Dave Keenan & Douglas Blumeyer's guide to RTT/Conventions for names, variables, units, and notations: Difference between revisions
Cmloegcmluin (talk | contribs) remove 1200, ๐, and ๐ฟ from computations sections of objects table; the first two never come up anymore, and the latter only comes up in all-interval tunings schemes, so move it to that section |
Dave Keenan (talk | contribs) โObjects: Replaced <math>\llzigzag and \rrzigzag with {{llzigzag}} and {{rrzigzag}} where the former did not display correctly ย |
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This is an appendix to [[Dave Keenan]] & [[Douglas Blumeyer]]'s guide to RTT | {{breadcrumb}}{{texops}}{{texmap}}{{texzz}} | ||
This is an appendix to [[Dave Keenan]] & [[Douglas Blumeyer]]'s guide to RTT. The tables in this article present our recommendations for communicating about [[regular temperament theory]] (RTT), in particular the names and notations for temperament matrices, tuning schemes, interval complexities, and measurement units. ย | |||
Our recommendations are designed to make this topic easy to learn for musicians who do not have technical backgrounds, though we have generally deferred to established mathematical, scientific, and engineering conventions for the benefit of those who do. | Our recommendations are designed to make this topic easy to learn for musicians who do not have technical backgrounds, though we have generally deferred to established mathematical, scientific, and engineering conventions for the benefit of those who do. | ||
See [[Extended bra-ket notation#Variant including curly and square brackets|here]] for more information on our variation on extended bra-ket notation. | |||
We've followed a | We've followed a variable styling convention, explained in the table below, which we hope serves as an aid to quickly identifying objects and remembering their properties and purposes, but at the least we hope our choices are unobtrusive. In short, the objects with simple units of primes, generators or cents, i.e. the things which are actually audible in our application, are distinguished by upright formatting, while other variables are italic as is conventional. This is crossed with the linear-algebra convention that objects of order-1 like vectors are bolded and order-2 like matrices are uppercased: | ||
{| class="wikitable center-all" ย | {| class="wikitable center-all" | ||
| | |- | ||
! | ! ย | ||
| | | Units → | ||
! rowspan="2" | | ! rowspan="2" | | ||
| | | Simple units | ||
| | | Compound or no units | ||
|- | |- | ||
| | | ↓ Order | ||
| | | ↓ Style → | ||
|upright | | Roman (upright) | ||
|'' | | ''Italic'' | ||
ย | |||
|- | |- | ||
! scope="col" height="8px" ! colspan="2" | | ! scope="col" height="8px" ! colspan="2" | | ||
| Line 25: | Line 25: | ||
! colspan="2" | | ! colspan="2" | | ||
|- | |- | ||
|0 | | 0 | ||
| | | lowercase | ||
! rowspan="3" | | ! rowspan="3" | | ||
|scalar with simple unit | | scalar (with simple unit) | ||
|''scalar'' with no unit | | ''scalar'' (with no unit) | ||
|- | |- | ||
|1 | | 1 | ||
|'''bold''' | | '''bold lowercase''' | ||
|'''vector''' | | '''vector''' | ||
|'''''map''''' ( | | '''''map''''' (row vector) | ||
|- | |- | ||
|2 | | 2 | ||
|UPPERCASE | | UPPERCASE | ||
|LIST | | BASIS or LIST (of vectors) | ||
| | | ''MATRIX'' | ||
|} | |} | ||
We present our conventions here in three separate sections, one for each level of this article series: '''basic''', '''intermediate''', and '''advanced'''. The basic section contains only information covered in the basic part of the series, the intermediate section contains both basic and intermediate, and the advanced section contains it all (that is to say, the sections are cumulative)<ref>The advanced section also contains conventions collected from other RTT-related articles Dave and Douglas have contributed to but are outside the main guide to RTT series.</ref>. We expect that for most readers, the basic tier will be the best reference (this is the reference designed primarily for musicians interested in RTT, as opposed to scientists, engineers, mathematicians, or theoreticians), and so we've left the other two sections initially collapsed. | We present our conventions here in three separate sections, one for each level of this article series: '''basic''', '''intermediate''', and '''advanced'''. The basic section contains only information covered in the basic part of the series, the intermediate section contains both basic and intermediate, and the advanced section contains it all (that is to say, the sections are cumulative)<ref group="note">The advanced section also contains conventions collected from other RTT-related articles Dave and Douglas have contributed to but are outside the main guide to RTT series.</ref>. We expect that for most readers, the basic tier will be the best reference (this is the reference designed primarily for musicians interested in RTT, as opposed to scientists, engineers, mathematicians, or theoreticians), and so we've left the other two sections initially collapsed. | ||
ย | |||
ย | |||
== Basic == | |||
=== Objects === | |||
{| class="wikitable mw-collapsible" | {| class="wikitable mw-collapsible" | ||
|+ | |+ style="font-size: 105%;" | ย | ||
|- | |- | ||
! | ! rowspan="2" | Equivalent<br />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 | |||
! Column | |||
! Diagonal | |||
! Entry | |||
|- | |- | ||
| | ! colspan="17" | Mapping | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>\textbf{i}</math> | ||
|[[ | | [[interval|(Just) interval]] | ||
| | | ย | ||
| | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (d, 1)</math> | ||
| | | Integer | ||
| | | Vector | ||
| | | ย | ||
| | | [...โฉ | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math> | | <math>\mathrm{i}_i</math> | ||
| | | Specific type: Vector ([[prime-count vector]] or PC-vector) | ||
Jargon name: Monzo | |||
|- | |- | ||
|<math>M\textbf{i}</math> | | | ||
|<math>\textbf{y}</math> | | <math>M</math> | ||
|[[ | | [[Mapping|(Temperament) mapping (matrix)]] | ||
|<math>\scriptsize ย | | | ||
| <math>\small ๐ด</math>/<math>\small ๐ฝ</math> | |||
| Generators per prime | |||
| | |||
| <math>\scriptsize (r, d)</math> | |||
| Integer | |||
| Matrix | |||
| [โจ...] ...} | |||
| โจ[...} ...] | |||
| <math>๐_i</math> | |||
| | |||
| | |||
| <math>m_{ij}</math> | |||
| Jargon name: Val list | |||
|- | |||
| <math>M\textbf{i}</math> | |||
| <math>\textbf{y}</math> | |||
| [[Mapped interval]] | |||
| <math>\scriptsize ย | |||
\begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\small ๐ด</math> | | <math>\small ๐ด</math> | ||
| | | Generators | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} ย | \begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} ย | \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} ย | ||
\!\! ย | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (r, 1)</math> | | <math>\scriptsize (r, 1)</math> | ||
| | | Integer | ||
| | | Vector | ||
| | | ย | ||
|[...} | | [...} | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | Specific type: [[Generator-count vector]] (GC-vector) | ||
Jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval | |||
|- | |- | ||
| | | ย | ||
|<math>๐</math> | | <math>๐</math> | ||
|[[map|( | | [[map|(Temperament) map]] | ||
| | | ย | ||
|<math>\small ๐ด</math>/<math>\small ๐ฝ</math> | | <math>\small ๐ด</math>/<math>\small ๐ฝ</math> | ||
| | | Generators per prime | ||
| | | ย | ||
|<math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Integer | ||
| | | Vector | ||
|โจ...] | | โจ...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>m_i</math> | | <math>m_i</math> | ||
| | | Jargon name: val | ||
|- | |- | ||
| | | ย | ||
|<math>d</math> | | <math>d</math> | ||
|[[dimensionality]] | | [[dimensionality]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math>r</math> | | <math>r</math> | ||
|[[ | | [[Rank]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Tuning | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>{\large\textbf{๐}}\hspace{2mu}</math> | ||
|[[ | | [[Log-prime map]] | ||
| | | ย | ||
|<math>\mathsf{ | | <math>\small\mathsf{oct}</math>/<math>\small ๐ฝ</math> | ||
| | | Octaves per prime | ||
| | | ย | ||
|<math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Real | ||
| | | Vector | ||
|โจ...] | | โจ...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math> | | <math>{\large ๐}\hspace{2mu}_i</math> | ||
| | | ย | ||
|- | |- | ||
| | | <math>1200ร{\large\textbf{๐}}\hspace{2mu}</math> | ||
|<math> | | <math>๐</math> | ||
|[[ | | [[just tuning map|Just(-prime) tuning map]] | ||
| | | ย | ||
|<math>\mathsf{ยข}</math>/<math>\small | | <math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | ||
| | | Cents per prime | ||
| | | ย | ||
|<math>\scriptsize (1, | | <math>\scriptsize (1, d)</math> | ||
| | | Real | ||
| | | Vector | ||
| | | โจ...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math> | | <math>j_i</math> | ||
| | | ย | ||
|- | |- | ||
|<math>๐M</math> | | | ||
|<math>๐</math> | | <math>๐</math> | ||
|[[tuning map|( | | [[Generator tuning map]] | ||
|<math>\scriptsize ย | | | ||
| <math>\mathsf{ยข}</math>/<math>\small ๐ด</math> | |||
| Cents per generator | |||
| | |||
| <math>\scriptsize (1, r)</math> | |||
| Real | |||
| Vector | |||
| {...] | |||
| | |||
| | |||
| | |||
| | |||
| <math>g_i</math> | |||
| | |||
|- | |||
| <math>๐M</math> | |||
| <math>๐</math> | |||
| [[tuning map|(Tempered-prime) tuning map]] | |||
| <math>\scriptsize ย | |||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ด} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ด} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} ๐ \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} ๐ฝ \end{array} ย | \begin{array} {c} ๐ \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} ๐ฝ \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | | <math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | ||
| | | Cents per prime | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{r}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{r}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array} ย | \begin{array} {c} M \\[-3pt] \left(\cancel{r}, d\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Real | ||
| | | Vector | ||
|โจ...] | | โจ...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>t_i</math> | | <math>t_i</math> | ||
| | | ย | ||
|- | |- | ||
|<math>๐ - ๐</math> | | <math>๐ - ๐</math> | ||
|<math>๐</math> | | <math>๐</math> | ||
|[[retuning map| | | [[retuning map|Retuning (or mistuning) map]] | ||
| | | ย | ||
|<math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | | <math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | ||
| | | Cents per prime | ||
| | | ย | ||
|<math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Real | ||
| | | Vector | ||
|โจ...] | | โจ...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>r_i</math> | | <math>r_i</math> | ||
| | | Previous name: prime error map | ||
|- | |- | ||
|<math>๐\textbf{i}</math> | | <math>๐\textbf{i}</math> | ||
|<math>\mathrm{o}</math> | | <math>\mathrm{o}</math> | ||
|[[interval span|( | | [[interval span|(Just) (interval) size]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | Mnemonic: <math>\mathrm{o}</math>riginal size | ||
|- | |- | ||
|<math>๐M\textbf{i} | | <math>๐M\textbf{i}</math><br /> | ||
๐\textbf{i}</math> | <math>๐\textbf{i}</math> | ||
|<math>\mathrm{a}</math> | | <math>\mathrm{a}</math> | ||
| | | {{subpage|tuning_fundamentals|uprev|s=Example 3|text=Tempered (interval) size}} | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | Mnemonic: <math>\mathrm{a}</math>ltered size | ||
|- | |- | ||
|<math>๐\textbf{i} - ๐\textbf{i} | | <math>๐\textbf{i} - ๐\textbf{i}</math><br /> | ||
a - o | <math>a - o</math><br /> | ||
๐\textbf{i}</math> | <math>๐\textbf{i}</math> | ||
|<math>\mathrm{e}</math> | | <math>\mathrm{e}</math> | ||
|[[error|( | | [[error|(Interval) error]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Optimization | ||
|- | |- | ||
| | | ย | ||
|<math>p</math> | | <math>p</math> | ||
|[[ | | [[Optimization power]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>\llangle\,ยท\,\rrangle_p</math> | ||
|[[ | | [[Power mean]] (<math>p</math>-mean) | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Damage | ||
|- | |- | ||
| | | ย | ||
|<math>c</math> | | <math>c</math> | ||
| | | {{subpage|tuning_fundamentals|uprev|s=Complexity}} | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{(C)}</math><ref>For educational purposes, we use the ๐ 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'' / | | <math>\small\mathsf{๐}\scriptsize\mathsf{(C)}</math><ref group="note">For educational purposes, we use the ๐ symbol here to represent the implicit [[Wikipedia:Dimensionless_quantity| dimensionless unit]] that the weighting annotation "(C)" is attached to. But this symbol should not be shown in the reduced result. Another way to understand how we arrive at a bare annotation for the units of this quantity is to consider that {{nowrap|''w'' {{=}} ''d''/{{!}}''e''{{!}}}} whose units are {{nowrap|ยข(W) / ยข}} and the cents cancel.</ref> | ||
|<math>\small\mathsf{(C)}</math> | | <math>\small\mathsf{(C)}</math> | ||
| | | Complexity weight | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<math> | | <math>\dfrac1c</math> | ||
|<math>s</math> | | <math>s</math> | ||
|[[ | | [[Simplicity]] | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{(S)}</math> | | <math>\small\mathsf{๐}\scriptsize\mathsf{(S)}</math> | ||
|<math>\small\mathsf{(S)}</math> | | <math>\small\mathsf{(S)}</math> | ||
| | | Simplicity weight | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<math>c</math> or <math>s</math> | | <math>c</math> or <math>s</math> | ||
|<math>w</math> | | <math>w</math> | ||
|[[ | | [[Weight]] | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{(C)}</math> or ๐<math>\small\mathsf{(S)}</math> | | <math>\small\mathsf{๐}\scriptsize\mathsf{(C)}</math> or ๐<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 | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<math> | | <math>\abs{\mathrm{e}} w</math> | ||
|<math>\mathrm{d}</math> | | <math>\mathrm{d}</math> | ||
|[[ | | [[Damage]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} | \begin{array} {c} \abs{\mathrm{e}} \\[-2pt] {\small\mathsf{ยข}} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} w \\[-2pt] \mathsf{(U, C, or\,S)} \end{array} ย | \begin{array} {c} w \\[-2pt] \mathsf{(U, C, \text{or}\,S)} \end{array} ย | ||
</math> | </math> | ||
| <math>\mathsf{ยข}\small\mathsf{(U)}</math> or <math>\mathsf{ยข}\small\mathsf{(C)}</math> or <math>\mathsf{ยข}\small\mathsf{(S)}</math> | | <math>\mathsf{ยข}\small\mathsf{(U)}</math> or <math>\mathsf{ยข}\small\mathsf{(C)}</math> or <math>\mathsf{ยข}\small\mathsf{(S)}</math> | ||
| ( | | (See damages table) | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} | \begin{array} {c} \abs{\mathrm{e}} \\[-3pt] \left(1, \cancel{1}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} w \\[-3pt] (\cancel{1}, 1) \end{array} | \begin{array} {c} w \\[-3pt] \left(\cancel{1}, 1\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Target-intervals | ||
|- | |- | ||
| | | ย | ||
|<math>\mathrm{T}</math> | | <math>\mathrm{T}</math> | ||
|[[ | | [[Target-interval list]] | ||
| | | ย | ||
|<math>\small ๐ฝ</math> | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | ย | ||
|<math>\scriptsize (d, k)</math> | | <math>\scriptsize (d, k)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | ย | ||
|[[...โฉ ...] | | [[...โฉ ...] | ||
| | | ย | ||
|<math>\textbf{t}_i</math> | | <math>\textbf{t}_i</math> | ||
| | | ย | ||
|<math>\mathrm{t}_{ij}</math> | | <math>\mathrm{t}_{ij}</math> | ||
| | | ย | ||
|- | |- | ||
