | EBK notation | ||
! colspan="4" | | ! colspan="4" | Subobjects | ||
! rowspan="2" | | ! rowspan="2" | Notes | ||
|- | |- | ||
! | ! Unreduced | ||
! | ! Reduced | ||
! | ! Read as | ||
! | ! Unreduced | ||
! | ! Reduced | ||
! | ! Numeric | ||
! | ! Structural | ||
! | ! Row-first | ||
! | ! Col-first | ||
! | ! Row | ||
! | ! Column | ||
! | ! Diagonal | ||
! | ! Entry | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Mapping | ||
|- | |- | ||
| | | | ||
| <math>\textbf{i}</math> | | <math>\textbf{i}</math> | ||
| [[interval| ( | | [[interval|(Just) interval]] | ||
| | | | ||
| <math>\small 𝗽</math> | | <math>\small 𝗽</math> | ||
| | | Primes | ||
| | | | ||
| <math>\scriptsize (d, 1)</math> | | <math>\scriptsize (d, 1)</math> | ||
| | | Integer | ||
| | | Vector | ||
| | | | ||
| [...⟩ | | [...⟩ | ||
| Line 2,191: | Line 2,191: | ||
| | | | ||
| <math>\mathrm{i}_i</math> | | <math>\mathrm{i}_i</math> | ||
| | | Specific type: vector ([[prime-count vector]] or PC-vector) | ||
Jargon name: monzo | |||
|- | |- | ||
| | | | ||
| <math>M</math> | | <math>M</math> | ||
| [[Mapping| ( | | [[Mapping|(Temperament) mapping (matrix)]] | ||
| | | | ||
| <math>\small 𝗴</math>/<math>\small 𝗽</math> | | <math>\small 𝗴</math>/<math>\small 𝗽</math> | ||
| | | Generators per prime | ||
| | | | ||
| <math>\scriptsize (r, d)</math> | | <math>\scriptsize (r, d)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| [⟨...] ...} | | [⟨...] ...} | ||
| ⟨[...} ...] | | ⟨[...} ...] | ||
| Line 2,210: | Line 2,210: | ||
| | | | ||
| <math>m_{ij}</math> | | <math>m_{ij}</math> | ||
| | | Jargon name: val list | ||
|- | |- | ||
| <math>M\textbf{i}</math> | | <math>M\textbf{i}</math> | ||
| <math>\textbf{y}</math> | | <math>\textbf{y}</math> | ||
| [[ | | [[Mapped interval]] | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} M \\[-2pt] 𝗴 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} M \\[-2pt] 𝗴 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 2,224: | Line 2,224: | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} | \begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\! \! | \! \! | ||
</math> | </math> | ||
| <math>\scriptsize (r, 1)</math> | | <math>\scriptsize (r, 1)</math> | ||
| | | Integer | ||
| | | Vector | ||
| | | | ||
| [...} | | [...} | ||
| Line 2,238: | Line 2,238: | ||
| | | | ||
| | | | ||
| | | Specific type: [[generator-count vector]] (GC-vector) | ||
Jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval | |||
|- | |- | ||
| | | | ||
| <math>𝒎</math> | | <math>𝒎</math> | ||
| [[map| ( | | [[map|(Temperament) map]] | ||
| | | | ||
| <math>\small 𝗴</math>/<math>\small 𝗽</math> | | <math>\small 𝗴</math>/<math>\small 𝗽</math> | ||
| | | Generators per prime | ||
| | | | ||
| <math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Integer | ||
| | | Vector | ||
| ⟨...] | | ⟨...] | ||
| | | | ||
| Line 2,257: | Line 2,257: | ||
| | | | ||
| <math>m_i</math> | | <math>m_i</math> | ||
| | | Jargon name: val | ||
|- | |- | ||
| <math>n + r</math> | | <math>n + r</math> | ||
| <math>d</math> | | <math>d</math> | ||
| [[ | | [[Dimensionality]] | ||
| | | | ||
| | | | ||
| Line 2,267: | Line 2,267: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,279: | Line 2,279: | ||
| <math>d - n</math> | | <math>d - n</math> | ||
| <math>r</math> | | <math>r</math> | ||
| [[ | | [[Rank]] | ||
| | | | ||
| | | | ||
| Line 2,285: | Line 2,285: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,297: | Line 2,297: | ||
| <math>d - r</math> | | <math>d - r</math> | ||
| <math>n</math> | | <math>n</math> | ||
| [[ | | [[Nullity]] | ||
| | | | ||
| | | | ||
| Line 2,303: | Line 2,303: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,313: | Line 2,313: | ||
| | | | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Tuning | ||
|- | |- | ||
| <math>\slant{\mathbf{1}}L</math> | | <math>\slant{\mathbf{1}}L</math> | ||
| <math>{\large\textbf{𝓁}}\hspace{2mu}</math> | | <math>{\large\textbf{𝓁}}\hspace{2mu}</math> | ||
| [[ | | [[Log-prime map]] | ||
| | | | ||
| <math>\small\mathsf{oct}</math>/<math>\small 𝗽</math> | | <math>\small\mathsf{oct}</math>/<math>\small 𝗽</math> | ||
| | | Octaves per prime | ||
| | | | ||
| <math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Real | ||
| | | Vector | ||
| ⟨...] | | ⟨...] | ||
| | | | ||
| Line 2,337: | Line 2,337: | ||
𝒈_{\text{j}}M_{\text{j}}</math> | 𝒈_{\text{j}}M_{\text{j}}</math> | ||
| <math>𝒋</math> | | <math>𝒋</math> | ||
| [[just tuning map| | | [[just tuning map|Just(-prime) tuning map]] | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | \begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | ||
| Line 2,351: | Line 2,351: | ||
</math> | </math> | ||
| <math>\mathsf{¢}</math>/<math>\small 𝗽</math> | | <math>\mathsf{¢}</math>/<math>\small 𝗽</math> | ||
| | | Cents per prime | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
\begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array} | \begin{array} {c} 1200 \\[-3pt] \left(1, \cancel{1}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} \slant{\mathbf{1}} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array} | \begin{array} {c} \slant{\mathbf{1}} \\[-3pt] \left(\cancel{1}, \cancel{d}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array} | \begin{array} {c} L \\[-3pt] \left(\cancel{d}, \cancel{d}) \end{array} | ||
\\ \scriptsize \quad | \\ \scriptsize \quad | ||
\! \! | \! \! | ||
