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

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remove 1200, ๐Ÿ, and ๐ฟ 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)
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โ†’Objects: Replaced <math>\llzigzag and \rrzigzag with {{llzigzag}} and {{rrzigzag}} where the former did not display correctly
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This is an appendix to [[Dave Keenan]] & [[Douglas Blumeyer]]'s guide to RTT, or "[[D&D's guide]]" for short. The tables in this article present our recommendations for communicating about [[regular temperament theory]] (RTT), in particular the names and notations for temperament matrices, tuning schemes, interval complexities, and measurement units. ย 
{{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.


For more information on our variation on extended bra-ket notation, please see [[Extended_bra-ket_notation#Variant_including_curly_and_square_brackets|Extended bra-ket notation: Variant including curly and square brackets]].
See [[Extended bra-ket notation#Variant including curly and square brackets|here]] for more information on our variation on extended bra-ket notation.


We've followed a symbol formatting pattern, explained by the table below, which we hope serves as an aid to quickly identifying objects and remembering their properties and purposes, but at the least we hope our choices are unobtrusive. In short, the objects with simple units of primes, generators or cents, i.e. the things which are actually audible in our application, are distinguished by upright formatting, while other variables are italic as is conventional. This is crossed with the mathematical convention that objects of order-1 like vectors are bolded and order-2 like matrices are uppercased:
We've followed a variable styling convention, explained in the table below, which we hope serves as an aid to quickly identifying objects and remembering their properties and purposes, but at the least we hope our choices are unobtrusive. In short, the objects with simple units of primes, generators or cents, i.e. the things which are actually audible in our application, are distinguished by upright formatting, while other variables are italic as is conventional. This is crossed with the linear-algebra convention that objects of order-1 like vectors are bolded and order-2 like matrices are uppercased:


{| class="wikitable center-all" ย 
{| class="wikitable center-all"
|+
|-
!
! ย 
| ย  units โ†’
| Units &rarr;
! rowspan="2" |
! rowspan="2" | &nbsp;
|simple units
| Simple units
|compound or no units
| Compound or no units
|-
|-
| โ†“ order
| &darr; Order
| โ†“ style โ†’
| &darr; Style &rarr;
|upright
| Roman (upright)
|''italic''
| ''Italic''
ย 
|-
|-
! scope="col" height="8px" ! colspan="2" |
! scope="col" height="8px" ! colspan="2" |
Line 25: Line 25:
! colspan="2" |
! colspan="2" |
|-
|-
|0
| 0
|plain
| lowercase
! rowspan="3" |
! rowspan="3" | &nbsp;
|scalar with simple unit
| scalar (with simple unit)
|''scalar'' with no unit
| ''scalar'' (with no unit)
|-
|-
|1
| 1
|'''bold'''
| '''bold lowercase'''
|'''vector'''
| '''vector'''
|'''''map''''' (covector)
| '''''map''''' (row vector)
|-
|-
|2
| 2
|UPPERCASE
| UPPERCASE
|LIST or BASIS
| BASIS or LIST (of vectors)
|true ''MATRIX''
| ''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 
==Basic==
ย 
===Objects===


== Basic ==
=== Objects ===
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
|+
|+ 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
! rowspan="2" | Equivalent<br />expressions
!reduced
! rowspan="2" | Variable
!read as
! rowspan="2" | Name
!unreduced
! colspan="3" | Units
!reduced
! colspan="2" | Shape
!numeric
! colspan="2" | Type
!structural
! colspan="2" | EBK notation
!row-first
! colspan="4" | Subobjects
!col-first
! rowspan="2" | Notes
!row
!col
!diag
!entry
|-
|-
! colspan="17" |mapping
! Unreduced
! Reduced
! Read as
! Unreduced
! Reduced
! Numeric
! Structural
! Row-first
! Col-first
! Row
! Column
! Diagonal
! Entry
|-
|-
|
! colspan="17" | Mapping
|<math>\textbf{i}</math>
|[[interval|(just) interval]]
|
|<math>\small ๐—ฝ</math>
|primes
|
|<math>\scriptsize (d, 1)</math>
|integer
|vector
|
|[...โŸฉ
|
|
|
|<math>\mathrm{i}_i</math>
|specific type: [[prime-count vector]] (PC-vector)
jargon name: monzo
|-
|-
|
| ย 
|<math>M</math>
| <math>\textbf{i}</math>
|[[Mapping|(temperament) mapping (matrix)]]
| [[interval|(Just) interval]]
|
| ย 
|<math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>
|generators per prime
| Primes
|
| ย 
|<math>\scriptsize (r, d)</math>
| <math>\scriptsize (d, 1)</math>
|integer
| Integer
|matrix
| Vector
|[โŸจ...] ...}
| ย 
|โŸจ[...} ...]
| [...โŸฉ
|<math>๐’Ž_i</math>
| ย 
|
| ย 
|
| ย 
|<math>m_{ij}</math>
| <math>\mathrm{i}_i</math>
|jargon name: val list
| Specific type: Vector ([[prime-count vector]] or PC-vector)
Jargon name: Monzo
|-
|-
|<math>M\textbf{i}</math>
|
|<math>\textbf{y}</math>
| <math>M</math>
|[[mapped interval]]
| [[Mapping|(Temperament) mapping (matrix)]]
|<math>\scriptsize ย 
|
| <math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
| Generators per prime
|
| <math>\scriptsize (r, d)</math>
| Integer
| Matrix
| [โŸจ...] ...}
| โŸจ[...} ...]
| <math>๐’Ž_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] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
</math>
</math>
|<math>\small ๐—ด</math>
| <math>\small ๐—ด</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] (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
| Integer
|vector
| Vector
|
| ย 
|[...}
| [...}
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|specific type: [[generator-count vector]] (GC-vector)
| Specific type: [[Generator-count vector]] (GC-vector)
jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval
Jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval
|-
|-
|
| ย 
|<math>๐’Ž</math>
| <math>๐’Ž</math>
|[[map|(temperament) map]]
| [[map|(Temperament) map]]
|
| ย 
|<math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
| <math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
|generators per prime
| Generators per prime
|
| ย 
|<math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
|integer
| Integer
|vector
| Vector
|โŸจ...]
| โŸจ...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>m_i</math>
| <math>m_i</math>
|jargon name: val
| 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
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|
| ย 
|<math>r</math>
| <math>r</math>
|[[rank]]
| [[Rank]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |tuning
! colspan="17" | Tuning
|-
|-
|
| ย 
|<math>๐’‹</math>
| <math>{\large\textbf{๐“}}\hspace{2mu}</math>
|[[just tuning map|just(-prime) tuning map]]
| [[Log-prime map]]
|
| ย 
|<math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
| <math>\small\mathsf{oct}</math>/<math>\small ๐—ฝ</math>
|cents per prime
| Octaves per prime
|
| ย 
|<math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
|real
| Real
|vector
| Vector
|โŸจ...]
| โŸจ...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>j_i</math>
| <math>{\large ๐“}\hspace{2mu}_i</math>
|
| ย 
|-
|-
|
| <math>1200ร—{\large\textbf{๐“}}\hspace{2mu}</math>
|<math>๐’ˆ</math>
| <math>๐’‹</math>
|[[generator tuning map]]
| [[just tuning map|Just(-prime) tuning map]]
|
| ย 
|<math>\mathsf{ยข}</math>/<math>\small ๐—ด</math>
| <math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
|cents per generator
| Cents per prime
|
| ย 
|<math>\scriptsize (1, r)</math>
| <math>\scriptsize (1, d)</math>
|real
| Real
|vector
| Vector
|{...]
| โŸจ...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>g_i</math>
| <math>j_i</math>
|
| ย 
|-
|-
|<math>๐’ˆM</math>
|
|<math>๐’•</math>
| <math>๐’ˆ</math>
|[[tuning map|(tempered-prime) tuning map]]
| [[Generator tuning map]]
|<math>\scriptsize ย 
|
| <math>\mathsf{ยข}</math>/<math>\small ๐—ด</math>
| Cents per generator
|
| <math>\scriptsize (1, r)</math>
| Real
| Vector
| {...]
|
|
|
|
| <math>g_i</math>
|
|-
| <math>๐’ˆM</math>
| <math>๐’•</math>
| [[tuning map|(Tempered-prime) tuning map]]
| <math>\scriptsize ย 
\begin{array} {c} ๐’ˆ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array} ย 
\begin{array} {c} ๐’ˆ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} ๐‘€ \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array} ย 
\begin{array} {c} ๐‘€ \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array} ย 
</math>
</math>
|<math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
| <math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
|cents per prime
| Cents per prime
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\!\! ย 
\! \! ย 
\begin{array} {c} ๐’ˆ \\[-3pt] (1, \cancel{r}) \end{array} ย 
\begin{array} {c} ๐’ˆ \\[-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
| Real
|vector
| Vector
|โŸจ...]
| โŸจ...]
|
| 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 (or mistuning) map]]
| [[retuning map|Retuning (or mistuning) map]]
|
| ย 
|<math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
| <math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
|cents per prime
| Cents per prime
|
| ย 
|<math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
|real
| Real
|vector
| Vector
|โŸจ...]
| โŸจ...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>r_i</math>
| <math>r_i</math>
|previous name: prime error map
| 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|(just) (interval) size]]
| [[interval span|(Just) (interval) size]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \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
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|mnemonic: <math>\mathrm{o}</math>riginal size
| Mnemonic: <math>\mathrm{o}</math>riginal size
|-
|-
|<math>๐’ˆM\textbf{i} \\
| <math>๐’ˆM\textbf{i}</math><br />
๐’•\textbf{i}</math>
<math>๐’•\textbf{i}</math>
|<math>\mathrm{a}</math>
| <math>\mathrm{a}</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Example_3|tempered (interval) size]]
| {{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{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \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
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|mnemonic: <math>\mathrm{a}</math>ltered size
| 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|(interval) error]]
| [[error|(Interval) error]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \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
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |optimization
! colspan="17" | Optimization
|-
|-
|
| ย 
|<math>p</math>
| <math>p</math>
|[[optimization power]]
| [[Optimization power]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|
| ย 
|<math>โŸช ยท โŸซ_p</math>
| <math>\llangle\,ยท\,\rrangle_p</math>
|[[power mean]] (<math>p</math>-mean)
| [[Power mean]] (<math>p</math>-mean)
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |damage
! colspan="17" | Damage
|-
|-
|<math>s^{-1}</math>
| ย 
|<math>c</math>
| <math>c</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity|complexity]]
| {{subpage|tuning_fundamentals|uprev|s=Complexity}}
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math><ref>For educational purposes, we use the ๐Ÿ™ symbol here to represent the implicit [[Wikipedia:Dimensionless_quantity|dimensionless unit]] that the weighting annotation "(C)" is attached to. But this symbol should not be shown in the reduced result. Another way to understand how we arrive at a bare annotation for the units of this quantity is to consider that ''w'' = ''d'' / |''e''| whose units are ยข(W) / ยข and the cents cancel.</ref>
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math><ref group="note">For educational purposes, we use the ๐Ÿ™ 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
| Complexity weight
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|<math>c^{-1}</math>
| <math>\dfrac1c</math>
|<math>s</math>
| <math>s</math>
|[[simplicity]]
| [[Simplicity]]
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)}</math>
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)}</math>
|<math>\small\mathsf{(S)}</math>
| <math>\small\mathsf{(S)}</math>
|simplicity weight
| Simplicity weight
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| 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]]
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math> or ๐Ÿ™<math>\small\mathsf{(S)}</math>
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math> or ๐Ÿ™<math>\small\mathsf{(S)}</math>
|<math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(S)}</math>
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(S)}</math>
|complexity weight or simplicity weight
| Complexity weight or simplicity weight
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|<math>|\mathrm{e}|w</math>
| <math>\abs{\mathrm{e}} w</math>
|<math>\mathrm{d}</math>
| <math>\mathrm{d}</math>
|[[damage]]
| [[Damage]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} |\mathrm{e}| \\[-2pt] {\small\mathsf{ยข}} \end{array} ย 
\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)
| (See damages table)
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\!\! ย 
\! \! ย 
\begin{array} {c} |\mathrm{e}| \\[-3pt] (1, \cancel{1}) \end{array} ย 
\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
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |target-intervals
! colspan="17" | Target-intervals
|-
|-
|
| ย 
|<math>\mathrm{T}</math>
| <math>\mathrm{T}</math>
|[[target-interval list]]
| [[Target-interval list]]
|
| ย 
|<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>
|primes
| Primes
|
| ย 
|<math>\scriptsize (d, k)</math>
| <math>\scriptsize (d, k)</math>
|integer
| Integer
|matrix
| Matrix
|
| ย 
|[[...โŸฉ ...]
| [[...โŸฉ ...]
|
| 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]]
| [[Mapped target-interval list]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
</math>
</math>
|<math>\small ๐—ด</math>
| <math>\small ๐—ด</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 
\! \! 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
| Integer
|matrix
| 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'
| 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>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Primes|target-interval (just) size list]]
| {{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{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \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} \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
| Real
|list
| List
|[...]
| [...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\mathrm{o}_i</math>
| <math>\mathrm{o}_i</math>
|mnemonic: <math>\textbf{o}</math>riginal size list
| Mnemonic: <math>\textbf{o}</math>riginal size list
|-
|-
|<math>๐’•\mathrm{T} \\
| <math>๐’•\mathrm{T}</math><br />
๐’ˆM\mathrm{T}</math>
<math>๐’ˆM\mathrm{T}</math>
|<math>\textbf{a}</math>
| <math>\textbf{a}</math>
|[[tempered target-interval size list]]
| [[Tempered target-interval size list]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \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} \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
| Real
|list
| List
|[...]
| [...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\mathrm{a}_i</math>
| <math>\mathrm{a}_i</math>
|mnemonic: <math>\textbf{a}</math>ltered size list
| Mnemonic: <math>\textbf{a}</math>ltered size list
|-
|-
|<math>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]]
|[[target-interval error list]]
| <math>\scriptsize ย 
|<math>\scriptsize ย 
\begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \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} \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
| Real
|list
| 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]]
| [[Target-interval weight matrix]]
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)}</math> or <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(U)}</math>
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)}</math> or <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(U)}</math>
|<math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(S)}</math> or <math>\small\mathsf{(U)}</math>
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(S)}</math> or <math>\small\mathsf{(U)}</math>
|complexity weight or simplicity weight
| Complexity weight or simplicity weight
|
| ย 
|<math>\scriptsize (k, k)</math>
| <math>\scriptsize (k, k)</math>
|real
| Real
|matrix
| Matrix
|
| ย 
|[[...] ...]
| [[...] ...]
|
| ย 
|
| ย 
|<math>๐’˜</math>
| <math>๐’˜</math>
|<math>w_i</math>
| <math>w_i</math>
|
| ย 
|-
|-
|<math>S^{-1}</math>
| ย 
|<math>C</math>
| <math>C</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity-weight_damage|target-interval complexity weight matrix]]
| {{subpage|tuning_fundamentals|uprev|s=complexity-weight_damage|text=Target-interval complexity weight matrix}}
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math>
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math>
|<math>\small\mathsf{(C)}</math>
| <math>\small\mathsf{(C)}</math>
|complexity weight
| Complexity weight
|
| ย 
|<math>\scriptsize (k, k)</math>
| <math>\scriptsize (k, k)</math>
|real
| Real
|matrix
| Matrix
|
| ย 
|[[...] ...]
| [[...] ...]
|
| ย 
|
| ย 
|<math>๐’„</math>
| <math>๐’„</math>
|<math>c_i</math>
| <math>c_i</math>
|
| ย 
|-
|-
|<math>C^{-1}</math>
| <math>\dfrac1C</math>
|<math>S</math>
| <math>S</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity-weight_damage|target-interval simplicity weight matrix]]
| {{subpage|tuning fundamentals|uprev|s=complexity-weight_damage|text=Target-interval simplicity weight matrix}}
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)}</math>
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)}</math>
|<math>\small\mathsf{(S)}</math>
| <math>\small\mathsf{(S)}</math>
|simplicity weight
| Simplicity weight
|
| ย 
|<math>\scriptsize (k, k)</math>
| <math>\scriptsize (k, k)</math>
|real
| Real
|matrix
| 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>|\textbf{e}|W</math>
| <math>\abs{\textbf{e}} W</math>
|<math>\textbf{d}</math>
| <math>\textbf{d}</math>
|[[target-interval damage list]]<ref>You may sometimes see annotated units without parentheses, such as "dBA", but this is not compliant with SI standards, so we always keep the parentheses.</ref>
| [[Target-interval damage list]]<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} |\textbf{e}| \\[-2pt] {\small\mathsf{ยข}} \end{array} ย 
\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
| Weighted cents
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\!\! ย 
\! \! ย 
\begin{array} {c} |\textbf{e}| \\[-3pt] (1, \cancel{k}) \end{array} ย 
\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
| Real
|list
| 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]]
| [[Target-interval count]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|mnemonic: <math>k</math>ount
| Mnemonic: <math>k</math>ount
|-
|-
! colspan="17" |unchanged-intervals
! colspan="17" | Held-intervals
|-
|-
|
| ย 
|<math>h</math>
| <math>\mathrm{H}</math>
|[[unchanged-interval count]]
| [[Held-interval basis]]
|
| ย 
|
| <math>\small ๐—ฝ</math>
|
| Primes
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (d, h)</math>
|integer
| ย 
|scalar
| Matrix
|
| ย 
|
| [[...โŸฉ ...]
|
| ย 
|
| <math>\textbf{h}_i</math>
|
| ย 
|
| <math>\mathrm{h}_{ij}</math>
|mnemonic: unc<math>h</math>anged interval count
|
|-
|-
! colspan="17" |exploring temperaments
| ย 
| <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 ๐—ฝ</math>
|primes
|
|<math>\scriptsize (d, n)</math>
|integer
|matrix
|
|[[...โŸฉ ...]
|
|<math>\textbf{c}_i</math>
|
|<math>\mathrm{c}_{ij}</math>
|jargon name: monzo list
|-
|-
|
|
|<math>\textbf{c}</math>
| <math>\mathrm{C}</math>
|[[comma]]
| [[Comma basis]]
|
|
|<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>
|primes
| Primes
|
|
|<math>\scriptsize (d, 1)</math>
| <math>\scriptsize (d, n)</math>
|integer
| Integer
|vector
| Matrix
|
|
|[...โŸฉ
| [[...โŸฉ ...]
|
|
|
| <math>\textbf{c}_i</math>
|
|
|<math>\mathrm{c}_i</math>
| <math>\mathrm{c}_{ij}</math>
|specific type: [[prime-count vector]] (PC-vector)
| Jargon name: monzo list
|-
| ย 
| <math>\textbf{c}</math>
| [[Comma]]
| ย 
| <math>\small ๐—ฝ</math>
| Primes
| ย 
| <math>\scriptsize (d, 1)</math>
| Integer
| Vector
| ย 
| [...โŸฉ
| 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 ๐—ด</math>
! Symbol
|generators
! Name
|yes
! Vectorized
|-
|-
|<math>\small ๐—ฝ</math>
| <math>\small ๐—ด</math>
|primes
| Generators
|yes
| Yes
|-
|-
|<math>\mathsf{ยข}</math><ref>It seems there is no standard symbol for a musical cent, except the word spelled in full (see https://en.wikipedia.org/wiki/Cent_(music)). But it seems unlikely anyone will interpret the cent currency symbol "ยข" following a number in a musical context as anything other than musical cents.</ref>
| <math>\small ๐—ฝ</math>
|cents
| Primes
|
| Yes
|-
|-
|<math>\mathsf{ยข}\small\mathsf{(U)}</math>
| <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>
|unity-weighted cents
| Cents
|
| ย 
|-
|-
|<math>\mathsf{ยข}\small\mathsf{(C)}</math>
| <math>\mathsf{ยข}\small\mathsf{(U)}</math>
|complexity-weighted cents
| Unity-weighted cents
|
| ย 
|-
|-
|<math>\mathsf{ยข}\small\mathsf{(S)}</math>
| <math>\mathsf{ยข}\small\mathsf{(C)}</math>
|simplicity-weighted cents
| Complexity-weighted cents
|
| ย 
|-
|-
|<math>\small\mathsf{oct}</math>
| <math>\mathsf{ยข}\small\mathsf{(S)}</math>
|octaves
| Simplicity-weighted cents
|
| ย 
|-
|-
|<math>\small\mathsf{(C)}</math>
| <math>\small\mathsf{oct}</math>
|complexity weight
| Octaves
|
| ย 
|-
|-
|<math>\small\mathsf{(S)}</math>
| <math>\small\mathsf{(C)}</math>
|simplicity weight
| 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 [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning fundamentals#Systematic tuning scheme names]].