|<math>M\mathrm{T}</math> | | <math>M\mathrm{T}</math> | ||
|<math>\mathrm{Y}</math> | | <math>\mathrm{Y}</math> | ||
|[[ | | [[Mapped target-interval list]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\small ๐ด</math> | | <math>\small ๐ด</math> | ||
| | | Generators | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} ย | \begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} ย | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (r, k)</math> | | <math>\scriptsize (r, k)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | ย | ||
|[[...} ...] | | [[...} ...] | ||
| | | ย | ||
|<math>\textbf{y}_i</math> | | <math>\textbf{y}_i</math> | ||
| | | ย | ||
|<math>\mathrm{y}_{ij}</math> | | <math>\mathrm{y}_{ij}</math> | ||
| | | Mnemonic: looks like bent-up 'T', or cross between 'M' and 'T' | ||
|- | |- | ||
|<math>๐\mathrm{T}</math> | | <math>๐\mathrm{T}</math> | ||
|<math>\textbf{o}</math> | | <math>\textbf{o}</math> | ||
| | | {{subpage|tuning_fundamentals|uprev|s=primes|text=Target-interval (just) size list}} | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
|[...] | | [...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\mathrm{o}_i</math> | | <math>\mathrm{o}_i</math> | ||
| | | Mnemonic: <math>\textbf{o}</math>riginal size list | ||
|- | |- | ||
|<math>๐\mathrm{T} | | <math>๐\mathrm{T}</math><br /> | ||
๐M\mathrm{T}</math> | <math>๐M\mathrm{T}</math> | ||
|<math>\textbf{a}</math> | | <math>\textbf{a}</math> | ||
|[[ | | [[Tempered target-interval size list]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
|[...] | | [...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\mathrm{a}_i</math> | | <math>\mathrm{a}_i</math> | ||
| | | Mnemonic: <math>\textbf{a}</math>ltered size list | ||
|- | |- | ||
|<math>๐\mathrm{T} - ๐\mathrm{T} | | <math>๐\mathrm{T} - ๐\mathrm{T}</math><br /> | ||
\textbf{a} - \textbf{o} | <math>\textbf{a} - \textbf{o}</math><br /> | ||
๐\mathrm{T} | <math>๐\mathrm{T}</math> | ||
</math> | | <math>\textbf{e}</math> | ||
|<math>\textbf{e}</math> | | [[Target-interval error list]] | ||
|[[ | | <math>\scriptsize ย | ||
|<math>\scriptsize ย | |||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
|[...] | | [...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\mathrm{e}_i</math> | | <math>\mathrm{e}_i</math> | ||
| | | ย | ||
|- | |- | ||
|<math>C</math> or <math>S</math> | | <math>C</math> or <math>S</math> | ||
|<math>W</math> | | <math>W</math> | ||
|[[ | | [[Target-interval weight matrix]] | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{๐}\scriptsize\mathsf{(S)}</math> or <math>\small\mathsf{๐}\scriptsize\mathsf{(U)}</math> | | <math>\small\mathsf{๐}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{๐}\scriptsize\mathsf{(S)}</math> or <math>\small\mathsf{๐}\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 | ||
| | | ย | ||
|<math>\scriptsize (k, k)</math> | | <math>\scriptsize (k, k)</math> | ||
| | | Real | ||
| | | Matrix | ||
| | | ย | ||
|[[...] ...] | | [[...] ...] | ||
| | | ย | ||
| | | ย | ||
|<math>๐</math> | | <math>๐</math> | ||
|<math>w_i</math> | | <math>w_i</math> | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math>C</math> | | <math>C</math> | ||
| | | {{subpage|tuning_fundamentals|uprev|s=complexity-weight_damage|text=Target-interval complexity weight matrix}} | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{(C)}</math> | | <math>\small\mathsf{๐}\scriptsize\mathsf{(C)}</math> | ||
|<math>\small\mathsf{(C)}</math> | | <math>\small\mathsf{(C)}</math> | ||
| | | Complexity weight | ||
| | | ย | ||
|<math>\scriptsize (k, k)</math> | | <math>\scriptsize (k, k)</math> | ||
| | | Real | ||
| | | Matrix | ||
| | | ย | ||
|[[...] ...] | | [[...] ...] | ||
| | | ย | ||
| | | ย | ||
|<math>๐</math> | | <math>๐</math> | ||
|<math>c_i</math> | | <math>c_i</math> | ||
| | | ย | ||
|- | |- | ||
|<math> | | <math>\dfrac1C</math> | ||
|<math>S</math> | | <math>S</math> | ||
| | | {{subpage|tuning fundamentals|uprev|s=complexity-weight_damage|text=Target-interval simplicity weight matrix}} | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{(S)}</math> | | <math>\small\mathsf{๐}\scriptsize\mathsf{(S)}</math> | ||
|<math>\small\mathsf{(S)}</math> | | <math>\small\mathsf{(S)}</math> | ||
| | | Simplicity weight | ||
| | | ย | ||
|<math>\scriptsize (k, k)</math> | | <math>\scriptsize (k, k)</math> | ||
| | | Real | ||
| | | Matrix | ||
| | | ย | ||
|[[...] ...] | | [[...] ...] | ||
| | | ย | ||
| | | ย | ||
|<math>๐</math> | | <math>๐</math> | ||
|<math>s_i</math> | | <math>s_i</math> | ||
| | | Entry-wise reciprocal of <math>C</math> | ||
|- | |- | ||
|<math> | | <math>\abs{\textbf{e}} W</math> | ||
|<math>\textbf{d}</math> | | <math>\textbf{d}</math> | ||
|[[ | | [[Target-interval damage list]]<ref group="note">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 ย | | <math>\scriptsize ย | ||
\begin{array} {c} | \begin{array} {c} \abs{\textbf{e}} \\[-2pt] {\small\mathsf{ยข}} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} W \\[-2pt] (\mathsf{U, C, or\,S}) \end{array} ย | \begin{array} {c} W \\[-2pt] (\mathsf{U, C, \text{or}\,S}) \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}\small\mathsf{(U)}</math>, <math>\mathsf{ยข}\small\mathsf{(C)}</math>, or <math>\mathsf{ยข}\small\mathsf{(S)}</math> | | <math>\mathsf{ยข}\small\mathsf{(U)}</math>, <math>\mathsf{ยข}\small\mathsf{(C)}</math>, or <math>\mathsf{ยข}\small\mathsf{(S)}</math> | ||
| | | Weighted cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} | \begin{array} {c} \abs{\textbf{e}} \\[-3pt] \left(1, \cancel{k}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} W \\[-3pt] (\cancel{k}, k) \end{array} | \begin{array} {c} W \\[-3pt] \left(\cancel{k}, k\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
|[...] | | [...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\mathrm{d}_i</math> | | <math>\mathrm{d}_i</math> | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math>k</math> | | <math>k</math> | ||
|[[ | | [[Target-interval count]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | Mnemonic: <math>k</math>ount | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Held-intervals | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>\mathrm{H}</math> | ||
|[[ | | [[Held-interval basis]] | ||
| | | ย | ||
| | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (d, h)</math> | ||
| | | ย | ||
| Matrix | |||
| | | ย | ||
| | | [[...โฉ ...] | ||
| | | ย | ||
| | | <math>\textbf{h}_i</math> | ||
| | | ย | ||
| | | <math>\mathrm{h}_{ij}</math> | ||
| | | | ||
|- | |- | ||
| ย | |||
| <math>h</math> | |||
| [[Held-interval count]] | |||
| | |||
| | |||
| | |||
| | |||
| <math>\scriptsize (1, 1)</math> | |||
| Integer | |||
| Scalar | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |- | ||
| | ! colspan="17" | Exploring temperaments | ||
|- | |- | ||
| | | | ||
|<math>\textbf{c}</math> | | <math>\mathrm{C}</math> | ||
|[[ | | [[Comma basis]] | ||
| | | | ||
|<math>\small ๐ฝ</math> | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | | ||
|<math>\scriptsize (d, 1)</math> | | <math>\scriptsize (d, n)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | | ||
|[...โฉ | | [[...โฉ ...] | ||
| | | | ||
| | | <math>\textbf{c}_i</math> | ||
| | | | ||
|<math>\mathrm{c}_i</math> | | <math>\mathrm{c}_{ij}</math> | ||
| | | Jargon name: monzo list | ||
|- | |||
| ย | |||
| <math>\textbf{c}</math> | |||
| [[Comma]] | |||
| ย | |||
| <math>\small ๐ฝ</math> | |||
| Primes | |||
| ย | |||
| <math>\scriptsize (d, 1)</math> | |||
| Integer | |||
| Vector | |||
| ย | |||
| [...โฉ | |||
| ย | |||
| ย | |||
| ย | |||
| <math>\mathrm{c}_i</math> | |||
| Specific type: vector ([[prime-count vector]] or PC-vector) | |||
|} | |} | ||
===Units=== | === Units === | ||
ย | We recommend using a narrow no-break space (U+202F) between quantities and their units.<ref group="note">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 | ||
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. | .</ref> For how to type this, see the [[#WinCompose|WinCompose]] section below. | ||
{| class="wikitable center-all mw-collapsible" | {| class="wikitable center-all mw-collapsible" | ||
|+ | |+ style="font-size: 105%;" | | ||
|- | |- | ||
! Symbol | |||
! Name | |||
! Vectorized | |||
|- | |- | ||
|<math>\small | | <math>\small ๐ด</math> | ||
| | | Generators | ||
| | | Yes | ||
|- | |- | ||
|<math>\ | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | Yes | ||
|- | |- | ||
|<math>\mathsf{ยข} | | <math>\mathsf{ยข}</math><ref group="note">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 "ยข" following a number in a musical context as anything other than musical cents.</ref> | ||
| | | Cents | ||
| | | ย | ||
|- | |- | ||
|<math>\mathsf{ยข}\small\mathsf{( | | <math>\mathsf{ยข}\small\mathsf{(U)}</math> | ||
| | | Unity-weighted cents | ||
| | | ย | ||
|- | |- | ||
|<math>\mathsf{ยข}\small\mathsf{( | | <math>\mathsf{ยข}\small\mathsf{(C)}</math> | ||
| | | Complexity-weighted cents | ||
| | | ย | ||
|- | |- | ||
|<math>\small\mathsf{ | | <math>\mathsf{ยข}\small\mathsf{(S)}</math> | ||
| | | Simplicity-weighted cents | ||
| | | ย | ||
|- | |- | ||
|<math>\small\mathsf{ | | <math>\small\mathsf{oct}</math> | ||
| | | Octaves | ||
| | | ย | ||
|- | |- | ||
|<math>\small\mathsf{(S)}</math> | | <math>\small\mathsf{(C)}</math> | ||
| | | Complexity weight | ||
| | | | ||
|- | |||
| <math>\small\mathsf{(S)}</math> | |||
| Simplicity weight | |||
| ย | |||
|} | |} | ||
===Tuning schemes=== | === Tuning schemes === | ||
ย | Copied from {{subpage|tuning fundamentals|uprev|s=Systematic tuning scheme names}}. | ||
Copied from | |||
{| class="wikitable center-all mw-collapsible" | {| class="wikitable center-all mw-collapsible" | ||
|+ | |+ style="font-size: 105%;" | | ||
|- | |- | ||
! Damage weight | |||
! Optimization power | |||
! Systematic name | |||
|- | |- | ||
|<none> | | <none> | ||
| rowspan="3" | | | rowspan="3" | ∞ | ||
| | | Minimax-U | ||
|- | |- | ||
| | | Complexity | ||
| | | Minimax-C | ||
|- | |- | ||
|1/ | | 1/Complexity | ||
| | | Minimax-S | ||
|- | |- | ||
|<none> | | <none> | ||
| rowspan="3" |2 | | rowspan="3" | 2 | ||
| | | MiniRMS-U | ||
|- | |- | ||
| | | Complexity | ||
| | | MiniRMS-C | ||
|- | |- | ||
|1/ | | 1/Complexity | ||
| | | MiniRMS-S | ||
|- | |- | ||
| | | <none> | ||
| rowspan="3" |1 | | rowspan="3" | 1 | ||
| | | Miniaverage-U | ||
|- | |- | ||
| | | Complexity | ||
| | | Miniaverage-C | ||
|- | |- | ||
|1/ | | 1/Complexity | ||
| | | Miniaverage-S | ||
|} | |} | ||
===Damages=== | === Damages === | ||
ย | |||
{| class="wikitable center-all mw-collapsible" | {| class="wikitable center-all mw-collapsible" | ||
|+ | |+ style="font-size: 105%;" | ย | ||
|- | |- | ||
! | ! colspan="2" | Quantity | ||
! | ! colspan="2" | Unit | ||
|- | |- | ||
! Abbreviation | |||
! Name | |||
! Symbol | |||
! Name | |||
|- | |- | ||
| | | U-damage | ||
| | | Unity-weight damage | ||
|<math>\mathsf{ยข}\small\mathsf{( | | <math>\mathsf{ยข}\small\mathsf{(U)}</math> | ||
| | | Unity-weighted cents | ||
|- | |- | ||
|S-damage | | C-damage | ||
| | | Complexity-weight damage | ||
|<math>\mathsf{ยข}\small\mathsf{(S)}</math> | | <math>\mathsf{ยข}\small\mathsf{(C)}</math> | ||
| | | Complexity-weighted cents | ||
|- | |||
| S-damage | |||
| Simplicity-weight damage | |||
| <math>\mathsf{ยข}\small\mathsf{(S)}</math> | |||
| Simplicity-weighted cents | |||
|} | |} | ||
===Complexity and simplicity=== | === Complexity and simplicity === | ||
ย | |||
{| class="wikitable center-all mw-collapsible" | {| class="wikitable center-all mw-collapsible" | ||
|+ | |+ style="font-size: 105%;" | | ||
! colspan="2" | | |- | ||
! colspan="2" | | ! colspan="2" | Quantity | ||
! colspan="2" | Unit | |||
|- | |- | ||
! | ! Abbreviation | ||
! | ! Name | ||
! | ! Symbol | ||
! | ! Name | ||
|- | |- | ||
|C | | C | ||
| | | Complexity | ||
|<math>\small\mathsf{(C)}</math> | | <math>\small\mathsf{(C)}</math> | ||
| | | Complexity weight | ||
|- | |- | ||
|S | | S | ||
| | | Simplicity | ||
|<math>\small\mathsf{(S)}</math> | | <math>\small\mathsf{(S)}</math> | ||
| | | Simplicity weight | ||
|} | |} | ||
==Intermediate== | == Intermediate == | ||
ย | === Objects === | ||
===Objects=== | |||
ย | |||
{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ | |+ style="font-size: 105%;" | ย | ||
|- | |- | ||
! | ! 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 | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>\textbf{i}</math> | ||
|[[ | | [[interval|(Just) interval]] | ||
| | | ย | ||
| | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (d, 1)</math> | ||
| | | Integer | ||
| | | Vector | ||
| | | ย | ||
| | | [...โฉ | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math> | | <math>\mathrm{i}_i</math> | ||
| | | Specific type: vector ([[prime-count vector]] or PC-vector) | ||
Jargon name: monzo | |||
|- | |- | ||
|<math>M\textbf{i}</math> | | | ||
|<math>\textbf{y}</math> | | <math>M</math> | ||
|[[ | | [[Mapping|(Temperament) mapping (matrix)]] | ||
|<math>\scriptsize ย | | | ||
| <math>\small ๐ด</math>/<math>\small ๐ฝ</math> | |||
| Generators per prime | |||
| | |||
| <math>\scriptsize (r, d)</math> | |||
| Integer | |||
| Matrix | |||
| [โจ...] ...} | |||
| โจ[...} ...] | |||
| <math>๐_i</math> | |||
| | |||
| | |||
| <math>m_{ij}</math> | |||
| Jargon name: val list | |||
|- | |||
| <math>M\textbf{i}</math> | |||
| <math>\textbf{y}</math> | |||
| [[Mapped interval]] | |||
| <math>\scriptsize ย | |||
\begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\small ๐ด</math> | | <math>\small ๐ด</math> | ||
| | | Generators | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} ย | \begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array} ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (r, 1)</math> | | <math>\scriptsize (r, 1)</math> | ||
| | | Integer | ||
| | | Vector | ||
| | | ย | ||
|[...} | | [...} | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | Specific type: [[generator-count vector]] (GC-vector) | ||
Jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval | |||
|- | |- | ||
| | | ย | ||
|<math>๐</math> | | <math>๐</math> | ||
|[[map|( | | [[map|(Temperament) map]] | ||
| | | ย | ||
|<math>\small ๐ด</math>/<math>\small ๐ฝ</math> | | <math>\small ๐ด</math>/<math>\small ๐ฝ</math> | ||
| | | Generators per prime | ||
| | | ย | ||
|<math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Integer | ||
| | | Vector | ||
|โจ...] | | โจ...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<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]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<math>d - n</math> | | <math>d - n</math> | ||
|<math>r</math> | | <math>r</math> | ||
|[[ | | [[Rank]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<math>d - r</math> | | <math>d - r</math> | ||
|<math>n</math> | | <math>n</math> | ||
|[[ | | [[Nullity]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Tuning | ||
|- | |- | ||
|<math> | | | ||
| <math>{\large\textbf{๐}}\hspace{2mu}</math> | |||
| [[Log-prime map]] | |||
| | |||
|[[ | | <math>\small\mathsf{oct}</math>/<math>\small ๐ฝ</math> | ||
|<math> | | Octaves per prime | ||
| | |||
| <math>\scriptsize (1, d)</math> | |||
\ | | Real | ||
| Vector | |||
| โจ...] | |||
| | |||
| | |||
| | |||
| | |||
| <math>{\large ๐}\hspace{2mu}_i</math> | |||
</math> | | | ||
|<math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | |- | ||
| | | <math>1200ร{\large\textbf{๐}}\hspace{2mu}</math> | ||
| | | <math>๐</math> | ||
| [[just tuning map|Just(-prime) tuning map]] | |||
| | |||
| <math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | |||
| Cents per prime | |||
| ย | |||
| <math>\scriptsize (1, d)</math> | |||
| Real | |||
| Vector | |||
| โจ...] | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