\begin{array} {c} G_{\text{j}} \\[-3pt] (\cancel{d}, \cancel{r}) \end{array} | \begin{array} {c} G_{\text{j}} \\[-3pt] \left(\cancel{d}, \cancel{r}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} M_{\text{j}} \\[-3pt] (\cancel{r}, d) \end{array} | \begin{array} {c} M_{\text{j}} \\[-3pt] \left(\cancel{r}, d\right) \end{array} | ||
\! \! | \! \! | ||
</math> | </math> | ||
| <math>\scriptsize (1, d_{\text{p}})</math> | | <math>\scriptsize \left(1, d_{\text{p}}\right)</math> | ||
| | | Real | ||
| | | Vector | ||
| ⟨...] | | ⟨...] | ||
| | | | ||
| Line 2,379: | Line 2,379: | ||
| <math>1200×\slant{\mathbf{1}}LG</math> | | <math>1200×\slant{\mathbf{1}}LG</math> | ||
| <math>𝒈</math> | | <math>𝒈</math> | ||
| [[ | | [[Generator tuning map]] | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | \begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | ||
| Line 2,391: | Line 2,391: | ||
</math> | </math> | ||
| <math>\mathsf{¢}</math>/<math>\small 𝗴</math> | | <math>\mathsf{¢}</math>/<math>\small 𝗴</math> | ||
| | | Cents per generator | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
| Line 2,405: | Line 2,405: | ||
</math> | </math> | ||
| <math>\scriptsize (1, r)</math> | | <math>\scriptsize (1, r)</math> | ||
| | | Real | ||
| | | Vector | ||
| {...] | | {...] | ||
| | | | ||
| Line 2,419: | Line 2,419: | ||
𝒈M</math> | 𝒈M</math> | ||
| <math>𝒕</math> | | <math>𝒕</math> | ||
| [[tuning map| ( | | [[tuning map|(Tempered-prime) tuning map]] | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | \begin{array} {c} 1200 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | ||
| Line 2,433: | Line 2,433: | ||
</math> | </math> | ||
| <math>\mathsf{¢}</math>/<math>\small 𝗽</math> | | <math>\mathsf{¢}</math>/<math>\small 𝗽</math> | ||
| | | Cents per prime | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
\begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array} | \begin{array} {c} 1200 \\[-3pt] \left(1, \cancel{1}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} \slant{\mathbf{1}} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array} | \begin{array} {c} \slant{\mathbf{1}} \\[-3pt] \left(\cancel{1}, \cancel{d}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array} | \begin{array} {c} L \\[-3pt] \left(\cancel{d}, \cancel{d}\right) \end{array} | ||
\\ \scriptsize \quad | \\ \scriptsize \quad | ||
\! \! | \! \! | ||
\begin{array} {c} G \\[-3pt] (\cancel{d}, \cancel{r}) \end{array} | \begin{array} {c} G \\[-3pt] \left(\cancel{d}, \cancel{r}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array} | \begin{array} {c} M \\[-3pt] \left(\cancel{r}, d\right) \end{array} | ||
\! \! | \! \! | ||
</math> | </math> | ||
| <math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Real | ||
| | | Vector | ||
| ⟨...] | | ⟨...] | ||
| | | | ||
| Line 2,462: | Line 2,462: | ||
1200×\slant{\mathbf{1}}L(P - I)</math> | 1200×\slant{\mathbf{1}}L(P - I)</math> | ||
| <math>𝒓</math> | | <math>𝒓</math> | ||
| [[retuning map| | | [[retuning map|Retuning (or mistuning) map]] | ||
| | | | ||
| <math>\mathsf{¢}</math>/<math>\small 𝗽</math> | | <math>\mathsf{¢}</math>/<math>\small 𝗽</math> | ||
| | | Cents per prime | ||
| | | | ||
| <math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Real | ||
| | | Vector | ||
| ⟨...] | | ⟨...] | ||
| | | | ||
| Line 2,476: | Line 2,476: | ||
| | | | ||
| <math>r_i</math> | | <math>r_i</math> | ||
| | | 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| ( | | [[interval span|(Just) (interval) size]] | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 2,487: | Line 2,487: | ||
</math> | </math> | ||
| <math>\mathsf{¢}</math> | | <math>\mathsf{¢}</math> | ||
| | | Cents | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
\begin{array} {c} 𝒋 \\[-3pt] (1, \cancel{d}) \end{array} | \begin{array} {c} 𝒋 \\[-3pt] \left(1, \cancel{d}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\! \! | \! \! | ||
</math> | </math> | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,504: | Line 2,504: | ||
| | | | ||
| | | | ||
| | | Mnemonic: <math>\mathrm{o}</math>riginal size | ||
|- | |- | ||
| <math>𝒈M\textbf{i} \\ | | <math>𝒈M\textbf{i} \\ | ||
𝒕\textbf{i}</math> | 𝒕\textbf{i}</math> | ||
| <math>\mathrm{a}</math> | | <math>\mathrm{a}</math> | ||
| {{subpage|tuning fundamentals|uprev|s=Example 3|text= | | {{subpage|tuning fundamentals|uprev|s=Example 3|text=Tempered (interval) size}} | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 2,519: | Line 2,519: | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
\begin{array} {c} 𝒕 \\[-3pt] (1, \cancel{d}) \end{array} | \begin{array} {c} 𝒕 \\[-3pt] \left(1, \cancel{d}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\! \! | \! \! | ||
</math> | </math> | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,533: | Line 2,533: | ||
| | | | ||
| | | | ||
| | | Mnemonic: <math>\mathrm{a}</math>ltered size | ||
|- | |- | ||
| <math>𝒕\textbf{i} - 𝒋\textbf{i} \\ | | <math>𝒕\textbf{i} - 𝒋\textbf{i} \\ | ||
| Line 2,539: | Line 2,539: | ||
𝒓\textbf{i}</math> | 𝒓\textbf{i}</math> | ||
| <math>\mathrm{e}</math> | | <math>\mathrm{e}</math> | ||
| [[error| ( | | [[error|(Interval) error]] | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 𝒓 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} 𝒓 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 2,546: | Line 2,546: | ||
</math> | </math> | ||
| <math>\mathsf{¢}</math> | | <math>\mathsf{¢}</math> | ||