{| class="wikitable center-all mw-collapsible"
{| class="wikitable center-all mw-collapsible"
|+
|+ style="font-size: 105%;" |
|-
|-
|'''damage weight'''
! Damage weight
|'''optimization power'''
! Optimization power
|'''systematic name'''
! Systematic name
|-
|-
|<none>
| <none>
| rowspan="3" |โˆž
| rowspan="3" | &infin;
|minimax-U
| Minimax-U
|-
|-
|complexity
| Complexity
|minimax-C
| Minimax-C
|-
|-
|1/complexity
| 1/Complexity
|minimax-S
| Minimax-S
|-
|-
|<none>
| <none>
| rowspan="3" |2
| rowspan="3" | 2
|miniRMS-U
| MiniRMS-U
|-
|-
|complexity
| Complexity
|miniRMS-C
| MiniRMS-C
|-
|-
|1/complexity
| 1/Complexity
|miniRMS-S
| MiniRMS-S
|-
|-
|<none>
| &lt;none&gt;
| rowspan="3" |1
| rowspan="3" | 1
|minimean-U
| Miniaverage-U
|-
|-
|complexity
| Complexity
|minimean-C
| Miniaverage-C
|-
|-
|1/complexity
| 1/Complexity
|minimean-S
| 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
! colspan="2" | Quantity
!name
! colspan="2" | Unit
!symbol
!name
|-
|-
|U-damage
! Abbreviation
|unity-weight damage
! Name
|<math>\mathsf{ยข}\small\mathsf{(U)}</math>
! Symbol
|unity-weighted cents
! Name
|-
|-
|C-damage
| U-damage
|complexity-weight damage
| Unity-weight damage
|<math>\mathsf{ยข}\small\mathsf{(C)}</math>
| <math>\mathsf{ยข}\small\mathsf{(U)}</math>
|complexity-weighted cents
| Unity-weighted cents
|-
|-
|S-damage
| C-damage
|simplicity-weight damage
| Complexity-weight damage
|<math>\mathsf{ยข}\small\mathsf{(S)}</math>
| <math>\mathsf{ยข}\small\mathsf{(C)}</math>
|simplicity-weighted cents
| 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" |quantity
|-
! colspan="2" |unit
! colspan="2" | Quantity
! colspan="2" | Unit
|-
|-
!abbreviation
! Abbreviation
!name
! Name
!symbol
! Symbol
!name
! Name
|-
|-
|C
| C
|complexity
| Complexity
|<math>\small\mathsf{(C)}</math>
| <math>\small\mathsf{(C)}</math>
|complexity weight
| Complexity weight
|-
|-
|S
| S
|simplicity
| Simplicity
|<math>\small\mathsf{(S)}</math>
| <math>\small\mathsf{(S)}</math>
|simplicity weight
| 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
! rowspan="2" | Equivalent expressions
!reduced
! rowspan="2" | Variable
!read as
! rowspan="2" | Name
!unreduced
! colspan="3" | Units
!reduced
! colspan="2" | Shape
!numeric
! colspan="2" | Type
!structural
! colspan="2" | EBK notation
!row-first
! colspan="4" | Subobjects
!col-first
! rowspan="2" | Notes
!row
!col
!diag
!entry
|-
|-
! colspan="17" |mapping
! Unreduced
! Reduced
! Read as
! Unreduced
! Reduced
! Numeric
! Structural
! Row-first
! Col-first
! Row
! Col
! Diag
! Entry
|-
|-
|
! colspan="17" | Mapping
|<math>\textbf{i}</math>
|[[interval|(just) interval]]
|
|<math>\small ๐—ฝ</math>
|primes
|
|<math>\scriptsize (d, 1)</math>
|integer
|vector
|
|[...โŸฉ
|
|
|
|<math>\mathrm{i}_i</math>
|specific type: [[prime-count vector]] (PC-vector)
jargon name: monzo
|-
|-
|
| ย 
|<math>M</math>
| <math>\textbf{i}</math>
|[[Mapping|(temperament) mapping (matrix)]]
| [[interval|(Just) interval]]
|
| ย 
|<math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>
|generators per prime
| Primes
|
| ย 
|<math>\scriptsize (r, d)</math>
| <math>\scriptsize (d, 1)</math>
|integer
| Integer
|matrix
| Vector
|[โŸจ...] ...}
| ย 
|โŸจ[...} ...]
| [...โŸฉ
|<math>๐’Ž_i</math>
| ย 
|
| ย 
|
| ย 
|<math>m_{ij}</math>
| <math>\mathrm{i}_i</math>
|jargon name: val list
| Specific type: vector ([[prime-count vector]] or PC-vector)
Jargon name: monzo
|-
|-
|<math>M\textbf{i}</math>
|
|<math>\textbf{y}</math>
| <math>M</math>
|[[mapped interval]]
| [[Mapping|(Temperament) mapping (matrix)]]
|<math>\scriptsize ย 
|
| <math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
| Generators per prime
|
| <math>\scriptsize (r, d)</math>
| Integer
| Matrix
| [โŸจ...] ...}
| โŸจ[...} ...]
| <math>๐’Ž_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] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
</math>
</math>
|<math>\small ๐—ด</math>
| <math>\small ๐—ด</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
| Integer
|vector
| Vector
|
| ย 
|[...}
| [...}
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|specific type: [[generator-count vector]] (GC-vector)
| Specific type: [[generator-count vector]] (GC-vector)
jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval
Jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval
|-
|-
|
| ย 
|<math>๐’Ž</math>
| <math>๐’Ž</math>
|[[map|(temperament) map]]
| [[map|(Temperament) map]]
|
| ย 
|<math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
| <math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
|generators per prime
| Generators per prime
|
| ย 
|<math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
|integer
| Integer
|vector
| Vector
|โŸจ...]
| โŸจ...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>m_i</math>
| <math>m_i</math>
|jargon name: val
| Jargon name: val
|-
|-
|<math>n + r</math>
| <math>n + r</math>
|<math>d</math>
| <math>d</math>
|[[dimensionality]]
| [[Dimensionality]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|<math>d - n</math>
| <math>d - n</math>
|<math>r</math>
| <math>r</math>
|[[rank]]
| [[Rank]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|<math>d - r</math>
| <math>d - r</math>
|<math>n</math>
| <math>n</math>
|[[nullity]]
| [[Nullity]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |tuning
! colspan="17" | Tuning
|-
|-
|<math>1200ร—\textbf{1}LG_{\text{j}}M_{\text{j}} \\
|
1200ร—\textbf{1}L \\
| <math>{\large\textbf{๐“}}\hspace{2mu}</math>
๐’ˆ_{\text{j}}M_{\text{j}}</math>
| [[Log-prime map]]
|<math>๐’‹</math>
|
|[[just tuning map|just(-prime) tuning map]]
| <math>\small\mathsf{oct}</math>/<math>\small ๐—ฝ</math>
|<math>\scriptsize
| Octaves per prime
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
|
\begin{array} {c} \\[-2pt] ยท \end{array}
| <math>\scriptsize (1, d)</math>
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
| Real
\begin{array} {c} \\[-2pt] ยท \end{array}
| Vector
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array}
| โŸจ...]
\begin{array} {c} \\[-2pt] ยท \end{array}
|
\\ \scriptsize \quad
|
\begin{array} {c} G_{\text{j}} \\[-2pt] \cancel{๐—ฝ} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array}
|
\begin{array} {c} \\[-2pt] ยท \end{array}
|
\begin{array} {c} M_{\text{j}} \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array}
| <math>{\large ๐“}\hspace{2mu}_i</math>
</math>
|
|<math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
|-
|cents per prime
| <math>1200ร—{\large\textbf{๐“}}\hspace{2mu}</math>
|<math>\scriptsize
| <math>๐’‹</math>
\!\!
| [[just tuning map|Just(-prime) tuning map]]
\begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array}
|
\!\!
| <math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
\begin{array} {c} \textbf{1} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array}
| Cents per prime
\!\!
| ย 
\begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array}
| <math>\scriptsize (1, d)</math>
\\ \scriptsize \quad
| Real
\!\!
| Vector
\begin{array} {c} G_{\text{j}} \\[-3pt] (\cancel{d}, \cancel{r}) \end{array}
| โŸจ...]
\!\!
| ย 
\begin{array} {c} M_{\text{j}} \\[-3pt] (\cancel{r}, d) \end{array}
| ย 
\!\!
| ย 
</math>
| ย 
|<math>\scriptsize (1, d)</math>
| <math>j_i</math>
|real
| ย 
|vector
|โŸจ...]
|
|
|
|
|<math>j_i</math>
|
|-
|-
|<math>1200ร—\textbf{1}LG</math>
| ย 
|<math>๐’ˆ</math>
| <math>๐’ˆ</math>
|[[generator tuning map]]
| [[Generator tuning map]]
|<math>\scriptsize
| ย 
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
| <math>\mathsf{ยข}</math>/<math>\small ๐—ด</math>
\begin{array} {c} \\[-2pt] ยท \end{array}
| Cents per generator
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
| ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
| <math>\scriptsize (1, r)</math>
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array}
| Real
\begin{array} {c} \\[-2pt] ยท \end{array}
| Vector
\\ \scriptsize \quad
| {...]
\begin{array} {c} G \\[-2pt] \cancel{๐—ฝ} \hspace{-2mu} / \hspace{-2mu} ๐—ด \end{array}
| ย 
</math>
| ย 
|<math>\mathsf{ยข}</math>/<math>\small ๐—ด</math>
| ย 
|cents per generator
| ย 
|<math>\scriptsize
| <math>g_i</math>
\!\!
| ย 
\begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array}
\!\!
\begin{array} {c} \textbf{1} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array}
\!\!
\begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array}
\\ \scriptsize \quad
\!\!
\begin{array} {c} G \\[-3pt] (\cancel{d}, r) \end{array}
\!\!
</math>
|<math>\scriptsize (1, r)</math>
|real
|vector
|{...]
|
|
|
|
|<math>g_i</math>
|
|-
|-
|<math>1200ร—\textbf{1}LGM \\
| ย 
1200ร—\textbf{1}LP \\
| <math>๐’•</math>
๐’ˆM</math>
| [[tuning map|(Tempered-prime) tuning map]]
|<math>๐’•</math>
| ย 
|[[tuning map|(tempered-prime) tuning map]]
| <math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
|<math>\scriptsize
| Cents per prime
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
| ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
| <math>\scriptsize (1, d)</math>
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
| Real
\begin{array} {c} \\[-2pt] ยท \end{array}
| Vector
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array}
| โŸจ...]
\begin{array} {c} \\[-2pt] ยท \end{array}
| ย 
\\ \scriptsize \quad
| ย 
\begin{array} {c} G \\[-2pt] \cancel{๐—ฝ} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array}
| ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
| ย 
\begin{array} {c} M \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array}
| <math>t_i</math>
</math>
| ย 
|<math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
|cents per prime
|<math>\scriptsize
\!\!
\begin{array} {c} 1200 \\[-3pt] (1ร—\cancel{1}) \end{array}
\!\!
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array}
\!\!
\begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array}
\\ \scriptsize \quad
\!\!
\begin{array} {c} G \\[-3pt] (\cancel{d}, \cancel{r}) \end{array}
\!\!
\begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array}
\!\!
</math>
|<math>\scriptsize (1, d)</math>
|real
|vector
|โŸจ...]
|
|
|
|
|<math>t_i</math>
|
|-
|-
|<math>๐’• - ๐’‹ \\
| <math>๐’• - ๐’‹</math><br />
1200ร—\textbf{1}L(P - I)</math>
<math>1200ร—\slant{\mathbf{1}}L(P - I)</math>
|<math>๐’“</math>
| <math>๐’“</math>
|[[retuning map|retuning (or mistuning) map]]
| [[retuning map|Retuning (or mistuning) map]]
|
| ย 
|<math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
| <math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
|cents per prime
| Cents per prime
|
| ย 
|<math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
|real
| Real
|vector
| Vector
|โŸจ...]
| โŸจ...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>r_i</math>
| <math>r_i</math>
|previous name: prime error map
| 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|(just) (interval) size]]
| [[interval span|(Just) (interval) size]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \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} \mathbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array}
\!\!
\! \! ย 
</math>
</math>
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|mnemonic: <math>\mathrm{o}</math>riginal size
| Mnemonic: <math>\mathrm{o}</math>riginal size
|-
|-
|<math>๐’ˆM\textbf{i} \\
| <math>๐’ˆM\textbf{i}</math><br />
๐’•\textbf{i}</math>
<math>๐’•\textbf{i}</math>
|<math>\mathrm{a}</math>
| <math>\mathrm{a}</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Example_3|tempered (interval) size]]
| {{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{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \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
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|mnemonic: <math>\mathrm{a}</math>ltered size
| 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|(interval) error]]
| [[error|(Interval) error]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \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
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |optimization
! colspan="17" | Optimization
|-
|-
|
| ย 
|<math>p</math>
| <math>p</math>
|[[optimization power]]
| [[Optimization power]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|
| ย 
|<math>โŸช ยท โŸซ_p</math>
| <math>\llangle\,ยท\,\rrangle_p</math>
|[[power mean]] (<math>p</math>-mean)
| [[Power mean]] (<math>p</math>-mean)
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |damage
! colspan="17" | Damage
|-
|-
|<math>s^{-1}</math>
| ย 
|<math>c</math>
| <math>c</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity|complexity]]
| {{subpage|tuning_fundamentals|uprev|s=Complexity}}
| colspan="3" |(see complexities section of complexities and simplicities table)
| colspan="3" | (See complexities section of complexities and simplicities table)
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|<math>c^{-1}</math>
| <math>\dfrac1c</math>
|<math>s</math>
| <math>s</math>
|[[simplicity]]
| [[Simplicity]]
| colspan="3" |(see simplicities section of complexities and simplicities table)
| colspan="3" | (See simplicities section of complexities and simplicities table)
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| 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" |(see complexities and simplicities table)
| colspan="3" | (See complexities and simplicities table)
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|<math>|\mathrm{e}|w</math>
| <math>\abs{\mathrm{e}} w</math>
|<math>\mathrm{d}</math>
| <math>\mathrm{d}</math>
|[[damage]]
| [[Damage]]
| colspan="3" |(see damages table)
| colspan="3" | (See damages table)
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |target-intervals
! colspan="17" | Target-intervals
|-
|-
|
| ย 
|<math>\mathrm{T}</math>
| <math>\mathrm{T}</math>
|[[target-interval list]]
| [[Target-interval list]]
|
| ย 
|<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>
|primes
| Primes
|
| ย 
|<math>\scriptsize (d, k)</math>
| <math>\scriptsize (d, k)</math>
|integer
| Integer
|matrix
| Matrix
|
| ย 
|[[...โŸฉ ...]
| [[...โŸฉ ...]
|
| 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]]
| [[Mapped target-interval list]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
</math>
</math>
|<math>\small ๐—ด</math>
| <math>\small ๐—ด</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 
\! \! 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
| Integer
|matrix
| 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'
| 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>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Primes|target-interval (just) size list]]
| {{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{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \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} \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
| Real
|list
| List
|[...]
| [...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\mathrm{o}_i</math>
| <math>\mathrm{o}_i</math>
|mnemonic: <math>\textbf{o}</math>riginal size list
| Mnemonic: <math>\textbf{o}</math>riginal size list
|-
|-
|<math>๐’•\mathrm{T}</math>
| <math>๐’•\mathrm{T}</math>
|<math>\textbf{a}</math>
| <math>\textbf{a}</math>
|[[tempered target-interval size list]]
| [[Tempered target-interval size list]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \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} \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
| Real
|list
| List
|[...]
| [...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\mathrm{a}_i</math>
| <math>\mathrm{a}_i</math>
|mnemonic: <math>\textbf{a}</math>ltered size list
| Mnemonic: <math>\textbf{a}</math>ltered size list
|-
|-
|<math>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{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \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} \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
| Real
|list
| 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]]
| [[Target-interval weight matrix]]
| colspan="3" |(see complexities and simplicities table)
| colspan="3" | (See complexities and simplicities table)
|
| ย 
|<math>\scriptsize (k, k)</math>
| <math>\scriptsize (k, k)</math>
|real
| Real
|matrix
| Matrix
|
| ย 
|[[...] ...]
| [[...] ...]
|
| ย 
|
| ย 
|<math>๐’˜</math>
| <math>๐’˜</math>
|<math>w_i</math>
| <math>w_i</math>
|
| ย 
|-
|-
|<math>S^{-1}</math>
| ย 
|<math>C</math>
| <math>C</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity-weight_damage|target-interval complexity weight matrix]]
| {{subpage|tuning_fundamentals|uprev|s=complexity-weight damage|text=Target-interval complexity weight matrix}}
| colspan="3" |(see complexities section of complexities and simplicities table)
| colspan="3" | (See complexities section of complexities and simplicities table)
|
| ย 
|<math>\scriptsize (k, k)</math>
| <math>\scriptsize (k, k)</math>
|real
| Real
|matrix
| Matrix
|
| ย 
|[[...] ...]
| [[...] ...]
|
| ย 
|
| ย 
|<math>๐’„</math>
| <math>๐’„</math>
|<math>c_i</math>
| <math>c_i</math>
|
| ย 
|-
|-
|<math>C^{-1}</math>
| <math>\dfrac1C</math>
|<math>S</math>
| <math>S</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity-weight_damage|target-interval simplicity weight matrix]]
| {{subpage|tuning_fundamentals|uprev|s=complexity-weight_damage|text=Target-interval simplicity weight matrix}}
| colspan="3" |(see simplicities section of complexities and simplicities table)
| colspan="3" | (See simplicities section of complexities and simplicities table)
|
| ย 
|<math>\scriptsize (k, k)</math>
| <math>\scriptsize (k, k)</math>
|real
| Real
|matrix
| 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>|\textbf{e}|W \\
| <math>\abs{\textbf{e}} W</math>
1200ร—\textbf{1}L|P - I|\mathrm{T}W</math>
| <math>\textbf{d}</math>
|<math>\textbf{d}</math>
| [[Target-interval damage list]]
|[[target-interval damage list]]
| colspan="3" | (See damages table)
| colspan="3" |(see damages table)
| ย 
|
| <math>\scriptsize (1, k)</math>
|<math>\scriptsize (1, k)</math>
| Real
|real
| List
|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]]
| [[Target-interval count]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|mnemonic: <math>k</math>ount
| Mnemonic: <math>k</math>ount
|-
|-
! colspan="17" |unchanged-intervals
! colspan="17" | Held-intervals
|-
|-
|
| ย 
|<math>\mathrm{U}</math>
| <math>\mathrm{H}</math>
|[[unchanged-interval basis]]
| [[Held-interval basis]]
|
| ย 
|<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>
|primes
| Primes
|
| ย 
|<math>\scriptsize (d, r)</math>
| <math>\scriptsize (d, h)</math>
|
| ย 
|matrix
| Matrix
|
| ย 
|[[...โŸฉ ...]
| [[...โŸฉ ...]
|
| ย 
|<math>\textbf{u}_i</math>
| <math>\textbf{h}_i</math>
|
| ย 
|<math>\mathrm{u}_{ij}</math>
| <math>\mathrm{h}_{ij}</math>
|jargon name: eigenmonzo list
| ย 
|-
|-
|
| ย 
|<math>h</math>
| <math>h</math>
|[[unchanged-interval count]]
| [[Held-interval count]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|mnemonic: unc<math>h</math>anged interval count
| ย 
|-
|-
! colspan="17" |exploring temperaments
! colspan="17" | Exploring temperaments
|-
|-
|
| ย 
|<math>\mathrm{C}</math>
| <math>\mathrm{C}</math>
|[[comma basis]]
| [[Comma basis]]
|
| ย 
|<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>
|primes
| Primes
|
| ย 
|<math>\scriptsize (d, n)</math>
| <math>\scriptsize (d, n)</math>
|integer
| Integer
|matrix
| Matrix
|
| ย 
|[[...โŸฉ ...]
| [[...โŸฉ ...]
|
| 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
| Jargon name: monzo list
|-
|-
|
| ย 
|<math>\textbf{c}</math>
| <math>\textbf{c}</math>
|[[comma]]
| [[Comma]]
|
| ย 
|<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>
|primes
| primes
|
| ย 
|<math>\scriptsize (d, 1)</math>
| <math>\scriptsize (d, 1)</math>
|integer
| Integer
|vector
| Vector
|
| ย 
|[...โŸฉ
| [...โŸฉ
|
| ย 
|
| ย 
|
| ย 
|<math>\mathrm{c}_i</math>
| <math>\mathrm{c}_i</math>
|specific type: [[prime-count vector]] (PC-vector)
| Specific type: vector ([[prime-count vector]] or PC-vector)
|-
|-
! colspan="17" |computation
! colspan="17" | Computation
|-
|-
|
| ย 
|<math>G</math>
| {{llzigzag}}<math>\,ยท\,</math>{{rrzigzag}}<math>_p</math>
|[[generator embedding matrix|generator embedding (matrix)]]
| [[Power sum]] (<math>p</math>-sum)
|
|
|<math>\small ๐—ฝ</math>/<math>\small ๐—ด</math>
|
|primes per generator
| ย 
|
| ย 
|<math>\scriptsize (d, r)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|matrix
| Scalar
|[{...] ...โŸฉ
| ย 
|{[...โŸฉ ...]
| ย 
|<math>๐’ˆ_i</math>
| ย 
|
| ย 
|
| ย 
|<math>g_{ij}</math>
| ย 
|
| ย 
|-
|-
|
! colspan="17" | All-interval tuning schemes
|<math>K</math>
|[[constraint (matrix)]]
|
|
|
|
|<math>\scriptsize (r, k)</math>
|<math>\scriptsize \{0, +1, -1\}</math>
|matrix
|[[...] ...]
|
|<math>๐’Œ_i</math>
|
|
|<math>k_{ij}</math>
|mnemonic: <math>K</math>onstraint
|-
|-
|
| <math>\mathrm{I}</math>
|<math>{\largeโง›}ยท{\largeโงš}_p</math>
| <math>\mathrm{T}_{\text{p}}</math>
|[[power sum]] (<math>p</math>-sum)
| [[Prime proxy target-interval list]]
|
|
|
| <math>\small ๐—ฝ</math>
|
| Primes
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (d, d)</math>
|real
| Integer
|scalar
| Matrix
|
| ย 
|
| โŸจ[...โŸฉ ...]
|
| ย 
|
| ย 
|
| <math>\mathbf{1}</math>
|
| ย 
|
| ย 
|-
|-
! colspan="17" |projection
| ย 
| <math>X</math>
| [[Complexity prescaler]]
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math>
| <math>\small\mathsf{(C)}</math>
| Complexity weight
|
| <math>\scriptsize (d, d)</math>
| Real
| Matrix
| [โŸจ...] ...โŸฉ
|
|
|
| <math>๐’™</math>
| <math>x_i</math>
|
|-
|-
|<math>GM</math>
| <math>\text{diag}({\large\textbf{๐“}}\hspace{2mu})</math>
|<math>P</math>
| <math>L</math>
|[[Projection matrix|projection (matrix)]]
| [[Log-prime matrix]]
|<math>\scriptsize
|
\begin{array} {c} G \\[-2pt] ๐—ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array}
| <math>\small\mathsf{oct}</math>/<math>\small ๐—ฝ</math>
\begin{array} {c} \\[-2pt] ยท \end{array}
| Octaves per prime
\begin{array} {c} M \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array}
| ย 
</math>
| <math>\scriptsize (d, d)</math>
|<math>\small ๐—ฝ</math>/<math>\small ๐—ฝ</math>
| Real
|primes per prime<ref>Note that "primes per prime" does not cancel out, because the first primes increment by rows while the second primes increment by columns. See [[Projection matrix#Units]] for details.</ref>
| Matrix
|<math>\scriptsize
| [โŸจ...] ...โŸฉ
\!\!
| โŸจ[...โŸฉ ...]
\begin{array} {c} G \\[-3pt] (d, \cancel{r}) \end{array}
| <math>{\large\textbf{๐“}}\hspace{2mu}_i</math>
\!\!
| ย 
\begin{array} {c} M \\[-3pt] (\cancel{r}, d) \end{array}
| <math>{\large\textbf{๐“}}\hspace{2mu}</math>
\!\!
| <math>{\large ๐“}\hspace{2mu}_{ij}</math>
</math>
| ย 
|<math>\scriptsize (d, d)</math>
|real
|matrix
|[โŸจ...] ...โŸฉ
|โŸจ[...โŸฉ ...]
|<math>๐’‘_i</math>
|
|
|<math>p_i</math>
|
|-
|-
|<math>GM\textbf{i}</math>
| ย 
|<math>P\textbf{i}</math>
| <math>q</math>
|[[projected interval]]
| {{subpage|all-interval_tuning_schemes|uprev|s=Dual norms|text=Interval complexity norm power}}
|<math>\scriptsize
|
\begin{array} {c} G \\[-2pt] ๐—ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array}
|
\begin{array} {c} \\[-2pt] ยท \end{array}
| ย 
\begin{array} {c} M \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array}
| ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
| <math>\scriptsize (1, 1)</math>
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array}
| Real
</math>
| Scalar
|<math>\small ๐—ฝ</math>
| ย 
|primes
| ย 
|<math>\scriptsize ย 
| ย 
\!\!
| ย 
\begin{array} {c} G \\[-3pt] (d, \cancel{r}) \end{array}
| ย 
\!\!
| ย 
\begin{array} {c} M \\[-3pt] (\cancel{r}, \cancel{d}) \end{array}
| ย 
\!\!
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array}
\!\!
</math>
|<math>\scriptsize (d, 1)</math>
|real
|vector
|
|[...โŸฉ
|
|
|
|
|specific type: [[prime-count vector]] (PC-vector)
|-
|-
! colspan="17" |JI equivalents
| ย 
| <math>\norm{ยท}_q</math>
| [[Power norm]] (<math>p</math>-norm)
|
|
|
|
| <math>\scriptsize (1, 1)</math>
| Real
| Scalar
|
|
|
|
|
|
|
|-
|-
|<math>I</math>
| <math>\dfrac1{1-\frac1q}</math>
|<math>M_{\text{j}}</math>
| <math>\text{dual}(q)</math>
|[[JI mapping (matrix)]]
| {{subpage|all-interval tuning schemes|uprev|s=Dual norms|text=Dual norm power}}
|
| ย 
|<math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
| ย 
|generators per prime
| ย 
|
| ย 
|<math>\scriptsize (d, d)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Real
|matrix
| Scalar
|[โŸจ...] ...}
| ย 
|โŸจ[...} ...]
| ย 
|
| ย 
|
| ย 
|<math>๐Ÿ</math>
| ย 
|
| ย 
|
| ย 
|-
|-
|<math>1200ร—\textbf{1}LG_{\text{j}}</math>
| ย 
|<math>๐’ˆ_{\text{j}}</math>
| <math>\norm{X\mathbf{i}}_q</math>
|[[JI generator tuning map]]
| [[interval complexity]]
|<math>\scriptsize
| ย 
\begin{array} {c} 1200 \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
| <math>\small\mathsf{(C)}</math>
\begin{array} {c} \\[-2pt] ยท \end{array}
| ย 
\begin{array} {c} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
| ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
| <math>\scriptsize (1, 1)</math>
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array}
| Real
\begin{array} {c} \\[-2pt] ยท \end{array}
| Scalar
\\ \scriptsize \quad
| ย 
\begin{array} {c} G_{\text{j}} \\[-2pt] \cancel{๐—ฝ} \hspace{-2mu} / \hspace{-2mu} ๐—ด \end{array}
| ย 
</math>
| ย 
|<math>\mathsf{ยข}</math>/<math>\small ๐—ด</math>
| ย 
|cents per generator
| ย 
|<math>\scriptsize
| ย 
\!\!
| ย 
\begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array}
\!\!
\begin{array} {c} \textbf{1} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array}
\!\!
\begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array}
\\ \scriptsize \quad
\!\!
\begin{array} {c} G_{\text{j}} \\[-3pt] (\cancel{d}, r) \end{array}
\!\!
</math>
|<math>\scriptsize (1, d)</math>
|real
|vector
|{...]
|
|
|
|
|<math>g_{\text{j}i}</math>
|
|-
|<math>I</math>
|<math>G_{\text{j}}</math>
|[[JI generator embedding matrix|JI generator embedding (matrix)]]
|
|<math>\small ๐—ฝ</math>/<math>\small ๐—ด</math>
|primes per generator
|
|<math>\scriptsize (d, d)</math>
|integer
|matrix
|[{...] ...โŸฉ
|{[...โŸฉ ...]
|
|
|<math>๐Ÿ</math>
|
|
|-
! colspan="17" |all-interval tuning schemes
|-
|<math>I</math>
|<math>\mathrm{T}_{\text{p}}</math>
|[[prime proxy target-interval (matrix)]]
|
|<math>\small ๐—ฝ</math>
|primes
|
|<math>\scriptsize (d, d)</math>
|integer
|matrix
|
|โŸจ[...โŸฉ ...]
|
|
|<math>๐Ÿ</math>
|
|
|-
|<math>S_{\text{p}}^{-1}</math>
|<math>C_{\text{p}}</math>
|[[complexity prescaler]]
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math>
|<math>\small\mathsf{(C)}</math>
|complexity weight
|
|<math>\scriptsize (d, d)</math>
|real
|matrix
|[โŸจ...] ...โŸฉ
|
|
|
|<math>๐’„_{\text{p}}</math>
|<math>c_{\text{p}i}</math>
|
|-
|<math>C_{\text{p}}^{-1}</math>
|<math>S_{\text{p}}</math>
|[[simplicity prescaler]]
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)}</math>
|<math>\small\mathsf{(S)}</math>
|simplicity weight
|
|<math>\scriptsize (d, d)</math>
|real
|matrix
|
|โŸจ[...โŸฉ ...]
|
|
|<math>๐’”_{\text{p}}</math>
|<math>s_{\text{p}i}</math>
|
|-
|
|<math>L</math>
|[[log-prime matrix]]
|
|<math>\small\mathsf{oct}</math>/<math>\small ๐—ฝ</math>
|octaves per prime
|
|<math>\scriptsize (d, d)</math>
|real
|matrix
|[โŸจ...] ...โŸฉ
|โŸจ[...โŸฉ ...]
|<math>\textbf{๐“}_i</math>
|
|<math>\textbf{๐“}</math>
|<math>๐“_{ij}</math>
|
|-
|-
|
| ย 
|<math>q</math>
| <math>\norm{๐’“X^{-1}}_{\text{dual}(q)}</math>
|[[interval complexity norm power]]
| [[Retuning magnitude]]
|
| ย 
|
| <math>\mathsf{ยข}\small\mathsf{(C^{-1})}</math>
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|
|<math>โ€– ยท โ€–_q</math>
|[[power norm]] (<math>q</math>-norm)
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|real
|scalar
|
|
|
|
|
|
|
|}
|}