|<math>\scriptsize (1, d)</math> | | <math>j_i</math> | ||
| | | ย | ||
| | |||
|โจ...] | |||
| | |||
| | |||
| | |||
| | |||
|<math>j_i</math> | |||
| | |||
|- | |- | ||
| | | ย | ||
|<math>๐</math> | | <math>๐</math> | ||
|[[ | | [[Generator tuning map]] | ||
| | | ย | ||
| <math>\mathsf{ยข}</math>/<math>\small ๐ด</math> | |||
| Cents per generator | |||
| ย | |||
| <math>\scriptsize (1, r)</math> | |||
| Real | |||
| Vector | |||
| {...] | |||
| ย | |||
| ย | |||
|<math>\mathsf{ยข}</math>/<math>\small ๐ด</math> | | ย | ||
| | | ย | ||
| | | <math>g_i</math> | ||
| ย | |||
|<math>\scriptsize (1, r)</math> | |||
| | |||
| | |||
|{...] | |||
| | |||
| | |||
| | |||
| | |||
|<math>g_i</math> | |||
| | |||
|- | |- | ||
| | | ย | ||
| <math>๐</math> | |||
| [[tuning map|(Tempered-prime) tuning map]] | |||
|<math>๐</math> | | ย | ||
|[[tuning map|( | | <math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | ||
| | | Cents per prime | ||
| ย | |||
| <math>\scriptsize (1, d)</math> | |||
| Real | |||
| Vector | |||
| โจ...] | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| <math>t_i</math> | |||
| ย | |||
|<math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | |||
| | |||
| | |||
|<math>\scriptsize (1, d)</math> | |||
| | |||
| | |||
|โจ...] | |||
| | |||
| | |||
| | |||
| | |||
|<math>t_i</math> | |||
| | |||
|- | |- | ||
|<math>๐ - ๐ | | <math>๐ - ๐</math><br /> | ||
1200ร\ | <math>1200ร\slant{\mathbf{1}}L(P - I)</math> | ||
|<math>๐</math> | | <math>๐</math> | ||
|[[retuning map| | | [[retuning map|Retuning (or mistuning) map]] | ||
| | | ย | ||
|<math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | | <math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | ||
| | | Cents per prime | ||
| | | ย | ||
|<math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Real | ||
| | | Vector | ||
|โจ...] | | โจ...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>r_i</math> | | <math>r_i</math> | ||
| | | Previous name: prime error map | ||
|- | |- | ||
|<math>๐\textbf{i}</math> | | <math>๐\textbf{i}</math> | ||
|<math>\mathrm{o}</math> | | <math>\mathrm{o}</math> | ||
|[[interval span|( | | [[interval span|(Just) (interval) size]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \ | \begin{array} {c} \mathbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | Mnemonic: <math>\mathrm{o}</math>riginal size | ||
|- | |- | ||
|<math>๐M\textbf{i} | | <math>๐M\textbf{i}</math><br /> | ||
๐\textbf{i}</math> | <math>๐\textbf{i}</math> | ||
|<math>\mathrm{a}</math> | | <math>\mathrm{a}</math> | ||
| | | {{subpage|tuning fundamentals|uprev|s=Example 3|text=Tempered (interval) size}} | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | Mnemonic: <math>\mathrm{a}</math>ltered size | ||
|- | |- | ||
|<math>๐\textbf{i} - ๐\textbf{i} | | <math>๐\textbf{i} - ๐\textbf{i}</math><br /> | ||
a - o | <math>a - o</math><br /> | ||
๐\textbf{i}</math> | <math>๐\textbf{i}</math> | ||
|<math>\mathrm{e}</math> | | <math>\mathrm{e}</math> | ||
|[[error|( | | [[error|(Interval) error]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Optimization | ||
|- | |- | ||
| | | ย | ||
|<math>p</math> | | <math>p</math> | ||
|[[ | | [[Optimization power]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>\llangle\,ยท\,\rrangle_p</math> | ||
|[[ | | [[Power mean]] (<math>p</math>-mean) | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Damage | ||
|- | |- | ||
| | | ย | ||
|<math>c</math> | | <math>c</math> | ||
| | | {{subpage|tuning_fundamentals|uprev|s=Complexity}} | ||
| colspan="3" |( | | colspan="3" | (See complexities section of complexities and simplicities table) | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<math> | | <math>\dfrac1c</math> | ||
|<math>s</math> | | <math>s</math> | ||
|[[ | | [[Simplicity]] | ||
| colspan="3" |( | | colspan="3" | (See simplicities section of complexities and simplicities table) | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<math>c</math> or <math>s</math> | | <math>c</math> or <math>s</math> | ||
|<math>w</math> | | <math>w</math> | ||
|[[weight]] | | [[weight]] | ||
| colspan="3" |( | | colspan="3" | (See complexities and simplicities table) | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<math> | | <math>\abs{\mathrm{e}} w</math> | ||
|<math>\mathrm{d}</math> | | <math>\mathrm{d}</math> | ||
|[[ | | [[Damage]] | ||
| colspan="3" |( | | colspan="3" | (See damages table) | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Target-intervals | ||
|- | |- | ||
| | | ย | ||
|<math>\mathrm{T}</math> | | <math>\mathrm{T}</math> | ||
|[[ | | [[Target-interval list]] | ||
| | | ย | ||
|<math>\small ๐ฝ</math> | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | ย | ||
|<math>\scriptsize (d, k)</math> | | <math>\scriptsize (d, k)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | ย | ||
|[[...โฉ ...] | | [[...โฉ ...] | ||
| | | ย | ||
|<math>\textbf{t}_i</math> | | <math>\textbf{t}_i</math> | ||
| | | ย | ||
|<math>\mathrm{t}_{ij}</math> | | <math>\mathrm{t}_{ij}</math> | ||
| | | ย | ||
|- | |- | ||
|<math>M\mathrm{T}</math> | | <math>M\mathrm{T}</math> | ||
|<math>\mathrm{Y}</math> | | <math>\mathrm{Y}</math> | ||
|[[ | | [[Mapped target-interval list]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\small ๐ด</math> | | <math>\small ๐ด</math> | ||
| | | Generators | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} ย | \begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} ย | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (r, k)</math> | | <math>\scriptsize (r, k)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | ย | ||
|[[...} ...] | | [[...} ...] | ||
| | | ย | ||
|<math>\textbf{y}_i</math> | | <math>\textbf{y}_i</math> | ||
| | | ย | ||
|<math>\mathrm{y}_{ij}</math> | | <math>\mathrm{y}_{ij}</math> | ||
| | | Mnemonic: looks like bent-up 'T', or cross between 'M' and 'T' | ||
|- | |- | ||
|<math>๐\mathrm{T}</math> | | <math>๐\mathrm{T}</math> | ||
|<math>\textbf{o}</math> | | <math>\textbf{o}</math> | ||
| | | {{subpage|tuning fundamentals|uprev|s=primes|text=Target-interval (just) size list}} | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
|[...] | | [...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\mathrm{o}_i</math> | | <math>\mathrm{o}_i</math> | ||
| | | Mnemonic: <math>\textbf{o}</math>riginal size list | ||
|- | |- | ||
|<math>๐\mathrm{T}</math> | | <math>๐\mathrm{T}</math> | ||
|<math>\textbf{a}</math> | | <math>\textbf{a}</math> | ||
|[[ | | [[Tempered target-interval size list]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
|[...] | | [...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\mathrm{a}_i</math> | | <math>\mathrm{a}_i</math> | ||
| | | Mnemonic: <math>\textbf{a}</math>ltered size list | ||
|- | |- | ||
|<math>๐\mathrm{T} - ๐\mathrm{T} | | <math>๐\mathrm{T} - ๐\mathrm{T}</math><br /> | ||
๐\mathrm{T} | <math>๐\mathrm{T}</math><br /> | ||
\textbf{a} - \textbf{o}</math> | <math>\textbf{a} - \textbf{o}</math> | ||
|<math>\textbf{e}</math> | | <math>\textbf{e}</math> | ||
|[[ | | [[Target-interval error list]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
|[...] | | [...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\mathrm{e}_i</math> | | <math>\mathrm{e}_i</math> | ||
| | | ย | ||
|- | |- | ||
|<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) | ||
| | | ย | ||
|<math>\scriptsize (k, k)</math> | | <math>\scriptsize (k, k)</math> | ||
| | | Real | ||
| | | Matrix | ||
| | | ย | ||
|[[...] ...] | | [[...] ...] | ||
| | | ย | ||
| | | ย | ||
|<math>๐</math> | | <math>๐</math> | ||
|<math>w_i</math> | | <math>w_i</math> | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math>C</math> | | <math>C</math> | ||
| | | {{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) | ||
| | | ย | ||
|<math>\scriptsize (k, k)</math> | | <math>\scriptsize (k, k)</math> | ||
| | | Real | ||
| | | Matrix | ||
| | | ย | ||
|[[...] ...] | | [[...] ...] | ||
| | | ย | ||
| | | ย | ||
|<math>๐</math> | | <math>๐</math> | ||
|<math>c_i</math> | | <math>c_i</math> | ||
| | | ย | ||
|- | |- | ||
|<math> | | <math>\dfrac1C</math> | ||
|<math>S</math> | | <math>S</math> | ||
| | | {{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) | ||
| | | ย | ||
|<math>\scriptsize (k, k)</math> | | <math>\scriptsize (k, k)</math> | ||
| | | Real | ||
| | | Matrix | ||
| | | ย | ||
|[[...] ...] | | [[...] ...] | ||
| | | ย | ||
| | | ย | ||
|<math>๐</math> | | <math>๐</math> | ||
|<math>s_i</math> | | <math>s_i</math> | ||
| | | Entry-wise reciprocal of <math>C</math> | ||
|- | |- | ||
|<math> | | <math>\abs{\textbf{e}} W</math> | ||
| <math>\textbf{d}</math> | |||
|<math>\textbf{d}</math> | | [[Target-interval damage list]] | ||
|[[ | | colspan="3" | (See damages table) | ||
| colspan="3" |( | | ย | ||
| | | <math>\scriptsize (1, k)</math> | ||
|<math>\scriptsize (1, k)</math> | | Real | ||
| | | List | ||
| | | [...] | ||
|[...] | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | <math>\mathrm{d}_i</math> | ||
|<math>\mathrm{d}_i</math> | | ย | ||
| | |||
|- | |- | ||
| | | ย | ||
|<math>k</math> | | <math>k</math> | ||
|[[ | | [[Target-interval count]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | Mnemonic: <math>k</math>ount | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Held-intervals | ||
|- | |- | ||
| | | ย | ||
|<math>\mathrm{ | | <math>\mathrm{H}</math> | ||
|[[ | | [[Held-interval basis]] | ||
| | | ย | ||
|<math>\small ๐ฝ</math> | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | ย | ||
|<math>\scriptsize (d, | | <math>\scriptsize (d, h)</math> | ||
| | | ย | ||
| | | Matrix | ||
| | | ย | ||
|[[...โฉ ...] | | [[...โฉ ...] | ||
| | | ย | ||
|<math>\textbf{ | | <math>\textbf{h}_i</math> | ||
| | | ย | ||
|<math>\mathrm{ | | <math>\mathrm{h}_{ij}</math> | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math>h</math> | | <math>h</math> | ||
|[[ | | [[Held-interval count]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Exploring temperaments | ||
|- | |- | ||
| | | ย | ||
|<math>\mathrm{C}</math> | | <math>\mathrm{C}</math> | ||
|[[ | | [[Comma basis]] | ||
| | | ย | ||
|<math>\small ๐ฝ</math> | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | ย | ||
|<math>\scriptsize (d, n)</math> | | <math>\scriptsize (d, n)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | ย | ||
|[[...โฉ ...] | | [[...โฉ ...] | ||
| | | ย | ||
|<math>\textbf{c}_i</math> | | <math>\textbf{c}_i</math> | ||
| | | ย | ||
|<math>\mathrm{c}_{ij}</math> | | <math>\mathrm{c}_{ij}</math> | ||
| | | Jargon name: monzo list | ||
|- | |- | ||
| | | ย | ||
|<math>\textbf{c}</math> | | <math>\textbf{c}</math> | ||
|[[ | | [[Comma]] | ||
| | | ย | ||
|<math>\small ๐ฝ</math> | | <math>\small ๐ฝ</math> | ||
|primes | | primes | ||
| | | ย | ||
|<math>\scriptsize (d, 1)</math> | | <math>\scriptsize (d, 1)</math> | ||
| | | Integer | ||
| | | Vector | ||
| | | ย | ||
|[...โฉ | | [...โฉ | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\mathrm{c}_i</math> | | <math>\mathrm{c}_i</math> | ||
| | | Specific type: vector ([[prime-count vector]] or PC-vector) | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Computation | ||
|- | |- | ||
| | | ย | ||
|<math> | | {{llzigzag}}<math>\,ยท\,</math>{{rrzigzag}}<math>_p</math> | ||
| [[Power sum]] (<math>p</math>-sum) | |||
| | |||
| | |||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| | ! colspan="17" | All-interval tuning schemes | ||
|- | |- | ||
| | | <math>\mathrm{I}</math> | ||
|<math>{ | | <math>\mathrm{T}_{\text{p}}</math> | ||
|[[ | | [[Prime proxy target-interval list]] | ||
| | |||
| <math>\small ๐ฝ</math> | |||
| | | Primes | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (d, d)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | ย | ||
| | | โจ[...โฉ ...] | ||
| | | ย | ||
| | | ย | ||
| | | <math>\mathbf{1}</math> | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| ย | |||
| <math>X</math> | |||
| [[Complexity prescaler]] | |||
| <math>\small\mathsf{๐}\scriptsize\mathsf{(C)}</math> | |||
| <math>\small\mathsf{(C)}</math> | |||
| Complexity weight | |||
| | |||
| <math>\scriptsize (d, d)</math> | |||
| Real | |||
| Matrix | |||
| [โจ...] ...โฉ | |||
| | |||
| | |||
| | |||
| <math>๐</math> | |||
| <math>x_i</math> | |||
| | |||
|- | |- | ||
|<math> | | <math>\text{diag}({\large\textbf{๐}}\hspace{2mu})</math> | ||
|<math> | | <math>L</math> | ||
|[[ | | [[Log-prime matrix]] | ||
|<math>\ | | | ||
| <math>\small\mathsf{oct}</math>/<math>\small ๐ฝ</math> | |||
\ | | Octaves per prime | ||
| ย | |||
| <math>\scriptsize (d, d)</math> | |||
| Real | |||
| | | Matrix | ||
| | | [โจ...] ...โฉ | ||
| โจ[...โฉ ...] | |||
| <math>{\large\textbf{๐}}\hspace{2mu}_i</math> | |||
| ย | |||
| <math>{\large\textbf{๐}}\hspace{2mu}</math> | |||
| <math>{\large ๐}\hspace{2mu}_{ij}</math> | |||
| ย | |||
|<math>\scriptsize (d, d)</math> | |||
| | |||
| | |||
|[โจ...] ...โฉ | |||
|โจ[...โฉ ...] | |||
|<math> | |||
| | |||
| | |||
|<math> | |||
| | |||
|- | |- | ||
| | | ย | ||
|<math> | | <math>q</math> | ||
| | | {{subpage|all-interval_tuning_schemes|uprev|s=Dual norms|text=Interval complexity norm power}} | ||
| | |||
| | |||
| ย | |||
| ย | |||
| <math>\scriptsize (1, 1)</math> | |||
| Real | |||
| Scalar | |||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize ย | | ย | ||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |- | ||
| ย | |||
| <math>\norm{ยท}_q</math> | |||
| [[Power norm]] (<math>p</math>-norm) | |||
| | |||
| | |||
| | |||
| | |||
| <math>\scriptsize (1, 1)</math> | |||
| Real | |||
| Scalar | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |- | ||
|<math> | | <math>\dfrac1{1-\frac1q}</math> | ||
|<math> | | <math>\text{dual}(q)</math> | ||
| | | {{subpage|all-interval tuning schemes|uprev|s=Dual norms|text=Dual norm power}} | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
| <math>\norm{X\mathbf{i}}_q</math> | |||
| [[interval complexity]] | |||
|<math>\ | | ย | ||
| <math>\small\mathsf{(C)}</math> | |||
| ย | |||
| ย | |||
| <math>\scriptsize (1, 1)</math> | |||
| Real | |||
| Scalar | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
|[[ | |||
| | |||
|<math>\small\mathsf{(C)}</math> | |||
| | |||
| | |||
|<math>\scriptsize ( | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |- | ||
| | | ย | ||
|<math>q</math> | | <math>\norm{๐X^{-1}}_{\text{dual}(q)}</math> | ||
|[[ | | [[Retuning magnitude]] | ||
| ย | |||
| <math>\mathsf{ยข}\small\mathsf{(C^{-1})}</math> | |||
| ย | |||
| | | ย | ||
|<math>\ | | <math>\scriptsize (1, 1)</math> | ||
| Real | |||
| Scalar | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| | |||
| | |||
|<math>\scriptsize (1, 1)</math> | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|} | |} | ||
===Units=== | === Units === | ||
ย | |||
Same as the basic level. ย | Same as the basic level. ย | ||
===Tuning schemes=== | === Tuning schemes === | ||
ย | |||
{| class="wikitable center-all mw-collapsible mw-collapsed" | {| class="wikitable center-all mw-collapsible mw-collapsed" | ||
|+ | |+ style="font-size: 105%;" | | ||
|- | |- | ||
! colspan="3" rowspan="3" | | ! colspan="3" rowspan="3" | Retuning (or mistuning) magnitude | ||
! colspan="9" | | ! colspan="9" | Damage | ||
! rowspan="4" | | ! rowspan="4" | Target<br />intervals | ||
ย | ! colspan="2" rowspan="3" | Systematic name | ||
intervals | ! rowspan="4" | Previously named tuning schemes that are specific types of this tuning scheme | ||
! colspan="2" rowspan="3" | | ! rowspan="4" | Of interest? | ||
! rowspan="4" | | |||
! rowspan="4" | | |||
|- | |- | ||
! colspan="6" | | ! colspan="6" | Weight | ||
! colspan="3" rowspan="1" | | ! colspan="3" rowspan="1" | Optimization | ||
|- | |- | ||
! colspan="3" | | ! colspan="3" | Interval complexity | ||
! colspan="3" rowspan="1" | | ! colspan="3" rowspan="1" | Slope | ||