| | | Cents | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
\begin{array} {c} 𝒓 \\[-3pt] (1, \cancel{d}) \end{array} | \begin{array} {c} 𝒓 \\[-3pt] \left(1, \cancel{d}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\! \! | \! \! | ||
</math> | </math> | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,565: | Line 2,565: | ||
| | | | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Optimization | ||
|- | |- | ||
| | | | ||
| <math>p</math> | | <math>p</math> | ||
| [[ | | [[Optimization power]] | ||
| | | | ||
| | | | ||
| Line 2,575: | Line 2,575: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,586: | Line 2,586: | ||
|- | |- | ||
| | | | ||
| <math> | | <math>\llangle\,\cdot\,\rrangle_p</math> | ||
| [[ | | [[Power mean]] (<math>p</math>-mean) | ||
| | | | ||
| | | | ||
| Line 2,593: | Line 2,593: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,603: | Line 2,603: | ||
| | | | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Damage | ||
|- | |- | ||
| | | | ||
| <math>c</math> | | <math>c</math> | ||
| {{subpage| | | {{subpage|Tuning_fundamentals|prev|s=complexity}} | ||
| colspan="3" | ( | | colspan="3" | (See complexities section of complexities and simplicities table) | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,623: | Line 2,623: | ||
| <math>\dfrac1c</math> | | <math>\dfrac1c</math> | ||
| <math>s</math> | | <math>s</math> | ||
| [[ | | [[Simplicity]] | ||
| colspan="3" | ( | | colspan="3" | (See simplicities section of complexities and simplicities table) | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,639: | Line 2,639: | ||
| <math>c</math> or <math>s</math> | | <math>c</math> or <math>s</math> | ||
| <math>w</math> | | <math>w</math> | ||
| [[ | | [[Weight]] | ||
| colspan="3" | ( | | colspan="3" | (See complexities and simplicities table) | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,653: | Line 2,653: | ||
| | | | ||
|- | |- | ||
| <math> | | <math>\abs{\mathrm{e}} w</math> | ||
| <math>\mathrm{d}</math> | | <math>\mathrm{d}</math> | ||
| [[ | | [[Damage]] | ||
| colspan="3" | ( | | colspan="3" | (See damages table) | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,669: | Line 2,669: | ||
| | | | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Target-intervals | ||
|- | |- | ||
| | | | ||
| <math>\mathrm{T}</math> | | <math>\mathrm{T}</math> | ||
| [[ | | [[Target-interval list]] | ||
| | | | ||
| <math>\small 𝗽</math> | | <math>\small 𝗽</math> | ||
| | | Primes | ||
| | | | ||
| <math>\scriptsize (d, k)</math> | | <math>\scriptsize (d, k)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | | ||
| [[...⟩ ...] | | [[...⟩ ...] | ||
| Line 2,691: | Line 2,691: | ||
| <math>M\mathrm{T}</math> | | <math>M\mathrm{T}</math> | ||
| <math>\mathrm{Y}</math> | | <math>\mathrm{Y}</math> | ||
| [[ | | [[Mapped target-interval list]] | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} M \\[-2pt] 𝗴 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} M \\[-2pt] 𝗴 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 2,698: | Line 2,698: | ||
</math> | </math> | ||
| <math>\small 𝗴</math> | | <math>\small 𝗴</math> | ||
| | | Generators | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} | \begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\! \! | \! \! | ||
</math> | </math> | ||
| <math>\scriptsize (r, k)</math> | | <math>\scriptsize (r, k)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | | ||
| [[...} ...] | | [[...} ...] | ||
| Line 2,715: | Line 2,715: | ||
| | | | ||
| <math>\mathrm{y}_{ij}</math> | | <math>\mathrm{y}_{ij}</math> | ||
| | | Mnemonic: looks like bent-up 'T', or cross between 'M' and 'T' | ||
|- | |- | ||
| <math>𝒋\mathrm{T}</math> | | <math>𝒋\mathrm{T}</math> | ||
| <math>\textbf{o}</math> | | <math>\textbf{o}</math> | ||
| {{subpage|tuning fundamentals|uprev|s=primes|text= | | {{subpage|tuning fundamentals|uprev|s=primes|text=Target-interval (just) size list}} | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 2,726: | Line 2,726: | ||
</math> | </math> | ||
| <math>\mathsf{¢}</math> | | <math>\mathsf{¢}</math> | ||
| | | Cents | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
\begin{array} {c} 𝒋 \\[-3pt] (1, \cancel{d}) \end{array} | \begin{array} {c} 𝒋 \\[-3pt] \left(1, \cancel{d}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\! \! | \! \! | ||
</math> | </math> | ||
| <math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
| [...] | | [...] | ||
| | | | ||
| Line 2,743: | Line 2,743: | ||
| | | | ||
| <math>\mathrm{o}_i</math> | | <math>\mathrm{o}_i</math> | ||
| | | Mnemonic: <math>\textbf{o}</math>riginal size list | ||
|- | |- | ||
| <math>𝒕\mathrm{T}</math> | | <math>𝒕\mathrm{T}</math> | ||
| <math>\textbf{a}</math> | | <math>\textbf{a}</math> | ||
| [[ | | [[Tempered target-interval size list]] | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 2,754: | Line 2,754: | ||
</math> | </math> | ||
| <math>\mathsf{¢}</math> | | <math>\mathsf{¢}</math> | ||
| | | Cents | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
\begin{array} {c} 𝒕 \\[-3pt] (1, \cancel{d}) \end{array} | \begin{array} {c} 𝒕 \\[-3pt] \left(1, \cancel{d}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\! \! | \! \! | ||
</math> | </math> | ||
| <math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
| [...] | | [...] | ||
| | | | ||
| Line 2,771: | Line 2,771: | ||
| | | | ||
| <math>\mathrm{a}_i</math> | | <math>\mathrm{a}_i</math> | ||
| | | Mnemonic: <math>\textbf{a}</math>ltered size list | ||