===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" |retuning (or mistuning) magnitude
! colspan="3" rowspan="3" | Retuning (or mistuning) magnitude
! colspan="9" |damage
! colspan="9" | Damage
! rowspan="4" |target
! 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" |systematic name
! rowspan="4" | Of interest?
! rowspan="4" |previously named tuning schemes that are specific types of this tuning scheme
! rowspan="4" |of interest?
|-
|-
! colspan="6" |weight
! colspan="6" | Weight
! colspan="3" rowspan="1" |optimization
! colspan="3" rowspan="1" | Optimization
|-
|-
! colspan="3" |interval complexity
! colspan="3" | Interval complexity
! colspan="3" rowspan="1" |slope
! colspan="3" rowspan="1" | Slope
! colspan="1" rowspan="2" |initial
! colspan="1" rowspan="2" | Initial
! colspan="1" rowspan="2" |name
! colspan="1" rowspan="2" | Name
! colspan="1" rowspan="2" |power
! colspan="1" rowspan="2" | Power
|-
|-
!initial
! Initial
!name
! Name
!power
! Power
!initial
! Initial
!name
! Name
!power
! Power
!initial
! Initial
!name
! Name
!multiplier
! Multiplier
! colspan="1" |abbreviated
! colspan="1" | Abbreviated
! colspan="1" |read ("____ tuning scheme")
! colspan="1" | Read ("____ tuning scheme")
|-
|-
|<n/a>
| <n/a>
|maximum
| Maximum
|โˆž
| &infin;
|(t)
| (t)
|taxicab
| Taxicab
|1
| 1
| rowspan="2" |S
| rowspan="2" | ''S''
| rowspan="2" |simplicity-weight
| rowspan="2" | Simplicity-weight
| rowspan="2" |1/complexity
| rowspan="2" | 1/Complexity
| rowspan="17" |<n/a>
| rowspan="17" | <n/a>
| rowspan="7" |minimax
| rowspan="7" | Minimax
| rowspan="7" |โˆž
| rowspan="7" | โˆž
| rowspan="2" |all
| rowspan="2" | All
|minimax-S
| Minimax-S
|minimax simplicity-weight damage
| Minimax simplicity-weight damage
|"[[TOP]]"/"[[T1]]"/"[[TIPTOP]]"*, "[[CTOP]]", "[[POTOP]]"/"[[POTT]]"*, "[[BOP tuning|BOP]]", "[[Weil Norms, Tenney-Weil Norms, and TWp Interval and Tuning Space|Weil]]", "[[Kees]]"
| "[[TOP]]"/"[[T1]]"/"[[TIPTOP]]"*, "[[CTOP]]", "[[POTOP]]"/"[[POTT]]"*, "[[BOP tuning|BOP]]", "[[Weil Norms, Tenney-Weil Norms, and TWp Interval and Tuning Space|Weil]]", "[[Kees]]"
|yes
| Yes
|-
|-
|<n/a>
| <n/a>
|Euclidean
| Euclidean
|2
| 2
|E
| E
|Euclidean
| Euclidean
|2
| 2
|minimax-ES
| Minimax-ES
|minimax Euclideanized-simplicity-weight damage
| Minimax Euclideanized-simplicity-weight damage
|"[[Tenney-Euclidean tuning|TE]]"/"[[T2]]"/"[[TOP-RMS]]", "[[CTE tuning|CTE]]", "[[POTE tuning|POTE]]", "[[Frobenius]]", "[[BE]]", "[[WE]]", "[[KE]]"
| "[[Tenney-Euclidean tuning|TE]]"/"[[T2]]"/"[[TOP-RMS]]", "[[CTE tuning|CTE]]", "[[POTE tuning|POTE]]", "[[Frobenius]]", "[[BE]]", "[[WE]]", "[[KE]]"
|
| 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
| Unity-weight
|<none>
| <none>
| rowspan="15" |<set>
| rowspan="15" | <set>
|<set> minimax-U
| <set> Minimax-U
|<set> minimax unity-weight-damage
| <set> Minimax unity-weight-damage
|"[[Minimax tuning|minimax]]"
| "[[Minimax tuning| minimax]]"
|yes
| yes
|-
|-
|(t)
| (t)
|taxicab
| taxicab
|1
| 1
| rowspan="2" |S
| rowspan="2" | S
| rowspan="2" |simplicity-weight
| rowspan="2" | Simplicity-weight
| rowspan="2" |1/complexity
| rowspan="2" | 1/Complexity
|<set> minimax-S
| <set> Minimax-S
|<set> minimax simplicity-weight damage
| <set> Minimax simplicity-weight damage
|
| ย 
|yes
| Yes
|-
|-
|E
| E
|Euclidean
| Euclidean
|2
| 2
|<set> minimax-ES
| <set> Minimax-ES
|<set> minimax Euclideanized-simplicity-weight damage
| <set> Minimax Euclideanized-simplicity-weight damage
|
| ย 
|
| ย 
|-
|-
|(t)
| (t)
|taxicab
| Taxicab
|1
| 1
| rowspan="2" |C
| rowspan="2" | C
| rowspan="2" |complexity-weight
| rowspan="2" | Complexity-weight
| rowspan="2" |complexity
| rowspan="2" | Complexity
|<set> minimax-C
| <set> Minimax-C
|<set> minimax complexity-weight damage
| <set> Minimax complexity-weight damage
|
| ย 
|yes
| Yes
|-
|-
|E
| E
|Euclidean
| Euclidean
|2
| 2
|<set> minimax-EC
| <set> Minimax-EC
|<set> minimax Euclideanized-complexity-weight damage
| <set> Minimax Euclideanized-complexity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="3" |<n/a>
| colspan="3" | <n/a>
|U
| U
|unity-weight
| Unity-weight
|<none>
| <none>
| rowspan="5" |miniRMS
| rowspan="5" | MiniRMS
| rowspan="5" |2
| rowspan="5" | 2
|<set> miniRMS-U
| <set> MiniRMS-U
|<set> miniRMS unity-weight damage
| <set> MiniRMS unity-weight damage
|"[[least squares]]"
| "[[Least squares]]"
|yes
| Yes
|-
|-
|(t)
| (t)
|taxicab
| Taxicab
|1
| 1
| rowspan="2" |S
| rowspan="2" | S
| rowspan="2" |simplicity-weight
| rowspan="2" | Simplicity-weight
| rowspan="2" |1/complexity
| rowspan="2" | 1/Complexity
|<set> miniRMS-S
| <set> MiniRMS-S
|<set> miniRMS simplicity-weight damage
| <set> MiniRMS simplicity-weight damage
|
| ย 
|yes
| Yes
|-
|-
|E
| E
|Euclidean
| Euclidean
|2
| 2
|<set> miniRMS-ES
| <set> MiniRMS-ES
|<set> miniRMS Euclideanized-simplicity-weight damage
| <set> MiniRMS Euclideanized-simplicity-weight damage
|
| ย 
|
| ย 
|-
|-
|(t)
| (t)
|taxicab
| Taxicab
|1
| 1
| rowspan="2" |C
| rowspan="2" | C
| rowspan="2" |complexity-weight
| rowspan="2" | Complexity-weight
| rowspan="2" |complexity
| rowspan="2" | Complexity
|<set> miniRMS-C
| <set> MiniRMS-C
|<set> miniRMS complexity-weight damage
| <set> MiniRMS complexity-weight damage
|
| ย 
|yes
| Yes
|-
|-
|E
| E
|Euclidean
| Euclidean
|2
| 2
|<set> miniRMS-EC
| <set> MiniRMS-EC
|<set> miniRMS Euclideanized-complexity-weight damage
| <set> MiniRMS Euclideanized-complexity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="3" |<n/a>
| colspan="3" | <n/a>
|U
| U
|unity-weight
| Unity-weight
|<none>
| <none>
| rowspan="5" |minimean
| rowspan="5" | Miniaverage
| rowspan="5" |1
| rowspan="5" | 1
|<set> minimean-U
| <set> Miniaverage-U
|<set> minimean unity-weight damage
| <set> Miniaverage unity-weight damage
|
| ย 
|yes
| Yes
|-
|-
|(t)
| (t)
|taxicab
| Taxicab
|1
| 1
| rowspan="2" |S
| rowspan="2" | S
| rowspan="2" |simplicity-weight
| rowspan="2" | Simplicity-weight
| rowspan="2" |1/complexity
| rowspan="2" | 1/Complexity
|<set> minimean-S
| <set> Miniaverage-S
|<set> minimean simplicity-weight damage
| <set> Miniaverage simplicity-weight damage
|
| ย 
|yes
| Yes
|-
|-
|E
| E
|Euclidean
| Euclidean
|2
| 2
|<set> minimean-ES
| <set> Miniaverage-ES
|<set> minimean Euclideanized-simplicity-weight damage
| <set> Miniaverage Euclideanized-simplicity-weight damage
|
| ย 
|
| ย 
|-
|-
|(t)
| (t)
|taxicab
| Taxicab
|1
| 1
| rowspan="2" |C
| rowspan="2" | C
| rowspan="2" |complexity-weight
| rowspan="2" | Complexity-weight
| rowspan="2" |complexity
| rowspan="2" | Complexity
|<set> minimean-C
| <set> Miniaverage-C
|<set> minimean complexity-weight damage
| <set> Miniaverage complexity-weight damage
|
| ย 
|yes
| Yes
|-
|-
|E
| E
|Euclidean
| Euclidean
|2
| 2
|<set> minimean-EC
| <set> Miniaverage-EC
|<set> minimean Euclideanized-complexity-weight damage
| <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" |quantity
! colspan="2" | Quantity
! colspan="2" |unit
! colspan="2" | Unit
|-
|-
!abbreviation
! Abbreviation
!name
! Name
!symbol
! Symbol
!name
! Name
|-
|-
|U-damage
| U-damage
|unity-weight damage
| Unity-weight damage
|<math>\mathsf{ยข}\small\mathsf{(U)}</math>
| <math>\mathsf{ยข}\small\mathsf{(U)}</math>
|unity-weighted cents
| Unity-weighted cents
|-
|-
|C-damage
| C-damage
|complexity-weight damage
| Complexity-weight damage
|<math>\mathsf{ยข}\small\mathsf{(C)}</math>
| <math>\mathsf{ยข}\small\mathsf{(C)}</math>
|complexity-weighted cents
| 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
| Simplicity-weight damage
|<math>\mathsf{ยข}\small\mathsf{(S)}</math>
| <math>\mathsf{ยข}\small\mathsf{(S)}</math>
|simplicity-weighted cents
| 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
! colspan="2" | Quantity
!name
! colspan="2" | Unit
!symbol
!name
|-
|-
|C
! Abbreviation
|complexity
! Name
|<math>\small\mathsf{(C)}</math>
! Symbol
|complexity weight
! Name
|-
|-
|EC
| C
|Euclideanized complexity
| Complexity
|<math>\small\mathsf{(EC)}</math>
| <math>\small\mathsf{(C)}</math>
|Euclideanized-complexity weight
| Complexity weight
|-
|-
|S
| EC
|simplicity
| Euclideanized complexity
|<math>\small\mathsf{(S)}</math>
| <math>\small\mathsf{(EC)}</math>
|simplicity weight
| 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" |equivalent expressions
|-
! rowspan="2" |variable
! rowspan="2" | Equivalent expressions
! rowspan="2" |name
! rowspan="2" | Variable
! colspan="3" |units
! rowspan="2" | Name
! colspan="2" |shape
! colspan="3" | Units
! colspan="2" |type
! colspan="2" | Shape
! colspan="2" |EBK notation
! colspan="2" | Type
! colspan="4" |subobjects
! colspan="2" | EBK notation
! rowspan="2" |notes
! colspan="4" | Subobjects
! rowspan="2" | Notes
|-
|-
!unreduced
! Unreduced
!reduced
! Reduced
!read as
! Read as
!unreduced
! Unreduced
!reduced
! Reduced
!numeric
! Numeric
!structural
! Structural
!row-first
! Row-first
!col-first
! Col-first
!row
! Row
!col
! Column
!diag
! Diagonal
!entry
! Entry
|-
|-
! colspan="17" |mapping
! colspan="17" | Mapping
|-
|-
|
| ย 
|<math>\textbf{i}</math>
| <math>\textbf{i}</math>
|[[interval|(just) interval]]
| [[interval|(Just) interval]]
|
| ย 
|<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>
|primes
| Primes
|
| ย 
|<math>\scriptsize (d, 1)</math>
| <math>\scriptsize (d, 1)</math>
|integer
| Integer
|vector
| Vector
|
| ย 
|[...โŸฉ
| [...โŸฉ
|
| ย 
|
| ย 
|
| ย 
|<math>\mathrm{i}_i</math>
| <math>\mathrm{i}_i</math>
|specific type: [[prime-count vector]] (PC-vector)
| Specific type: vector ([[prime-count vector]] or PC-vector)
jargon name: monzo
Jargon name: monzo
|-
|-
|
| ย 
|<math>M</math>
| <math>M</math>
|[[Mapping|(temperament) mapping (matrix)]]
| [[Mapping|(Temperament) mapping (matrix)]]
|
| ย 
|<math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
| <math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
|generators per prime
| Generators per prime
|
| ย 
|<math>\scriptsize (r, d)</math>
| <math>\scriptsize (r, d)</math>
|integer
| Integer
|matrix
| Matrix
|[โŸจ...] ...}
| [โŸจ...] ...}
|โŸจ[...} ...]
| โŸจ[...} ...]
|<math>๐’Ž_i</math>
| <math>๐’Ž_i</math>
|
| ย 
|
| ย 
|<math>m_{ij}</math>
| <math>m_{ij}</math>
|jargon name: val list
| Jargon name: val list
|-
|-
|<math>M\textbf{i}</math>
| <math>M\textbf{i}</math>
|<math>\textbf{y}</math>
| <math>\textbf{y}</math>
|[[mapped interval]]
| [[Mapped interval]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
</math>
</math>
|<math>\small ๐—ด</math>
| <math>\small ๐—ด</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
| Integer
|vector
| Vector
|
| ย 
|[...}
| [...}
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|specific type: [[generator-count vector]] (GC-vector)
| Specific type: [[generator-count vector]] (GC-vector)
jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval
Jargon name: tmonzo; mnemonic: <math>\textbf{y}</math>nterval
|-
|-
|
| ย 
|<math>๐’Ž</math>
| <math>๐’Ž</math>
|[[map|(temperament) map]]
| [[map|(Temperament) map]]
|
| ย 
|<math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
| <math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
|generators per prime
| Generators per prime
|
| ย 
|<math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
|integer
| Integer
|vector
| Vector
|โŸจ...]
| โŸจ...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>m_i</math>
| <math>m_i</math>
|jargon name: val
| Jargon name: val
|-
|-
|<math>n + r</math>
| <math>n + r</math>
|<math>d</math>
| <math>d</math>
|[[dimensionality]]
| [[Dimensionality]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|<math>d - n</math>
| <math>d - n</math>
|<math>r</math>
| <math>r</math>
|[[rank]]
| [[Rank]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|<math>d - r</math>
| <math>d - r</math>
|<math>n</math>
| <math>n</math>
|[[nullity]]
| [[Nullity]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |tuning
! colspan="17" | Tuning
|-
|-
|<math>1200ร—\textbf{1}LG_{\text{j}}M_{\text{j}} \\
| <math>\slant{\mathbf{1}}L</math>
1200ร—\textbf{1}L \\
| <math>{\large\textbf{๐“}}\hspace{2mu}</math>
๐’ˆ_{\text{j}}M_{\text{j}}</math>
| [[Log-prime map]]
|<math>๐’‹</math>
|
|[[just tuning map|just(-prime) tuning map]]
| <math>\small\mathsf{oct}</math>/<math>\small ๐—ฝ</math>
|<math>\scriptsize ย 
| Octaves per prime
|
| <math>\scriptsize (1, d)</math>
| Real
| Vector
| โŸจ...]
|
|
|
|
| <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’ˆ_{\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} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array}
Line 2,508: Line 2,359:
\begin{array} {c} M_{\text{j}} \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array}
\begin{array} {c} M_{\text{j}} \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array}
</math>
</math>
|<math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
| <math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
|cents per prime
| 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} \textbf{1} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array}
\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
| Real
|vector
| Vector
|โŸจ...]
| โŸจ...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>j_i</math>
| <math>j_i</math>
|
| ย 
|-
|-
|<math>1200ร—\textbf{1}LG</math>
| <math>1200ร—\slant{\mathbf{1}}LG</math>
|<math>๐’ˆ</math>
| <math>๐’ˆ</math>
|[[generator tuning map]]
| [[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} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array}
Line 2,548: Line 2,399:
\begin{array} {c} G \\[-2pt] \cancel{๐—ฝ} \hspace{-2mu} / \hspace{-2mu} ๐—ด \end{array}
\begin{array} {c} G \\[-2pt] \cancel{๐—ฝ} \hspace{-2mu} / \hspace{-2mu} ๐—ด \end{array}
</math>
</math>
|<math>\mathsf{ยข}</math>/<math>\small ๐—ด</math>
| <math>\mathsf{ยข}</math>/<math>\small ๐—ด</math>
|cents per generator
| 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} \textbf{1} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array}
\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
| Real
|vector
| Vector
|{...]
| {...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>g_i</math>
| <math>g_i</math>
|
| ย 
|-
|-
|<math>1200ร—\textbf{1}LGM \\
| <math>1200ร—\slant{\mathbf{1}}LGM</math><br />
1200ร—\textbf{1}LP \\
<math>1200ร—\slant{\mathbf{1}}LP</math><br />
๐’ˆM</math>
<math>๐’ˆM</math>
|<math>๐’•</math>
| <math>๐’•</math>
|[[tuning map|(tempered-prime) 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} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \slant{\mathbf{1}} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array}
Line 2,590: Line 2,441:
\begin{array} {c} M \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array}
\begin{array} {c} M \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array}
</math>
</math>
|<math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
| <math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
|cents per prime
| Cents per prime
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\!\!
\! \! ย 
\begin{array} {c} 1200 \\[-3pt] (1ร—\cancel{1}) \end{array}
\begin{array} {c} 1200 \\[-3pt] \left(1, \cancel{1}\right) \end{array}
\!\!
\! \! ย 
\begin{array} {c} \textbf{i} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array}
\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
| Real
|vector
| Vector
|โŸจ...]
| โŸจ...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>t_i</math>
| <math>t_i</math>
|
| ย 
|-
|-
|<math>๐’• - ๐’‹ \\
| <math>๐’• - ๐’‹</math><br />
1200ร—\textbf{1}L(P - I)</math>
<math>1200ร—\slant{\mathbf{1}}L(P - I)</math>
|<math>๐’“</math>
| <math>๐’“</math>
|[[retuning map|retuning (or mistuning) map]]
| [[retuning map|Retuning (or mistuning) map]]
|
| ย 
|<math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
| <math>\mathsf{ยข}</math>/<math>\small ๐—ฝ</math>
|cents per prime
| Cents per prime
|
| ย 
|<math>\scriptsize (1, d)</math>
| <math>\scriptsize (1, d)</math>
|real
| Real
|vector
| Vector
|โŸจ...]
| โŸจ...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>r_i</math>
| <math>r_i</math>
|previous name: prime error map
| 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|(just) (interval) size]]
| [[interval span|(Just) (interval) size]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \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
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|mnemonic: <math>\mathrm{o}</math>riginal size
| Mnemonic: <math>\mathrm{o}</math>riginal size
|-
|-
|<math>๐’ˆM\textbf{i} \\
| <math>๐’ˆM\textbf{i}</math><br />
๐’•\textbf{i}</math>
<math>๐’•\textbf{i}</math>
|<math>\mathrm{a}</math>
| <math>\mathrm{a}</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Example_3|tempered (interval) size]]
| {{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{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \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
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|mnemonic: <math>\mathrm{a}</math>ltered size
| 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|(interval) error]]
| [[error|(Interval) error]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \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
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |optimization
! colspan="17" | Optimization
|-
|-
|
| ย 
|<math>p</math>
| <math>p</math>
|[[optimization power]]
| [[Optimization power]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|
| ย 
|<math>โŸช ยท โŸซ_p</math>
| <math>\llangle\,ยท\,\rrangle_p</math>
|[[power mean]] (<math>p</math>-mean)
| [[Power mean]] (<math>p</math>-mean)
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |damage
! colspan="17" | Damage
|-
|-
|<math>s^{-1}</math>
| ย 
|<math>c</math>
| <math>c</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity|complexity]]
| {{subpage|Tuning_fundamentals|prev|s=complexity}}
| colspan="3" |(see complexities section of complexities and simplicities table)
| colspan="3" | (See complexities section of complexities and simplicities table)
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|<math>c^{-1}</math>
| <math>\dfrac1c</math>
|<math>s</math>
| <math>s</math>
|[[simplicity]]
| [[Simplicity]]
| colspan="3" |(see simplicities section of complexities and simplicities table)
| colspan="3" | (See simplicities section of complexities and simplicities table)
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| 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" |(see complexities and simplicities table)
| colspan="3" | (See complexities and simplicities table)
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|<math>|\mathrm{e}|w</math>
| <math>\abs{\mathrm{e}} w</math>
|<math>\mathrm{d}</math>
| <math>\mathrm{d}</math>
|[[damage]]
| [[Damage]]
| colspan="3" |(see damages table)
| colspan="3" | (See damages table)
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|real
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |target-intervals
! colspan="17" | Target-intervals
|-
|-
|
| ย 
|<math>\mathrm{T}</math>
| <math>\mathrm{T}</math>
|[[target-interval list]]
| [[Target-interval list]]
|
| ย 
|<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>
|primes
| Primes
|
| ย 
|<math>\scriptsize (d, k)</math>
| <math>\scriptsize (d, k)</math>
|integer
| Integer
|matrix
| Matrix
|
| ย 
|[[...โŸฉ ...]
| [[...โŸฉ ...]
|
| 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]]
| [[Mapped target-interval list]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
</math>
</math>
|<math>\small ๐—ด</math>
| <math>\small ๐—ด</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 
\! \! 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
| Integer
|matrix
| 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'
| 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>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Primes|target-interval (just) size list]]
| {{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{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \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} \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
| Real
|list
| List
|[...]
| [...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\mathrm{o}_i</math>
| <math>\mathrm{o}_i</math>
|mnemonic: <math>\textbf{o}</math>riginal size list
| Mnemonic: <math>\textbf{o}</math>riginal size list
|-
|-
|<math>๐’•\mathrm{T}</math>
| <math>๐’•\mathrm{T}</math>
|<math>\textbf{a}</math>
| <math>\textbf{a}</math>
|[[tempered target-interval size list]]
| [[Tempered target-interval size list]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \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} \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
| Real
|list
| List
|[...]
| [...]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\mathrm{a}_i</math>
| <math>\mathrm{a}_i</math>
|mnemonic: <math>\textbf{a}</math>ltered size list
| Mnemonic: <math>\textbf{a}</math>ltered size list
|-
|-
|<math>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{๐—ฝ} \end{array} ย 
\begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} ย 
\begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \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} \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
| Real
|list
| List
|[...]
| [...]
|
|
|
|
|
|
|
|
|<math>\mathrm{e}_i</math>
| <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 ๐—ฝ</math>
| Primes
|
| <math>\scriptsize (d, h)</math>
|
| Matrix
|
| [[...โŸฉ ...]
|
| <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 ๐—ฝ</math>
| Primes
|
| <math>\scriptsize (d, n)</math>
| Integer
| Matrix
|
| [[...โŸฉ ...]
|
| <math>\textbf{c}_i</math>
|
| <math>\mathrm{c}_{ij}</math>
| Jargon name: monzo list
|-
|
| <math>\textbf{c}</math>
| [[Comma]]
|
| <math>\small ๐—ฝ</math>
| Primes
|
| <math>\scriptsize (d, 1)</math>
| Integer
| Vector
|
| [...โŸฉ
|
|
|
| <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>C</math> or <math>S</math>
| <math>\mathrm{I}</math>
|<math>W</math>
| <math>\mathrm{T}_{\text{p}}</math>