! colspan="1" rowspan="2" | | ! colspan="1" rowspan="2" | Initial | ||
! colspan="1" rowspan="2" | | ! colspan="1" rowspan="2" | Name | ||
! colspan="1" rowspan="2" | | ! colspan="1" rowspan="2" | Power | ||
|- | |- | ||
! | ! Initial | ||
! | ! Name | ||
! | ! Power | ||
! | ! Initial | ||
! | ! Name | ||
! | ! Power | ||
! | ! Initial | ||
! | ! Name | ||
! | ! Multiplier | ||
! colspan="1" | | ! colspan="1" | Abbreviated | ||
! colspan="1" | | ! colspan="1" | Read ("____ tuning scheme") | ||
|- | |- | ||
|<n/a> | | <n/a> | ||
| | | Maximum | ||
| | | ∞ | ||
|(t) | | (t) | ||
| | | Taxicab | ||
|1 | | 1 | ||
| rowspan="2" |S | | rowspan="2" | ''S'' | ||
| rowspan="2" | | | rowspan="2" | Simplicity-weight | ||
| rowspan="2" |1/ | | rowspan="2" | 1/Complexity | ||
| rowspan="17" |<n/a> | | rowspan="17" | <n/a> | ||
| rowspan="7" | | | rowspan="7" | Minimax | ||
| rowspan="7" |โ | | rowspan="7" | โ | ||
| rowspan="2" | | | rowspan="2" | All | ||
| | | Minimax-S | ||
| | | 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 | ||
|- | |- | ||
|<n/a> | | <n/a> | ||
|Euclidean | | Euclidean | ||
|2 | | 2 | ||
|E | | E | ||
|Euclidean | | Euclidean | ||
|2 | | 2 | ||
| | | Minimax-ES | ||
| | | 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]]" | ||
| | | ย | ||
|- | |- | ||
| colspan="3" rowspan="15" |<n/a> | | colspan="3" rowspan="15" | <n/a> | ||
| colspan="3" |<n/a> | | colspan="3" | <n/a> | ||
|U | | U | ||
| | | Unity-weight | ||
|<none> | | <none> | ||
| rowspan="15" |<set> | | rowspan="15" | <set> | ||
|<set> | | <set> Minimax-U | ||
|<set> | | <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" | | | rowspan="2" | Simplicity-weight | ||
| rowspan="2" |1/ | | rowspan="2" | 1/Complexity | ||
|<set> | | <set> Minimax-S | ||
|<set> | | <set> Minimax simplicity-weight damage | ||
| | | ย | ||
| | | Yes | ||
|- | |- | ||
|E | | E | ||
|Euclidean | | Euclidean | ||
|2 | | 2 | ||
|<set> | | <set> Minimax-ES | ||
|<set> | | <set> Minimax Euclideanized-simplicity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|(t) | | (t) | ||
| | | Taxicab | ||
|1 | | 1 | ||
| rowspan="2" |C | | rowspan="2" | C | ||
| rowspan="2" | | | rowspan="2" | Complexity-weight | ||
| rowspan="2" | | | rowspan="2" | Complexity | ||
|<set> | | <set> Minimax-C | ||
|<set> | | <set> Minimax complexity-weight damage | ||
| | | ย | ||
| | | Yes | ||
|- | |- | ||
|E | | E | ||
|Euclidean | | Euclidean | ||
|2 | | 2 | ||
|<set> | | <set> Minimax-EC | ||
|<set> | | <set> Minimax Euclideanized-complexity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<n/a> | | colspan="3" | <n/a> | ||
|U | | U | ||
| | | Unity-weight | ||
|<none> | | <none> | ||
| rowspan="5" | | | rowspan="5" | MiniRMS | ||
| rowspan="5" |2 | | rowspan="5" | 2 | ||
|<set> | | <set> MiniRMS-U | ||
|<set> | | <set> MiniRMS unity-weight damage | ||
|"[[ | | "[[Least squares]]" | ||
| | | Yes | ||
|- | |- | ||
|(t) | | (t) | ||
| | | Taxicab | ||
|1 | | 1 | ||
| rowspan="2" |S | | rowspan="2" | S | ||
| rowspan="2" | | | rowspan="2" | Simplicity-weight | ||
| rowspan="2" |1/ | | rowspan="2" | 1/Complexity | ||
|<set> | | <set> MiniRMS-S | ||
|<set> | | <set> MiniRMS simplicity-weight damage | ||
| | | ย | ||
| | | Yes | ||
|- | |- | ||
|E | | E | ||
|Euclidean | | Euclidean | ||
|2 | | 2 | ||
|<set> | | <set> MiniRMS-ES | ||
|<set> | | <set> MiniRMS Euclideanized-simplicity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|(t) | | (t) | ||
| | | Taxicab | ||
|1 | | 1 | ||
| rowspan="2" |C | | rowspan="2" | C | ||
| rowspan="2" | | | rowspan="2" | Complexity-weight | ||
| rowspan="2" | | | rowspan="2" | Complexity | ||
|<set> | | <set> MiniRMS-C | ||
|<set> | | <set> MiniRMS complexity-weight damage | ||
| | | ย | ||
| | | Yes | ||
|- | |- | ||
|E | | E | ||
|Euclidean | | Euclidean | ||
|2 | | 2 | ||
|<set> | | <set> MiniRMS-EC | ||
|<set> | | <set> MiniRMS Euclideanized-complexity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<n/a> | | colspan="3" | <n/a> | ||
|U | | U | ||
| | | Unity-weight | ||
|<none> | | <none> | ||
| rowspan="5" | | | rowspan="5" | Miniaverage | ||
| rowspan="5" |1 | | rowspan="5" | 1 | ||
|<set> | | <set> Miniaverage-U | ||
|<set> | | <set> Miniaverage unity-weight damage | ||
| | | ย | ||
| | | Yes | ||
|- | |- | ||
|(t) | | (t) | ||
| | | Taxicab | ||
|1 | | 1 | ||
| rowspan="2" |S | | rowspan="2" | S | ||
| rowspan="2" | | | rowspan="2" | Simplicity-weight | ||
| rowspan="2" |1/ | | rowspan="2" | 1/Complexity | ||
|<set> | | <set> Miniaverage-S | ||
|<set> | | <set> Miniaverage simplicity-weight damage | ||
| | | ย | ||
| | | Yes | ||
|- | |- | ||
|E | | E | ||
|Euclidean | | Euclidean | ||
|2 | | 2 | ||
|<set> | | <set> Miniaverage-ES | ||
|<set> | | <set> Miniaverage Euclideanized-simplicity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|(t) | | (t) | ||
| | | Taxicab | ||
|1 | | 1 | ||
| rowspan="2" |C | | rowspan="2" | C | ||
| rowspan="2" | | | rowspan="2" | Complexity-weight | ||
| rowspan="2" | | | rowspan="2" | Complexity | ||
|<set> | | <set> Miniaverage-C | ||
|<set> | | <set> Miniaverage complexity-weight damage | ||
| | | ย | ||
| | | Yes | ||
|- | |- | ||
|E | | E | ||
|Euclidean | | Euclidean | ||
|2 | | 2 | ||
|<set> | | <set> Miniaverage-EC | ||
|<set> | | <set> Miniaverage Euclideanized-complexity-weight damage | ||
| | | ย | ||
| | | ย | ||
|} | |} | ||
===Damages=== | === Damages === | ||
ย | |||
{| class="wikitable center-all mw-collapsible mw-collapsed" | {| class="wikitable center-all mw-collapsible mw-collapsed" | ||
|+ | |+ style="font-size: 105%;" | | ||
|- | |- | ||
! colspan="2" | | ! colspan="2" | Quantity | ||
! colspan="2" | | ! colspan="2" | Unit | ||
|- | |- | ||
! | ! Abbreviation | ||
! | ! Name | ||
! | ! Symbol | ||
! | ! Name | ||
|- | |- | ||
|U-damage | | U-damage | ||
| | | Unity-weight damage | ||
|<math>\mathsf{ยข}\small\mathsf{(U)}</math> | | <math>\mathsf{ยข}\small\mathsf{(U)}</math> | ||
| | | Unity-weighted cents | ||
|- | |- | ||
|C-damage | | C-damage | ||
| | | Complexity-weight damage | ||
|<math>\mathsf{ยข}\small\mathsf{(C)}</math> | | <math>\mathsf{ยข}\small\mathsf{(C)}</math> | ||
| | | Complexity-weighted cents | ||
|- | |- | ||
|EC-damage | | EC-damage | ||
|Euclideanized-complexity-weight damage | | Euclideanized-complexity-weight damage | ||
|<math>\mathsf{ยข}</math><math>\small\mathsf{(EC)}</math> | | <math>\mathsf{ยข}</math><math>\small\mathsf{(EC)}</math> | ||
|Euclideanized-complexity-weighted cents | | Euclideanized-complexity-weighted cents | ||
|- | |- | ||
|S-damage | | S-damage | ||
| | | Simplicity-weight damage | ||
|<math>\mathsf{ยข}\small\mathsf{(S)}</math> | | <math>\mathsf{ยข}\small\mathsf{(S)}</math> | ||
| | | Simplicity-weighted cents | ||
|- | |- | ||
|ES-damage | | ES-damage | ||
|Euclideanized-simplicity-weight damage | | Euclideanized-simplicity-weight damage | ||
|<math>\mathsf{ยข}</math><math>\small\mathsf{(ES)}</math> | | <math>\mathsf{ยข}</math><math>\small\mathsf{(ES)}</math> | ||
|Euclideanized-simplicity-weighted cents | | Euclideanized-simplicity-weighted cents | ||
|} | |} | ||
===Complexity and simplicity=== | === Complexity and simplicity === | ||
ย | |||
{| class="wikitable center-all mw-collapsible mw-collapsed" | {| class="wikitable center-all mw-collapsible mw-collapsed" | ||
|+ | |+ style="font-size: 105%;" | ย | ||
|- | |- | ||
! | ! colspan="2" | Quantity | ||
! | ! colspan="2" | Unit | ||
|- | |- | ||
! Abbreviation | |||
! Name | |||
! Symbol | |||
! Name | |||
|- | |- | ||
| | | C | ||
| | | Complexity | ||
|<math>\small\mathsf{( | | <math>\small\mathsf{(C)}</math> | ||
| | | Complexity weight | ||
|- | |- | ||
| | | EC | ||
| | | Euclideanized complexity | ||
|<math>\small\mathsf{( | | <math>\small\mathsf{(EC)}</math> | ||
| | | Euclideanized-complexity weight | ||
|- | |- | ||
|ES | | S | ||
|Euclideanized simplicity | | Simplicity | ||
|<math>\small\mathsf{(ES)}</math> | | <math>\small\mathsf{(S)}</math> | ||
|Euclideanized-simplicity weight | | Simplicity weight | ||
|- | |||
| ES | |||
| Euclideanized simplicity | |||
| <math>\small\mathsf{(ES)}</math> | |||
| Euclideanized-simplicity weight | |||
|} | |} | ||
==Advanced== | == Advanced == | ||
ย | === Objects === | ||
===Objects=== | |||
ย | |||
{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ | |+ style="font-size: 105%;" | | ||
! rowspan="2" | | |- | ||
! rowspan="2" | | ! rowspan="2" | Equivalent expressions | ||
! rowspan="2" | | ! rowspan="2" | Variable | ||
! colspan="3" | | ! rowspan="2" | Name | ||
! colspan="2" | | ! colspan="3" | Units | ||
! colspan="2" | | ! colspan="2" | Shape | ||
! colspan="2" |EBK notation | ! colspan="2" | Type | ||
! colspan="4" | | ! colspan="2" | EBK notation | ||
! rowspan="2" | | ! colspan="4" | Subobjects | ||
! rowspan="2" | Notes | |||
|- | |- | ||
! | ! Unreduced | ||
! | ! Reduced | ||
! | ! Read as | ||
! | ! Unreduced | ||
! | ! Reduced | ||
! | ! Numeric | ||
! | ! Structural | ||
! | ! Row-first | ||
! | ! Col-first | ||
! | ! Row | ||
! | ! Column | ||
! | ! Diagonal | ||
! | ! Entry | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Mapping | ||
|- | |- | ||
| | | ย | ||
|<math>\textbf{i}</math> | | <math>\textbf{i}</math> | ||
|[[interval|( | | [[interval|(Just) interval]] | ||
| | | ย | ||
|<math>\small ๐ฝ</math> | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | ย | ||
|<math>\scriptsize (d, 1)</math> | | <math>\scriptsize (d, 1)</math> | ||
| | | Integer | ||
| | | Vector | ||
| | | ย | ||
|[...โฉ | | [...โฉ | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\mathrm{i}_i</math> | | <math>\mathrm{i}_i</math> | ||
| | | Specific type: vector ([[prime-count vector]] or PC-vector) | ||
Jargon name: monzo | |||
|- | |- | ||
| | | ย | ||
|<math>M</math> | | <math>M</math> | ||
|[[Mapping|( | | [[Mapping|(Temperament) mapping (matrix)]] | ||
| | | ย | ||
|<math>\small ๐ด</math>/<math>\small ๐ฝ</math> | | <math>\small ๐ด</math>/<math>\small ๐ฝ</math> | ||
| | | Generators per prime | ||
| | | ย | ||
|<math>\scriptsize (r, d)</math> | | <math>\scriptsize (r, d)</math> | ||
| | | Integer | ||
| | | Matrix | ||
|[โจ...] ...} | | [โจ...] ...} | ||
|โจ[...} ...] | | โจ[...} ...] | ||
|<math>๐_i</math> | | <math>๐_i</math> | ||
| | | ย | ||
| | | ย | ||
|<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 ย | | <math>\scriptsize ย | ||
\begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\small ๐ด</math> | | <math>\small ๐ด</math> | ||
|generators | | generators | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} ย | \begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array} ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (r, 1)</math> | | <math>\scriptsize (r, 1)</math> | ||
| | | Integer | ||
| | | Vector | ||
| | | ย | ||
|[...} | | [...} | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | Specific type: [[generator-count vector]] (GC-vector) | ||
Jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval | |||
|- | |- | ||
| | | ย | ||
|<math>๐</math> | | <math>๐</math> | ||
|[[map|( | | [[map|(Temperament) map]] | ||
| | | ย | ||
|<math>\small ๐ด</math>/<math>\small ๐ฝ</math> | | <math>\small ๐ด</math>/<math>\small ๐ฝ</math> | ||
| | | Generators per prime | ||
| | | ย | ||
|<math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Integer | ||
| | | Vector | ||
|โจ...] | | โจ...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<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]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<math>d - n</math> | | <math>d - n</math> | ||
|<math>r</math> | | <math>r</math> | ||
|[[ | | [[Rank]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<math>d - r</math> | | <math>d - r</math> | ||
|<math>n</math> | | <math>n</math> | ||
|[[ | | [[Nullity]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Tuning | ||
|- | |- | ||
|<math>1200ร\ | | <math>\slant{\mathbf{1}}L</math> | ||
1200ร\ | | <math>{\large\textbf{๐}}\hspace{2mu}</math> | ||
๐_{\text{j}}M_{\text{j}}</math> | | [[Log-prime map]] | ||
|<math>๐</math> | | | ||
|[[just tuning map| | | <math>\small\mathsf{oct}</math>/<math>\small ๐ฝ</math> | ||
|<math>\scriptsize ย | | Octaves per prime | ||
| | |||
| <math>\scriptsize (1, d)</math> | |||
| Real | |||
| Vector | |||
| โจ...] | |||
| | |||
| | |||
| | |||
| | |||
| <math>{\large ๐}\hspace{2mu}_i</math> | |||
| | |||
|- | |||
| <math>1200ร\slant{\mathbf{1}}LG_{\text{j}}M_{\text{j}}</math><br /> | |||
<math>1200ร\slant{\mathbf{1}}L</math><br /> | |||
<math>๐_{\text{j}}M_{\text{j}}</math> | |||
| <math>๐</math> | |||
| [[just tuning map|Just(-prime) tuning map]] | |||
| <math>\scriptsize ย | |||
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | \begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \ | \begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} | \begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} | ||
| Line 2,508: | Line 2,359: | ||
\begin{array} {c} M_{\text{j}} \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} ๐ฝ \end{array} | \begin{array} {c} M_{\text{j}} \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} ๐ฝ \end{array} | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | | <math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | ||
| | | Cents per prime | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array} | \begin{array} {c} 1200 \\[-3pt] \left(1, \cancel{1}\right) \end{array} | ||
\!\! | \! \! ย | ||
\begin{array} {c} \ | \begin{array} {c} \slant{\mathbf{1}} \\[-3pt] \left(\cancel{1}, \cancel{d}\right) \end{array} | ||
\!\! | \! \! ย | ||
\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 ย | \\ \scriptsize \quad ย | ||
\!\! | \! \! ย | ||
\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} | ||
\!\! | \! \! ย | ||
\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} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, d_{\text{p}})</math> | | <math>\scriptsize \left(1, d_{\text{p}}\right)</math> | ||
| | | Real | ||
| | | Vector | ||
|โจ...] | | โจ...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>j_i</math> | | <math>j_i</math> | ||
| | | ย | ||
|- | |- | ||
|<math>1200ร\ | | <math>1200ร\slant{\mathbf{1}}LG</math> | ||
|<math>๐</math> | | <math>๐</math> | ||
|[[ | | [[Generator tuning map]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | \begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \ | \begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} | \begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} | ||
| Line 2,548: | Line 2,399: | ||
\begin{array} {c} G \\[-2pt] \cancel{๐ฝ} \hspace{-2mu} / \hspace{-2mu} ๐ด \end{array} | \begin{array} {c} G \\[-2pt] \cancel{๐ฝ} \hspace{-2mu} / \hspace{-2mu} ๐ด \end{array} | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math>/<math>\small ๐ด</math> | | <math>\mathsf{ยข}</math>/<math>\small ๐ด</math> | ||
| | | Cents per generator | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array} | \begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array} | ||
\!\! | \! \! ย | ||
\begin{array} {c} \ | \begin{array} {c} \slant{\mathbf{1}} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array} | ||
\!\! | \! \! ย | ||
\begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array} | \begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array} | ||
\\ \scriptsize \quad ย | \\ \scriptsize \quad ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} G \\[-3pt] (\cancel{d}, r) \end{array} | \begin{array} {c} G \\[-3pt] (\cancel{d}, r) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, r)</math> | | <math>\scriptsize (1, r)</math> | ||
| | | Real | ||
| | | Vector | ||
|{...] | | {...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>g_i</math> | | <math>g_i</math> | ||
| | | ย | ||
|- | |- | ||