|- | |- | ||
| <math>𝒕\mathrm{T} - 𝒋\mathrm{T} \\ | | <math>𝒕\mathrm{T} - 𝒋\mathrm{T} \\ | ||
| Line 2,784: | Line 2,784: | ||
</math> | </math> | ||
| <math>\mathsf{¢}</math> | | <math>\mathsf{¢}</math> | ||
| | | Cents | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
\begin{array} {c} | \begin{array} {c} 𝒕 \\[-3pt] \left(1, \cancel{d}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\! \! | \! \! | ||
</math> | </math> | ||
| <math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
| [...] | | [...] | ||
| | | | ||
| Line 2,805: | Line 2,805: | ||
| <math>C</math> or <math>S</math> | | <math>C</math> or <math>S</math> | ||
| <math>W</math> | | <math>W</math> | ||
| [[ | | [[Target-interval weight matrix]] | ||
| colspan="3" | ( | | colspan="3" | (See complexities and simplicities table) | ||
| | | | ||
| <math>\scriptsize (k, k)</math> | | <math>\scriptsize (k, k)</math> | ||
| | | Real | ||
| | | Matrix | ||
| | | | ||
| [[...] ...] | | [[...] ...] | ||
| Line 2,821: | Line 2,821: | ||
| | | | ||
| <math>C</math> | | <math>C</math> | ||
| {{subpage|tuning fundamentals|uprev|s=complexity-weight damage|text= | | {{subpage|tuning fundamentals|uprev|s=complexity-weight damage|text=Target-interval complexity weight matrix}} | ||
| colspan="3" | ( | | colspan="3" | (See complexities section of complexities and simplicities table) | ||
| | | | ||
| <math>\scriptsize (k, k)</math> | | <math>\scriptsize (k, k)</math> | ||
| | | Real | ||
| | | Matrix | ||
| | | | ||
| [[...] ...] | | [[...] ...] | ||
| Line 2,837: | Line 2,837: | ||
| <math>\dfrac1C</math> | | <math>\dfrac1C</math> | ||
| <math>S</math> | | <math>S</math> | ||
| {{subpage|tuning fundamentals|uprev|s=complexity-weight_damage|text= | | {{subpage|tuning fundamentals|uprev|s=complexity-weight_damage|text=Target-interval simplicity weight matrix}} | ||
| colspan="3" | ( | | colspan="3" | (See simplicities section of complexities and simplicities table) | ||
| | | | ||
| <math>\scriptsize (k, k)</math> | | <math>\scriptsize (k, k)</math> | ||
| | | Real | ||
| | | Matrix | ||
| | | | ||
| [[...] ...] | | [[...] ...] | ||
| Line 2,849: | Line 2,849: | ||
| <math>𝒔</math> | | <math>𝒔</math> | ||
| <math>s_i</math> | | <math>s_i</math> | ||
| | | Entry-wise reciprocal of <math>C</math> | ||
|- | |- | ||
| <math> | | <math>\abs{\textbf{e}} W \\ | ||
1200×\slant{\mathbf{1}}L | 1200×\slant{\mathbf{1}}L\abs{P - I} \mathrm{T}W</math> | ||
| <math>\textbf{d}</math> | | <math>\textbf{d}</math> | ||
| [[ | | [[Target-interval damage list]] | ||
| colspan="3" | ( | | colspan="3" | (See damages table) | ||
| | | | ||
| <math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
| [...] | | [...] | ||
| | | | ||
| Line 2,870: | Line 2,870: | ||
| | | | ||
| <math>k</math> | | <math>k</math> | ||
| [[ | | [[Target-interval count]] | ||
| | | | ||
| | | | ||
| Line 2,876: | Line 2,876: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,884: | Line 2,884: | ||
| | | | ||
| | | | ||
| | | Mnemonic: <math>k</math>ount | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Held-intervals | ||
|- | |- | ||
| | | | ||
| <math>\mathrm{H}</math> | | <math>\mathrm{H}</math> | ||
| [[ | | [[Held-interval basis]] | ||
| | | | ||
| <math>\small 𝗽</math> | | <math>\small 𝗽</math> | ||
| | | Primes | ||
| | | | ||
| <math>\scriptsize (d, h)</math> | | <math>\scriptsize (d, h)</math> | ||
| | | | ||
| | | Matrix | ||
| | | | ||
| [[...⟩ ...] | | [[...⟩ ...] | ||
| Line 2,908: | Line 2,908: | ||
| | | | ||
| <math>h</math> | | <math>h</math> | ||
| [[ | | [[Held-interval count]] | ||
| | | | ||
| | | | ||
| Line 2,914: | Line 2,914: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,924: | Line 2,924: | ||
| | | | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Exploring temperaments | ||
|- | |- | ||
| | | | ||
| <math>\mathrm{C}</math> | | <math>\mathrm{C}</math> | ||
| [[ | | [[Comma basis]] | ||
| | | | ||
| <math>\small 𝗽</math> | | <math>\small 𝗽</math> | ||
| | | Primes | ||
| | | | ||
| <math>\scriptsize (d, n)</math> | | <math>\scriptsize (d, n)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | | ||
| [[...⟩ ...] | | [[...⟩ ...] | ||
| Line 2,942: | Line 2,942: | ||
| | | | ||
| <math>\mathrm{c}_{ij}</math> | | <math>\mathrm{c}_{ij}</math> | ||
| | | Jargon name: monzo list | ||
|- | |- | ||
| | | | ||
| <math>\textbf{c}</math> | | <math>\textbf{c}</math> | ||
| [[ | | [[Comma]] | ||
| | | | ||
| <math>\small 𝗽</math> | | <math>\small 𝗽</math> | ||
| | | Primes | ||
| | | | ||
| <math>\scriptsize (d, 1)</math> | | <math>\scriptsize (d, 1)</math> | ||
| | | Integer | ||
| | | Vector | ||
| | | | ||
| [...⟩ | | [...⟩ | ||
| Line 2,960: | Line 2,960: | ||
| | | | ||
| <math>\mathrm{c}_i</math> | | <math>\mathrm{c}_i</math> | ||
| | | Specific type: vector ([[prime-count vector]] or PC-vector) | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Computation | ||
|- | |- | ||
| | | | ||
| <math>\llzigzag·\,\rrzigzag\! _p</math> | | <math>\llzigzag·\,\rrzigzag\! _p</math> | ||
| [[ | | [[Power sum]] (<math>p</math>-sum) | ||
| | | | ||
| | | | ||
| Line 2,972: | Line 2,972: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 2,982: | Line 2,982: | ||
| | | | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | All-interval tuning schemes | ||
|- | |- | ||
| <math>\mathrm{I}</math> | | <math>\mathrm{I}</math> | ||
| <math>\mathrm{T}_{\text{p}}</math> | | <math>\mathrm{T}_{\text{p}}</math> | ||