|[[target-interval weight matrix]]
| [[Prime proxy target-interval list]]
| colspan="3" |(see complexities and simplicities table)
| ย 
|
| <math>\small ๐—ฝ</math>
|<math>\scriptsize (k, k)</math>
| Primes
|real
| ย 
|matrix
| <math>\scriptsize (d, d)</math>
|
| Integer
|[[...] ...]
| Matrix
|
| ย 
|
| โŸจ[...โŸฉ ...]
|<math>๐’˜</math>
| ย 
|<math>w_i</math> or [math]w_{ij}[/math]
| ย 
|
| <math>\slant{\mathbf{1}}</math>
| ย 
| ย 
|-
|-
|<math>S^{-1}</math>
| ย 
|<math>C</math>
| <math>X</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity-weight_damage|target-interval complexity weight matrix]]
| {{subpage|alternative complexities|uprev|s=Prescaling_vs._pretransforming|text=Complexity pretransformer}}
| colspan="3" |(see complexities section of complexities and simplicities table)
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math><ref group="note">In these tables, "alternative" means any complexity other than the default of log-product complexity, and "alt" stands for its abbreviation.</ref>
|
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math>
|<math>\scriptsize (k, k)</math>
| Complexity weight or <alternative>-complexity weight
|real
| ย 
|matrix
| <math>\scriptsize (d, d)</math> or <math>\scriptsize (d+1, d+1)</math>
|
| Real
|[[...] ...]
| Matrix
|
| [โŸจ...] ...โŸฉ
|
|
|<math>๐’„</math>
| <math>๐’™_i</math>
|<math>c_i</math>
| ย 
|
| <math>๐’™</math>
| <math>x_i</math> or <math>x_{ij}</math>
| ย 
|-
|-
|<math>C^{-1}</math>
| <math>\text{diag}({\large\textbf{๐“}}\hspace{2mu})</math>
|<math>S</math>
| <math>L</math>
|[[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#Complexity-weight_damage|target-interval simplicity weight matrix]]
| [[Log-prime matrix]]
| colspan="3" |(see simplicities section of complexities and simplicities table)
| ย 
|
| <math>\small\mathsf{oct}</math>/<math>\small ๐—ฝ</math>
|<math>\scriptsize (k, k)</math>
| Octaves per prime
|real
| ย 
|matrix
| <math>\scriptsize (d, d)</math>
|
| Real
|[[...] ...]
| Matrix
|
| [โŸจ...] ...โŸฉ
|
| โŸจ[...โŸฉ ...]
|<math>๐’”</math>
| <math>{\large\textbf{๐“}}\hspace{2mu}_i</math>
|<math>s_i</math>
| ย 
|
| <math>{\large\textbf{๐“}}\hspace{2mu}</math>
| <math>{\large ๐“}\hspace{2mu}_{ij}</math>
| ย 
|-
|-
|<math>|\textbf{e}|W \\
|
1200ร—\textbf{1}L|P - I|\mathrm{T}W</math>
| <math>q</math>
|<math>\textbf{d}</math>
| {{subpage|all-interval_tuning_schemes|uprev|s=dual_norms|text=Interval complexity norm power}}
|[[target-interval damage list]]
| ย 
| colspan="3" |(see damages table)
| ย 
|
| ย 
|<math>\scriptsize (1, k)</math>
| ย 
|real
| <math>\scriptsize (1, 1)</math>
|list
| Real
|[...]
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\mathrm{d}_i</math>
| ย 
|
| ย 
| ย 
|-
|-
|
| ย 
|<math>k</math>
| <math>\norm{ยท}_q</math>
|[[target-interval count]]
| [[Power norm]] (<math>p</math>-norm)
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|mnemonic: <math>k</math>ount
| ย 
|-
|-
! colspan="17" |unchanged-intervals
| <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>\mathrm{U}</math>
| <math>\norm{X\mathbf{i}}_q</math>
|[[unchanged-interval basis]]
| [[Interval complexity]]
|
| ย 
|<math>\small ๐—ฝ</math>
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math>
|primes
| ย 
|
| ย 
|<math>\scriptsize (d, r)</math>
| <math>\scriptsize (1, 1)</math>
|
| Real
|matrix
| Scalar
|
| ย 
|[[...โŸฉ ...]
| ย 
|
| ย 
|<math>\textbf{u}_i</math>
| ย 
|
| ย 
|<math>\mathrm{u}_{ij}</math>
| ย 
|jargon name: eigenmonzo list
| ย 
|-
|-
|
| ย 
|<math>h</math>
| <math>\norm{๐’“X^{-1}}_{\text{dual}(q)}</math>
|[[unchanged-interval count]]
| [[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>
|integer
| Real
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |exploring temperaments
! colspan="17" | Alternative complexities
|-
|-
|
| ย 
|<math>\mathrm{C}</math>
| <math>๐’‘</math>
|[[comma basis]]
| {{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 
|<math>\small ๐—ฝ</math>
|
|primes
|
|
| ย 
|<math>\scriptsize (d, n)</math>
| <math>\scriptsize (1, d)</math>
|integer
| Integer
|matrix
| List
|
| [...]
|[[...โŸฉ ...]
| ย 
|
| ย 
|<math>\textbf{c}_i</math>
|
|
| ย 
|<math>\mathrm{c}_{ij}</math>
| <math>p_i</math>
|jargon name: monzo list
| ย 
|-
|-
|
| ย 
|<math>\textbf{c}</math>
| <math>\slant{\mathbf{1}}</math>
|[[comma]]
| {{subpage|alternative complexities|uprev|s=proportionality to size|text=Summation map}}
|
| ย 
|<math>\small ๐—ฝ</math>
| ย 
|primes
| ย 
|
| ย 
|<math>\scriptsize (d, 1)</math>
| <math>\scriptsize (1, d)</math>
|integer
| Integer
|vector
| Vector
|
| โŸจ...]
|[...โŸฉ
|
|
| ย 
|
| ย 
|
| ย 
|<math>\mathrm{c}_i</math>
| <math>1</math>
|specific type: [[prime-count vector]] (PC-vector)
| ย 
|-
|-
! colspan="17" |computation
|
| <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>G</math>
| <math>Z</math>
|[[generator embedding matrix|generator embedding (matrix)]]
| {{subpage|alternative complexities|uprev|s=Normifying: size-sensitizing matrix|text=Size-sensitizing matrix}}
|
| ย 
|<math>\small ๐—ฝ</math>/<math>\small ๐—ด</math>
| ย 
|primes per generator
| ย 
|
| ย 
|<math>\scriptsize (d, r)</math>
| <math>\scriptsize (d+1, d)</math>
|real
| Real
|matrix
| Matrix
|[{...] ...โŸฉ
| [โŸจโ€ฆ]...]
|{[...โŸฉ ...]
| ย 
|<math>๐’ˆ_i</math>
| <math>๐’›_i</math>
|
| ย 
|
| ย 
|<math>g_{ij}</math>
| <math>z_{ij}</math>
|
| ย 
|-
|-
|
! colspan="17" | Non-standard domain bases
|<math>K</math>
|[[constraint (matrix)]]
|
|
|
|
|<math>\scriptsize (r, k)</math>
|<math>\scriptsize \{0, +1, -1\}</math>
|matrix
|[[...] ...]
|
|<math>๐’Œ_i</math>
|
|
|<math>k_{ij}</math>
|mnemonic: <math>K</math>onstraint
|-
|-
|
| rowspan="2" | ย 
|<math>{\largeโง›}ยท{\largeโงš}_p</math>
| <math>B_s</math>
|[[power sum]] (<math>p</math>-sum)
| rowspan="2" | [[Domain_basis#Basis_matrix_conversion|(Domain) basis (change) matrix]]
|
| rowspan="2" |
|
| <math>\small ๐—ฝ</math>/<math>\small ๐—ฏ</math>
|
| Primes per nonprime basis elements
|
| rowspan="2" | ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (d_p, d_b)</math>
|real
| rowspan="2" | Integer
|scalar
| 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 
|-
|-
! colspan="17" |projection
| <math>B_{Ls}</math>
| <math>\small ๐—•</math>/<math>\small ๐—ฏ</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|projection (matrix)]]
| <math>G</math>
|<math>\scriptsize ย 
| [[generator embedding matrix|Generator embedding (matrix)]]
|
| <math>\small ๐—ฝ</math>/<math>\small ๐—ด</math>
| Primes per generator
|
| <math>\scriptsize (d, r)</math>
| Real
| Matrix
| [{...] ...โŸฉ
| {[...โŸฉ ...]
| <math>๐’ˆ_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] ๐—ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array}
\begin{array} {c} G \\[-2pt] ๐—ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} M \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array}
\begin{array} {c} M \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array}
</math>
</math>
|<math>\small ๐—ฝ</math>/<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>/<math>\small ๐—ฝ</math>
|primes per prime
| 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
| Real
|matrix
| Matrix
|[โŸจ...] ...โŸฉ
| [โŸจ...] ...โŸฉ
|โŸจ[...โŸฉ ...]
| โŸจ[...โŸฉ ...]
|<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]]
| [[Projected interval]]
|<math>\scriptsize ย 
| <math>\scriptsize ย 
\begin{array} {c} G \\[-2pt] ๐—ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array}
\begin{array} {c} G \\[-2pt] ๐—ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \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{๐—ฝ} \end{array}
\begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array}
</math>
</math>
|<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>
|primes
| 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) \end{array}
\begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array}
\!\!
\! \! ย 
</math>
|<math>\scriptsize (d, 1)</math>
|real
|vector
|
|[...โŸฉ
|
|
|
|
|specific type: [[prime-count vector]] (PC-vector)
|-
|
|<math>\textit{ฮ›}</math>
|[[scaling factor (eigenvalue) matrix|scaling factor matrix]]
|
|
|
|
|<math>\scriptsize (d, d)</math>
|
|matrix
|[โŸจโ€ฆ] โ€ฆโŸฉ
|โŸจ[โ€ฆโŸฉ โ€ฆ]
|
|
|<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
|-
|
|<math>\mathrm{V}</math>
|[[unrotated vector (eigenvector) list|unrotated vector list]]
|
|<math>\small ๐—ฝ</math>
|primes
|
|<math>\scriptsize (d, d)</math>
|
|matrix
|
|โŸจ[...โŸฉ ...]
|
|<math>\textbf{v}_i</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
|-
|
|<math>F</math>
|[[generator form matrix]]
|
|
|
|
|<math>\scriptsize (r, r)</math>
|
|matrix
|[{...] โ€ฆ}
|
|
|<math>๐’‡_i</math>
|
|<math>f_{ij}</math>
|
|-
! colspan="17" |JI equivalents
|-
|<math>I</math>
|<math>M_{\text{j}}</math>
|[[JI mapping (matrix)]]
|
|<math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
|generators per prime
|
|<math>\scriptsize (d, d)</math>
|integer
|matrix
|[โŸจ...] ...}
|โŸจ[...} ...]
|
|
|<math>๐Ÿ</math>
|
|
|-
|<math>1200ร—\textbf{1}LG_{\text{j}}</math>
|<math>๐’ˆ_{\text{j}}</math>
|[[JI generator tuning map]]
|<math>\scriptsize
\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} \textbf{1} \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{\mathsf{oct}} \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\begin{array} {c} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array}
\begin{array} {c} \\[-2pt] ยท \end{array}
\\ \scriptsize \quad
\begin{array} {c} G_{\text{j}} \\[-2pt] \cancel{๐—ฝ} \hspace{-2mu} / \hspace{-2mu} ๐—ด \end{array}
</math>
|<math>\mathsf{ยข}</math>/<math>\small ๐—ด</math>
|cents per generator
|<math>\scriptsize
\!\!
\begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array}
\!\!
\begin{array} {c} \textbf{1} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array}
\!\!
\begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array}
\\ \scriptsize \quad
\!\!
\begin{array} {c} G_{\text{j}} \\[-3pt] (\cancel{d}, r) \end{array}
\!\!
</math>
</math>
|<math>\scriptsize (1, d)</math>
| <math>\scriptsize (d, 1)</math>
|real
| Real
|vector
| Vector
|{...]
| ย 
|
| [...โŸฉ
|
| ย 
|
| ย 
|
| ย 
|<math>g_{\text{j}i}</math>
| ย 
|
| Specific type: vector ([[prime-count vector]] or PC-vector)
|-
|<math>I</math>
|<math>G_{\text{j}}</math>
|[[JI generator embedding matrix|JI generator embedding (matrix)]]
|
|<math>\small ๐—ฝ</math>/<math>\small ๐—ด</math>
|primes per generator
|
|<math>\scriptsize (d, d)</math>
|integer
|matrix
|[{...] ...โŸฉ
|{[...โŸฉ ...]
|
|
|<math>๐Ÿ</math>
|
|
|-
|-
! colspan="17" |all-interval tuning schemes
| ย 
| <math>\mathrm{U}</math>
| [[Unchanged-interval basis]]
|
| <math>\small ๐—ฝ</math>
| Primes
|
| <math>\scriptsize (d, r)</math>
|
| Matrix
|
| [[...โŸฉ ...]
|
| <math>\textbf{u}_i</math>
|
| <math>\mathrm{u}_{ij}</math>
| Jargon name: eigenmonzo list
|-
|-
|<math>I</math>
| ย 
|<math>\mathrm{T}_{\text{p}}</math>
| <math>\textit{ฮ›}</math>
|[[prime proxy target-interval (matrix)]]
| [[scaling factor matrix|Scaling factor (eigenvalue) matrix]]
|
| ย 
|<math>\small ๐—ฝ</math>
| ย 
|primes
| ย 
|
| ย 
|<math>\scriptsize (d, d)</math>
| <math>\scriptsize (d, d)</math>
|integer
| ย 
|matrix
| Matrix
|
| [โŸจโ€ฆ] โ€ฆโŸฉ
|โŸจ[...โŸฉ ...]
| โŸจ[โ€ฆโŸฉ โ€ฆ]
|
| ย 
|
| ย 
|<math>๐Ÿ</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>C_{\text{p}}</math>
| <math>\mathrm{V}</math>
|[[complexity pretransformer]]
| [[unrotated vector list|Unrotated vector (eigenvector) list]]
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math><ref>In these tables, "alternative" means any complexity other than the default of log-product complexity, and "alt" stands for its abbreviation.</ref>
| ย 
|<math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math>
| <math>\small ๐—ฝ</math>
|complexity weight or <alternative>-complexity weight
| Primes
|
| ย 
|<math>\scriptsize (d, d)</math> or <math>\scriptsize (d+1, d+1)</math>
| <math>\scriptsize (d, d)</math>
|real
| ย 
|matrix
| Matrix
|[โŸจ...] ...โŸฉ
|
|
| โŸจ[...โŸฉ ...]
|<math>๐’„_{\text{p}_i}</math>
| ย 
|
| <math>\textbf{v}_i</math>
|<math>๐’„_{\text{p}}</math>
| ย 
|<math>c_{\text{p}i}</math> or [math]c_{\text{p}ij}[/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>S_{\text{p}}</math>
| <math>F</math>
|[[simplicity pretransformer]]
| [[Generator form matrix]]
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)}</math> or <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{S)}</math>
| ย 
|<math>\small\mathsf{(S)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math>
| ย 
|simplicity weight or <alternative>-simplicity weight
| ย 
|
| ย 
|<math>\scriptsize (d, d)</math> or <math>\scriptsize (d+1, d+1)</math>
| <math>\scriptsize (r, r)</math>
|real
| ย 
|matrix
| Matrix
|
| [{...] โ€ฆ}
|โŸจ[...โŸฉ ...]
| ย 
|<math>๐’”_{\text{p}i}</math>
| ย 
|
| <math>๐’‡_i</math>
|<math>๐’”_{\text{p}}</math>
|
|<math>s_{\text{p}i}</math> or [math]s_{\text{p}ij}[/math]
| <math>f_{ij}</math>
|
| ย 
|-
|-
|<math>\text{diag}(\log_2(\textbf{p}))</math>
| <math>I</math>
|<math>L</math>
| <math>M_{\text{j}}</math>
|[[log-prime matrix]]
| [[Generator_embedding_optimization#Algebraic_setup|JI mapping (matrix)]]
|
| ย 
|<math>\small\mathsf{oct}</math>/<math>\small ๐—ฝ</math>
| <math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
|octaves per prime
| Generators per prime
|
| ย 
|<math>\scriptsize (d, d)</math>
| <math>\scriptsize (d, d)</math>
|real
| Integer
|matrix
| Matrix
|[โŸจ...] ...โŸฉ
| [โŸจ...] ...}
|โŸจ[...โŸฉ ...]
| โŸจ[...} ...]
|<math>\textbf{๐“}_i</math>
| ย 
|
| ย 
|<math>\textbf{๐“}</math>
| <math>\slant{\mathbf{1}}</math>
|<math>๐“_{ij}</math>
| ย 
|
| ย 
|-
|-
|
| <math>I</math>
|<math>q</math>
| <math>G_{\text{j}}</math>
|[[interval complexity norm power]]
| [[Generator_embedding_optimization#Algebraic_setup|JI generator embedding (matrix)]]
|
| ย 
|
| <math>\small ๐—ฝ</math>/<math>\small ๐—ด</math>
|
| Primes per generator
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (d, d)</math>
|real
| Integer
|scalar
| Matrix
|
| [{...] ...โŸฉ
|
| {[...โŸฉ ...]
|
| ย 
|
| ย 
|
| <math>\slant{\mathbf{1}}</math>
|
| ย 
|
| ย 
|-
|-
|
| ย 
|<math>โ€– ยท โ€–_q</math>
| <math>K</math>
|[[power norm]] (<math>q</math>-norm)
| [[Generator_embedding_optimization#How_to_build_constraint_matrices|Constraint (matrix)]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (k, r)</math>
|real
| <math>\scriptsize \{0, +1, -1\}</math>
|scalar
| Matrix
|
| [[...] ...]
|
| ย 
|
| <math>๐’Œ_i</math>
|
| ย 
|
| ย 
|
| <math>k_{ij}</math>
|
| Mnemonic: <math>K</math>onstraint
|-
|-
! colspan="17" |alternative complexities
| ย 
| <math>๐’ƒ</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>๐’‘</math>
| <math>B</math>
|[[prime list]]<ref>May be used for a prime-limit or for any prime-only list.</ref>
| [[Generator embedding optimization#How to identify tunings|(Generator tuning map) blend matrix]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, d)</math>
| <math>\scriptsize (d, ฯ„-1)</math>
|integer
| Real
|list
| Matrix
|[...]
| [[...โŸฉ...]
|
| ย 
|
| ย 
|
| <math>๐’ƒ_{i}</math>
|
| ย 
|<math>p_i</math>
| <math>b_{ij}</math>
|
| ย 
|-
|-
|
| ย 
|<math>Z</math>
| <math>D</math>
|[[size-sensitizing matrix]]
| [[Generator embedding optimization#The deltas matrix|(Generator tuning map) deltas matrix]]
|
| ย 
|
| <math>\mathsf{ยข}</math>/<math>\small ๐—ด</math>
|
| Cents per generator
|
| ย 
|<math>\scriptsize (d+1, d)</math>
| <math>\scriptsize (ฯ„-1,r)</math>
|real
| Real
|matrix
| Matrix
|[โŸจโ€ฆ]...]
| [{...] ...]
|
| ย 
|<math>๐’›_i</math>
| <math>๐œน_i</math>
|
| ย 
|
| ย 
|<math>z_{ij}</math>
| <math>๐›ฟ_{ij}</math>
|
| ย 
|-
|-
! colspan="17" |non-standard domain bases
| ย 
| <math>ฯ„</math>
| [[Generator embedding optimization#The deltas matrix|Tied basic minimax tuning count]]
|
|
|
|
|
| Integer
| Scalar
|
|
|
|
|
|
|
|-
|-
| rowspan="2" |
! colspan="17" | Exterior algebra
|<math>B_s</math>
| rowspan="2" |[[(domain) basis (change) matrix]]
| rowspan="2" |
|<math>\small ๐—ฝ</math>/<math>\small ๐—ฏ</math>
|primes per nonprime basis elements
| rowspan="2" |
|<math>\scriptsize (d_p, d_b)</math>
| rowspan="2" |integer
| rowspan="2" |matrix
| rowspan="2" |[[...] ...]
| rowspan="2" |[[...] ...]
| rowspan="2" |
| rowspan="2" |<math>b_i</math>
| rowspan="2" |
| rowspan="2" |<math>b_{ij}</math>
| rowspan="2" |
|-
|-
|<math>B_{Ls}</math>
|
|<math>\small ๐—•</math>/<math>\small ๐—ฏ</math>
| <math>๐•ž</math>
|superspace basis elements per (subspace) basis elements
| [[Multimap]]
|<math>\scriptsize (d_L, d_s)</math>
|
| <math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
| Generators per prime
|
| <math>\scriptsize (1, d)</math>
| Integer
| Multivector
| โŸจ...] or โŸจโŸจ...]] or โŸจโŸจโŸจ...]]] ...
|
|
|
|
| <math>๐•ž_i</math>
|
|-
|-
|
| ย 
|<math>ฮฆ</math>
| <math>๐•”</math>
|[[generator preimage transversal]]
| [[Multicomma]]
|
| ย 
|<math>\small ๐—ฝ</math>/<math>\small ๐—ด</math>
| <math>\small ๐—ฝ</math>
|primes per generator
| Primes
|
| ย 
|<math>\scriptsize (d, r)</math>
| <math>\scriptsize (1, n)</math>
|integer
| Integer
|matrix
| Multivector
|
| ย 
|{[...โŸฉ ...]
| [...โŸฉ or [[...โŸฉโŸฉ or [[[...โŸฉโŸฉโŸฉ ...
|
| ย 
|<math>\mathbf{ฯ•}_i</math>
| ย 
|
| ย 
|<math>\mathrm{ฯ•}_{ij}</math>
| <math>๐•”_i</math>
|<math>Mฮฆ = I</math>
| ย 
|-
|-
! colspan="17" |exterior algebra
| ย 
| <math>๐•ง</math>
| (Generic temperament multivector)
|
|
|
|
| <math>\scriptsize (1, {{d}\choose{r}})</math> or <math>\scriptsize (1, {{d}\choose{n}})</math>
| Integer
| Multivector
| โŸจ...] or โŸจโŸจ...]] or โŸจโŸจโŸจ...]]] ...
| [...โŸฉ or [[...โŸฉโŸฉ or [[[...โŸฉโŸฉโŸฉ ...
|
|
|
| <math>๐•ง_i</math>
|
|-
|-
|
| ย 
|<math>๐•ž</math>
| <math>A</math>
|[[multimap]]
| (Generic temperament matrix)
|
|
|<math>\small ๐—ด</math>/<math>\small ๐—ฝ</math>
|
|generators per prime
|
|
| ย 
|<math>\scriptsize (1, d)</math>
| <math>\scriptsize (g, d)</math> or <math>\scriptsize (d, g)</math>
|integer
| Integer
|multivector
| Matrix
|โŸจ...] or โŸจโŸจ...]] or โŸจโŸจโŸจ...]]] ...
| [โŸจ...] ...}
|
| โŸจ[...} ...] or [[...โŸฉ ...]
|
| <math>๐’‚_i</math>
|
| <math>๐’‚_i</math>
|
| <math>๐’‚</math>
|<math>๐•ž_i</math>
| <math>a_{ij}</math>
|
| ย 
|-
|-
|
| ย 
|<math>๐•”</math>
| <math>v</math>
|[[multicomma]]
| [[Variance]]
|
| ย 
|<math>\small ๐—ฝ</math>
| ย 
|primes
| ย 
|
| ย 
|<math>\scriptsize (1, n)</math>
| ย 
|integer
| ย 
|multivector
| ย 
|
| ย 
|[...โŸฉ or [[...โŸฉโŸฉ or [[[...โŸฉโŸฉโŸฉ ...
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>๐•”_i</math>
| ย 
|
| ย 
|-
|-
|
| ย 
|<math>๐•ง</math>
| <math>g</math>
|[[(generic temperament multivector)]]
| [[Grade]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, {{d}\choose{r}})</math> or <math>\scriptsize (1, {{d}\choose{n}})</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|multivector
| Scalar
|โŸจ...] or โŸจโŸจ...]] or โŸจโŸจโŸจ...]]] ...
| ย 
|[...โŸฉ or [[...โŸฉโŸฉ or [[[...โŸฉโŸฉโŸฉ ...
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>๐•ง_i</math>
| ย 
|
| ย 
|-
|-
|
! colspan="17" | Temperament addition
|<math>A</math>
|[[(generic temperament matrix)]]
|
|
|
|
|<math>\scriptsize (g, d)</math> or <math>\scriptsize (d, g)</math>
|integer
|matrix
|[โŸจ...] ...}
|โŸจ[...} ...] or [[...โŸฉ ...]
|<math>๐’‚_i</math>
|<math>๐’‚_i</math>
|<math>๐’‚</math>
|<math>a_{ij}</math>
|
|-
|-
|
| <math>\min(r, n)</math>
|<math>v</math>
| <math>g_\text{min}</math>
|[[variance]]
| [[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>g</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
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
! colspan="17" |temperament addition
| 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
| [โŸจ...]] or [[...] ...โŸฉ
| โŸจ[...]] or [[...โŸฉ ...]
| <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>
|
|-
|-
|<math>\min(r, n)</math>
| ย 
|<math>g_\text{min}</math>
| <math>L_\text{ind}</math>
|[[min-grade]]
| [[Temperament_addition#Glossary|Linear-independence basis]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize \left(l_\text{ind}, d\right)</math> or <math>\scriptsize \left(d, l_\text{ind}\right)</math>
|integer
| Integer
|scalar
| Matrix
|
| [โŸจ...]] or [[...] ...โŸฉ
|
| โŸจ[...]] or [[...โŸฉ ...]
|
| <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>\max(r, n)</math>
| <math>\dim(L_\text{dep})</math>
|<math>g_\text{max}</math>
| <math>l_\text{dep}</math>
|[[max-grade]]
| [[Temperament_addition#3._Linear_independence_between_temperaments|Linear-dependence]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (1, 1)</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|scalar
| Scalar
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|-
|-
|
| <math>\dim(L_\text{ind})</math>
|<math>L_\text{dep}</math>
| <math>l_\text{ind}</math>
|[[linear-dependence basis]]
| [[Temperament_addition#3._Linear_independence_between_temperaments|Linear-independence]]
|
| ย 
|
| ย 
|
| ย 
|
| ย 
|<math>\scriptsize (l_\text{dep}, d)</math> or <math>\scriptsize (d, l_\text{dep})</math>
| <math>\scriptsize (1, 1)</math>
|integer
| Integer
|matrix
| Scalar
|[โŸจ...]] or [[...] ...โŸฉ
| ย 
|โŸจ[...]] or [[...โŸฉ ...]
| ย 
|<math>\textbf{๐“}_{\text{dep}i}</math>
| ย 
|<math>\textbf{๐“}_{\text{dep}i}</math>
| ย 
|<math>\textbf{๐“}_\text{dep}</math>
| ย 
|<math>๐“_{\text{dep}ij}</math>
| ย 
|
| ย 
|-
|
|<math>L_\text{ind}</math>
|[[linear-independence basis]]
|
|
|
|
|<math>\scriptsize (l_\text{ind}, d)</math> or <math>\scriptsize (d, l_\text{ind})</math>
|integer
|matrix
|[โŸจ...]] or [[...] ...โŸฉ
|โŸจ[...]] or [[...โŸฉ ...]
|<math>\textbf{๐“}_{\text{ind}i}</math>
|<math>\textbf{๐“}_{\text{ind}i}</math>
|<math>\textbf{๐“}_\text{ind}</math>
|<math>๐“_{\text{ind}ij}</math>
|
|-
|<math>\dim(L_\text{dep})</math>
|<math>l_\text{dep}</math>
|[[linear-dependence]]
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|integer
|scalar
|
|
|
|
|
|
|
|-
|<math>\dim(L_\text{ind})</math>
|<math>l_\text{ind}</math>
|[[linear-independence]]
|
|
|
|
|<math>\scriptsize (1, 1)</math>
|integer
|scalar
|
|
|
|
|
|
|
|}
|}