|<math>1200ร\ | | <math>1200ร\slant{\mathbf{1}}LGM</math><br /> | ||
1200ร\ | <math>1200ร\slant{\mathbf{1}}LP</math><br /> | ||
๐M</math> | <math>๐M</math> | ||
|<math>๐</math> | | <math>๐</math> | ||
|[[tuning map|( | | [[tuning map|(Tempered-prime) tuning map]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | \begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \ | \begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array} | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} | \begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} | ||
| Line 2,590: | Line 2,441: | ||
\begin{array} {c} M \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} ๐ฝ \end{array} | \begin{array} {c} M \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} ๐ฝ \end{array} | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | | <math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | ||
| | | Cents per prime | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} 1200 \\[-3pt] ( | \begin{array} {c} 1200 \\[-3pt] \left(1, \cancel{1}\right) \end{array} | ||
\!\! | \! \! ย | ||
\begin{array} {c} \ | \begin{array} {c} \slant{\mathbf{1}} \\[-3pt] \left(\cancel{1}, \cancel{d}\right) \end{array} | ||
\!\! | \! \! ย | ||
\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 ย | \\ \scriptsize \quad ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} G \\[-3pt] (\cancel{d}, \cancel{r}) \end{array} | \begin{array} {c} G \\[-3pt] \left(\cancel{d}, \cancel{r}\right) \end{array} | ||
\!\! | \! \! ย | ||
\begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array} | \begin{array} {c} M \\[-3pt] \left(\cancel{r}, d\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Real | ||
| | | Vector | ||
|โจ...] | | โจ...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>t_i</math> | | <math>t_i</math> | ||
| | | ย | ||
|- | |- | ||
|<math>๐ - ๐ | | <math>๐ - ๐</math><br /> | ||
1200ร\ | <math>1200ร\slant{\mathbf{1}}L(P - I)</math> | ||
|<math>๐</math> | | <math>๐</math> | ||
|[[retuning map| | | [[retuning map|Retuning (or mistuning) map]] | ||
| | | ย | ||
|<math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | | <math>\mathsf{ยข}</math>/<math>\small ๐ฝ</math> | ||
| | | Cents per prime | ||
| | | ย | ||
|<math>\scriptsize (1, d)</math> | | <math>\scriptsize (1, d)</math> | ||
| | | Real | ||
| | | Vector | ||
|โจ...] | | โจ...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>r_i</math> | | <math>r_i</math> | ||
| | | Previous name: prime error map | ||
|- | |- | ||
|<math>๐\textbf{i}</math> | | <math>๐\textbf{i}</math> | ||
|<math>\mathrm{o}</math> | | <math>\mathrm{o}</math> | ||
|[[interval span|( | | [[interval span|(Just) (interval) size]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | Mnemonic: <math>\mathrm{o}</math>riginal size | ||
|- | |- | ||
|<math>๐M\textbf{i} | | <math>๐M\textbf{i}</math><br /> | ||
๐\textbf{i}</math> | <math>๐\textbf{i}</math> | ||
|<math>\mathrm{a}</math> | | <math>\mathrm{a}</math> | ||
| | | {{subpage|tuning fundamentals|uprev|s=Example 3|text=Tempered (interval) size}} | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
|cents | | cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | Mnemonic: <math>\mathrm{a}</math>ltered size | ||
|- | |- | ||
|<math>๐\textbf{i} - ๐\textbf{i} | | <math>๐\textbf{i} - ๐\textbf{i}</math><br /> | ||
a - o | <math>a - o</math><br /> | ||
๐\textbf{i}</math> | <math>๐\textbf{i}</math> | ||
|<math>\mathrm{e}</math> | | <math>\mathrm{e}</math> | ||
|[[error|( | | [[error|(Interval) error]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Optimization | ||
|- | |- | ||
| | | ย | ||
|<math>p</math> | | <math>p</math> | ||
|[[ | | [[Optimization power]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>\llangle\,ยท\,\rrangle_p</math> | ||
|[[ | | [[Power mean]] (<math>p</math>-mean) | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Damage | ||
|- | |- | ||
| | | ย | ||
|<math>c</math> | | <math>c</math> | ||
| | | {{subpage|Tuning_fundamentals|prev|s=complexity}} | ||
| colspan="3" |( | | colspan="3" | (See complexities section of complexities and simplicities table) | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<math> | | <math>\dfrac1c</math> | ||
|<math>s</math> | | <math>s</math> | ||
|[[ | | [[Simplicity]] | ||
| colspan="3" |( | | colspan="3" | (See simplicities section of complexities and simplicities table) | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<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) | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
|<math> | | <math>\abs{\mathrm{e}} w</math> | ||
|<math>\mathrm{d}</math> | | <math>\mathrm{d}</math> | ||
|[[ | | [[Damage]] | ||
| colspan="3" |( | | colspan="3" | (See damages table) | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Target-intervals | ||
|- | |- | ||
| | | ย | ||
|<math>\mathrm{T}</math> | | <math>\mathrm{T}</math> | ||
|[[ | | [[Target-interval list]] | ||
| | | ย | ||
|<math>\small ๐ฝ</math> | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | ย | ||
|<math>\scriptsize (d, k)</math> | | <math>\scriptsize (d, k)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | ย | ||
|[[...โฉ ...] | | [[...โฉ ...] | ||
| | | ย | ||
|<math>\textbf{t}_i</math> | | <math>\textbf{t}_i</math> | ||
| | | ย | ||
|<math>\mathrm{t}_{ij}</math> | | <math>\mathrm{t}_{ij}</math> | ||
| | | ย | ||
|- | |- | ||
|<math>M\mathrm{T}</math> | | <math>M\mathrm{T}</math> | ||
|<math>\mathrm{Y}</math> | | <math>\mathrm{Y}</math> | ||
|[[ | | [[Mapped target-interval list]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} M \\[-2pt] ๐ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\small ๐ด</math> | | <math>\small ๐ด</math> | ||
| | | Generators | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} ย | \begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} ย | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (r, k)</math> | | <math>\scriptsize (r, k)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | ย | ||
|[[...} ...] | | [[...} ...] | ||
| | | ย | ||
|<math>\textbf{y}_i</math> | | <math>\textbf{y}_i</math> | ||
| | | ย | ||
|<math>\mathrm{y}_{ij}</math> | | <math>\mathrm{y}_{ij}</math> | ||
| | | Mnemonic: looks like bent-up 'T', or cross between 'M' and 'T' | ||
|- | |- | ||
|<math>๐\mathrm{T}</math> | | <math>๐\mathrm{T}</math> | ||
|<math>\textbf{o}</math> | | <math>\textbf{o}</math> | ||
| | | {{subpage|tuning fundamentals|uprev|s=primes|text=Target-interval (just) size list}} | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
|[...] | | [...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\mathrm{o}_i</math> | | <math>\mathrm{o}_i</math> | ||
| | | Mnemonic: <math>\textbf{o}</math>riginal size list | ||
|- | |- | ||
|<math>๐\mathrm{T}</math> | | <math>๐\mathrm{T}</math> | ||
|<math>\textbf{a}</math> | | <math>\textbf{a}</math> | ||
|[[ | | [[Tempered target-interval size list]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} ๐ \\[-3pt] (1, \cancel{d}) \end{array} ย | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
| | | List | ||
|[...] | | [...] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\mathrm{a}_i</math> | | <math>\mathrm{a}_i</math> | ||
| | | Mnemonic: <math>\textbf{a}</math>ltered size list | ||
|- | |- | ||
|<math>๐\mathrm{T} - ๐\mathrm{T} | | <math>๐\mathrm{T} - ๐\mathrm{T}</math><br /> | ||
๐\mathrm{T} | <math>๐\mathrm{T}</math><br /> | ||
\textbf{a} - \textbf{o}</math> | <math>\textbf{a} - \textbf{o}</math> | ||
|<math>\textbf{e}</math> | | <math>\textbf{e}</math> | ||
|[[target-interval error list]] | | [[target-interval error list]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | \begin{array} {c} ๐ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐ฝ} \end{array} ย | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐ฝ} \end{array} ย | ||
</math> | </math> | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} | \begin{array} {c} ๐ \\[-3pt] \left(1, \cancel{d}\right) \end{array} ย | ||
\!\! ย | \! \! ย | ||
\begin{array} {c} \mathrm{T} \\[-3pt] (\cancel{d}, k) \end{array} | \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (1, k)</math> | | <math>\scriptsize (1, k)</math> | ||
| | | Real | ||
|list | | List | ||
|[...] | | [...] | ||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
|<math>\mathrm{ | | <math>\mathrm{e}_i</math> | ||
| | | | ||
|- | |||
| <math>C</math> or <math>S</math> | |||
| <math>W</math> | |||
| [[Target-interval weight matrix]] | |||
| colspan="3" | (See complexities and simplicities table) | |||
| | |||
| <math>\scriptsize (k, k)</math> | |||
| Real | |||
| Matrix | |||
| | |||
| [[...] ...] | |||
| | |||
| | |||
| <math>๐</math> | |||
| <math>w_i</math> or <math>w_{ij}</math> | |||
| | |||
|- | |||
| | |||
| <math>C</math> | |||
| {{subpage|tuning fundamentals|uprev|s=complexity-weight damage|text=Target-interval complexity weight matrix}} | |||
| colspan="3" | (See complexities section of complexities and simplicities table) | |||
| | |||
| <math>\scriptsize (k, k)</math> | |||
| Real | |||
| Matrix | |||
| | |||
| [[...] ...] | |||
| | |||
| | |||
| <math>๐</math> | |||
| <math>c_i</math> | |||
| | |||
|- | |||
| <math>\dfrac1C</math> | |||
| <math>S</math> | |||
| {{subpage|tuning fundamentals|uprev|s=complexity-weight_damage|text=Target-interval simplicity weight matrix}} | |||
| colspan="3" | (See simplicities section of complexities and simplicities table) | |||
| | |||
| <math>\scriptsize (k, k)</math> | |||
| Real | |||
| Matrix | |||
| | |||
| [[...] ...] | |||
| | |||
| | |||
| <math>๐</math> | |||
| <math>s_i</math> | |||
| Entry-wise reciprocal of <math>C</math> | |||
|- | |||
| <math>\abs{\textbf{e}} W</math><br /> | |||
<math>1200ร\slant{\mathbf{1}}L\abs{P - I} \mathrm{T}W</math> | |||
| <math>\textbf{d}</math> | |||
| [[Target-interval damage list]] | |||
| colspan="3" | (See damages table) | |||
| | |||
| <math>\scriptsize (1, k)</math> | |||
| Real | |||
| List | |||
| [...] | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| <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 ๐ฝ</math> | |||
| Primes | |||
| | |||
| <math>\scriptsize (d, h)</math> | |||
| | |||
| Matrix | |||
| | |||
| [[...โฉ ...] | |||
| | |||
| <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 ๐ฝ</math> | |||
| Primes | |||
| | |||
| <math>\scriptsize (d, n)</math> | |||
| Integer | |||
| Matrix | |||
| | |||
| [[...โฉ ...] | |||
| | |||
| <math>\textbf{c}_i</math> | |||
| | |||
| <math>\mathrm{c}_{ij}</math> | |||
| Jargon name: monzo list | |||
|- | |||
| | |||
| <math>\textbf{c}</math> | |||
| [[Comma]] | |||
| | |||
| <math>\small ๐ฝ</math> | |||
| Primes | |||
| | |||
| <math>\scriptsize (d, 1)</math> | |||
| Integer | |||
| Vector | |||
| | |||
| [...โฉ | |||
| | |||
| | |||
| | |||
| <math>\mathrm{c}_i</math> | |||
| Specific type: vector ([[prime-count vector]] or PC-vector) | |||
|- | |||
! colspan="17" | Computation | |||
|- | |||
| | |||
| {{llzigzag}}<math>\,ยท\,</math>{{rrzigzag}}<math>_p</math> | |||
| [[Power sum]] (<math>p</math>-sum) | |||
| | |||
| | |||
| | |||
| | |||
| <math>\scriptsize (1, 1)</math> | |||
| Real | |||
| Scalar | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
! colspan="17" | All-interval tuning schemes | |||
|- | |- | ||
|<math> | | <math>\mathrm{I}</math> | ||
|<math> | | <math>\mathrm{T}_{\text{p}}</math> | ||
|[[target-interval | | [[Prime proxy target-interval list]] | ||
| | | ย | ||
| | | <math>\small ๐ฝ</math> | ||
|<math>\scriptsize ( | | Primes | ||
| | | ย | ||
| | | <math>\scriptsize (d, d)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | ย | ||
| | | โจ[...โฉ ...] | ||
|<math> | | ย | ||
| | | ย | ||
| | | <math>\slant{\mathbf{1}}</math> | ||
| ย | |||
| ย | |||
|- | |- | ||
|<math> | | ย | ||
|<math>C</math> | | <math>X</math> | ||
| | | {{subpage|alternative complexities|uprev|s=Prescaling_vs._pretransforming|text=Complexity pretransformer}} | ||
| | | <math>\small\mathsf{๐}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{๐}\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> | |||
| Complexity weight or <alternative>-complexity weight | |||
| | | ย | ||
| <math>\scriptsize (d, d)</math> or <math>\scriptsize (d+1, d+1)</math> | |||
| | | Real | ||
|[ | | Matrix | ||
| | | [โจ...] ...โฉ | ||
| | | | ||
|<math> | | <math>๐_i</math> | ||
|<math> | | ย | ||
| | | <math>๐</math> | ||
| <math>x_i</math> or <math>x_{ij}</math> | |||
| ย | |||
|- | |- | ||
|<math> | | <math>\text{diag}({\large\textbf{๐}}\hspace{2mu})</math> | ||
|<math> | | <math>L</math> | ||
|[[ | | [[Log-prime matrix]] | ||
| | | ย | ||
| | | <math>\small\mathsf{oct}</math>/<math>\small ๐ฝ</math> | ||
|<math>\scriptsize ( | | Octaves per prime | ||
| | | ย | ||
| | | <math>\scriptsize (d, d)</math> | ||
| | | Real | ||
| | | Matrix | ||
| | | [โจ...] ...โฉ | ||
| | | โจ[...โฉ ...] | ||
|<math> | | <math>{\large\textbf{๐}}\hspace{2mu}_i</math> | ||
|<math> | | ย | ||
| | | <math>{\large\textbf{๐}}\hspace{2mu}</math> | ||
| <math>{\large ๐}\hspace{2mu}_{ij}</math> | |||
| ย | |||
|- | |- | ||
|<math>| | | | ||
| <math>q</math> | |||
| | | {{subpage|all-interval_tuning_schemes|uprev|s=dual_norms|text=Interval complexity norm power}} | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, | | ย | ||
| | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| ย | |||
|- | |- | ||
| | | ย | ||
|<math> | | <math>\norm{ยท}_q</math> | ||
|[[ | | [[Power norm]] (<math>p</math>-norm) | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| <math>\dfrac1{1-\frac1q}</math> | |||
| <math>\text{dual}(q)</math> | |||
| {{subpage|all-interval tuning schemes|uprev|s=dual_norms|text=Dual norm power}} | |||
| | |||
| | |||
| | |||
| | |||
| <math>\scriptsize (1, 1)</math> | |||
| Real | |||
| Scalar | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| ย | |||
|- | |- | ||
| | | ย | ||
|<math>\ | | <math>\norm{X\mathbf{i}}_q</math> | ||
|[[ | | [[Interval complexity]] | ||
| | | ย | ||
|<math>\small | | <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math> | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>\norm{๐X^{-1}}_{\text{dual}(q)}</math> | ||
|[[ | | [[Retuning magnitude]] | ||
| | | ย | ||
| | | <math>\mathsf{ยข}\small\mathsf{(C^{-1})}</math> or <math>\mathsf{ยข}\small\mathsf{(}</math><alt>-<math>\small\mathsf{C^{-1})}</math> | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Real | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
! colspan="17" | | ! colspan="17" | Alternative complexities | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>๐</math> | ||
| | | {{subpage|alternative complexities|uprev|s=formulas|text=Prime list}}<ref group="note">May be used for a prime-limit or for any prime-only list.</ref> | ||
| | | ย | ||
|< | | | ||
| | | | ||
| | | ย | ||
|<math>\scriptsize (d | | <math>\scriptsize (1, d)</math> | ||
| | | Integer | ||
| List | |||
| | | [...] | ||
| | | ย | ||
| | | ย | ||
| | | | ||
| | | ย | ||
|<math> | | <math>p_i</math> | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math>\ | | <math>\slant{\mathbf{1}}</math> | ||
| | | {{subpage|alternative complexities|uprev|s=proportionality to size|text=Summation map}} | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (d | | <math>\scriptsize (1, d)</math> | ||
| | | Integer | ||
| | | Vector | ||
| | | โจ...] | ||
| | |||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math> | | <math>1</math> | ||
| | | ย | ||
|- | |- | ||
| | |||
| <math>1200</math> | |||
| {{subpage|alternative complexities|uprev|s=Proportionality to size|text=Octaves-to-cents conversion}} | |||
| | |||
| ยข/oct | |||
| Cents per octave | |||
| | |||
| <math>\scriptsize (1, 1)</math> | |||
| Integer | |||
| Scalar | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| ย | |||
|- | |- | ||
| | | ย | ||
|<math> | | <math>Z</math> | ||