| [[ | | [[Prime proxy target-interval list]] | ||
| | | | ||
| <math>\small 𝗽</math> | | <math>\small 𝗽</math> | ||
| | | Primes | ||
| | | | ||
| <math>\scriptsize (d, d)</math> | | <math>\scriptsize (d, d)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | | ||
| ⟨[...⟩ ...] | | ⟨[...⟩ ...] | ||
| Line 3,004: | Line 3,004: | ||
| | | | ||
| <math>X</math> | | <math>X</math> | ||
| {{subpage|alternative complexities|uprev|s=Prescaling_vs._pretransforming|text= | | {{subpage|alternative complexities|uprev|s=Prescaling_vs._pretransforming|text=Complexity pretransformer}} | ||
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math><ref group="note">In these tables, "alternative" means any complexity other than the default of log-product complexity, and "alt" stands for its abbreviation.</ref> | | <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math><ref group="note">In these tables, "alternative" means any complexity other than the default of log-product complexity, and "alt" stands for its abbreviation.</ref> | ||
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math> | | <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math> | ||
| | | Complexity weight or <alternative>-complexity weight | ||
| | | | ||
| <math>\scriptsize (d, d)</math> or <math>\scriptsize (d+1, d+1)</math> | | <math>\scriptsize (d, d)</math> or <math>\scriptsize (d+1, d+1)</math> | ||
| | | Real | ||
| | | Matrix | ||
| [⟨...] ...⟩ | | [⟨...] ...⟩ | ||
| | | | ||
| Line 3,022: | Line 3,022: | ||
| <math>\text{diag}({\large\textbf{𝓁}}\hspace{2mu})</math> | | <math>\text{diag}({\large\textbf{𝓁}}\hspace{2mu})</math> | ||
| <math>L</math> | | <math>L</math> | ||
| [[ | | [[Log-prime matrix]] | ||
| | | | ||
| <math>\small\mathsf{oct}</math>/<math>\small 𝗽</math> | | <math>\small\mathsf{oct}</math>/<math>\small 𝗽</math> | ||
| | | Octaves per prime | ||
| | | | ||
| <math>\scriptsize (d, d)</math> | | <math>\scriptsize (d, d)</math> | ||
| | | Real | ||
| | | Matrix | ||
| [⟨...] ...⟩ | | [⟨...] ...⟩ | ||
| ⟨[...⟩ ...] | | ⟨[...⟩ ...] | ||
| Line 3,040: | Line 3,040: | ||
| | | | ||
| <math>q</math> | | <math>q</math> | ||
| {{subpage|all-interval_tuning_schemes|uprev|s=dual_norms|text= | | {{subpage|all-interval_tuning_schemes|uprev|s=dual_norms|text=Interval complexity norm power}} | ||
| | | | ||
| | | | ||
| Line 3,046: | Line 3,046: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 3,057: | Line 3,057: | ||
|- | |- | ||
| | | | ||
| <math> | | <math>\norm{\cdot}_q</math> | ||
| [[ | | [[Power norm]] (<math>p</math>-norm) | ||
| | | | ||
| | | | ||
| Line 3,064: | Line 3,064: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 3,076: | Line 3,076: | ||
| <math>\dfrac1{1-\frac1q}</math> | | <math>\dfrac1{1-\frac1q}</math> | ||
| <math>\text{dual}(q)</math> | | <math>\text{dual}(q)</math> | ||
| {{subpage|all-interval tuning schemes|uprev|s=dual_norms|text= | | {{subpage|all-interval tuning schemes|uprev|s=dual_norms|text=Dual norm power}} | ||
| | | | ||
| | | | ||
| Line 3,082: | Line 3,082: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 3,093: | Line 3,093: | ||
|- | |- | ||
| | | | ||
| <math> | | <math>\norm{X\mathbf{i}}_q</math> | ||
| [[ | | [[Interval complexity]] | ||
| | | | ||
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math> | | <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math> | ||
| Line 3,100: | Line 3,100: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 3,111: | Line 3,111: | ||
|- | |- | ||
| | | | ||
| <math> | | <math>\norm{𝒓X^{-1}}_{\text{dual}(q)}</math> | ||
| [[ | | [[Retuning magnitude]] | ||
| | | | ||
| <math>\mathsf{¢}\small\mathsf{(C^{-1})}</math> or <math>\mathsf{¢}\small\mathsf{(}</math><alt>-<math>\small\mathsf{C^{-1})}</math> | | <math>\mathsf{¢}\small\mathsf{(C^{-1})}</math> or <math>\mathsf{¢}\small\mathsf{(}</math><alt>-<math>\small\mathsf{C^{-1})}</math> | ||
| Line 3,118: | Line 3,118: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 3,128: | Line 3,128: | ||
| | | | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Alternative complexities | ||
|- | |- | ||
| | | | ||
| <math>𝒑</math> | | <math>𝒑</math> | ||
| {{subpage|alternative complexities|uprev|s=formulas|text= | | {{subpage|alternative complexities|uprev|s=formulas|text=Prime list}}<ref group="note">May be used for a prime-limit or for any prime-only list.</ref> | ||
| | | | ||
| | | | ||
| Line 3,138: | Line 3,138: | ||
| | | | ||
| <math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Integer | ||
| | | List | ||
| [...] | | [...] | ||
| | | | ||
| Line 3,150: | Line 3,150: | ||
| | | | ||
| <math>\slant{\mathbf{1}}</math> | | <math>\slant{\mathbf{1}}</math> | ||
| {{subpage|alternative complexities|uprev|s=proportionality to size|text= | | {{subpage|alternative complexities|uprev|s=proportionality to size|text=Summation map}} | ||
| | | | ||
| | | | ||
| Line 3,156: | Line 3,156: | ||
| | | | ||
| <math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Integer | ||
| | | Vector | ||
| ⟨...] | | ⟨...] | ||
| | | | ||
| Line 3,168: | Line 3,168: | ||
| | | | ||
| <math>1200</math> | | <math>1200</math> | ||
| {{subpage|alternative complexities|uprev|s=Proportionality to size|text= | | {{subpage|alternative complexities|uprev|s=Proportionality to size|text=Octaves-to-cents conversion}} | ||
| | | | ||
| ¢/oct | | ¢/oct | ||
| | | Cents per octave | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 3,186: | Line 3,186: | ||