===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
! Symbol
!vectorized
! Name
! Vectorized
|-
|-
|<math>\small ๐—ด</math>
| <math>\small ๐—ด</math>
|generators
| Generators
|yes
| Yes
|-
|-
|<math>\small ๐—ฝ</math>
| <math>\small ๐—ฝ</math>
|primes
| Primes
|yes
| Yes
|-
|-
|<math>\small ๐—ฏ</math>
| <math>\small ๐—ฏ</math>
|(subspace) basis elements
| (Subspace) basis elements
|yes
| Yes
|-
|-
|<math>\small ๐—•</math>
| <math>\small ๐—•</math>
|superspace basis elements
| Superspace basis elements
|yes
| Yes
|-
|-
|<math>\mathsf{ยข}</math>
| <math>\mathsf{ยข}</math>
|cents
| Cents
|
| ย 
|-
|-
|<math>\mathsf{ยข}\small{(}</math><weight><math>\small\mathsf{)}</math>
| <math>\mathsf{ยข}\small{(}</math><weight><math>\small\mathsf{)}</math>
|weighted cents
| Weighted cents
|
| ย 
|-
|-
|<math>\small\mathsf{oct}</math>
| <math>\small\mathsf{oct}</math>
|octaves
| 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" |retuning (or mistuning) magnitude
|-
! colspan="12" rowspan="1" |damage
! colspan="6" rowspan="3" | Retuning (or mistuning) magnitude
! rowspan="5" |target
! colspan="12" rowspan="1" | Damage
ย 
! rowspan="5" | Target<br />intervals
intervals
! colspan="2" rowspan="4" | Systematic name
! colspan="2" rowspan="4" |systematic name
! rowspan="5" | Previously named tuning schemes that are specific types of this tuning scheme
! rowspan="5" |previously named tuning schemes that are specific types of this tuning scheme
! rowspan="5" | Of interest?
! rowspan="5" |of interest?
|-
|-
! colspan="9" rowspan="1" |weight
! colspan="9" rowspan="1" | Weight
! colspan="3" rowspan="1" |optimization
! colspan="3" rowspan="1" | Optimization
|-
|-
! colspan="6" rowspan="1" |interval complexity
! colspan="6" rowspan="1" | Interval complexity
! colspan="3" rowspan="1" |slope
! colspan="3" rowspan="1" | Slope
! colspan="1" rowspan="3" |initial
! colspan="1" rowspan="3" | Initial
! colspan="1" rowspan="3" |name
! colspan="1" rowspan="3" | Name
! colspan="1" rowspan="3" |power
! colspan="1" rowspan="3" | Power
|-
|-
! colspan="3" rowspan="1" |norm pretransformer
! colspan="3" rowspan="1" | Norm pretransformer
! colspan="3" rowspan="1" |norm power
! colspan="3" rowspan="1" | Norm power
! colspan="3" rowspan="1" |norm pretransformer
! colspan="3" rowspan="1" | Norm pretransformer
! colspan="3" rowspan="1" |norm power
! colspan="3" rowspan="1" | Norm power
! colspan="1" rowspan="2" |initial
! colspan="1" rowspan="2" | Initial
! colspan="1" rowspan="2" |name
! colspan="1" rowspan="2" | Name
! colspan="1" rowspan="2" |multiplier
! colspan="1" rowspan="2" | Multiplier
|-
|-
!initial
! Initial
!name
! Name
!multiplier
! Multiplier
!initial
! Initial
!name
! Name
!power
! Power
!initial
! Initial
!name
! Name
!multiplier
! Multiplier
!initial
! Initial
!name
! Name
!power
! Power
! colspan="1" |abbreviatedย  ย  ย  ย 
! colspan="1" | Abbreviated
! colspan="1" |read ("____ tuning scheme") ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย  ย 
! colspan="1" | Read ("____ tuning scheme")
|-
|-
| colspan="3" |<none>
| colspan="3" | <none>
| rowspan="4" |<n/a>
| rowspan="4" | <n/a>
| rowspan="2" |maximum
| rowspan="2" | Maximum
| rowspan="2" |โˆž
| rowspan="2" | &infin;
| colspan="3" |<none>
| colspan="3" | <none>
| rowspan="2" |(t)
| rowspan="2" | (t)
| rowspan="2" |taxicab
| rowspan="2" | Taxicab
| rowspan="2" |1
| rowspan="2" | 1
| rowspan="4" |S
| rowspan="4" | S
| rowspan="4" |simplicity-weight
| rowspan="4" | Simplicity-weight
| rowspan="4" |1/complexity
| rowspan="4" | 1/Complexity
| rowspan="31" |<n/a>
| rowspan="31" | <n/a>
| rowspan="13" |minimax
| rowspan="13" | Minimax
| rowspan="13" |โˆž
| rowspan="13" | &infin;
| rowspan="4" |all
| rowspan="4" | All
|minimax-S
| Minimax-S
|minimax simplicity-weight damage
| 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-<alt>-S
|minimax <alternative>-simplicity-weight damage
| 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-ES
|minimax Euclideanized-simplicity-weight damage
| 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-E-<alt>-S
|minimax Euclideanized-<alternative>-simplicity-weight damage
| Minimax Euclideanized-<alternative>-simplicity-weight damage
|"[[Frobenius]]", "[[BE]]", "[[WE]]", "[[KE]]"
| "[[Frobenius]]", "[[BE]]", "[[WE]]", "[[KE]]"
|yes
| 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
| Unity-weight
|<none>
| <none>
| rowspan="27" |<set>
| rowspan="27" | <set>
|<set> minimax-U
| <set> Minimax-U
|<set> minimax unity-weight damage
| <set> Minimax unity-weight damage
|"[[Minimax tuning|minimax]]"
| "[[Minimax tuning|Minimax]]"
|yes
| yes
|-
|-
| colspan="3" |<none>
| colspan="3" | <none>
| rowspan="2" |(t)
| rowspan="2" | (t)
| rowspan="2" |taxicab
| rowspan="2" | Taxicab
| rowspan="2" |1
| rowspan="2" | 1
| rowspan="4" |S
| rowspan="4" | S
| rowspan="4" |simplicity-weight
| rowspan="4" | Simplicity-weight
| rowspan="4" |1/complexity
| rowspan="4" | 1/Complexity
|<set> minimax-S
| <set> Minimax-S
|<set> minimax simplicity-weight damage
| <set> Minimax simplicity-weight damage
|
| ย 
|yes
| Yes
|-
|-
| colspan="3" |<various>
| colspan="3" | <various>
|<set> minimax-<alt>-S
| <set> Minimax-<alt>-S
|<set> minimax <alternative>-simplicity-weight damage
| <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> minimax-ES
| <set> Minimax-ES
|<set> minimax Euclideanized-simplicity-weight damage
| <set> Minimax Euclideanized-simplicity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="3" |<various>
| colspan="3" | <various>
|<set> minimax-E-<alt>-S
| <set> Minimax-E-<alt>-S
|<set> minimax Euclideanized-<alternative>-simplicity-weight damage
| <set> Minimax Euclideanized-<alternative>-simplicity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="3" |<none>
| colspan="3" | <none>
| rowspan="2" |(t)
| rowspan="2" | (t)
| rowspan="2" |taxicab
| rowspan="2" | Taxicab
| rowspan="2" |1
| rowspan="2" | 1
| rowspan="4" |C
| rowspan="4" | C
| rowspan="4" |complexity-weight
| rowspan="4" | Complexity-weight
| rowspan="4" |complexity
| rowspan="4" | Complexity
|<set> minimax-C
| <set> Cinimax-C
|<set> minimax complexity-weight damage
| <set> Cinimax complexity-weight damage
|
| ย 
|yes
| Yes
|-
|-
| colspan="3" |<various>
| colspan="3" | <various>
|<set> minimax-<alt>-C
| <set> Minimax-<alt>-C
|<set> minimax <alternative>-complexity-weight damage
| <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> minimax-EC
| <set> Minimax-EC
|<set> minimax Euclideanized-complexity-weight damage
| <set> Minimax Euclideanized-complexity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="3" |<various>
| colspan="3" | <various>
|<set> minimax-E-<alt>-C
| <set> Minimax-E-<alt>-C
|<set> minimax Euclideanized-<alternative>-complexity-weight damage
| <set> Minimax Euclideanized-<alternative>-complexity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="6" |<n/a>
| colspan="6" | <n/a>
|U
| U
|unity-weight
| Unity-weight
|<none>
| <none>
| rowspan="9" |miniRMS
| rowspan="9" | MiniRMS
| rowspan="9" |2
| rowspan="9" | 2
|<set> miniRMS-U
| <set> MiniRMS-U
|<set> miniRMS unity-weight damage
| <set> MiniRMS unity-weight damage
|"[[least squares]]"
| "[[Least squares]]"
|yes
| yes
|-
|-
| colspan="3" |<none>
| colspan="3" | <none>
| rowspan="2" |(t)
| rowspan="2" | (t)
| rowspan="2" |taxicab
| rowspan="2" | Taxicab
| rowspan="2" |1
| rowspan="2" | 1
| rowspan="4" |S
| rowspan="4" | S
| rowspan="4" |simplicity-weight
| rowspan="4" | Simplicity-weight
| rowspan="4" |1/complexity
| rowspan="4" | 1/Complexity
|<set> miniRMS-S
| <set> MiniRMS-S
|<set> miniRMS simplicity-weight damage
| <set> MiniRMS simplicity-weight damage
|
| ย 
|yes
| Yes
|-
|-
| colspan="3" |<various>
| colspan="3" | <various>
|<set> miniRMS-<alt>-S
| <set> MiniRMS-<alt>-S
|<set> miniRMS <alternative>-simplicity-weight damage
| <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> miniRMS-ES
| <set> MiniRMS-ES
|<set> miniRMS Euclideanized-simplicity-weight damage
| <set> MiniRMS Euclideanized-simplicity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="3" |<various>
| colspan="3" | <various>
|<set> miniRMS-E-<alt>-S
| <set> MiniRMS-E-<alt>-S
|<set> miniRMS Euclideanized-<alternative>-simplicity-weight damage
| <set> MiniRMS Euclideanized-<alternative>-simplicity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="3" |<none>
| colspan="3" | <none>
| rowspan="2" |(t)
| rowspan="2" | (t)
| rowspan="2" |taxicab
| rowspan="2" | Taxicab
| rowspan="2" |1
| rowspan="2" | 1
| rowspan="4" |C
| rowspan="4" | C
| rowspan="4" |complexity-weight
| rowspan="4" | Complexity-weight
| rowspan="4" |complexity
| rowspan="4" | Complexity
|<set> miniRMS-C
| <set> MiniRMS-C
|<set> miniRMS complexity-weight damage
| <set> MiniRMS complexity-weight damage
|
| ย 
|yes
| yes
|-
|-
| colspan="3" |<various>
| colspan="3" | <various>
|<set> miniRMS-<alt>-C
| <set> MiniRMS-<alt>-C
|<set> miniRMS <alternative>-complexity-weight damage
| <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> miniRMS-EC
| <set> MiniRMS-EC
|<set> miniRMS Euclideanized-complexity-weight damage
| <set> MiniRMS Euclideanized-complexity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="3" |<various>
| colspan="3" | <various>
|<set> miniRMS-E-<alt>-C
| <set> MiniRMS-E-<alt>-C
|<set> miniRMS Euclideanized-<alternative>-complexity-weight damage
| <set> MiniRMS Euclideanized-<alternative>-complexity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="6" |<n/a>
| colspan="6" | <n/a>
|U
| U
|unity-weight
| Unity-weight
|<none>
| <none>
| rowspan="9" |minimean
| rowspan="9" | Miniaverage
| rowspan="9" |1
| rowspan="9" | 1
|<set> minimean-U
| <set> Miniaverage-U
|<set> minimean unity-weight damage
| <set> Miniaverage unity-weight damage
|
| ย 
|yes
| yes
|-
|-
| colspan="3" |<none>
| colspan="3" | <none>
| rowspan="2" |(t)
| rowspan="2" | (t)
| rowspan="2" |taxicab
| rowspan="2" | Taxicab
| rowspan="2" |1
| rowspan="2" | 1
| rowspan="4" |S
| rowspan="4" | S
| rowspan="4" |simplicity-weight
| rowspan="4" | Simplicity-weight
| rowspan="4" |1/complexity
| rowspan="4" | 1/Complexity
|<set> minimean-S
| <set> Miniaverage-S
|<set> minimean simplicity-weight damage
| <set> Miniaverage simplicity-weight damage
|
| ย 
|yes
| Yes
|-
|-
| colspan="3" |<various>
| colspan="3" | <various>
|<set> minimean-<alt>-S
| <set> Miniaverage-<alt>-S
|<set> minimean <alternative>-simplicity-weight damage
| <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> minimean-ES
| <set> Miniaverage-ES
|<set> minimean Euclideanized-simplicity-weight damage
| <set> Miniaverage Euclideanized-simplicity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="3" |<various>
| colspan="3" | <various>
|<set> minimean-E-<alt>-S
| <set> Miniaverage-E-<alt>-S
|<set> minimean Euclideanized-<alternative>-simplicity-weight damage
| <set> Miniaverage Euclideanized-<alternative>-simplicity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="3" |<none>
| colspan="3" | <none>
| rowspan="2" |(t)
| rowspan="2" | (t)
| rowspan="2" |taxicab
| rowspan="2" | Taxicab
| rowspan="2" |1
| rowspan="2" | 1
| rowspan="4" |C
| rowspan="4" | C
| rowspan="4" |complexity-weight
| rowspan="4" | Complexity-weight
| rowspan="4" |complexity
| rowspan="4" | Complexity
|<set> minimean-C
| <set> Miniaverage-C
|<set> minimean complexity-weight damage
| <set> Miniaverage complexity-weight damage
|
| ย 
|yes
| Yes
|-
|-
| colspan="3" |<various>
| colspan="3" | <various>
|<set> minimean-<alt>-C
| <set> Miniaverage-<alt>-C
|<set> minimean <alternative>-complexity-weight damage
| <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> minimean-EC
| <set> Miniaverage-EC
|<set> minimean Euclideanized-complexity-weight damage
| <set> Miniaverage Euclideanized-complexity-weight damage
|
| ย 
|
| ย 
|-
|-
| colspan="3" |<various>
| colspan="3" | <various>
|<set> minimean-E-<alt>-C
| <set> Miniaverage-E-<alt>-C
|<set> minimean Euclideanized-<alternative>-complexity-weight damage
| <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" |quantity
|-
! colspan="2" |unit
! colspan="2" | Quantity
! colspan="2" | Unit
|-
|-
!abbreviation
! Abbreviation
!name
! Name
!symbol
! Symbol
!name
! Name
|-
|-
|U-damage
| U-damage
|unity-weight damage
| Unity-weight damage
|<math>\mathsf{ยข}\small\mathsf{(U)}</math>
| <math>\mathsf{ยข}\small\mathsf{(U)}</math>
|unity-weighted cents
| Unity-weighted cents
|-
|-
|C-damage
| C-damage
|complexity-weight damage
| Complexity-weight damage
|<math>\mathsf{ยข}\small\mathsf{(C)}</math>
| <math>\mathsf{ยข}\small\mathsf{(C)}</math>
|complexity-weighted cents
| 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
| Simplicity-weight damage
|<math>\mathsf{ยข}\small\mathsf{(S)}</math>
| <math>\mathsf{ยข}\small\mathsf{(S)}</math>
|simplicity-weighted cents
| 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
! colspan="2" | Quantity
!name
! colspan="2" | Unit
!unit
!name
|-
|-
|C
! Abbreviation
|complexity
! Name
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math> = <math>\small\mathsf{(C)}</math>
! Unit
|complexity weight
! Name
|-
|-
|<alt>-C
| C
|<alternative> complexity
| Complexity
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math> = <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math>
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)}</math> = <math>\small\mathsf{(C)}</math>
|<alternative>-complexity weight
| Complexity weight
|-
|-
|EC
| <alt>-C
|Euclideanized complexity
| <alternative> complexity
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(EC)}</math> = <math>\small\mathsf{(EC)}</math>
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math> = <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math>
|Euclideanized-complexity weight
| <alternative>-complexity weight
|-
|-
|E-<alt>-C
| EC
|Euclideanized-<alternative> complexity
| Euclideanized complexity
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(E}</math>-<alt>-<math>\scriptsize\mathsf{C)}</math> = <math>\small\mathsf{(E}</math>-<alt>-<math>\small\mathsf{C)}</math>
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(EC)}</math> = <math>\small\mathsf{(EC)}</math>
|Euclideanized-<alternative>-complexity weight
| Euclideanized-complexity weight
|-
|-
|S
| E-<alt>-C
|simplicity
| Euclideanized-<alternative> complexity
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)}</math> = <math>\small\mathsf{(S)}</math>
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(E}</math>-<alt>-<math>\scriptsize\mathsf{C)}</math> = <math>\small\mathsf{(E}</math>-<alt>-<math>\small\mathsf{C)}</math>
|simplicity weight
| Euclideanized-<alternative>-complexity weight
|-
|-
|<alt>-S
| S
|<alternative> simplicity
| Simplicity
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{S)}</math> = <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math>
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)}</math> = <math>\small\mathsf{(S)}</math>
|<alternative>-simplicity weight
| Simplicity weight
|-
|-
|ES
| <alt>-S
|Euclideanized simplicity
| <alternative> simplicity
|<math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(ES)}</math> = <math>\small\mathsf{(ES)}</math>
| <math>\small\mathsf{๐Ÿ™}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{S)}</math> = <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{S)}</math>
|Euclideanized-simplicity weight
| <alternative>-simplicity weight
|-
|-
|E-<alt>-S
| ES
|Euclideanized-<alternative> simplicity
| Euclideanized simplicity
|<math>\small\mathsf{๐Ÿ™}\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Ÿ™}\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Ÿ™}\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 โ™ฏ, โ™ญ, ยข, โˆš, ยฐ, โ‚‚, ร—, {{inv}}, โŸฉ, โˆž, 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 โŽ„. So for example, you type {{nowrap|โ™ฏ as <code>โŽ„##</code>|โ™ญ as <code>โŽ„bb</code>|ยข as <code>โŽ„c/</code>|โˆš as <code>โŽ„v/</code>|ยฐ as <code>โŽ„00</code>|โ‚‚ as <code>โŽ„-2</code>|ร— as <code>โŽ„xx</code>|{{inv}} as <code>โŽ„11</code>|โŸฉ as <code>โŽ„&gt;&gt;</code>|โˆž as <code>โŽ„88</code>|and ฯ• as <code>โŽ„8f</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 โ™ฏ, โ™ญ, ยข, โˆš, ยฐ, โ‚‚, ร—, โปยน, โŸฉ, โˆž, 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 โŽ„. So for example, you type โ™ฏ as โŽ„##, โ™ญ as โŽ„bb, ยข as โŽ„c/, โˆš as โŽ„v/, ยฐ as โŽ„00, โ‚‚ as โŽ„-2, ร— as โŽ„xx, โปยน as โŽ„11, โŸฉ as โŽ„>>, โˆž as โŽ„88, and ฯ• as โŽ„8f.
ย 
For Windows users, install WinCompose then copy-paste the contents of this file: https://dkeenan.com/XCompose.txt into your user sequences (Show sequences โ†’ User-defined sequences โ†’ 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 these instructions (from the author of WinCompose) for how to set up Compose-key sequences in Mac OS: http://sam.hocevar.net/blog/category/osx/
For Windows users, install WinCompose then copy-paste the contents of this file: https://dkeenan.com/XCompose.txt into your user sequences (Show sequences &rarr; User-defined sequences &rarr; Edit). Then save and reload. You can always choose to override or add alternatives to our sequences if you find others to be more intuitive.