| | | {{subpage|alternative complexities|uprev|s=Normifying: size-sensitizing matrix|text=Size-sensitizing matrix}} | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (d, | | <math>\scriptsize (d+1, d)</math> | ||
| | | Real | ||
| | | Matrix | ||
|[ | | [โจโฆ]...] | ||
| | | ย | ||
|<math> | | <math>๐_i</math> | ||
| | | ย | ||
| | | ย | ||
|<math> | | <math>z_{ij}</math> | ||
| | | ย | ||
|- | |- | ||
| | ! colspan="17" | Non-standard domain bases | ||
|- | |- | ||
| | | rowspan="2" | ย | ||
|<math> | | <math>B_s</math> | ||
|[[ | | rowspan="2" | [[Domain_basis#Basis_matrix_conversion|(Domain) basis (change) matrix]] | ||
| rowspan="2" | | |||
| | | <math>\small ๐ฝ</math>/<math>\small ๐ฏ</math> | ||
| | | Primes per nonprime basis elements | ||
| | | rowspan="2" | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (d_p, d_b)</math> | ||
| | | rowspan="2" | Integer | ||
| | | rowspan="2" | Matrix | ||
| | | rowspan="2" | [[...] ...] | ||
| | | rowspan="2" | [[...] ...] | ||
| | | rowspan="2" | ย | ||
| | | rowspan="2" | <math>b_i</math> | ||
| | | rowspan="2" | ย | ||
| | | rowspan="2" | <math>b_{ij}</math> | ||
| | | rowspan="2" | ย | ||
|- | |- | ||
| <math>B_{Ls}</math> | |||
| <math>\small ๐</math>/<math>\small ๐ฏ</math> | |||
| Superspace basis elements per (subspace) basis elements | |||
| <math>\scriptsize (d_L, d_s)</math> | |||
|- | |- | ||
|<math>G_cF^{-1}FM_c | ! colspan="17" | Embedding and projection | ||
\mathrm{V}\textit{ฮ}\mathrm{V}^{-1}</math> | |- | ||
|<math>P</math> | | | ||
|[[Projection matrix| | | <math>G</math> | ||
|<math>\scriptsize ย | | [[generator embedding matrix|Generator embedding (matrix)]] | ||
| | |||
| <math>\small ๐ฝ</math>/<math>\small ๐ด</math> | |||
| Primes per generator | |||
| | |||
| <math>\scriptsize (d, r)</math> | |||
| Real | |||
| Matrix | |||
| [{...] ...โฉ | |||
| {[...โฉ ...] | |||
| <math>๐_i</math> | |||
| | |||
| | |||
| <math>g_{ij}</math> | |||
| | |||
|- | |||
| <math>G_cF^{-1}FM_c</math><br /> | |||
<math>\mathrm{V}\textit{ฮ}\mathrm{V}^{-1}</math> | |||
| <math>P</math> | |||
| [[Projection matrix|Projection (matrix)]] | |||
| <math>\scriptsize ย | |||
\begin{array} {c} G \\[-2pt] ๐ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐ด} \end{array} | \begin{array} {c} G \\[-2pt] ๐ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐ด} \end{array} | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
\begin{array} {c} M \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} ๐ฝ \end{array} | \begin{array} {c} M \\[-2pt] \cancel{๐ด} \hspace{-2mu} / \hspace{-2mu} ๐ฝ \end{array} | ||
</math> | </math> | ||
|<math>\small ๐ฝ</math>/<math>\small ๐ฝ</math> | | <math>\small ๐ฝ</math>/<math>\small ๐ฝ</math> | ||
| | | Primes per prime | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} G \\[-3pt] (d, \cancel{r}) \end{array} | \begin{array} {c} G \\[-3pt] \left(d, \cancel{r}\right) \end{array} | ||
\!\! | \! \! ย | ||
\begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array} | \begin{array} {c} M \\[-3pt] \left(\cancel{r}, d\right) \end{array} | ||
\!\! | \! \! ย | ||
</math> | </math> | ||
|<math>\scriptsize (d, d)</math> | | <math>\scriptsize (d, d)</math> | ||
| | | Real | ||
| | | Matrix | ||
|[โจ...] ...โฉ | | [โจ...] ...โฉ | ||
|โจ[...โฉ ...] | | โจ[...โฉ ...] | ||
|<math>๐_i</math> | | <math>๐_i</math> | ||
| | | ย | ||
| | | ย | ||
|<math>p_i</math> | | <math>p_i</math> | ||
| | | ย | ||
|- | |- | ||
|<math>GM\textbf{i}</math> | | <math>GM\textbf{i}</math> | ||
|<math>P\textbf{i}</math> | | <math>P\textbf{i}</math> | ||
|[[ | | [[Projected interval]] | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\begin{array} {c} G \\[-2pt] ๐ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐ด} \end{array} | \begin{array} {c} G \\[-2pt] ๐ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐ด} \end{array} | ||
\begin{array} {c} \\[-2pt] ยท \end{array} | \begin{array} {c} \\[-2pt] ยท \end{array} | ||
| Line 3,217: | Line 3,295: | ||
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} | \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐ฝ} \end{array} | ||
</math> | </math> | ||
|<math>\small ๐ฝ</math> | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
|<math>\scriptsize ย | | <math>\scriptsize ย | ||
\!\! | \! \! ย | ||
\begin{array} {c} G \\[-3pt] (d, \cancel{r}) \end{array} | \begin{array} {c} G \\[-3pt] \left(d, \cancel{r}\right) \end{array} | ||
\!\! | \! \! ย | ||
\begin{array} {c} M \\[-3pt] (\cancel{r}, \cancel{d}) \end{array} | \begin{array} {c} M \\[-3pt] \left(\cancel{r}, \cancel{d}\right) \end{array} | ||
\!\! | \! \! ย | ||
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1 | \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} | ||
\! \! ย | |||
\!\! | |||
</math> | </math> | ||
|<math>\scriptsize (1 | | <math>\scriptsize (d, 1)</math> | ||
| | | Real | ||
| | | Vector | ||
| | | ย | ||
| | | [...โฉ | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| Specific type: vector ([[prime-count vector]] or PC-vector) | |||
|- | |- | ||
| ย | |||
| <math>\mathrm{U}</math> | |||
| [[Unchanged-interval basis]] | |||
| | |||
| <math>\small ๐ฝ</math> | |||
| Primes | |||
| | |||
| <math>\scriptsize (d, r)</math> | |||
| | |||
| Matrix | |||
| | |||
| [[...โฉ ...] | |||
| | |||
| <math>\textbf{u}_i</math> | |||
| | |||
| <math>\mathrm{u}_{ij}</math> | |||
| Jargon name: eigenmonzo list | |||
|- | |- | ||
| | | ย | ||
|<math>\ | | <math>\textit{ฮ}</math> | ||
|[[ | | [[scaling factor matrix|Scaling factor (eigenvalue) matrix]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (d, d)</math> | | <math>\scriptsize (d, d)</math> | ||
| | | ย | ||
| | | Matrix | ||
| | | [โจโฆ] โฆโฉ | ||
|โจ[ | | โจ[โฆโฉ โฆ] | ||
| | | ย | ||
| | | ย | ||
|<math> | | <math>๐</math> | ||
| | | <math>ฮป_i</math> | ||
| | | Mnemonic: <math>\mathrm{V}</math> is mirrored of <math>\textit{ฮ}</math> which it combines with to create the projection matrix; previous name: eigenvalue matrix | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>\mathrm{V}</math> | ||
|[[ | | [[unrotated vector list|Unrotated vector (eigenvector) list]] | ||
| | | ย | ||
|<math>\small | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | ย | ||
|<math>\scriptsize (d, d)</math> | | <math>\scriptsize (d, d)</math> | ||
| | | ย | ||
| | | Matrix | ||
|[ | | | ||
| | | โจ[...โฉ ...] | ||
|<math> | | ย | ||
| | | <math>\textbf{v}_i</math> | ||
|<math> | | ย | ||
|<math> | | <math>\mathrm{v}_{ij}</math> | ||
| Mnemonic: <math>\mathrm{V}</math> is mirrored of <math>\textit{ฮ}</math> which it combines with to create the projection matrix; jargon name: eigenmonzo and comma list | |||
|- | |- | ||
| | | ย | ||
|<math> | | <math>F</math> | ||
|[[ | | [[Generator form matrix]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (r, r)</math> | ||
| | | ย | ||
| | | Matrix | ||
| | | [{...] โฆ} | ||
| ย | |||
| | | ย | ||
| | | <math>๐_i</math> | ||
|<math> | | | ||
|<math> | | <math>f_{ij}</math> | ||
| | | ย | ||
|- | |- | ||
|<math> | | <math>I</math> | ||
|<math> | | <math>M_{\text{j}}</math> | ||
|[[ | | [[Generator_embedding_optimization#Algebraic_setup|JI mapping (matrix)]] | ||
| | | ย | ||
|<math>\small | | <math>\small ๐ด</math>/<math>\small ๐ฝ</math> | ||
| | | Generators per prime | ||
| | | ย | ||
|<math>\scriptsize (d, d)</math> | | <math>\scriptsize (d, d)</math> | ||
| | | Integer | ||
| | | Matrix | ||
|[โจ...] ... | | [โจ...] ...} | ||
|โจ[... | | โจ[...} ...] | ||
| | | ย | ||
| | | ย | ||
|<math>\ | | <math>\slant{\mathbf{1}}</math> | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| | | <math>I</math> | ||
|<math> | | <math>G_{\text{j}}</math> | ||
|[[ | | [[Generator_embedding_optimization#Algebraic_setup|JI generator embedding (matrix)]] | ||
| | | ย | ||
| | | <math>\small ๐ฝ</math>/<math>\small ๐ด</math> | ||
| | | Primes per generator | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (d, d)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | [{...] ...โฉ | ||
| | | {[...โฉ ...] | ||
| | | ย | ||
| | | ย | ||
| | | <math>\slant{\mathbf{1}}</math> | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>K</math> | ||
|[[ | | [[Generator_embedding_optimization#How_to_build_constraint_matrices|Constraint (matrix)]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (k, r)</math> | ||
| | | <math>\scriptsize \{0, +1, -1\}</math> | ||
| | | Matrix | ||
| | | [[...] ...] | ||
| | | ย | ||
| | | <math>๐_i</math> | ||
| | | ย | ||
| | | ย | ||
| | | <math>k_{ij}</math> | ||
| | | Mnemonic: <math>K</math>onstraint | ||
|- | |- | ||
| ย | |||
| <math>๐</math> | |||
| [[Generator embedding optimization#Generalizing to higher dimensions: The blend map|(Generator tuning map) blend map]] | |||
| | |||
| | |||
| | |||
| | |||
| <math>\scriptsize (1, ฯ-1)</math> | |||
| Real | |||
| Vector | |||
| [...] | |||
| | |||
| | |||
| | |||
| | |||
| <math>b_i</math> | |||
| | |||
|- | |- | ||
| | | ย | ||
|<math> | | <math>B</math> | ||
|[[ | | [[Generator embedding optimization#How to identify tunings|(Generator tuning map) blend matrix]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1 | | <math>\scriptsize (d, ฯ-1)</math> | ||
| | | Real | ||
| | | Matrix | ||
|[...] | | [[...โฉ...] | ||
| | | ย | ||
| | | ย | ||
| | | <math>๐_{i}</math> | ||
| | | ย | ||
|<math> | | <math>b_{ij}</math> | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>D</math> | ||
|[[ | | [[Generator embedding optimization#The deltas matrix|(Generator tuning map) deltas matrix]] | ||
| | | ย | ||
| | | <math>\mathsf{ยข}</math>/<math>\small ๐ด</math> | ||
| | | Cents per generator | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (ฯ-1,r)</math> | ||
| | | Real | ||
| | | Matrix | ||
|[ | | [{...] ...] | ||
| | | ย | ||
|<math> | | <math>๐น_i</math> | ||
| | | ย | ||
| | | ย | ||
|<math> | | <math>๐ฟ_{ij}</math> | ||
| | | ย | ||
|- | |- | ||
| ย | |||
| <math>ฯ</math> | |||
| [[Generator embedding optimization#The deltas matrix|Tied basic minimax tuning count]] | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| Integer | |||
| Scalar | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |- | ||
! colspan="17" | Exterior algebra | |||
|- | |- | ||
|<math> | | | ||
|<math>\small | | <math>๐</math> | ||
| | | [[Multimap]] | ||
|<math>\scriptsize ( | | | ||
| <math>\small ๐ด</math>/<math>\small ๐ฝ</math> | |||
| Generators per prime | |||
| | |||
| <math>\scriptsize (1, d)</math> | |||
| Integer | |||
| Multivector | |||
| โจ...] or โจโจ...]] or โจโจโจ...]]] ... | |||
| | |||
| | |||
| | |||
| | |||
| <math>๐_i</math> | |||
| | |||
|- | |- | ||
| | | ย | ||
|<math> | | <math>๐</math> | ||
|[[ | | [[Multicomma]] | ||
| | | ย | ||
|<math>\small ๐ฝ | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize (1, n)</math> | ||
| | | Integer | ||
| | | Multivector | ||
| | | ย | ||
| | | [...โฉ or [[...โฉโฉ or [[[...โฉโฉโฉ ... | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math> | | <math>๐_i</math> | ||
| | | ย | ||
|- | |- | ||
| ย | |||
| <math>๐ง</math> | |||
| (Generic temperament multivector) | |||
| | |||
| | |||
| | |||
| | |||
| <math>\scriptsize (1, {{d}\choose{r}})</math> or <math>\scriptsize (1, {{d}\choose{n}})</math> | |||
| Integer | |||
| Multivector | |||
| โจ...] or โจโจ...]] or โจโจโจ...]]] ... | |||
| [...โฉ or [[...โฉโฉ or [[[...โฉโฉโฉ ... | |||
| | |||
| | |||
| | |||
| <math>๐ง_i</math> | |||
| | |||
|- | |- | ||
| | | ย | ||
|<math> | | <math>A</math> | ||
| | | (Generic temperament matrix) | ||
| | | | ||
|<math>\ | | | ||
| | |||
| ย | |||
| <math>\scriptsize (g, d)</math> or <math>\scriptsize (d, g)</math> | |||
| | | Integer | ||
| | | Matrix | ||
|โจ...] | | [โจ...] ...} | ||
| | | โจ[...} ...] or [[...โฉ ...] | ||
| | | <math>๐_i</math> | ||
| | | <math>๐_i</math> | ||
| <math>๐</math> | |||
|<math> | | <math>a_{ij}</math> | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>v</math> | ||
|[[ | | [[Variance]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| | | ย | ||
|<math> | | <math>g</math> | ||
|[[ | | [[Grade]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| | ! colspan="17" | Temperament addition | ||
|- | |- | ||
| | | <math>\min(r, n)</math> | ||
|<math> | | <math>g_\text{min}</math> | ||
|[[ | | [[Temperament_addition#Introductory_examples|Min-grade]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| | | <math>\max(r, n)</math> | ||
|<math> | | <math>g_\text{max}</math> | ||
|[[grade]] | | [[Temperament_addition#Introductory_examples|Max-grade]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| ย | |||
| <math>L_\text{dep}</math> | |||
| [[Temperament_addition#1._Find_the_.5Bmath.5DL_.7B.5Ctext.7Bdep.7D.7D.5B.2Fmath.5D|Linear-dependence basis]] | |||
| | |||
| | |||
| | |||
| | |||
| <math>\scriptsize \left(l_\text{dep}, d\right)</math> or <math>\scriptsize \left(d, l_\text{dep}\right)</math> | |||
| Integer | |||
| Matrix | |||
| [โจ...]] or [[...] ...โฉ | |||
| โจ[...]] or [[...โฉ ...] | |||
| <math>{\large\textbf{๐}}\hspace{2mu}_{\text{dep}i}</math> | |||
| <math>{\large\textbf{๐}}\hspace{2mu}_{\text{dep}i}</math> | |||
| <math>{\large\textbf{๐}}\hspace{2mu}_\text{dep}</math> | |||
| <math>{\large ๐}\hspace{2mu}_{\text{dep}ij}</math> | |||
| | |||
|- | |- | ||
| | | ย | ||
|<math> | | <math>L_\text{ind}</math> | ||
|[[ | | [[Temperament_addition#Glossary|Linear-independence basis]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize ( | | <math>\scriptsize \left(l_\text{ind}, d\right)</math> or <math>\scriptsize \left(d, l_\text{ind}\right)</math> | ||
| | | Integer | ||
| | | Matrix | ||
| | | [โจ...]] or [[...] ...โฉ | ||
| | | โจ[...]] or [[...โฉ ...] | ||
| | | <math>{\large\textbf{๐}}\hspace{2mu}_{\text{ind}i}</math> | ||
| | | <math>{\large\textbf{๐}}\hspace{2mu}_{\text{ind}i}</math> | ||
| | | <math>{\large\textbf{๐}}\hspace{2mu}_\text{ind}</math> | ||
| | | <math>{\large ๐}\hspace{2mu}_{\text{ind}ij}</math> | ||
| | | ย | ||
|- | |- | ||
|<math>\ | | <math>\dim(L_\text{dep})</math> | ||
|<math> | | <math>l_\text{dep}</math> | ||
|[[ | | [[Temperament_addition#3._Linear_independence_between_temperaments|Linear-dependence]] | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|<math>\scriptsize (1, 1)</math> | | <math>\scriptsize (1, 1)</math> | ||
| | | Integer | ||
| | | Scalar | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| <math>\dim(L_\text{ind})</math> | |||
|<math> | | <math>l_\text{ind}</math> | ||
| [[Temperament_addition#3._Linear_independence_between_temperaments|Linear-independence]] | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| <math>\scriptsize (1, 1)</math> | |||
| Integer | |||
| Scalar | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| ย | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|<math>\scriptsize (1, 1)</math> | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|} | |} | ||
===Units=== | === Units === | ||
ย | |||
{| class="wikitable center-all mw-collapsible mw-collapsed" | {| class="wikitable center-all mw-collapsible mw-collapsed" | ||
|+ | |+ style="font-size: 105%;" | | ||
! | |- | ||
! | ! Symbol | ||
! | ! Name | ||
! Vectorized | |||
|- | |- | ||
|<math>\small ๐ด</math> | | <math>\small ๐ด</math> | ||
| | | Generators | ||
| | | Yes | ||
|- | |- | ||
|<math>\small ๐ฝ</math> | | <math>\small ๐ฝ</math> | ||
| | | Primes | ||
| | | Yes | ||
|- | |- | ||
|<math>\small ๐ฏ</math> | | <math>\small ๐ฏ</math> | ||
|( | | (Subspace) basis elements | ||
| | | Yes | ||
|- | |- | ||
|<math>\small ๐</math> | | <math>\small ๐</math> | ||
| | | Superspace basis elements | ||
| | | Yes | ||
|- | |- | ||
|<math>\mathsf{ยข}</math> | | <math>\mathsf{ยข}</math> | ||
| | | Cents | ||
| | | ย | ||
|- | |- | ||
|<math>\mathsf{ยข}\small{(}</math><weight><math>\small\mathsf{)}</math> | | <math>\mathsf{ยข}\small{(}</math><weight><math>\small\mathsf{)}</math> | ||