| | | | ||
| <math>Z</math> | | <math>Z</math> | ||
| {{subpage|alternative complexities|uprev|s=Normifying: size-sensitizing matrix|text= | | {{subpage|alternative complexities|uprev|s=Normifying: size-sensitizing matrix|text=Size-sensitizing matrix}} | ||
| | | | ||
| | | | ||
| Line 3,192: | Line 3,192: | ||
| | | | ||
| <math>\scriptsize (d+1, d)</math> | | <math>\scriptsize (d+1, d)</math> | ||
| | | Real | ||
| | | Matrix | ||
| [⟨…]...] | | [⟨…]...] | ||
| | | | ||
| Line 3,202: | Line 3,202: | ||
| | | | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Non-standard domain bases | ||
|- | |- | ||
| rowspan="2" | | | rowspan="2" | | ||
| <math>B_s</math> | | <math>B_s</math> | ||
| rowspan="2" | [[Domain_basis#Basis_matrix_conversion| ( | | rowspan="2" | [[Domain_basis#Basis_matrix_conversion|(Domain) basis (change) matrix]] | ||
| rowspan="2" | | | rowspan="2" | | ||
| <math>\small 𝗽</math>/<math>\small 𝗯</math> | | <math>\small 𝗽</math>/<math>\small 𝗯</math> | ||
| | | Primes per nonprime basis elements | ||
| rowspan="2" | | | rowspan="2" | | ||
| <math>\scriptsize (d_p, d_b)</math> | | <math>\scriptsize (d_p, d_b)</math> | ||
| rowspan="2" | | | rowspan="2" | Integer | ||
| rowspan="2" | | | rowspan="2" | Matrix | ||
| rowspan="2" | [[...] ...] | | rowspan="2" | [[...] ...] | ||
| rowspan="2" | [[...] ...] | | rowspan="2" | [[...] ...] | ||
| Line 3,224: | Line 3,224: | ||
| <math>B_{Ls}</math> | | <math>B_{Ls}</math> | ||
| <math>\small 𝗕</math>/<math>\small 𝗯</math> | | <math>\small 𝗕</math>/<math>\small 𝗯</math> | ||
| | | Superspace basis elements per (subspace) basis elements | ||
| <math>\scriptsize (d_L, d_s)</math> | | <math>\scriptsize (d_L, d_s)</math> | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Embedding and projection | ||
|- | |- | ||
| | | | ||
| <math>G</math> | | <math>G</math> | ||
| [[generator embedding matrix| | | [[generator embedding matrix|Generator embedding (matrix)]] | ||
| | | | ||
| <math>\small 𝗽</math>/<math>\small 𝗴</math> | | <math>\small 𝗽</math>/<math>\small 𝗴</math> | ||
| | | Primes per generator | ||
| | | | ||
| <math>\scriptsize (d, r)</math> | | <math>\scriptsize (d, r)</math> | ||
| | | Real | ||
| | | Matrix | ||
| [{...] ...⟩ | | [{...] ...⟩ | ||
| {[...⟩ ...] | | {[...⟩ ...] | ||
| Line 3,250: | Line 3,250: | ||
\mathrm{V}\textit{Λ}\mathrm{V}^{-1}</math> | \mathrm{V}\textit{Λ}\mathrm{V}^{-1}</math> | ||
| <math>P</math> | | <math>P</math> | ||
| [[Projection matrix| | | [[Projection matrix|Projection (matrix)]] | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} G \\[-2pt] 𝗽 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array} | \begin{array} {c} G \\[-2pt] 𝗽 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array} | ||
| Line 3,257: | Line 3,257: | ||
</math> | </math> | ||
| <math>\small 𝗽</math>/<math>\small 𝗽</math> | | <math>\small 𝗽</math>/<math>\small 𝗽</math> | ||
| | | Primes per prime | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
\begin{array} {c} G \\[-3pt] (d, \cancel{r}) \end{array} | \begin{array} {c} G \\[-3pt] \left(d, \cancel{r}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array} | \begin{array} {c} M \\[-3pt] \left(\cancel{r}, d\right) \end{array} | ||
\! \! | \! \! | ||
</math> | </math> | ||
| <math>\scriptsize (d, d)</math> | | <math>\scriptsize (d, d)</math> | ||
| | | Real | ||
| | | Matrix | ||
| [⟨...] ...⟩ | | [⟨...] ...⟩ | ||
| ⟨[...⟩ ...] | | ⟨[...⟩ ...] | ||
| Line 3,278: | Line 3,278: | ||
| <math>GM\textbf{i}</math> | | <math>GM\textbf{i}</math> | ||
| <math>P\textbf{i}</math> | | <math>P\textbf{i}</math> | ||
| [[ | | [[Projected interval]] | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} G \\[-2pt] 𝗽 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array} | \begin{array} {c} G \\[-2pt] 𝗽 \hspace{-2mu} / \hspace{-2mu} \cancel{𝗴} \end{array} | ||
| Line 3,287: | Line 3,287: | ||
</math> | </math> | ||
| <math>\small 𝗽</math> | | <math>\small 𝗽</math> | ||
| | | Primes | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\! \! | \! \! | ||
\begin{array} {c} G \\[-3pt] (d, \cancel{r}) \end{array} | \begin{array} {c} G \\[-3pt] \left(d, \cancel{r}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} M \\[-3pt] (\cancel{r}, \cancel{d}) \end{array} | \begin{array} {c} M \\[-3pt] \left(\cancel{r}, \cancel{d}\right) \end{array} | ||
\! \! | \! \! | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\! \! | \! \! | ||
</math> | </math> | ||
| <math>\scriptsize (d, 1)</math> | | <math>\scriptsize (d, 1)</math> | ||
| | | Real | ||
| | | Vector | ||
| | | | ||
| [...⟩ | | [...⟩ | ||
| Line 3,306: | Line 3,306: | ||
| | | | ||
| | | | ||
| | | Specific type: vector ([[prime-count vector]] or PC-vector) | ||
|- | |- | ||
| | | | ||
| <math>\mathrm{U}</math> | | <math>\mathrm{U}</math> | ||
| [[ | | [[Unchanged-interval basis]] | ||
| | | | ||
| <math>\small 𝗽</math> | | <math>\small 𝗽</math> | ||
| | | Primes | ||
| | | | ||
| <math>\scriptsize (d, r)</math> | | <math>\scriptsize (d, r)</math> | ||
| | | | ||
| | | Matrix | ||
| | | | ||
| [[...⟩ ...] | | [[...⟩ ...] | ||
| Line 3,324: | Line 3,324: | ||
| | | | ||
| <math>\mathrm{u}_{ij}</math> | | <math>\mathrm{u}_{ij}</math> | ||
| | | Jargon name: eigenmonzo list | ||
|- | |- | ||
| | | | ||
| <math>\textit{Λ}</math> | | <math>\textit{Λ}</math> | ||
| [[scaling factor matrix| | | [[scaling factor matrix|Scaling factor (eigenvalue) matrix]] | ||