===Table of noteworthy sequences===
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" width="130px" | Compose-key sequence
|- style="white-space: nowrap;"
! scope="col" width="75px" | resulting text
! scope="col" style="width: 130px;" | Compose-key sequence
!description
! scope="col" style="width: 75px;" | Resulting text
|-
! Description
! colspan="3" rowspan="1" |Keyboard key symbols
|- style="white-space: nowrap;"
! colspan="3" | Keyboard key symbols
|-
|-
|โŽ„โŽ„โŽ„
| โŽ„โŽ„โŽ„
|โŽ„
| โŽ„
|compose key symbol (the right alt key by default)
| Compose key symbol (the right alt key by default)
|-
|-
|โŽ„\โฃ
| โŽ„\โฃ
|โฃ
| โฃ
|spacebar symbol
| Spacebar symbol
|-
|-
|โŽ„\โ–ถ๏ธŽ etc.
| โŽ„\โ–ถ๏ธŽ etc.
|โ–ถ๏ธŽ etc.
| โ–ถ๏ธŽ etc.
|right etc. arrow key symbols
| Right etc. arrow key symbols
|-
|-
|โŽ„\A or โŽ„\O
| โŽ„\A or โŽ„\O
|โŒฅ
| โŒฅ
|alt or option key symbol
| Alt or option key symbol
|-
|-
|โŽ„\B
| โŽ„\B
|โŒซ
| โŒซ
|backspace key symbol
| Backspace key symbol
|-
|-
|โŽ„\C
| โŽ„\C
|โœฒ
| โœฒ
|control key symbol
| Control key symbol
|-
|-
|โŽ„\D
| โŽ„\D
|โŒฆ
| โŒฆ
|delete key symbol
| Delete key symbol
|-
|-
|โŽ„\E
| โŽ„\E
|โŽ‹
| โŽ‹
|escape key symbol
| Escape key symbol
|-
|-
|โŽ„\L
| โŽ„\L
|โ‡ช
| โ‡ช
|caps lock key symbol
| Caps lock key symbol
|-
|-
|โŽ„\R or โŽ„\.E or โŽ„\\
| โŽ„\R or โŽ„\.E
|โŽ
| โŽ
|return or enter key symbol
| Return or enter key symbol
|-
|-
|โŽ„\S
| โŽ„\S
|โ‡ง
| โ‡ง
|shift key symbol
| Shift key symbol
|-
|-
|โŽ„\T
| โŽ„\T
|โญพ
| โญพ
|tab key symbol
| Tab key symbol
|-
|-
|โŽ„()
| โŽ„()
|โ—Œ
| โ—Œ
|dotted circle, represents any character (such as the character preceding a combining mark)
| Dotted circle, represents any character (such as the character preceding a combining mark)
|-
|-
! colspan="3" rowspan="1" |Double key sequences
! colspan="3" style="white-space: nowrap;" | Double key sequences
|-
|-
|โŽ„โฃโฃ
| โŽ„โฃโฃ
|โ€ฏ
| โ€ฏ
|narrow no-break space (used between quantities and their units)
| Narrow no-break space (used between quantities and their units)
|-
|-
|โŽ„..
| โŽ„..
|ยท
| ยท
|middle dot (used to multiply units when juxtaposition is ambiguous)
| Middle dot (used to multiply units when juxtaposition is ambiguous)
|-
|-
|โŽ„::
| โŽ„::
|รท
| รท
|divide sign
| Divide sign
|-
|-
|โŽ„;;
| โŽ„;;
|โ—Œฬฒฬ…
| โ—Œฬฒฬ…
|combining overline and low line (undirected value)
| Combining overline and low line (undirected value)
|-
|-
|<nowiki>โŽ„||</nowiki>
| โŽ„{{pipe}} {{pipe}}
|โ€–
| โ€–
|power norm bracket
| Power norm bracket
|-
|-
|โŽ„\\
| โŽ„<<
|โŽ
| โŸจ
|return or enter key symbol
| Left angle bracket
|-
|-
|โŽ„<<
| โŽ„>>
|โŸจ
| โŸฉ
|left angle bracket
| Right angle bracket
|-
|-
|โŽ„>>
| โŽ„~~
|โŸฉ
| โ‰ˆ
|right angle bracket
| Approximately equal
|-
|-
|โŽ„~~
| โŽ„**
|โ‰ˆ
| โ˜…
|approximately equal
| Black star
|-
|-
|โŽ„**
| โŽ„&#39;&#39;
|โ˜…
| โ€ฒ
|black star
| prime mark
|-
|-
|โŽ„<nowiki>''</nowiki>
| โŽ„11
|โ€ฒ
| โปยน
|prime mark
| Power of &minus;1 or inverse
|-
|-
|โŽ„11
| โŽ„22 through โŽ„77
|โปยน
| ยฒ ยณ โด โต โถ โท
|power of -1 or inverse
| Squared, cubed, fourth through seventh power
|-
|-
|โŽ„22 through โŽ„77
| โŽ„88
|ยฒ ยณ โด โต โถ โท
| โˆž
|squared, cubed, fourth through seventh power
| Infinity
|-
|-
|โŽ„88
| โŽ„00
|โˆž
| ยฐ
|infinity
| Degree sign
|-
|-
|โŽ„00
| โŽ„nn
|ยฐ
| โฟ
|degree sign
| Superscript small n
|-
|-
|โŽ„nn
| โŽ„--
|โฟ
| โ‚‹
|superscript small n
| Subscript minus sign
|-
|-
|โŽ„--
| โŽ„__
|โ‚‹
| โ—Œฬฒ
|subscript minus sign
| Combining low line (underline)
|-
|-
|โŽ„__
| โŽ„==
|โ—Œฬฒ
| โ‰ก
|combining low line (underline)
| Modular congruence
|-
|-
|โŽ„==
| โŽ„//
|โ‰ก
| โ„
|modular congruence
| Fraction slash (use with super and subscripts to create fractions)
|-
|-
|โŽ„//
| โŽ„##
|โ„
| โ™ฏ
|fraction slash (use with super and subscripts to create fractions)
| Musical sharp
|-
|-
|โŽ„##
| โŽ„bb
|โ™ฏ
| โ™ญ
|musical sharp
| Musical flat
|-
|-
|โŽ„bb
| โŽ„dd
|โ™ญ
| โˆ‚
|musical flat
| Partial derivative
|-
|-
|โŽ„dd
| โŽ„ff
|โˆ‚
| ฯ•
|partial derivative
| Small phi symbol
|-
|-
|โŽ„ff
| โŽ„gg
|ฯ•
| ษก
|small phi symbol
| Single-storey (opentail) small g
|-
|-
|โŽ„gg
| โŽ„ll
|ษก
| โ„“
|single-storey (opentail) small g
| Script small L
|-
|-
|โŽ„ll
| โŽ„uu
|โ„“
| ยต
|script small L
| Micro sign
|-
|-
|โŽ„uu
| โŽ„xx
|ยต
| ร—
|micro sign
| Multiplication sign
|-
|-
|โŽ„xx
| โŽ„DD
|ร—
| โˆ†
|multiplication sign
| Delta (small difference) operator
|-
|-
|โŽ„DD
| โŽ„FF
|โˆ†
| ฮฆ
|delta (small difference) operator
| Greek capital phi
|-
|-
|โŽ„FF
| โŽ„QQ
|ฮฆ
| ฯ˜
|Greek capital phi
| Greek capital letter archaic qoppa (small quotient operator)
|-
|-
|โŽ„QQ
| โŽ„TT
|ฯ˜
| แต€
|Greek capital letter archaic qoppa (small quotient operator)
| Superscript capital T (matrix transpose)
|-
|-
|โŽ„TT
| โŽ„++
|แต€
| โบ
|superscript capital T (matrix transpose)
| Superscript plus sign (matrix pseudoinverse)
|-
|-
|โŽ„++
| โŽ„โ–ถ๏ธŽโ–ถ๏ธŽ etc.
|โบ
| โ†’ etc.
|superscript plus sign (matrix pseudoinverse)
| Right etc. arrows
|-
|-
|โŽ„โ–ถ๏ธŽโ–ถ๏ธŽ etc.
! colspan="3" style="white-space: nowrap;" | Multiplication operators
|โ†’ etc.
|right etc. arrows
|-
|-
! colspan="3" rowspan="1" |Multiplication operators
| โŽ„xx
| ร—
| Multiplication sign
|-
|-
|โŽ„xx
| โŽ„Xx or โŽ„xX
|ร—
| โจฏ
|multiplication sign
| Vector or cross product (barely distinguishable from multiplication sign)
|-
|-
|โŽ„Xx or โŽ„xX
| โŽ„XX
|โจฏ
| โœ•
|vector or cross product (barely distinguishable from multiplication sign)
| Large multiplication sign (a better symbol for cross product)
|-
|-
|โŽ„XX
| โŽ„x*
|โœ•
| โ‹†
|large multiplication sign (a better symbol for cross product)
| Star operator (prefix: tensor complement, Hodge)
|-
|-
|โŽ„x*
| โŽ„X*
|โ‹†
| โˆ—
|star operator (prefix: tensor complement, Hodge)
| Asterisk operator (infix: scalar product, Dorst)
|-
|-
|โŽ„X*
| โŽ„x.
|โˆ—
| โ‹…
|asterisk operator (infix: scalar product, Dorst)
| Dot (product) operator
|-
|-
|โŽ„x.
| โŽ„X.
|โ‹…
| โ€ข
|dot (product) operator
| Bullet (infix: fat dot product, Dorst)
|-
|-
|โŽ„X.
! colspan="3" style="white-space: nowrap;" | Other operators
|โ€ข
|bullet (infix: fat dot product, Dorst)
|-
|-
! colspan="3" rowspan="1" |Other operators
| โŽ„v/
| โˆš
| Square root sign
|-
|-
|โŽ„v/
| โŽ„3v/
|โˆš
| โˆ›
|square root sign
| Cube root sign
|-
|-
|โŽ„3v/
| โŽ„4v/
|โˆ›
| โˆœ
|cube root sign
| Fourth root sign
|-
|-
|โŽ„4v/
| โŽ„-+
|โˆœ
| โ‚Š
|fourth root sign
| Subscript plus sign
|-
|-
|โŽ„-+
| โŽ„--
|โ‚Š
| โ‚‹
|subscript plus sign
| Subscript minus sign
|-
|-
|โŽ„--
| โŽ„-=
|โ‚‹
| โ‚Œ
|subscript minus sign
| Subscript equals sign
|-
|-
|โŽ„-=
| โŽ„++
|โ‚Œ
| โบ
|subscript equals sign
| Superscript plus sign (matrix pseudoinverse)
|-
|-
|โŽ„++
| โŽ„+- or โŽ„+=
|โบ
| ยฑ
|superscript plus sign (matrix pseudoinverse)
| Plus or minus sign
|-
|-
|โŽ„+- or โŽ„+=
| โŽ„=+
|ยฑ
| โˆ“
|plus or minus sign
| Minus or plus sign
|-
|-
|โŽ„=+
| โŽ„=-
|โˆ“
| โˆ’
|minus or plus sign
| Minus sign
|-
|-
|โŽ„=-
| โŽ„==
|โˆ’
| โ‰ก
|minus sign
| Modular congruence
|-
|-
|โŽ„==
| โŽ„/\
|โ‰ก
| โˆง
|modular congruence
| Logical AND, wedge product, progressive product
|-
|-
|โŽ„/\
| โŽ„\/
|โˆง
| โˆจ
|logical AND, wedge product, progressive product
| Logical OR, vee product, regressive product
|-
|-
|โŽ„\/
| โŽ„โŽ„/\
|โˆจ
| โ‹€
|logical OR, vee product, regressive product
| Larger logical AND, wedge product, progressive product
|-
|-
|โŽ„โŽ„/\
| โŽ„โŽ„\/
|โ‹€
| โ‹
|larger logical AND, wedge product, progressive product
| Larger logical OR, vee product, regressive product
|-
|-
|โŽ„โŽ„\/
| <nowiki>โŽ„| _</nowiki>
|โ‹
| โŒŠ
|larger logical OR, vee product, regressive product
| Left floor (infix: right contraction, Dorst)
|-
|-
|<nowiki>โŽ„|_</nowiki>
| <nowiki>โŽ„_| </nowiki>
|โŒŠ
| โŒ‹
|left floor (infix: right contraction, Dorst)
| Right floor (infix: left contraction, Dorst)
|-
|-
|<nowiki>โŽ„_|</nowiki>
| <nowiki>โŽ„| ^</nowiki>
|โŒ‹
| โŒˆ
|right floor (infix: left contraction, Dorst)
| Left ceiling
|-
|-
|<nowiki>โŽ„|^</nowiki>
| <nowiki>โŽ„^| </nowiki>
|โŒˆ
| โŒ‰
|left ceiling
| Right ceiling
|-
|-
|<nowiki>โŽ„^|</nowiki>
| โŽ„'-
|โŒ‰
| โจฝ
|right ceiling
| Righthand interior product
|-
|-
|โŽ„'-
| โŽ„-'
|โจฝ
| โจผ
|righthand interior product
| (Left-hand) interior product
|-
|-
|โŽ„-'
| โŽ„-,
|โจผ
| ยฌ
|(lefthand) interior product
| Not sign (prefix: multivector complement)
|-
|-
|โŽ„-,
| โŽ„โŽ„<>
|ยฌ
| โ‹„
|not sign (prefix: multivector complement)
| Diamond operator (prefix: multivector dual)
|-
|-
|โŽ„โŽ„<>
| โŽ„(.)
|โ‹„
| โจ€
|diamond operator (prefix: multivector dual)
| Entry-wise vector multiplication operator
|-
|-
|โŽ„(.)
| โŽ„(..)
|โจ€
| โŠ™
|entrywise vector multiplication operator
| Alternative entry-wise vector multiplication operator
|-
|-
|โŽ„(..)
| โŽ„(/)
|โŠ™
| โŠ˜
|alternative entrywise vector multiplication operator
| Entry-wise vector division operator
|-
|-
|โŽ„(/)
! colspan="3" | Mathematical letter and digit prefixes
|โŠ˜
|entrywise vector division operator
|-
|-
! colspan="3" |Mathematical letter and digit prefixes
| โŽ„3โ—Œ
| ั
| Cyrillic, โŽ„3q is ya (example)
|-
|-
|โŽ„3โ—Œ
| โŽ„4โ—Œ
|ั
| โ„ต
|cyrillic, โŽ„3q is ya (example)
| Hebrew, โŽ„4a is aleph (example)
|-
|-
|โŽ„4โ—Œ
| โŽ„5โ—Œ
|โ„ต
| ๐”ž
|hebrew, โŽ„4a is aleph (only a b g d)
| Fraktur, โŽ„5a
|-
|-
|โŽ„5โ—Œ
| โŽ„6โ—Œ
|๐”ž
| แตƒ ยน โ€ฏแชฒย  โธ
|fraktur, โŽ„5a
| Superscripts, โŽ„6a โŽ„61 โŽ„688 โŽ„68โฃ (not all letters, some only approximate) (same key as ^ but without shift)
|-
|-
|โŽ„6โ—Œ
| โŽ„68โ—Œ
|แตƒ ยน โ€ฏแชฒย  โธ
| แต
|superscripts, โŽ„6a โŽ„61 โŽ„688 โŽ„68โฃ (not all letters, some only approximate) (same key as ^ but without shift)
| Superscript greek, โŽ„68b is superscript beta (only a few)
|-
|-
|โŽ„68โ—Œ
| โŽ„7โ—Œ
|แต
| ๐’ถ
|superscript greek, โŽ„68b is superscript beta (only a few)
| Script, โŽ„7a
|-
|-
|โŽ„7โ—Œ
| โŽ„8โ—Œ
|๐’ถ
| ฮฑ
|script, โŽ„7a
| Greek, โŽ„8a is alpha (by sound where possible otherwise letter-shape)
|-
|-
|โŽ„8โ—Œ
| โŽ„8.โ—Œ
|ฮฑ
| ฯ‚
|greek, โŽ„8a is alpha (by sound where possible otherwise letter-shape)
| Greek variants, โŽ„8.s is final sigma
|-
|-
|โŽ„8.โ—Œ
| โŽ„9โ—Œ
|ฯ‚
| ๐š ๐Ÿ ๐Ÿ“ ๐Ÿ• ๐Ÿ– ๐ŸŽ
|greek variants, โŽ„8.s is final sigma
| Bold, โŽ„9a โŽ„91 โŽ„95โฃ โŽ„97โฃ โŽ„98โฃ โŽ„90โฃ
|-
|-
|โŽ„9โ—Œ
| โŽ„95โ—Œ
|๐š ๐Ÿ ๐Ÿ“ ๐Ÿ• ๐Ÿ– ๐ŸŽ
| ๐–†
|bold, โŽ„9a โŽ„91 โŽ„95โฃ โŽ„97โฃ โŽ„98โฃ โŽ„90โฃ
| Bold fraktur, โŽ„95a
|-
|-
|โŽ„95โ—Œ
| โŽ„97โ—Œ
|๐–†
| ๐“ช
|bold fraktur, โŽ„95a
| Bold script, โŽ„97a
|-
|-
|โŽ„97โ—Œ
| โŽ„98โ—Œ
|๐“ช
| ๐›‚
|bold script, โŽ„97a
| Bold greek, โŽ„98a is bold alpha
|-
|-
|โŽ„98โ—Œ
| โŽ„90โ—Œ
|๐›‚
| ๐’‚
|bold greek, โŽ„98a is bold alpha
| Bold italic, โŽ„90a
|-
|-
|โŽ„90โ—Œ
| โŽ„908โ—Œ
|๐’‚
| ๐œถ
|bold italic, โŽ„90a
| Bold italic greek, โŽ„908a is bold italic alpha
|-
|-
|โŽ„908โ—Œ
| โŽ„0โ—Œ
|๐œถ
| ๐‘Ž
|bold italic greek, โŽ„908a is bold italic alpha
| Italic, โŽ„0a
|-
|-
|โŽ„0โ—Œ
| โŽ„08โ—Œ
|๐‘Ž
| ๐›ผ
|italic, โŽ„0a
| Italic greek, โŽ„08a is italic alpha
|-
|-
|โŽ„08โ—Œ
| โŽ„-โ—Œ
|๐›ผ
| โ‚ แด€ย  โ€ฏอš โ‚ˆ
|italic greek, โŽ„08a is italic alpha
| Subscripts and small caps, โŽ„-a โŽ„-A โŽ„-88 โŽ„-8โฃ (not all letters, some only approximate) (same key as _ but without shift)
|-
|-
|โŽ„-โ—Œ
| โŽ„-8โ—Œ
|โ‚ แด€ย  โ€ฏอš โ‚ˆ
| แตฆ
|subscripts and small caps, โŽ„-a โŽ„-A โŽ„-88 โŽ„-8โฃ (not all letters, some only approximate) (same key as _ but without shift)
| Subscript greek, โŽ„-8b is subscript beta (only a few)
|-
|-
|โŽ„-8โ—Œ
| โŽ„{โ—Œ
|แตฆ
| ๐–บ ๐Ÿฃ ๐Ÿซ
|subscript greek, โŽ„-8b is subscript beta (only a few)
| Sans-serif, โŽ„{a โŽ„{1 โŽ„{9โฃ
|-
|-
|โŽ„{โ—Œ
| โŽ„{9โ—Œ
|๐–บ ๐Ÿฃ ๐Ÿซ
| ๐—ฎ ๐Ÿญ
|sans-serif, โŽ„{a โŽ„{1 โŽ„{9โฃ
| Sans-serif bold, โŽ„{9a โŽ„{91
|-
|-
|โŽ„{9โ—Œ
| โŽ„}โ—Œ
|๐—ฎ ๐Ÿญ
| ๐šŠ ๐Ÿท
|sans-serif bold, โŽ„{9a โŽ„{91
| Monospace, โŽ„}a โŽ„}1
|-
|-
|โŽ„}โ—Œ
| โŽ„{{pipe}} โ—Œ
|๐šŠ ๐Ÿท
| ๐•’ ๐Ÿ™ ๐Ÿ  ๐Ÿ˜
|monospace, โŽ„}a โŽ„}1
| Double-struck, โŽ„{{pipe}} a โŽ„{{pipe}} 1 โŽ„{{pipe}} 8โฃ โŽ„{{pipe}} 0โฃ
|-
|-
|<nowiki>โŽ„|โ—Œ</nowiki>
| โŽ„{{pipe}} 8โ—Œ
|๐•’ ๐Ÿ™ ๐Ÿ  ๐Ÿ˜
| โ„ผ
|<nowiki>double-struck, โŽ„|a โŽ„|1 โŽ„|8โฃ โŽ„|0โฃ</nowiki>
| Double-struck greek, โŽ„{{pipe}} 8p (only a few)
|-
|-
|<nowiki>โŽ„|8โ—Œ</nowiki>
| โŽ„{{pipe}} 0โ—Œ
|โ„ผ
| โ…‡ โ…ˆ
|<nowiki>double-struck greek, โŽ„|8p (only a few)</nowiki>
| Double-struck italic, โŽ„{{pipe}} 0e โŽ„{{pipe}} i (only a few)
|-
|-
|<nowiki>โŽ„|0โ—Œ</nowiki>
! colspan="3" style="white-space: nowrap;" | Power statistics brackets
|โ…‡ โ…ˆ
|<nowiki>double-struck italic, โŽ„|0e โŽ„|i (only a few)</nowiki>
|-
|-
! colspan="3" rowspan="1" |Power statistics brackets
| โŽ„{{pipe}} {{pipe}}
| โ€–
| Power-norm bracket
|-
|-
|<nowiki>โŽ„โŽ„|| or โŽ„||</nowiki>
| โŽ„{{pipe}}-1
|โ€–
| โ€–โ‚
|power-norm bracket
| 1-Norm right bracket
|-
|-
|<nowiki>โŽ„|-1</nowiki>
| โŽ„{{pipe}}-2
|โ€–โ‚
| โ€–โ‚‚
|1-norm right bracket
| 2-Norm right bracket
|-
|-
|<nowiki>โŽ„|-2</nowiki>
| โŽ„{{pipe}}-8
|โ€–โ‚‚
| โ€–โ€ฏอš
|2-norm right bracket
| โˆž-Norm right bracket
|-
|-
|<nowiki>โŽ„|-8</nowiki>
| โŽ„โŽ„<<
|โ€–โ€ฏอš
| โŸช
|โˆž-norm right bracket
| Left power-mean bracket
|-
|-
|โŽ„โŽ„<<
| โŽ„โŽ„>>
|โŸช
| โŸซ
|left power-mean bracket
| Right power-mean bracket
|-
|-
|โŽ„โŽ„>>
| โŽ„โŽ„{{((}}
|โŸซ
| โง›
|right power-mean bracket
| Left power-sum bracket (substitute for {{llzz}} when HTML is not available)
|-
|-
|<nowiki>โŽ„โŽ„{{</nowiki>
| โŽ„โŽ„{{))}}
|โง›
| โงš
|left power-sum bracket
| Right power-sum bracket (substitute for {{rrzz}} when HTML is not available)
|-
|-
|<nowiki>โŽ„โŽ„}}</nowiki>
! colspan="3" style="white-space: nowrap;" | Combining marks
|โงš
|right power-sum bracket
|-
|-
! colspan="3" rowspan="1" |Combining marks
| โŽ„\-
| โ—Œฬถ
| Combining strike-thru
|-
|-
|โŽ„\-
| โŽ„^_
|โ—Œฬถ
| โ—Œฬ…
|combining strike-thru
| Combining overline
|-
|-
|โŽ„^_
| โŽ„__
|โ—Œฬ…
| โ—Œฬฒ
|combining overline
| Combining low line
|-
|-
|โŽ„__
| โŽ„;; or โŽ„-_ or โŽ„_^
|โ—Œฬฒ
| โ—Œฬฒฬ…
|combining low line
| Combining overline and low line (undirected value)
|-
|โŽ„-_ or โŽ„_- or โŽ„_^
|โ—Œฬฒฬ…
|combining overline and low line (undirected value)
|}
ย 
===Keyboard map===
ย 
{| class="wikitable mw-collapsible mw-collapsed"
|+
| [[File:WinCompose keyboard map.png|1000px]]
|}
|}