| | | Weighted cents | ||
| | | ย | ||
|- | |- | ||
|<math>\small\mathsf{oct}</math> | | <math>\small\mathsf{oct}</math> | ||
| | | Octaves | ||
| | | ย | ||
|} | |} | ||
===Tuning schemes=== | === Tuning schemes === | ||
ย | |||
{| class="wikitable center-all mw-collapsible mw-collapsed" | {| class="wikitable center-all mw-collapsible mw-collapsed" | ||
|+ | |+ style="font-size: 105%;" | | ||
! colspan="6" rowspan="3" | | |- | ||
! colspan="12" rowspan="1" | | ! colspan="6" rowspan="3" | Retuning (or mistuning) magnitude | ||
! rowspan="5" | | ! colspan="12" rowspan="1" | Damage | ||
ย | ! rowspan="5" | Target<br />intervals | ||
intervals | ! colspan="2" rowspan="4" | Systematic name | ||
! colspan="2" rowspan="4" | | ! rowspan="5" | Previously named tuning schemes that are specific types of this tuning scheme | ||
! rowspan="5" | | ! rowspan="5" | Of interest? | ||
! rowspan="5" | | |||
|- | |- | ||
! colspan="9" rowspan="1" | | ! colspan="9" rowspan="1" | Weight | ||
! colspan="3" rowspan="1" | | ! colspan="3" rowspan="1" | Optimization | ||
|- | |- | ||
! colspan="6" rowspan="1" | | ! colspan="6" rowspan="1" | Interval complexity | ||
! colspan="3" rowspan="1" | | ! colspan="3" rowspan="1" | Slope | ||
! colspan="1" rowspan="3" | | ! colspan="1" rowspan="3" | Initial | ||
! colspan="1" rowspan="3" | | ! colspan="1" rowspan="3" | Name | ||
! colspan="1" rowspan="3" | | ! colspan="1" rowspan="3" | Power | ||
|- | |- | ||
! colspan="3" rowspan="1" | | ! colspan="3" rowspan="1" | Norm pretransformer | ||
! colspan="3" rowspan="1" | | ! colspan="3" rowspan="1" | Norm power | ||
! colspan="3" rowspan="1" | | ! colspan="3" rowspan="1" | Norm pretransformer | ||
! colspan="3" rowspan="1" | | ! colspan="3" rowspan="1" | Norm power | ||
! colspan="1" rowspan="2" | | ! colspan="1" rowspan="2" | Initial | ||
! colspan="1" rowspan="2" | | ! colspan="1" rowspan="2" | Name | ||
! colspan="1" rowspan="2" | | ! colspan="1" rowspan="2" | Multiplier | ||
|- | |- | ||
! | ! Initial | ||
! | ! Name | ||
! | ! Multiplier | ||
! | ! Initial | ||
! | ! Name | ||
! | ! Power | ||
! | ! Initial | ||
! | ! Name | ||
! | ! Multiplier | ||
! | ! Initial | ||
! | ! Name | ||
! | ! Power | ||
! colspan="1" | | ! colspan="1" | Abbreviated | ||
! colspan="1" | | ! colspan="1" | Read ("____ tuning scheme") | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="4" |<n/a> | | rowspan="4" | <n/a> | ||
| rowspan="2" | | | rowspan="2" | Maximum | ||
| rowspan="2" | | | rowspan="2" | ∞ | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |(t) | | rowspan="2" | (t) | ||
| rowspan="2" | | | rowspan="2" | Taxicab | ||
| rowspan="2" |1 | | rowspan="2" | 1 | ||
| rowspan="4" |S | | rowspan="4" | S | ||
| rowspan="4" | | | rowspan="4" | Simplicity-weight | ||
| rowspan="4" |1/ | | rowspan="4" | 1/Complexity | ||
| rowspan="31" |<n/a> | | rowspan="31" | <n/a> | ||
| rowspan="13" | | | rowspan="13" | Minimax | ||
| rowspan="13" | | | rowspan="13" | ∞ | ||
| rowspan="4" | | | rowspan="4" | All | ||
| | | Minimax-S | ||
| | | Minimax simplicity-weight damage | ||
|"[[TOP]]"/"[[T1]]"/"[[TIPTOP]]"*, "[[CTOP]]", "[[POTOP]]"/"[[POTT]]"* | | "[[TOP]]"/"[[T1]]"/"[[TIPTOP]]"*, "[[CTOP]]", "[[POTOP]]"/"[[POTT]]"* | ||
|yes | | yes | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
| | | Minimax-<alt>-S | ||
| | | Minimax <alternative>-simplicity-weight damage | ||
|"[[BOP tuning|BOP]]", "[[Weil Norms, Tenney-Weil Norms, and TWp Interval and Tuning Space|Weil]]", "[[Kees]]" | | "[[BOP tuning|BOP]]", "[[Weil Norms, Tenney-Weil Norms, and TWp Interval and Tuning Space|Weil]]", "[[Kees]]" | ||
|yes | | yes | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |Euclidean | | rowspan="2" | Euclidean | ||
| rowspan="2" |2 | | rowspan="2" | 2 | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |E | | rowspan="2" | E | ||
| rowspan="2" |Euclidean | | rowspan="2" | Euclidean | ||
| rowspan="2" |2 | | rowspan="2" | 2 | ||
| | | Minimax-ES | ||
| | | Minimax Euclideanized-simplicity-weight damage | ||
|"[[Tenney-Euclidean tuning|TE]]"/"[[T2]]"/"[[TOP-RMS]]", "[[CTE tuning|CTE]]", "[[POTE tuning|POTE]]" | | "[[Tenney-Euclidean tuning|TE]]"/"[[T2]]"/"[[TOP-RMS]]", "[[CTE tuning|CTE]]", "[[POTE tuning|POTE]]" | ||
|yes | | yes | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
| | | Minimax-E-<alt>-S | ||
| | | Minimax Euclideanized-<alternative>-simplicity-weight damage | ||
|"[[Frobenius]]", "[[BE]]", "[[WE]]", "[[KE]]" | | "[[Frobenius]]", "[[BE]]", "[[WE]]", "[[KE]]" | ||
| | | Yes | ||
|- | |- | ||
| colspan="6" rowspan="27" |<n/a> | | colspan="6" rowspan="27" | <n/a> | ||
| colspan="6" |<n/a> | | colspan="6" | <n/a> | ||
|U | | U | ||
| | | Unity-weight | ||
|<none> | | <none> | ||
| rowspan="27" |<set> | | rowspan="27" | <set> | ||
|<set> | | <set> Minimax-U | ||
|<set> | | <set> Minimax unity-weight damage | ||
|"[[Minimax tuning| | | "[[Minimax tuning|Minimax]]" | ||
|yes | | yes | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |(t) | | rowspan="2" | (t) | ||
| rowspan="2" | | | rowspan="2" | Taxicab | ||
| rowspan="2" |1 | | rowspan="2" | 1 | ||
| rowspan="4" |S | | rowspan="4" | S | ||
| rowspan="4" | | | rowspan="4" | Simplicity-weight | ||
| rowspan="4" |1/ | | rowspan="4" | 1/Complexity | ||
|<set> | | <set> Minimax-S | ||
|<set> | | <set> Minimax simplicity-weight damage | ||
| | | ย | ||
| | | Yes | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
|<set> | | <set> Minimax-<alt>-S | ||
|<set> | | <set> Minimax <alternative>-simplicity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |E | | rowspan="2" | E | ||
| rowspan="2" |Euclidean | | rowspan="2" | Euclidean | ||
| rowspan="2" |2 | | rowspan="2" | 2 | ||
|<set> | | <set> Minimax-ES | ||
|<set> | | <set> Minimax Euclideanized-simplicity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
|<set> | | <set> Minimax-E-<alt>-S | ||
|<set> | | <set> Minimax Euclideanized-<alternative>-simplicity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |(t) | | rowspan="2" | (t) | ||
| rowspan="2" | | | rowspan="2" | Taxicab | ||
| rowspan="2" |1 | | rowspan="2" | 1 | ||
| rowspan="4" |C | | rowspan="4" | C | ||
| rowspan="4" | | | rowspan="4" | Complexity-weight | ||
| rowspan="4" | | | rowspan="4" | Complexity | ||
|<set> | | <set> Cinimax-C | ||
|<set> | | <set> Cinimax complexity-weight damage | ||
| | | ย | ||
| | | Yes | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
|<set> | | <set> Minimax-<alt>-C | ||
|<set> | | <set> Minimax <alternative>-complexity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |E | | rowspan="2" | E | ||
| rowspan="2" |Euclidean | | rowspan="2" | Euclidean | ||
| rowspan="2" |2 | | rowspan="2" | 2 | ||
|<set> | | <set> Minimax-EC | ||
|<set> | | <set> Minimax Euclideanized-complexity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
|<set> | | <set> Minimax-E-<alt>-C | ||
|<set> | | <set> Minimax Euclideanized-<alternative>-complexity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="6" |<n/a> | | colspan="6" | <n/a> | ||
|U | | U | ||
| | | Unity-weight | ||
|<none> | | <none> | ||
| rowspan="9" | | | rowspan="9" | MiniRMS | ||
| rowspan="9" |2 | | rowspan="9" | 2 | ||
|<set> | | <set> MiniRMS-U | ||
|<set> | | <set> MiniRMS unity-weight damage | ||
|"[[ | | "[[Least squares]]" | ||
|yes | | yes | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |(t) | | rowspan="2" | (t) | ||
| rowspan="2" | | | rowspan="2" | Taxicab | ||
| rowspan="2" |1 | | rowspan="2" | 1 | ||
| rowspan="4" |S | | rowspan="4" | S | ||
| rowspan="4" | | | rowspan="4" | Simplicity-weight | ||
| rowspan="4" |1/ | | rowspan="4" | 1/Complexity | ||
|<set> | | <set> MiniRMS-S | ||
|<set> | | <set> MiniRMS simplicity-weight damage | ||
| | | ย | ||
| | | Yes | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
|<set> | | <set> MiniRMS-<alt>-S | ||
|<set> | | <set> MiniRMS <alternative>-simplicity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |E | | rowspan="2" | E | ||
| rowspan="2" |Euclidean | | rowspan="2" | Euclidean | ||
| rowspan="2" |2 | | rowspan="2" | 2 | ||
|<set> | | <set> MiniRMS-ES | ||
|<set> | | <set> MiniRMS Euclideanized-simplicity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
|<set> | | <set> MiniRMS-E-<alt>-S | ||
|<set> | | <set> MiniRMS Euclideanized-<alternative>-simplicity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |(t) | | rowspan="2" | (t) | ||
| rowspan="2" | | | rowspan="2" | Taxicab | ||
| rowspan="2" |1 | | rowspan="2" | 1 | ||
| rowspan="4" |C | | rowspan="4" | C | ||
| rowspan="4" | | | rowspan="4" | Complexity-weight | ||
| rowspan="4" | | | rowspan="4" | Complexity | ||
|<set> | | <set> MiniRMS-C | ||
|<set> | | <set> MiniRMS complexity-weight damage | ||
| | | ย | ||
|yes | | yes | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
|<set> | | <set> MiniRMS-<alt>-C | ||
|<set> | | <set> MiniRMS <alternative>-complexity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |E | | rowspan="2" | E | ||
| rowspan="2" |Euclidean | | rowspan="2" | Euclidean | ||
| rowspan="2" |2 | | rowspan="2" | 2 | ||
|<set> | | <set> MiniRMS-EC | ||
|<set> | | <set> MiniRMS Euclideanized-complexity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
|<set> | | <set> MiniRMS-E-<alt>-C | ||
|<set> | | <set> MiniRMS Euclideanized-<alternative>-complexity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="6" |<n/a> | | colspan="6" | <n/a> | ||
|U | | U | ||
| | | Unity-weight | ||
|<none> | | <none> | ||
| rowspan="9" | | | rowspan="9" | Miniaverage | ||
| rowspan="9" |1 | | rowspan="9" | 1 | ||
|<set> | | <set> Miniaverage-U | ||
|<set> | | <set> Miniaverage unity-weight damage | ||
| | | ย | ||
|yes | | yes | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |(t) | | rowspan="2" | (t) | ||
| rowspan="2" | | | rowspan="2" | Taxicab | ||
| rowspan="2" |1 | | rowspan="2" | 1 | ||
| rowspan="4" |S | | rowspan="4" | S | ||
| rowspan="4" | | | rowspan="4" | Simplicity-weight | ||
| rowspan="4" |1/ | | rowspan="4" | 1/Complexity | ||
|<set> | | <set> Miniaverage-S | ||
|<set> | | <set> Miniaverage simplicity-weight damage | ||
| | | ย | ||
| | | Yes | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
|<set> | | <set> Miniaverage-<alt>-S | ||
|<set> | | <set> Miniaverage <alternative>-simplicity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |E | | rowspan="2" | E | ||
| rowspan="2" |Euclidean | | rowspan="2" | Euclidean | ||
| rowspan="2" |2 | | rowspan="2" | 2 | ||
|<set> | | <set> Miniaverage-ES | ||
|<set> | | <set> Miniaverage Euclideanized-simplicity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
|<set> | | <set> Miniaverage-E-<alt>-S | ||
|<set> | | <set> Miniaverage Euclideanized-<alternative>-simplicity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |(t) | | rowspan="2" | (t) | ||
| rowspan="2" | | | rowspan="2" | Taxicab | ||
| rowspan="2" |1 | | rowspan="2" | 1 | ||
| rowspan="4" |C | | rowspan="4" | C | ||
| rowspan="4" | | | rowspan="4" | Complexity-weight | ||
| rowspan="4" | | | rowspan="4" | Complexity | ||
|<set> | | <set> Miniaverage-C | ||
|<set> | | <set> Miniaverage complexity-weight damage | ||
| | | ย | ||
| | | Yes | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
|<set> | | <set> Miniaverage-<alt>-C | ||
|<set> | | <set> Miniaverage <alternative>-complexity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<none> | | colspan="3" | <none> | ||
| rowspan="2" |E | | rowspan="2" | E | ||
| rowspan="2" |Euclidean | | rowspan="2" | Euclidean | ||
| rowspan="2" |2 | | rowspan="2" | 2 | ||
|<set> | | <set> Miniaverage-EC | ||
|<set> | | <set> Miniaverage Euclideanized-complexity-weight damage | ||
| | | ย | ||
| | | ย | ||
|- | |- | ||
| colspan="3" |<various> | | colspan="3" | <various> | ||
|<set> | | <set> Miniaverage-E-<alt>-C | ||
|<set> | | <set> Miniaverage Euclideanized-<alternative>-complexity-weight damage | ||
| | | ย | ||
| | | ย | ||
|} | |} | ||
===Damages=== | === Damages === | ||
ย | |||
{| class="wikitable center-all mw-collapsible mw-collapsed" | {| class="wikitable center-all mw-collapsible mw-collapsed" | ||
|+ | |+ style="font-size: 105%;" | | ||
! colspan="2" | | |- | ||
! colspan="2" | | ! colspan="2" | Quantity | ||
! colspan="2" | Unit | |||
|- | |- | ||
! | ! Abbreviation | ||
! | ! Name | ||
! | ! Symbol | ||
! | ! Name | ||
|- | |- | ||
|U-damage | | U-damage | ||
| | | Unity-weight damage | ||
|<math>\mathsf{ยข}\small\mathsf{(U)}</math> | | <math>\mathsf{ยข}\small\mathsf{(U)}</math> | ||
| | | Unity-weighted cents | ||
|- | |- | ||
|C-damage | | C-damage | ||
| | | Complexity-weight damage | ||
|<math>\mathsf{ยข}\small\mathsf{(C)}</math> | | <math>\mathsf{ยข}\small\mathsf{(C)}</math> | ||
| | | Complexity-weighted cents | ||
|- | |- | ||
|<alt>-C-damage | | <alt>-C-damage | ||
|<alternative>-complexity-weight damage | | <alternative>-complexity-weight damage | ||
|<math>\mathsf{ยข}</math><math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math> | | <math>\mathsf{ยข}</math><math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math> | ||
|<alternative>-complexity-weighted cents | | <alternative>-complexity-weighted cents | ||
|- | |- | ||
|EC-damage | | EC-damage | ||
|Euclideanized-complexity-weight damage | | Euclideanized-complexity-weight damage | ||
|<math>\mathsf{ยข}</math><math>\small\mathsf{(EC)}</math> | | <math>\mathsf{ยข}</math><math>\small\mathsf{(EC)}</math> | ||
|Euclideanized-complexity-weighted cents | | Euclideanized-complexity-weighted cents | ||
|- | |- | ||
|E-<alt>-C-damage | | E-<alt>-C-damage | ||
|Euclideanized-<alternative>-complexity-weight damage | | Euclideanized-<alternative>-complexity-weight damage | ||
|<math>\mathsf{ยข}</math><math>\small\mathsf{(E}</math>-<alt>-<math>\small\mathsf{C)}</math> | | <math>\mathsf{ยข}</math><math>\small\mathsf{(E}</math>-<alt>-<math>\small\mathsf{C)}</math> | ||
|Euclideanized-<alternative>-complexity-weighted cents | | Euclideanized-<alternative>-complexity-weighted cents | ||
|- | |- | ||
|S-damage | | S-damage | ||
| | | Simplicity-weight damage | ||
|<math>\mathsf{ยข}\small\mathsf{(S)}</math> | | <math>\mathsf{ยข}\small\mathsf{(S)}</math> | ||
| | | Simplicity-weighted cents | ||
|- | |- | ||
|<alt>-S-damage | | <alt>-S-damage | ||
|<alternative>-simplicity-weight damage | | <alternative>-simplicity-weight damage | ||
|<math>\mathsf{ยข}</math><math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math> | | <math>\mathsf{ยข}</math><math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math> | ||
|<alternative>-simplicity-weighted cents | | <alternative>-simplicity-weighted cents | ||
|- | |- | ||
|ES-damage | | ES-damage | ||
|Euclideanized-simplicity-weight damage | | Euclideanized-simplicity-weight damage | ||
|<math>\mathsf{ยข}</math><math>\small\mathsf{(ES)}</math> | | <math>\mathsf{ยข}</math><math>\small\mathsf{(ES)}</math> | ||
|Euclideanized-simplicity-weighted cents | | Euclideanized-simplicity-weighted cents | ||
|- | |- | ||
|E-<alt>-S-damage | | E-<alt>-S-damage | ||
|Euclideanized-<alternative>-simplicity-weight damage | | Euclideanized-<alternative>-simplicity-weight damage | ||
|<math>\mathsf{ยข}</math><math>\small\mathsf{(E}</math>-<alt>-<math>\small\mathsf{S)}</math> | | <math>\mathsf{ยข}</math><math>\small\mathsf{(E}</math>-<alt>-<math>\small\mathsf{S)}</math> | ||
|Euclideanized-<alternative>-simplicity-weighted cents | | Euclideanized-<alternative>-simplicity-weighted cents | ||
|} | |} | ||
===Complexity and simplicity=== | === Complexity and simplicity === | ||
ย | |||
{| class="wikitable center-all mw-collapsible mw-collapsed" | {| class="wikitable center-all mw-collapsible mw-collapsed" | ||
|+ | |+ style="font-size: 105%;" | ย | ||
|- | |- | ||
! | ! colspan="2" | Quantity | ||
! | ! colspan="2" | Unit | ||
|- | |- | ||
! Abbreviation | |||
! Name | |||