| | | | ||
| | | | ||
| Line 3,335: | Line 3,335: | ||
| <math>\scriptsize (d, d)</math> | | <math>\scriptsize (d, d)</math> | ||
| | | | ||
| | | Matrix | ||
| [⟨…] …⟩ | | [⟨…] …⟩ | ||
| ⟨[…⟩ …] | | ⟨[…⟩ …] | ||
| Line 3,342: | Line 3,342: | ||
| <math>𝝀</math> | | <math>𝝀</math> | ||
| <math>λ_i</math> | | <math>λ_i</math> | ||
| | | Mnemonic: <math>\mathrm{V}</math> is mirrored of <math>\textit{Λ}</math> which it combines with to create the projection matrix; previous name: eigenvalue matrix | ||
|- | |- | ||
| | | | ||
| <math>\mathrm{V}</math> | | <math>\mathrm{V}</math> | ||
| [[unrotated vector list| | | [[unrotated vector list|Unrotated vector (eigenvector) list]] | ||
| | | | ||
| <math>\small 𝗽</math> | | <math>\small 𝗽</math> | ||
| | | Primes | ||
| | | | ||
| <math>\scriptsize (d, d)</math> | | <math>\scriptsize (d, d)</math> | ||
| | | | ||
| | | Matrix | ||
| | | | ||
| ⟨[...⟩ ...] | | ⟨[...⟩ ...] | ||
| Line 3,360: | Line 3,360: | ||
| | | | ||
| <math>\mathrm{v}_{ij}</math> | | <math>\mathrm{v}_{ij}</math> | ||
| | | Mnemonic: <math>\mathrm{V}</math> is mirrored of <math>\textit{Λ}</math> which it combines with to create the projection matrix; jargon name: eigenmonzo and comma list | ||
|- | |- | ||
| | | | ||
| <math>F</math> | | <math>F</math> | ||
| [[ | | [[Generator form matrix]] | ||
| | | | ||
| | | | ||
| Line 3,371: | Line 3,371: | ||
| <math>\scriptsize (r, r)</math> | | <math>\scriptsize (r, r)</math> | ||
| | | | ||
| | | Matrix | ||
| [{...] …} | | [{...] …} | ||
| | | | ||
| Line 3,382: | Line 3,382: | ||
| <math>I</math> | | <math>I</math> | ||
| <math>M_{\text{j}}</math> | | <math>M_{\text{j}}</math> | ||
| [[Generator_embedding_optimization#Algebraic_setup| JI mapping (matrix)]] | | [[Generator_embedding_optimization#Algebraic_setup|JI mapping (matrix)]] | ||
| | | | ||
| <math>\small 𝗴</math>/<math>\small 𝗽</math> | | <math>\small 𝗴</math>/<math>\small 𝗽</math> | ||
| | | Generators per prime | ||
| | | | ||
| <math>\scriptsize (d, d)</math> | | <math>\scriptsize (d, d)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| [⟨...] ...} | | [⟨...] ...} | ||
| ⟨[...} ...] | | ⟨[...} ...] | ||
| Line 3,400: | Line 3,400: | ||
| <math>I</math> | | <math>I</math> | ||
| <math>G_{\text{j}}</math> | | <math>G_{\text{j}}</math> | ||
| [[Generator_embedding_optimization#Algebraic_setup| JI generator embedding (matrix)]] | | [[Generator_embedding_optimization#Algebraic_setup|JI generator embedding (matrix)]] | ||
| | | | ||
| <math>\small 𝗽</math>/<math>\small 𝗴</math> | | <math>\small 𝗽</math>/<math>\small 𝗴</math> | ||
| | | Primes per generator | ||
| | | | ||
| <math>\scriptsize (d, d)</math> | | <math>\scriptsize (d, d)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| [{...] ...⟩ | | [{...] ...⟩ | ||
| {[...⟩ ...] | | {[...⟩ ...] | ||
| Line 3,418: | Line 3,418: | ||
| | | | ||
| <math>K</math> | | <math>K</math> | ||
| [[Generator_embedding_optimization#How_to_build_constraint_matrices| | | [[Generator_embedding_optimization#How_to_build_constraint_matrices|Constraint (matrix)]] | ||
| | | | ||
| | | | ||
| Line 3,425: | Line 3,425: | ||
| <math>\scriptsize (k, r)</math> | | <math>\scriptsize (k, r)</math> | ||
| <math>\scriptsize \{0, +1, -1\}</math> | | <math>\scriptsize \{0, +1, -1\}</math> | ||
| | | Matrix | ||
| [[...] ...] | | [[...] ...] | ||
| | | | ||
| Line 3,432: | Line 3,432: | ||
| | | | ||
| <math>k_{ij}</math> | | <math>k_{ij}</math> | ||
| | | Mnemonic: <math>K</math>onstraint | ||
|- | |- | ||
| | | | ||
| <math>𝒃</math> | | <math>𝒃</math> | ||
| [[Generator embedding optimization#Generalizing to higher dimensions: | | [[Generator embedding optimization#Generalizing to higher dimensions: The blend map|(Generator tuning map) blend map]] | ||
| | | | ||
| | | | ||
| Line 3,442: | Line 3,442: | ||
| | | | ||
| <math>\scriptsize (1, τ-1)</math> | | <math>\scriptsize (1, τ-1)</math> | ||
| | | Real | ||
| | | Vector | ||
| [...] | | [...] | ||
| | | | ||
| Line 3,454: | Line 3,454: | ||
| | | | ||
| <math>B</math> | | <math>B</math> | ||
| [[Generator embedding optimization#How to identify tunings| ( | | [[Generator embedding optimization#How to identify tunings|(Generator tuning map) blend matrix]] | ||
| | | | ||
| | | | ||
| Line 3,460: | Line 3,460: | ||
| | | | ||
| <math>\scriptsize (d, τ-1)</math> | | <math>\scriptsize (d, τ-1)</math> | ||
| | | Real | ||
| | | Matrix | ||
| [[...⟩...] | | [[...⟩...] | ||
| | | | ||
| Line 3,472: | Line 3,472: | ||
| | | | ||
| <math>D</math> | | <math>D</math> | ||
| [[Generator embedding optimization#The deltas matrix| ( | | [[Generator embedding optimization#The deltas matrix|(Generator tuning map) deltas matrix]] | ||
| | | | ||
| <math>\mathsf{¢}</math>/<math>\small 𝗴</math> | | <math>\mathsf{¢}</math>/<math>\small 𝗴</math> | ||
| | | Cents per generator | ||
| | | | ||
| <math>\scriptsize (τ-1,r)</math> | | <math>\scriptsize (τ-1,r)</math> | ||
| | | Real | ||
| | | Matrix | ||
| [{...] ...] | | [{...] ...] | ||
| | | | ||
| Line 3,490: | Line 3,490: | ||
| | | | ||
| <math>τ</math> | | <math>τ</math> | ||
| [[Generator embedding optimization#The deltas matrix| | | [[Generator embedding optimization#The deltas matrix|Tied basic minimax tuning count]] | ||
| | | | ||
| | | | ||
| Line 3,496: | Line 3,496: | ||
| | | | ||
| | | | ||
| | | Integer | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 3,506: | Line 3,506: | ||
| | | | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Exterior algebra | ||
|- | |- | ||
| | | | ||
| <math>𝕞</math> | | <math>𝕞</math> | ||
| [[ | | [[Multimap]] | ||
| | | | ||
| <math>\small 𝗴</math>/<math>\small 𝗽</math> | | <math>\small 𝗴</math>/<math>\small 𝗽</math> | ||
| | | Generators per prime | ||
| | | | ||
| <math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Integer | ||
| | | Multivector | ||
| ⟨...] or ⟨⟨...]] or ⟨⟨⟨...]]] ... | | ⟨...] or ⟨⟨...]] or ⟨⟨⟨...]]] ... | ||
| | | | ||
| Line 3,528: | Line 3,528: | ||
| | | | ||
| <math>𝕔</math> | | <math>𝕔</math> | ||
| [[ | | [[Multicomma]] | ||
| | | | ||
| <math>\small 𝗽</math> | | <math>\small 𝗽</math> | ||
| | | Primes | ||
| | | | ||
| <math>\scriptsize (1, n)</math> | | <math>\scriptsize (1, n)</math> | ||
| | | Integer | ||
| | | Multivector | ||
| | | | ||
| [...⟩ or [[...⟩⟩ or [[[...⟩⟩⟩ ... | | [...⟩ or [[...⟩⟩ or [[[...⟩⟩⟩ ... | ||
| Line 3,546: | Line 3,546: | ||
| | | | ||
| <math>𝕧</math> | | <math>𝕧</math> | ||
| ( | | (Generic temperament multivector) | ||
| | | | ||
| | | | ||
| Line 3,552: | Line 3,552: | ||
| | | | ||
| <math>\scriptsize (1, {{d}\choose{r}})</math> or <math>\scriptsize (1, {{d}\choose{n}})</math> | | <math>\scriptsize (1, {{d}\choose{r}})</math> or <math>\scriptsize (1, {{d}\choose{n}})</math> | ||
| | | Integer | ||
| | | Multivector | ||
| ⟨...] or ⟨⟨...]] or ⟨⟨⟨...]]] ... | | ⟨...] or ⟨⟨...]] or ⟨⟨⟨...]]] ... | ||
| [...⟩ or [[...⟩⟩ or [[[...⟩⟩⟩ ... | | [...⟩ or [[...⟩⟩ or [[[...⟩⟩⟩ ... | ||
| Line 3,564: | Line 3,564: | ||
| | | | ||
| <math>A</math> | | <math>A</math> | ||
| ( | | (Generic temperament matrix) | ||
| | | | ||
| | | | ||
| Line 3,570: | Line 3,570: | ||
| | | | ||
| <math>\scriptsize (g, d)</math> or <math>\scriptsize (d, g)</math> | | <math>\scriptsize (g, d)</math> or <math>\scriptsize (d, g)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| [⟨...] ...} | | [⟨...] ...} | ||
| ⟨[...} ...] or [[...⟩ ...] | | ⟨[...} ...] or [[...⟩ ...] | ||
| Line 3,582: | Line 3,582: | ||
| | | | ||
| <math>v</math> | | <math>v</math> | ||
| [[ | | [[Variance]] | ||
| | | | ||
| | | | ||
| Line 3,600: | Line 3,600: | ||
| | | | ||
| <math>g</math> | | <math>g</math> | ||
| [[ | | [[Grade]] | ||
| | | | ||
| | | | ||
| Line 3,606: | Line 3,606: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 3,616: | Line 3,616: | ||
| | | | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Temperament addition | ||
|- | |- | ||
| <math>\min(r, n)</math> | | <math>\min(r, n)</math> | ||
| <math>g_\text{min}</math> | | <math>g_\text{min}</math> | ||
| [[Temperament_addition#Introductory_examples| | | [[Temperament_addition#Introductory_examples|Min-grade]] | ||
| | | | ||
| | | | ||
| Line 3,626: | Line 3,626: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 3,638: | Line 3,638: | ||
| <math>\max(r, n)</math> | | <math>\max(r, n)</math> | ||
| <math>g_\text{max}</math> | | <math>g_\text{max}</math> | ||
| [[Temperament_addition#Introductory_examples| | | [[Temperament_addition#Introductory_examples|Max-grade]] | ||
| | | | ||
| | | | ||
| Line 3,644: | Line 3,644: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 3,656: | Line 3,656: | ||
| | | | ||
| <math>L_\text{dep}</math> | | <math>L_\text{dep}</math> | ||
| [[Temperament_addition#1._Find_the_.5Bmath.5DL_.7B.5Ctext.7Bdep.7D.7D.5B.2Fmath.5D| | | [[Temperament_addition#1._Find_the_.5Bmath.5DL_.7B.5Ctext.7Bdep.7D.7D.5B.2Fmath.5D|Linear-dependence basis]] | ||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
| <math>\scriptsize (l_\text{dep}, d)</math> or <math>\scriptsize (d, l_\text{dep})</math> | | <math>\scriptsize \left(l_\text{dep}, d\right)</math> or <math>\scriptsize \left(d, l_\text{dep}\right)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| [⟨...]] or [[...] ...⟩ | | [⟨...]] or [[...] ...⟩ | ||
| ⟨[...]] or [[...⟩ ...] | | ⟨[...]] or [[...⟩ ...] | ||
| Line 3,674: | Line 3,674: | ||
| | | | ||
| <math>L_\text{ind}</math> | | <math>L_\text{ind}</math> | ||
| [[Temperament_addition#Glossary| | | [[Temperament_addition#Glossary|Linear-independence basis]] | ||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
| <math>\scriptsize (l_\text{ind}, d)</math> or <math>\scriptsize (d, l_\text{ind})</math> | | <math>\scriptsize \left(l_\text{ind}, d\right)</math> or <math>\scriptsize \left(d, l_\text{ind}\right)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| [⟨...]] or [[...] ...⟩ | | [⟨...]] or [[...] ...⟩ | ||
| ⟨[...]] or [[...⟩ ...] | | ⟨[...]] or [[...⟩ ...] | ||
| Line 3,692: | Line 3,692: | ||
| <math>\dim(L_\text{dep})</math> | | <math>\dim(L_\text{dep})</math> | ||
| <math>l_\text{dep}</math> | | <math>l_\text{dep}</math> | ||
| [[Temperament_addition#3._Linear_independence_between_temperaments| | | [[Temperament_addition#3._Linear_independence_between_temperaments|Linear-dependence]] | ||
| | | | ||
| | | | ||
| Line 3,698: | Line 3,698: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | | ||
| | | | ||
| Line 3,710: | Line 3,710: | ||
| <math>\dim(L_\text{ind})</math> | | <math>\dim(L_\text{ind})</math> | ||
| <math>l_\text{ind}</math> | | <math>l_\text{ind}</math> | ||
| [[Temperament_addition#3._Linear_independence_between_temperaments| | | [[Temperament_addition#3._Linear_independence_between_temperaments|Linear-independence]] | ||
| | | | ||
| | | | ||
| Line 3,716: | Line 3,716: | ||
| | | | ||
| <math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | | ||
| | | | ||