==Footnotes==
=== Keyboard map ===
[[File:WinCompose keyboard map.png|1000px]]


<references />
== Footnotes ==
<references group="note" />


[[Category:Regular temperament theory]]
[[Category:Dave Keenan & Douglas Blumeyer's guide to RTT]]
[[Category:Tuning]]
[[Category:Tuning]]

Latest revision as of 01:44, 7 August 2025

< Dave Keenan & Douglas Blumeyer's guide to RTT

[math]\displaystyle{ \def\hs{\hspace{-3px}} \def\lvsp{{}\mkern-5.5mu}{} \def\rvsp{{}\mkern-2.5mu}{} \def\llangle{\left\langle\lvsp\left\langle} \def\lllangle{\left\langle\lvsp\left\langle\lvsp\left\langle} \def\llllangle{\left\langle\lvsp\left\langle\lvsp\left\langle\lvsp\left\langle} \def\llbrack{\left[\left[} \def\lllbrack{\left[\left[\left[} \def\llllbrack{\left[\left[\left[\left[} \def\llvert{\left\vert\left\vert} \def\lllvert{\left\vert\left\vert\left\vert} \def\llllvert{\left\vert\left\vert\left\vert\left\vert} \def\rrangle{\right\rangle\rvsp\right\rangle} \def\rrrangle{\right\rangle\rvsp\right\rangle\rvsp\right\rangle} \def\rrrrangle{\right\rangle\rvsp\right\rangle\rvsp\right\rangle\rvsp\right\rangle} \def\rrbrack{\right]\right]} \def\rrrbrack{\right]\right]\right]} \def\rrrrbrack{\right]\right]\right]\right]} \def\rrvert{\right\vert\right\vert} \def\rrrvert{\right\vert\right\vert\right\vert} \def\rrrrvert{\right\vert\right\vert\right\vert\right\vert} }[/math][math]\displaystyle{ \def\abs#1{\left|{#1}\right|} \def\norm#1{\left\|{#1}\right\|} \def\floor#1{\left\lfloor{#1}\right\rfloor} \def\ceil#1{\left\lceil{#1}\right\rceil} \def\round#1{\left\lceil{#1}\right\rfloor} \def\rround#1{\left\lfloor{#1}\right\rceil} }[/math][math]\displaystyle{ \def\hs{\hspace{-3px}} \def\lvsp{{}\mkern-5.5mu}{} \def\rvsp{{}\mkern-2.5mu}{} \def\llangle{\left\langle\lvsp\left\langle} \def\lllangle{\left\langle\lvsp\left\langle\lvsp\left\langle} \def\llllangle{\left\langle\lvsp\left\langle\lvsp\left\langle\lvsp\left\langle} \def\llbrack{\left[\left[} \def\lllbrack{\left[\left[\left[} \def\llllbrack{\left[\left[\left[\left[} \def\llvert{\left\vert\left\vert} \def\lllvert{\left\vert\left\vert\left\vert} \def\llllvert{\left\vert\left\vert\left\vert\left\vert} \def\rrangle{\right\rangle\rvsp\right\rangle} \def\rrrangle{\right\rangle\rvsp\right\rangle\rvsp\right\rangle} \def\rrrrangle{\right\rangle\rvsp\right\rangle\rvsp\right\rangle\rvsp\right\rangle} \def\rrbrack{\right]\right]} \def\rrrbrack{\right]\right]\right]} \def\rrrrbrack{\right]\right]\right]\right]} \def\rrvert{\right\vert\right\vert} \def\rrrvert{\right\vert\right\vert\right\vert} \def\rrrrvert{\right\vert\right\vert\right\vert\right\vert} }[/math][math]\displaystyle{ \def\val#1{\left\langle\begin{matrix}#1\end{matrix}\right]} \def\tval#1{\left\langle\begin{matrix}#1\end{matrix}\right\vert} \def\bival#1{\llangle\begin{matrix}#1\end{matrix}\rrbrack} \def\bitval#1{\llangle\begin{matrix}#1\end{matrix}\rrvert} \def\trival#1{\lllangle\begin{matrix}#1\end{matrix}\rrrbrack} \def\tritval#1{\lllangle\begin{matrix}#1\end{matrix}\rrrvert} \def\quadval#1{\llllangle\begin{matrix}#1\end{matrix}\rrrrbrack} \def\quadtval#1{\llllangle\begin{matrix}#1\end{matrix}\rrrrvert} \def\monzo#1{\left[\begin{matrix}#1\end{matrix}\right\rangle} \def\tmonzo#1{\left\vert\begin{matrix}#1\end{matrix}\right\rangle} \def\bimonzo#1{\llbrack\begin{matrix}#1\end{matrix}\rrangle} \def\bitmonzo#1{\llvert\begin{matrix}#1\end{matrix}\rrangle} \def\trimonzo#1{\lllbrack\begin{matrix}#1\end{matrix}\rrrangle} \def\tritmonzo#1{\lllvert\begin{matrix}#1\end{matrix}\rrrangle} \def\quadmonzo#1{\llllbrack\begin{matrix}#1\end{matrix}\rrrrangle} \def\quadtmonzo#1{\llllvert\begin{matrix}#1\end{matrix}\rrrrangle} \def\rbra#1{\left\{\begin{matrix}#1\end{matrix}\right]} \def\rket#1{\left[\begin{matrix}#1\end{matrix}\right\}} \def\vmp#1#2{\left\langle\begin{matrix}#1\end{matrix}\,\vert\,\begin{matrix}#2\end{matrix}\right\rangle} \def\wmp#1#2{\llangle\begin{matrix}#1\end{matrix}\,\vert\vert\,\begin{matrix}#2\end{matrix}\rrangle} }[/math][math]\displaystyle{ \def\slant#1{\style{display: inline-block; margin: -.05em; transform: skew(-14deg) translateX(.03em);}{#1}} \def\smallLLzigzag{\hspace{-1.4mu}\style{display: inline-block; transform: scale(.62, 1.24) translateY(.05em); font-family: sans-serif;}{๊—จ\hspace{-2.6mu}๊—จ}\hspace{-1.4mu}} \def\smallRRzigzag{\hspace{-1.4mu}\style{display: inline-block; transform: scale(-.62, 1.24) translateY(.05em); font-family: sans-serif;}{๊—จ\hspace{-2.6mu}๊—จ}\hspace{-1.4mu}} \def\llzigzag{\hspace{-1.6mu}\style{display: inline-block; transform: scale(.62, 1.24) translateY(.07em); font-family: sans-serif;}{๊—จ\hspace{-3mu}๊—จ}\hspace{-1.6mu}} \def\rrzigzag{\hspace{-1.6mu}\style{display: inline-block; transform: scale(-.62, 1.24) translateY(.07em); font-family: sans-serif;}{๊—จ\hspace{-3mu}๊—จ}\hspace{-1.6mu}} \def\largeLLzigzag{\hspace{-1.8mu}\style{display: inline-block; transform: scale(.62, 1.24) translateY(.09em); font-family: sans-serif;}{๊—จ\hspace{-3.5mu}๊—จ}\hspace{-1.8mu}} \def\largeRRzigzag{\hspace{-1.8mu}\style{display: inline-block; transform: scale(-.62, 1.24) translateY(.09em); font-family: sans-serif;}{๊—จ\hspace{-3.5mu}๊—จ}\hspace{-1.8mu}} \def\LargeLLzigzag{\hspace{-2.5mu}\style{display: inline-block; transform: scale(.62, 1.24) translateY(.1em); font-family: sans-serif;}{๊—จ\hspace{-4.5mu}๊—จ}\hspace{-2.5mu}} \def\LargeRRzigzag{\hspace{-2.5mu}\style{display: inline-block; transform: scale(-.62, 1.24) translateY(.1em); font-family: sans-serif;}{๊—จ\hspace{-4.5mu}๊—จ}\hspace{-2.5mu}} }[/math]

This is an appendix to Dave Keenan & Douglas Blumeyer's guide to RTT. The tables in this article present our recommendations for communicating about regular temperament theory (RTT), in particular the names and notations for temperament matrices, tuning schemes, interval complexities, and measurement units.

Our recommendations are designed to make this topic easy to learn for musicians who do not have technical backgrounds, though we have generally deferred to established mathematical, scientific, and engineering conventions for the benefit of those who do.

See here for more information on our variation on extended bra-ket notation.

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:

Units →   Simple units Compound or no units
↓ Order ↓ Style → Roman (upright) Italic
0 lowercase   scalar (with simple unit) scalar (with no unit)
1 bold lowercase vector map (row vector)
2 UPPERCASE 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)[note 1]. 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.

Basic

Objects

Equivalent
expressions 		Variable Name Units Shape Type EBK notation Subobjects Notes
Unreduced Reduced Read as Unreduced Reduced Numeric Structural Row-first Col-first Row Column Diagonal Entry
Mapping
[math]\displaystyle{ \textbf{i} }[/math] (Just) interval [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, 1) }[/math] Integer Vector [...โŸฉ [math]\displaystyle{ \mathrm{i}_i }[/math] Specific type: Vector (prime-count vector or PC-vector)

Jargon name: Monzo

[math]\displaystyle{ M }[/math] (Temperament) mapping (matrix) [math]\displaystyle{ \small ๐—ด }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Generators per prime [math]\displaystyle{ \scriptsize (r, d) }[/math] Integer Matrix [โŸจ...] ...} โŸจ[...} ...] [math]\displaystyle{ ๐’Ž_i }[/math] [math]\displaystyle{ m_{ij} }[/math] Jargon name: Val list
[math]\displaystyle{ M\textbf{i} }[/math] [math]\displaystyle{ \textbf{y} }[/math] Mapped interval [math]\displaystyle{ \scriptsize \begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \small ๐—ด }[/math] Generators [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} M \\[-3pt] (r, \cancel{d}) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] (\cancel{d}, 1) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (r, 1) }[/math] Integer Vector [...} Specific type: Generator-count vector (GC-vector)

Jargon name: tmonzo; mnemonic: [math]\displaystyle{ \textbf{y} }[/math]nterval

[math]\displaystyle{ ๐’Ž }[/math] (Temperament) map [math]\displaystyle{ \small ๐—ด }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Generators per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Integer Vector โŸจ...] [math]\displaystyle{ m_i }[/math] Jargon name: val
[math]\displaystyle{ d }[/math] dimensionality [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
[math]\displaystyle{ r }[/math] Rank [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
Tuning
[math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu} }[/math] Log-prime map [math]\displaystyle{ \small\mathsf{oct} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Octaves per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Real Vector โŸจ...] [math]\displaystyle{ {\large ๐“}\hspace{2mu}_i }[/math]
[math]\displaystyle{ 1200ร—{\large\textbf{๐“}}\hspace{2mu} }[/math] [math]\displaystyle{ ๐’‹ }[/math] Just(-prime) tuning map [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Cents per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Real Vector โŸจ...] [math]\displaystyle{ j_i }[/math]
[math]\displaystyle{ ๐’ˆ }[/math] Generator tuning map [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ด }[/math] Cents per generator [math]\displaystyle{ \scriptsize (1, r) }[/math] Real Vector {...] [math]\displaystyle{ g_i }[/math]
[math]\displaystyle{ ๐’ˆM }[/math] [math]\displaystyle{ ๐’• }[/math] (Tempered-prime) tuning map [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’ˆ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} ๐‘€ \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Cents per prime [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’ˆ \\[-3pt] \left(1, \cancel{r}\right) \end{array} \! \! \begin{array} {c} M \\[-3pt] \left(\cancel{r}, d\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, d) }[/math] Real Vector โŸจ...] [math]\displaystyle{ t_i }[/math]
[math]\displaystyle{ ๐’• - ๐’‹ }[/math] [math]\displaystyle{ ๐’“ }[/math] Retuning (or mistuning) map [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Cents per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Real Vector โŸจ...] [math]\displaystyle{ r_i }[/math] Previous name: prime error map
[math]\displaystyle{ ๐’‹\textbf{i} }[/math] [math]\displaystyle{ \mathrm{o} }[/math] (Just) (interval) size [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’‹ \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar Mnemonic: [math]\displaystyle{ \mathrm{o} }[/math]riginal size
[math]\displaystyle{ ๐’ˆM\textbf{i} }[/math]

[math]\displaystyle{ ๐’•\textbf{i} }[/math]

[math]\displaystyle{ \mathrm{a} }[/math] Tempered (interval) size [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’• \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar Mnemonic: [math]\displaystyle{ \mathrm{a} }[/math]ltered size
[math]\displaystyle{ ๐’•\textbf{i} - ๐’‹\textbf{i} }[/math]

[math]\displaystyle{ a - o }[/math]
[math]\displaystyle{ ๐’“\textbf{i} }[/math]

[math]\displaystyle{ \mathrm{e} }[/math] (Interval) error [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’“ \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
Optimization
[math]\displaystyle{ p }[/math] Optimization power [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \llangle\,ยท\,\rrangle_p }[/math] Power mean ([math]\displaystyle{ p }[/math]-mean) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
Damage
[math]\displaystyle{ c }[/math] Complexity [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)} }[/math][note 2] [math]\displaystyle{ \small\mathsf{(C)} }[/math] Complexity weight [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \dfrac1c }[/math] [math]\displaystyle{ s }[/math] Simplicity [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)} }[/math] [math]\displaystyle{ \small\mathsf{(S)} }[/math] Simplicity weight [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ c }[/math] or [math]\displaystyle{ s }[/math] [math]\displaystyle{ w }[/math] Weight [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)} }[/math] or ๐Ÿ™[math]\displaystyle{ \small\mathsf{(S)} }[/math] [math]\displaystyle{ \small\mathsf{(C)} }[/math] or [math]\displaystyle{ \small\mathsf{(S)} }[/math] Complexity weight or simplicity weight [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \abs{\mathrm{e}} w }[/math] [math]\displaystyle{ \mathrm{d} }[/math] Damage [math]\displaystyle{ \scriptsize \begin{array} {c} \abs{\mathrm{e}} \\[-2pt] {\small\mathsf{ยข}} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} w \\[-2pt] \mathsf{(U, C, \text{or}\,S)} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(U)} }[/math] or [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(C)} }[/math] or [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(S)} }[/math] (See damages table) [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} \abs{\mathrm{e}} \\[-3pt] \left(1, \cancel{1}\right) \end{array} \! \! \begin{array} {c} w \\[-3pt] \left(\cancel{1}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
Target-intervals
[math]\displaystyle{ \mathrm{T} }[/math] Target-interval list [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, k) }[/math] Integer Matrix [[...โŸฉ ...] [math]\displaystyle{ \textbf{t}_i }[/math] [math]\displaystyle{ \mathrm{t}_{ij} }[/math]
[math]\displaystyle{ M\mathrm{T} }[/math] [math]\displaystyle{ \mathrm{Y} }[/math] Mapped target-interval list [math]\displaystyle{ \scriptsize \begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \small ๐—ด }[/math] Generators [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (r, k) }[/math] Integer Matrix [[...} ...] [math]\displaystyle{ \textbf{y}_i }[/math] [math]\displaystyle{ \mathrm{y}_{ij} }[/math] Mnemonic: looks like bent-up 'T', or cross between 'M' and 'T'
[math]\displaystyle{ ๐’‹\mathrm{T} }[/math] [math]\displaystyle{ \textbf{o} }[/math] Target-interval (just) size list [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’‹ \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, k) }[/math] Real List [...] [math]\displaystyle{ \mathrm{o}_i }[/math] Mnemonic: [math]\displaystyle{ \textbf{o} }[/math]riginal size list
[math]\displaystyle{ ๐’•\mathrm{T} }[/math]

[math]\displaystyle{ ๐’ˆM\mathrm{T} }[/math]

[math]\displaystyle{ \textbf{a} }[/math] Tempered target-interval size list [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’• \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, k) }[/math] Real List [...] [math]\displaystyle{ \mathrm{a}_i }[/math] Mnemonic: [math]\displaystyle{ \textbf{a} }[/math]ltered size list
[math]\displaystyle{ ๐’•\mathrm{T} - ๐’‹\mathrm{T} }[/math]