! Unit | |||
! Name | |||
|- | |- | ||
| | | C | ||
| | | Complexity | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{( | | <math>\small\mathsf{๐}\scriptsize\mathsf{(C)}</math> = <math>\small\mathsf{(C)}</math> | ||
| | | Complexity weight | ||
|- | |- | ||
| | | <alt>-C | ||
| | | <alternative> complexity | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{( | | <math>\small\mathsf{๐}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math> = <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math> | ||
| | | <alternative>-complexity weight | ||
|- | |- | ||
| | | EC | ||
|Euclideanized | | Euclideanized complexity | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{( | | <math>\small\mathsf{๐}\scriptsize\mathsf{(EC)}</math> = <math>\small\mathsf{(EC)}</math> | ||
|Euclideanized | | Euclideanized-complexity weight | ||
|- | |- | ||
| | | E-<alt>-C | ||
| | | Euclideanized-<alternative> complexity | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{( | | <math>\small\mathsf{๐}\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 | ||
|- | |- | ||
| | | S | ||
| | | Simplicity | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{( | | <math>\small\mathsf{๐}\scriptsize\mathsf{(S)}</math> = <math>\small\mathsf{(S)}</math> | ||
| | | Simplicity weight | ||
|- | |- | ||
| | | <alt>-S | ||
| | | <alternative> simplicity | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{( | | <math>\small\mathsf{๐}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{S)}</math> = <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math> | ||
| | | <alternative>-simplicity weight | ||
|- | |- | ||
|E-<alt>-S | | ES | ||
|Euclideanized-<alternative> simplicity | | Euclideanized simplicity | ||
|<math>\small\mathsf{๐}\scriptsize\mathsf{(E}</math>-<alt>-<math>\scriptsize\mathsf{S)}</math> = <math>\small\mathsf{(E}</math>-<alt>-<math>\small\mathsf{S)}</math> | | <math>\small\mathsf{๐}\scriptsize\mathsf{(ES)}</math> = <math>\small\mathsf{(ES)}</math> | ||
|Euclideanized-<alternative>-simplicity weight | | Euclideanized-simplicity weight | ||
|- | |||
| E-<alt>-S | |||
| Euclideanized-<alternative> simplicity | |||
| <math>\small\mathsf{๐}\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 | |||
|} | |} | ||
==WinCompose== | == WinCompose == | ||
ย | Are you tired of every time web-searching for and copy-pasting special characters that you use over and over in RTT discussions, or would like to use if only it were easy, such as โฏ, โญ, ยข, โ, ยฐ, โ, ร, {{inv}}, โฉ, โ, and ฯ? Well, try [http://wincompose.info/ WinCompose]! This tool lets you communicate about these ideas without disrupting your train of thought, by typing these characters with simple and memorable key sequences. These sequences always begin with your chosen Compose-key, which defaults to being your right Alt key. When describing these sequences we represent this key with the symbol โ. So for example, you type {{nowrap|โฏ as <code>โ##</code>|โญ as <code>โbb</code>|ยข as <code>โc/</code>|โ as <code>โv/</code>|ยฐ as <code>โ00</code>|โ as <code>โ-2</code>|ร as <code>โxx</code>|{{inv}} as <code>โ11</code>|โฉ as <code>โ>></code>|โ as <code>โ88</code>|and ฯ as <code>โ8f</code>}}. ย | ||
Are you tired of every time web-searching for and copy-pasting special characters that you use over and over in RTT discussions, or would like to use if only it were easy, such as โฏ, โญ, ยข, โ, ยฐ, โ, ร, | |||
ย | |||
For | For Windows users, install WinCompose then copy-paste the contents of this file: https://dkeenan.com/XCompose.txt into your user sequences (Show sequences → User-defined sequences → Edit). Then save and reload. You can always choose to override or add alternatives to our sequences if you find others to be more intuitive. | ||
For Mac users, we refer you to this repo, which gives tools and instructions for setting up key bindings as compose rules in Mac OS, and even comes pre-packaged with our rules: https://github.com/cmloegcmluin/compose2keybindings | |||
=== Table of noteworthy sequences === | |||
{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ | |+ style="font-size: 105%; white-space: nowrap;" | Dave Keenan & Douglas Blumeyer's compose-key sequences | ||
! scope="col" | |- style="white-space: nowrap;" | ||
! scope="col" | ! scope="col" style="width: 130px;" | Compose-key sequence | ||
! | ! scope="col" style="width: 75px;" | Resulting text | ||
|- | ! Description | ||
! colspan="3 | |- style="white-space: nowrap;" | ||
! colspan="3" | Keyboard key symbols | |||
|- | |- | ||
|โโโ | | โโโ | ||
|โ | | โ | ||
| | | Compose key symbol (the right alt key by default) | ||
|- | |- | ||
|โ\โฃ | | โ\โฃ | ||
|โฃ | | โฃ | ||
| | | Spacebar symbol | ||
|- | |- | ||
|โ\โถ๏ธ etc. | | โ\โถ๏ธ etc. | ||
|โถ๏ธ etc. | | โถ๏ธ etc. | ||
| | | Right etc. arrow key symbols | ||
|- | |- | ||
|โ\A or โ\O | | โ\A or โ\O | ||
|โฅ | | โฅ | ||
| | | Alt or option key symbol | ||
|- | |- | ||
|โ\B | | โ\B | ||
|โซ | | โซ | ||
| | | Backspace key symbol | ||
|- | |- | ||
|โ\C | | โ\C | ||
|โฒ | | โฒ | ||
| | | Control key symbol | ||
|- | |- | ||
|โ\D | | โ\D | ||
|โฆ | | โฆ | ||
| | | Delete key symbol | ||
|- | |- | ||
|โ\E | | โ\E | ||
|โ | | โ | ||
| | | Escape key symbol | ||
|- | |- | ||
|โ\L | | โ\L | ||
|โช | | โช | ||
| | | Caps lock key symbol | ||
|- | |- | ||
|โ\R or โ\.E | | โ\R or โ\.E | ||
|โ | | โ | ||
| | | Return or enter key symbol | ||
|- | |- | ||
|โ\S | | โ\S | ||
|โง | | โง | ||
| | | Shift key symbol | ||
|- | |- | ||
|โ\T | | โ\T | ||
|โญพ | | โญพ | ||
| | | Tab key symbol | ||
|- | |- | ||
|โ() | | โ() | ||
|โ | | โ | ||
| | | Dotted circle, represents any character (such as the character preceding a combining mark) | ||
|- | |- | ||
! colspan="3" | ! colspan="3" style="white-space: nowrap;" | Double key sequences | ||
|- | |- | ||
|โโฃโฃ | | โโฃโฃ | ||
|โฏ | | โฏ | ||
| | | Narrow no-break space (used between quantities and their units) | ||
|- | |- | ||
|โ.. | | โ.. | ||
|ยท | | ยท | ||
| | | Middle dot (used to multiply units when juxtaposition is ambiguous) | ||
|- | |- | ||
|โ:: | | โ:: | ||
|รท | | รท | ||
| | | Divide sign | ||
|- | |- | ||
|โ;; | | โ;; | ||
|โฬฒฬ
| | โฬฒฬ
| ||
| | | Combining overline and low line (undirected value) | ||
|- | |- | ||
| | | โ{{pipe}} {{pipe}} | ||
|โ | | โ | ||
| | | Power norm bracket | ||
|- | |- | ||
|โ | | โ<< | ||
| | | โจ | ||
| | | Left angle bracket | ||
|- | |- | ||
|โ | | โ>> | ||
| | | โฉ | ||
| | | Right angle bracket | ||
|- | |- | ||
|โ | | โ~~ | ||
| | | โ | ||
| | | Approximately equal | ||
|- | |- | ||
|โ | | โ** | ||
| | | โ
| ||
| | | Black star | ||
|- | |- | ||
|โ | | โ'' | ||
| | | โฒ | ||
| | | prime mark | ||
|- | |- | ||
| | | โ11 | ||
| | | โปยน | ||
| | | Power of −1 or inverse | ||
|- | |- | ||
| | | โ22 through โ77 | ||
| | | ยฒ ยณ โด โต โถ โท | ||
|power | | Squared, cubed, fourth through seventh power | ||
|- | |- | ||
| | | โ88 | ||
| | | โ | ||
| | | Infinity | ||
|- | |- | ||
| | | โ00 | ||
| | | ยฐ | ||
| | | Degree sign | ||
|- | |- | ||
| | | โnn | ||
| | | โฟ | ||
| | | Superscript small n | ||
|- | |- | ||
| | | โ-- | ||
| | | โ | ||
| | | Subscript minus sign | ||
|- | |- | ||
| | | โ__ | ||
| | | โฬฒ | ||
| | | Combining low line (underline) | ||
|- | |- | ||
| | | โ== | ||
| | | โก | ||
| | | Modular congruence | ||
|- | |- | ||
|โ | | โ// | ||
| | | โ | ||
| | | Fraction slash (use with super and subscripts to create fractions) | ||
|- | |- | ||
|โ | | โ## | ||
| | | โฏ | ||
| | | Musical sharp | ||
|- | |- | ||
| | | โbb | ||
| | | โญ | ||
| | | Musical flat | ||
|- | |- | ||
| | | โdd | ||
| | | โ | ||
| | | Partial derivative | ||
|- | |- | ||
| | | โff | ||
| | | ฯ | ||
| | | Small phi symbol | ||
|- | |- | ||
| | | โgg | ||
| | | ษก | ||
|small | | Single-storey (opentail) small g | ||
|- | |- | ||
| | | โll | ||
| | | โ | ||
| | | Script small L | ||
|- | |- | ||
| | | โuu | ||
| | | ยต | ||
| | | Micro sign | ||
|- | |- | ||
| | | โxx | ||
| | | ร | ||
| | | Multiplication sign | ||
|- | |- | ||
| | | โDD | ||
| | | โ | ||
| | | Delta (small difference) operator | ||
|- | |- | ||
| | | โFF | ||
| | | ฮฆ | ||
| | | Greek capital phi | ||
|- | |- | ||
| | | โQQ | ||
| | | ฯ | ||
|Greek capital | | Greek capital letter archaic qoppa (small quotient operator) | ||
|- | |- | ||
| | | โTT | ||
| | | แต | ||
| | | Superscript capital T (matrix transpose) | ||
|- | |- | ||
| | | โ++ | ||
| | | โบ | ||
| | | Superscript plus sign (matrix pseudoinverse) | ||
|- | |- | ||
| | | โโถ๏ธโถ๏ธ etc. | ||
| | | โ etc. | ||
| | | Right etc. arrows | ||
|- | |- | ||
| | ! colspan="3" style="white-space: nowrap;" | Multiplication operators | ||
|- | |- | ||
| โxx | |||
| ร | |||
| Multiplication sign | |||
|- | |- | ||
| | | โXx or โxX | ||
| | | โจฏ | ||
|multiplication sign | | Vector or cross product (barely distinguishable from multiplication sign) | ||
|- | |- | ||
| | | โXX | ||
| | | โ | ||
| | | Large multiplication sign (a better symbol for cross product) | ||
|- | |- | ||
| | | โx* | ||
| | | โ | ||
| | | Star operator (prefix: tensor complement, Hodge) | ||
|- | |- | ||
| | | โX* | ||
| | | โ | ||
| | | Asterisk operator (infix: scalar product, Dorst) | ||
|- | |- | ||
| | | โx. | ||
| | | โ
| ||
| | | Dot (product) operator | ||
|- | |- | ||
| | | โX. | ||
| | | โข | ||
|dot | | Bullet (infix: fat dot product, Dorst) | ||
|- | |- | ||
| | ! colspan="3" style="white-space: nowrap;" | Other operators | ||
|- | |- | ||
| โv/ | |||
| โ | |||
| Square root sign | |||
|- | |- | ||
| | | โ3v/ | ||
| | | โ | ||
| | | Cube root sign | ||
|- | |- | ||
| | | โ4v/ | ||
| | | โ | ||
| | | Fourth root sign | ||
|- | |- | ||
| | | โ-+ | ||
| | | โ | ||
| | | Subscript plus sign | ||
|- | |- | ||
|โ- | | โ-- | ||
| | | โ | ||
| | | Subscript minus sign | ||
|- | |- | ||
|โ- | | โ-= | ||
| | | โ | ||
| | | Subscript equals sign | ||
|- | |- | ||
|โ | | โ++ | ||
| | | โบ | ||
| | | Superscript plus sign (matrix pseudoinverse) | ||
|- | |- | ||
|โ++ | | โ+- or โ+= | ||
| | | ยฑ | ||
| | | Plus or minus sign | ||
|- | |- | ||
|โ+ | | โ=+ | ||
| | | โ | ||
|plus | | Minus or plus sign | ||
|- | |- | ||
|โ= | | โ=- | ||
| | | โ | ||
| | | Minus sign | ||
|- | |- | ||
|โ= | | โ== | ||
| | | โก | ||
| | | Modular congruence | ||
|- | |- | ||
|โ | | โ/\ | ||
| | | โง | ||
| | | Logical AND, wedge product, progressive product | ||
|- | |- | ||
|โ/ | | โ\/ | ||
| | | โจ | ||
| | | Logical OR, vee product, regressive product | ||
|- | |- | ||
| | | โโ/\ | ||
| | | โ | ||
|logical | | Larger logical AND, wedge product, progressive product | ||
|- | |- | ||
|โโ/ | | โโ\/ | ||
| | | โ | ||
| | | Larger logical OR, vee product, regressive product | ||
|- | |- | ||
| | | <nowiki>โ| _</nowiki> | ||
| | | โ | ||
| | | Left floor (infix: right contraction, Dorst) | ||
|- | |- | ||
|<nowiki> | | <nowiki>โ_| </nowiki> | ||
| | | โ | ||
| | | Right floor (infix: left contraction, Dorst) | ||
|- | |- | ||
|<nowiki> | | <nowiki>โ| ^</nowiki> | ||
| | | โ | ||
| | | Left ceiling | ||
|- | |- | ||
|<nowiki>โ| | | <nowiki>โ^| </nowiki> | ||
| | | โ | ||
| | | Right ceiling | ||
|- | |- | ||
| | | โ'- | ||
| | | โจฝ | ||
| | | Righthand interior product | ||
|- | |- | ||
|โ' | | โ-' | ||
| | | โจผ | ||
| | | (Left-hand) interior product | ||
|- | |- | ||
|โ- | | โ-, | ||
| | | ยฌ | ||
|( | | Not sign (prefix: multivector complement) | ||
|- | |- | ||
| | | โโ<> | ||
| | | โ | ||
| | | Diamond operator (prefix: multivector dual) | ||
|- | |- | ||
| | | โ(.) | ||
| | | โจ | ||
| | | Entry-wise vector multiplication operator | ||
|- | |- | ||
|โ(.) | | โ(..) | ||
| | | โ | ||
| | | Alternative entry-wise vector multiplication operator | ||
|- | |- | ||
|โ( | | โ(/) | ||
| | | โ | ||
| | | Entry-wise vector division operator | ||
|- | |- | ||
| | ! colspan="3" | Mathematical letter and digit prefixes | ||
|- | |- | ||
| โ3โ | |||
| ั | |||
| Cyrillic, โ3q is ya (example) | |||
|- | |- | ||
| | | โ4โ | ||
| | | โต | ||
| | | Hebrew, โ4a is aleph (example) | ||
|- | |- | ||
| | | โ5โ | ||
| | | ๐ | ||
| | | Fraktur, โ5a | ||
|- | |- | ||
| | | โ6โ | ||
| | | แต ยน โฏแชฒย โธ | ||
| | | Superscripts, โ6a โ61 โ688 โ68โฃ (not all letters, some only approximate) (same key as ^ but without shift) | ||
|- | |- | ||
| | | โ68โ | ||
| | | แต | ||
| | | Superscript greek, โ68b is superscript beta (only a few) | ||
|- | |- | ||
| | | โ7โ | ||
| | | ๐ถ | ||
| | | Script, โ7a | ||
|- | |- | ||
| | | โ8โ | ||
| | | ฮฑ | ||
| | | Greek, โ8a is alpha (by sound where possible otherwise letter-shape) | ||
|- | |- | ||
| | | โ8.โ | ||
| | | ฯ | ||
| | | Greek variants, โ8.s is final sigma | ||
|- | |- | ||
| | | โ9โ | ||
| | | ๐ ๐ ๐ ๐ ๐ ๐ | ||
| | | Bold, โ9a โ91 โ95โฃ โ97โฃ โ98โฃ โ90โฃ | ||
|- | |- | ||
| | | โ95โ | ||
| | | ๐ | ||
| | | Bold fraktur, โ95a | ||
|- | |- | ||
| | | โ97โ | ||
| | | ๐ช | ||
| | | Bold script, โ97a | ||
|- | |- | ||
| | | โ98โ | ||
| | | ๐ | ||
|bold | | Bold greek, โ98a is bold alpha | ||
|- | |- | ||
| | | โ90โ | ||
| | | ๐ | ||
| | | Bold italic, โ90a | ||
|- | |- | ||
| | | โ908โ | ||
| | | ๐ถ | ||
|bold italic | | Bold italic greek, โ908a is bold italic alpha | ||
|- | |- | ||
| | | โ0โ | ||
| | | ๐ | ||
| | | Italic, โ0a | ||
|- | |- | ||
| | | โ08โ | ||
| | | ๐ผ | ||
|italic | | Italic greek, โ08a is italic alpha | ||
|- | |- | ||
| | | โ-โ | ||
| | | โ แดย โฏอ โ | ||
| | | Subscripts and small caps, โ-a โ-A โ-88 โ-8โฃ (not all letters, some only approximate) (same key as _ but without shift) | ||
|- | |- | ||
|โ- | | โ-8โ | ||
| | | แตฆ | ||
| | | Subscript greek, โ-8b is subscript beta (only a few) | ||
|- | |- | ||
|โ | | โ{โ | ||
| | | ๐บ ๐ฃ ๐ซ | ||
| | | Sans-serif, โ{a โ{1 โ{9โฃ | ||
|- | |- | ||
|โ{ | | โ{9โ | ||
| | | ๐ฎ ๐ญ | ||
| | | Sans-serif bold, โ{9a โ{91 | ||
|- | |- | ||
|โ | | โ}โ | ||
| | | ๐ ๐ท | ||
| | | Monospace, โ}a โ}1 | ||
|- | |- | ||
|โ}โ | | โ{{pipe}} โ | ||
| | | ๐ ๐ ๐ ๐ | ||
| | | Double-struck, โ{{pipe}} a โ{{pipe}} 1 โ{{pipe}} 8โฃ โ{{pipe}} 0โฃ | ||
|- | |- | ||
| | | โ{{pipe}} 8โ | ||
| | | โผ | ||
| | | Double-struck greek, โ{{pipe}} 8p (only a few) | ||
|- | |- | ||
| | | โ{{pipe}} 0โ | ||
| | | โ
โ
| ||
| | | Double-struck italic, โ{{pipe}} 0e โ{{pipe}} i (only a few) | ||
|- | |- | ||
! colspan="3" style="white-space: nowrap;" | Power statistics brackets | |||
|- | |- | ||
| โ{{pipe}} {{pipe}} | |||
| โ | |||
| Power-norm bracket | |||
|- | |- | ||
| | | โ{{pipe}}-1 | ||
| | | โโ | ||
| | | 1-Norm right bracket | ||
|- | |- | ||
| | | โ{{pipe}}-2 | ||
| | | โโ | ||
| | | 2-Norm right bracket | ||
|- | |- | ||
| | | โ{{pipe}}-8 | ||
| | | โโฏอ | ||
| | | โ-Norm right bracket | ||
|- | |- | ||
|< | | โโ<< | ||
| | | โช | ||
| | | Left power-mean bracket | ||
|- | |- | ||
|โโ | | โโ>> | ||
| | | โซ | ||
| | | Right power-mean bracket | ||
|- | |- | ||
|โโ | | โโ{{((}} | ||
| | | โง | ||
| | | Left power-sum bracket (substitute for {{llzz}} when HTML is not available) | ||
|- | |- | ||
| | | โโ{{))}} | ||
| | | โง | ||
| | | Right power-sum bracket (substitute for {{rrzz}} when HTML is not available) | ||
|- | |- | ||
| | ! colspan="3" style="white-space: nowrap;" | Combining marks | ||
|- | |- | ||
| โ\- | |||
| โฬถ | |||
| Combining strike-thru | |||
|- | |- | ||
|โ | | โ^_ | ||
| | | โฬ
| ||
| | | Combining overline | ||
|- | |- | ||
| | | โ__ | ||
| | | โฬฒ | ||
| | | Combining low line | ||
|- | |- | ||
| | | โ;; or โ-_ or โ_^ | ||
| โฬฒฬ
| |||
| Combining overline and low line (undirected value) | |||
|โฬฒฬ
| |||
| | |||
ย | |||
ย | |||
|} | |} | ||
== | === Keyboard map === | ||
[[File:WinCompose keyboard map.png|1000px]] | |||
<references /> | == Footnotes == | ||
<references group="note" /> | |||
[[Category: | [[Category:Dave Keenan & Douglas Blumeyer's guide to RTT]] | ||
[[Category:Tuning]] | [[Category:Tuning]] | ||