[math]\displaystyle{ \textbf{a} - \textbf{o} }[/math]
[math]\displaystyle{ ๐’“\mathrm{T} }[/math]

[math]\displaystyle{ \textbf{e} }[/math] Target-interval error list [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’“ \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, k) }[/math] Real List [...] [math]\displaystyle{ \mathrm{e}_i }[/math]
[math]\displaystyle{ C }[/math] or [math]\displaystyle{ S }[/math] [math]\displaystyle{ W }[/math] Target-interval weight matrix [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)} }[/math] or [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)} }[/math] or [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(U)} }[/math] [math]\displaystyle{ \small\mathsf{(C)} }[/math] or [math]\displaystyle{ \small\mathsf{(S)} }[/math] or [math]\displaystyle{ \small\mathsf{(U)} }[/math] Complexity weight or simplicity weight [math]\displaystyle{ \scriptsize (k, k) }[/math] Real Matrix [[...] ...] [math]\displaystyle{ ๐’˜ }[/math] [math]\displaystyle{ w_i }[/math]
[math]\displaystyle{ C }[/math] Target-interval complexity weight matrix [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)} }[/math] [math]\displaystyle{ \small\mathsf{(C)} }[/math] Complexity weight [math]\displaystyle{ \scriptsize (k, k) }[/math] Real Matrix [[...] ...] [math]\displaystyle{ ๐’„ }[/math] [math]\displaystyle{ c_i }[/math]
[math]\displaystyle{ \dfrac1C }[/math] [math]\displaystyle{ S }[/math] Target-interval simplicity weight matrix [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)} }[/math] [math]\displaystyle{ \small\mathsf{(S)} }[/math] Simplicity weight [math]\displaystyle{ \scriptsize (k, k) }[/math] Real Matrix [[...] ...] [math]\displaystyle{ ๐’” }[/math] [math]\displaystyle{ s_i }[/math] Entry-wise reciprocal of [math]\displaystyle{ C }[/math]
[math]\displaystyle{ \abs{\textbf{e}} W }[/math] [math]\displaystyle{ \textbf{d} }[/math] Target-interval damage list[note 3] [math]\displaystyle{ \scriptsize \begin{array} {c} \abs{\textbf{e}} \\[-2pt] {\small\mathsf{ยข}} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} W \\[-2pt] (\mathsf{U, C, \text{or}\,S}) \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(U)} }[/math], [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(C)} }[/math], or [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(S)} }[/math] Weighted cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} \abs{\textbf{e}} \\[-3pt] \left(1, \cancel{k}\right) \end{array} \! \! \begin{array} {c} W \\[-3pt] \left(\cancel{k}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, k) }[/math] Real List [...] [math]\displaystyle{ \mathrm{d}_i }[/math]
[math]\displaystyle{ k }[/math] Target-interval count [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar Mnemonic: [math]\displaystyle{ k }[/math]ount
Held-intervals
[math]\displaystyle{ \mathrm{H} }[/math] Held-interval basis [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, h) }[/math] Matrix [[...โŸฉ ...] [math]\displaystyle{ \textbf{h}_i }[/math] [math]\displaystyle{ \mathrm{h}_{ij} }[/math]
[math]\displaystyle{ h }[/math] Held-interval count [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
Exploring temperaments
[math]\displaystyle{ \mathrm{C} }[/math] Comma basis [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, n) }[/math] Integer Matrix [[...โŸฉ ...] [math]\displaystyle{ \textbf{c}_i }[/math] [math]\displaystyle{ \mathrm{c}_{ij} }[/math] Jargon name: monzo list
[math]\displaystyle{ \textbf{c} }[/math] Comma [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, 1) }[/math] Integer Vector [...โŸฉ [math]\displaystyle{ \mathrm{c}_i }[/math] Specific type: vector (prime-count vector or PC-vector)

Units

We recommend using a narrow no-break space (U+202F) between quantities and their units.[note 4] For how to type this, see the WinCompose section below.

Symbol Name Vectorized
[math]\displaystyle{ \small ๐—ด }[/math] Generators Yes
[math]\displaystyle{ \small ๐—ฝ }[/math] Primes Yes
[math]\displaystyle{ \mathsf{ยข} }[/math][note 5] Cents
[math]\displaystyle{ \mathsf{ยข}\small\mathsf{(U)} }[/math] Unity-weighted cents
[math]\displaystyle{ \mathsf{ยข}\small\mathsf{(C)} }[/math] Complexity-weighted cents
[math]\displaystyle{ \mathsf{ยข}\small\mathsf{(S)} }[/math] Simplicity-weighted cents
[math]\displaystyle{ \small\mathsf{oct} }[/math] Octaves
[math]\displaystyle{ \small\mathsf{(C)} }[/math] Complexity weight
[math]\displaystyle{ \small\mathsf{(S)} }[/math] Simplicity weight

Tuning schemes

Copied from Systematic tuning scheme names.

Damage weight Optimization power Systematic name
<none> ∞ Minimax-U
Complexity Minimax-C
1/Complexity Minimax-S
<none> 2 MiniRMS-U
Complexity MiniRMS-C
1/Complexity MiniRMS-S
<none> 1 Miniaverage-U
Complexity Miniaverage-C
1/Complexity Miniaverage-S

Damages

Quantity Unit
Abbreviation Name Symbol Name
U-damage Unity-weight damage [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(U)} }[/math] Unity-weighted cents
C-damage Complexity-weight damage [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(C)} }[/math] Complexity-weighted cents
S-damage Simplicity-weight damage [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(S)} }[/math] Simplicity-weighted cents

Complexity and simplicity

Quantity Unit
Abbreviation Name Symbol Name
C Complexity [math]\displaystyle{ \small\mathsf{(C)} }[/math] Complexity weight
S Simplicity [math]\displaystyle{ \small\mathsf{(S)} }[/math] Simplicity weight

Intermediate

Objects

Equivalent expressions Variable Name Units Shape Type EBK notation Subobjects Notes
Unreduced Reduced Read as Unreduced Reduced Numeric Structural Row-first Col-first Row Col Diag Entry
Mapping
[math]\displaystyle{ \textbf{i} }[/math] (Just) interval [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, 1) }[/math] Integer Vector [...โŸฉ [math]\displaystyle{ \mathrm{i}_i }[/math] Specific type: vector (prime-count vector or PC-vector)

Jargon name: monzo

[math]\displaystyle{ M }[/math] (Temperament) mapping (matrix) [math]\displaystyle{ \small ๐—ด }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Generators per prime [math]\displaystyle{ \scriptsize (r, d) }[/math] Integer Matrix [โŸจ...] ...} โŸจ[...} ...] [math]\displaystyle{ ๐’Ž_i }[/math] [math]\displaystyle{ m_{ij} }[/math] Jargon name: val list
[math]\displaystyle{ M\textbf{i} }[/math] [math]\displaystyle{ \textbf{y} }[/math] Mapped interval [math]\displaystyle{ \scriptsize \begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \small ๐—ด }[/math] Generators [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (r, 1) }[/math] Integer Vector [...} Specific type: generator-count vector (GC-vector)

Jargon name: tmonzo; mnemonic: [math]\displaystyle{ \textbf{y} }[/math]nterval

[math]\displaystyle{ ๐’Ž }[/math] (Temperament) map [math]\displaystyle{ \small ๐—ด }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Generators per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Integer Vector โŸจ...] [math]\displaystyle{ m_i }[/math] Jargon name: val
[math]\displaystyle{ n + r }[/math] [math]\displaystyle{ d }[/math] Dimensionality [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
[math]\displaystyle{ d - n }[/math] [math]\displaystyle{ r }[/math] Rank [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
[math]\displaystyle{ d - r }[/math] [math]\displaystyle{ n }[/math] Nullity [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
Tuning
[math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu} }[/math] Log-prime map [math]\displaystyle{ \small\mathsf{oct} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Octaves per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Real Vector โŸจ...] [math]\displaystyle{ {\large ๐“}\hspace{2mu}_i }[/math]
[math]\displaystyle{ 1200ร—{\large\textbf{๐“}}\hspace{2mu} }[/math] [math]\displaystyle{ ๐’‹ }[/math] Just(-prime) tuning map [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Cents per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Real Vector โŸจ...] [math]\displaystyle{ j_i }[/math]
[math]\displaystyle{ ๐’ˆ }[/math] Generator tuning map [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ด }[/math] Cents per generator [math]\displaystyle{ \scriptsize (1, r) }[/math] Real Vector {...] [math]\displaystyle{ g_i }[/math]
[math]\displaystyle{ ๐’• }[/math] (Tempered-prime) tuning map [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Cents per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Real Vector โŸจ...] [math]\displaystyle{ t_i }[/math]
[math]\displaystyle{ ๐’• - ๐’‹ }[/math]

[math]\displaystyle{ 1200ร—\slant{\mathbf{1}}L(P - I) }[/math]

[math]\displaystyle{ ๐’“ }[/math] Retuning (or mistuning) map [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Cents per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Real Vector โŸจ...] [math]\displaystyle{ r_i }[/math] Previous name: prime error map
[math]\displaystyle{ ๐’‹\textbf{i} }[/math] [math]\displaystyle{ \mathrm{o} }[/math] (Just) (interval) size [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’‹ \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar Mnemonic: [math]\displaystyle{ \mathrm{o} }[/math]riginal size
[math]\displaystyle{ ๐’ˆM\textbf{i} }[/math]

[math]\displaystyle{ ๐’•\textbf{i} }[/math]

[math]\displaystyle{ \mathrm{a} }[/math] Tempered (interval) size [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’• \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar Mnemonic: [math]\displaystyle{ \mathrm{a} }[/math]ltered size
[math]\displaystyle{ ๐’•\textbf{i} - ๐’‹\textbf{i} }[/math]

[math]\displaystyle{ a - o }[/math]
[math]\displaystyle{ ๐’“\textbf{i} }[/math]

[math]\displaystyle{ \mathrm{e} }[/math] (Interval) error [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’“ \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
Optimization
[math]\displaystyle{ p }[/math] Optimization power [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \llangle\,ยท\,\rrangle_p }[/math] Power mean ([math]\displaystyle{ p }[/math]-mean) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
Damage
[math]\displaystyle{ c }[/math] Complexity (See complexities section of complexities and simplicities table) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \dfrac1c }[/math] [math]\displaystyle{ s }[/math] Simplicity (See simplicities section of complexities and simplicities table) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ c }[/math] or [math]\displaystyle{ s }[/math] [math]\displaystyle{ w }[/math] weight (See complexities and simplicities table) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \abs{\mathrm{e}} w }[/math] [math]\displaystyle{ \mathrm{d} }[/math] Damage (See damages table) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
Target-intervals
[math]\displaystyle{ \mathrm{T} }[/math] Target-interval list [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, k) }[/math] Integer Matrix [[...โŸฉ ...] [math]\displaystyle{ \textbf{t}_i }[/math] [math]\displaystyle{ \mathrm{t}_{ij} }[/math]
[math]\displaystyle{ M\mathrm{T} }[/math] [math]\displaystyle{ \mathrm{Y} }[/math] Mapped target-interval list [math]\displaystyle{ \scriptsize \begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \small ๐—ด }[/math] Generators [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (r, k) }[/math] Integer Matrix [[...} ...] [math]\displaystyle{ \textbf{y}_i }[/math] [math]\displaystyle{ \mathrm{y}_{ij} }[/math] Mnemonic: looks like bent-up 'T', or cross between 'M' and 'T'
[math]\displaystyle{ ๐’‹\mathrm{T} }[/math] [math]\displaystyle{ \textbf{o} }[/math] Target-interval (just) size list [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’‹ \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, k) }[/math] Real List [...] [math]\displaystyle{ \mathrm{o}_i }[/math] Mnemonic: [math]\displaystyle{ \textbf{o} }[/math]riginal size list
[math]\displaystyle{ ๐’•\mathrm{T} }[/math] [math]\displaystyle{ \textbf{a} }[/math] Tempered target-interval size list [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’• \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, k) }[/math] Real List [...] [math]\displaystyle{ \mathrm{a}_i }[/math] Mnemonic: [math]\displaystyle{ \textbf{a} }[/math]ltered size list
[math]\displaystyle{ ๐’•\mathrm{T} - ๐’‹\mathrm{T} }[/math]

[math]\displaystyle{ ๐’“\mathrm{T} }[/math]
[math]\displaystyle{ \textbf{a} - \textbf{o} }[/math]

[math]\displaystyle{ \textbf{e} }[/math] Target-interval error list [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’“ \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, k) }[/math] Real List [...] [math]\displaystyle{ \mathrm{e}_i }[/math]
[math]\displaystyle{ C }[/math] or [math]\displaystyle{ S }[/math] [math]\displaystyle{ W }[/math] Target-interval weight matrix (See complexities and simplicities table) [math]\displaystyle{ \scriptsize (k, k) }[/math] Real Matrix [[...] ...] [math]\displaystyle{ ๐’˜ }[/math] [math]\displaystyle{ w_i }[/math]
[math]\displaystyle{ C }[/math] Target-interval complexity weight matrix (See complexities section of complexities and simplicities table) [math]\displaystyle{ \scriptsize (k, k) }[/math] Real Matrix [[...] ...] [math]\displaystyle{ ๐’„ }[/math] [math]\displaystyle{ c_i }[/math]
[math]\displaystyle{ \dfrac1C }[/math] [math]\displaystyle{ S }[/math] Target-interval simplicity weight matrix (See simplicities section of complexities and simplicities table) [math]\displaystyle{ \scriptsize (k, k) }[/math] Real Matrix [[...] ...] [math]\displaystyle{ ๐’” }[/math] [math]\displaystyle{ s_i }[/math] Entry-wise reciprocal of [math]\displaystyle{ C }[/math]
[math]\displaystyle{ \abs{\textbf{e}} W }[/math] [math]\displaystyle{ \textbf{d} }[/math] Target-interval damage list (See damages table) [math]\displaystyle{ \scriptsize (1, k) }[/math] Real List [...] [math]\displaystyle{ \mathrm{d}_i }[/math]
[math]\displaystyle{ k }[/math] Target-interval count [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar Mnemonic: [math]\displaystyle{ k }[/math]ount
Held-intervals
[math]\displaystyle{ \mathrm{H} }[/math] Held-interval basis [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, h) }[/math] Matrix [[...โŸฉ ...] [math]\displaystyle{ \textbf{h}_i }[/math] [math]\displaystyle{ \mathrm{h}_{ij} }[/math]
[math]\displaystyle{ h }[/math] Held-interval count [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
Exploring temperaments
[math]\displaystyle{ \mathrm{C} }[/math] Comma basis [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, n) }[/math] Integer Matrix [[...โŸฉ ...] [math]\displaystyle{ \textbf{c}_i }[/math] [math]\displaystyle{ \mathrm{c}_{ij} }[/math] Jargon name: monzo list
[math]\displaystyle{ \textbf{c} }[/math] Comma [math]\displaystyle{ \small ๐—ฝ }[/math] primes [math]\displaystyle{ \scriptsize (d, 1) }[/math] Integer Vector [...โŸฉ [math]\displaystyle{ \mathrm{c}_i }[/math] Specific type: vector (prime-count vector or PC-vector)
Computation
ꗨ⁠ ⁠ꗨ[math]\displaystyle{ \,ยท\, }[/math]ꗨ⁠ ⁠ꗨ[math]\displaystyle{ _p }[/math] Power sum ([math]\displaystyle{ p }[/math]-sum) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
All-interval tuning schemes
[math]\displaystyle{ \mathrm{I} }[/math] [math]\displaystyle{ \mathrm{T}_{\text{p}} }[/math] Prime proxy target-interval list [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, d) }[/math] Integer Matrix โŸจ[...โŸฉ ...] [math]\displaystyle{ \mathbf{1} }[/math]
[math]\displaystyle{ X }[/math] Complexity prescaler [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)} }[/math] [math]\displaystyle{ \small\mathsf{(C)} }[/math] Complexity weight [math]\displaystyle{ \scriptsize (d, d) }[/math] Real Matrix [โŸจ...] ...โŸฉ [math]\displaystyle{ ๐’™ }[/math] [math]\displaystyle{ x_i }[/math]
[math]\displaystyle{ \text{diag}({\large\textbf{๐“}}\hspace{2mu}) }[/math] [math]\displaystyle{ L }[/math] Log-prime matrix [math]\displaystyle{ \small\mathsf{oct} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Octaves per prime [math]\displaystyle{ \scriptsize (d, d) }[/math] Real Matrix [โŸจ...] ...โŸฉ โŸจ[...โŸฉ ...] [math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu}_i }[/math] [math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu} }[/math] [math]\displaystyle{ {\large ๐“}\hspace{2mu}_{ij} }[/math]
[math]\displaystyle{ q }[/math] Interval complexity norm power [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \norm{ยท}_q }[/math] Power norm ([math]\displaystyle{ p }[/math]-norm) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \dfrac1{1-\frac1q} }[/math] [math]\displaystyle{ \text{dual}(q) }[/math] Dual norm power [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \norm{X\mathbf{i}}_q }[/math] interval complexity [math]\displaystyle{ \small\mathsf{(C)} }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \norm{๐’“X^{-1}}_{\text{dual}(q)} }[/math] Retuning magnitude [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(C^{-1})} }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar

Units

Same as the basic level.

Tuning schemes

Retuning (or mistuning) magnitude Damage Target
intervals 		Systematic name Previously named tuning schemes that are specific types of this tuning scheme Of interest?
Weight Optimization
Interval complexity Slope Initial Name Power
Initial Name Power Initial Name Power Initial Name Multiplier Abbreviated Read ("____ tuning scheme")
<n/a> Maximum ∞ (t) Taxicab 1 S Simplicity-weight 1/Complexity <n/a> Minimax โˆž All Minimax-S Minimax simplicity-weight damage "TOP"/"T1"/"TIPTOP"*, "CTOP", "POTOP"/"POTT"*, "BOP", "Weil", "Kees" Yes
<n/a> Euclidean 2 E Euclidean 2 Minimax-ES Minimax Euclideanized-simplicity-weight damage "TE"/"T2"/"TOP-RMS", "CTE", "POTE", "Frobenius", "BE", "WE", "KE"
<n/a> <n/a> U Unity-weight <none> <set> <set> Minimax-U <set> Minimax unity-weight-damage " minimax" yes
(t) taxicab 1 S Simplicity-weight 1/Complexity <set> Minimax-S <set> Minimax simplicity-weight damage Yes
E Euclidean 2 <set> Minimax-ES <set> Minimax Euclideanized-simplicity-weight damage
(t) Taxicab 1 C Complexity-weight Complexity <set> Minimax-C <set> Minimax complexity-weight damage Yes
E Euclidean 2 <set> Minimax-EC <set> Minimax Euclideanized-complexity-weight damage
<n/a> U Unity-weight <none> MiniRMS 2 <set> MiniRMS-U <set> MiniRMS unity-weight damage "Least squares" Yes
(t) Taxicab 1 S Simplicity-weight 1/Complexity <set> MiniRMS-S <set> MiniRMS simplicity-weight damage Yes
E Euclidean 2 <set> MiniRMS-ES <set> MiniRMS Euclideanized-simplicity-weight damage
(t) Taxicab 1 C Complexity-weight Complexity <set> MiniRMS-C <set> MiniRMS complexity-weight damage Yes
E Euclidean 2 <set> MiniRMS-EC <set> MiniRMS Euclideanized-complexity-weight damage
<n/a> U Unity-weight <none> Miniaverage 1 <set> Miniaverage-U <set> Miniaverage unity-weight damage Yes
(t) Taxicab 1 S Simplicity-weight 1/Complexity <set> Miniaverage-S <set> Miniaverage simplicity-weight damage Yes
E Euclidean 2 <set> Miniaverage-ES <set> Miniaverage Euclideanized-simplicity-weight damage
(t) Taxicab 1 C Complexity-weight Complexity <set> Miniaverage-C <set> Miniaverage complexity-weight damage Yes
E Euclidean 2 <set> Miniaverage-EC <set> Miniaverage Euclideanized-complexity-weight damage

Damages

Quantity Unit
Abbreviation Name Symbol Name
U-damage Unity-weight damage [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(U)} }[/math] Unity-weighted cents
C-damage Complexity-weight damage [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(C)} }[/math] Complexity-weighted cents
EC-damage Euclideanized-complexity-weight damage [math]\displaystyle{ \mathsf{ยข} }[/math][math]\displaystyle{ \small\mathsf{(EC)} }[/math] Euclideanized-complexity-weighted cents
S-damage Simplicity-weight damage [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(S)} }[/math] Simplicity-weighted cents
ES-damage Euclideanized-simplicity-weight damage [math]\displaystyle{ \mathsf{ยข} }[/math][math]\displaystyle{ \small\mathsf{(ES)} }[/math] Euclideanized-simplicity-weighted cents

Complexity and simplicity

Quantity Unit
Abbreviation Name Symbol Name
C Complexity [math]\displaystyle{ \small\mathsf{(C)} }[/math] Complexity weight
EC Euclideanized complexity [math]\displaystyle{ \small\mathsf{(EC)} }[/math] Euclideanized-complexity weight
S Simplicity [math]\displaystyle{ \small\mathsf{(S)} }[/math] Simplicity weight
ES Euclideanized simplicity [math]\displaystyle{ \small\mathsf{(ES)} }[/math] Euclideanized-simplicity weight

Advanced

Objects

Equivalent expressions Variable Name Units Shape Type EBK notation Subobjects Notes
Unreduced Reduced Read as Unreduced Reduced Numeric Structural Row-first Col-first Row Column Diagonal Entry
Mapping
[math]\displaystyle{ \textbf{i} }[/math] (Just) interval [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, 1) }[/math] Integer Vector [...โŸฉ [math]\displaystyle{ \mathrm{i}_i }[/math] Specific type: vector (prime-count vector or PC-vector)

Jargon name: monzo

[math]\displaystyle{ M }[/math] (Temperament) mapping (matrix) [math]\displaystyle{ \small ๐—ด }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Generators per prime [math]\displaystyle{ \scriptsize (r, d) }[/math] Integer Matrix [โŸจ...] ...} โŸจ[...} ...] [math]\displaystyle{ ๐’Ž_i }[/math] [math]\displaystyle{ m_{ij} }[/math] Jargon name: val list
[math]\displaystyle{ M\textbf{i} }[/math] [math]\displaystyle{ \textbf{y} }[/math] Mapped interval [math]\displaystyle{ \scriptsize \begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \small ๐—ด }[/math] generators [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (r, 1) }[/math] Integer Vector [...} Specific type: generator-count vector (GC-vector)

Jargon name: tmonzo; mnemonic: [math]\displaystyle{ \textbf{y} }[/math]nterval

[math]\displaystyle{ ๐’Ž }[/math] (Temperament) map [math]\displaystyle{ \small ๐—ด }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Generators per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Integer Vector โŸจ...] [math]\displaystyle{ m_i }[/math] Jargon name: val
[math]\displaystyle{ n + r }[/math] [math]\displaystyle{ d }[/math] Dimensionality [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
[math]\displaystyle{ d - n }[/math] [math]\displaystyle{ r }[/math] Rank [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
[math]\displaystyle{ d - r }[/math] [math]\displaystyle{ n }[/math] Nullity [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
Tuning
[math]\displaystyle{ \slant{\mathbf{1}}L }[/math] [math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu} }[/math] Log-prime map [math]\displaystyle{ \small\mathsf{oct} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Octaves per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Real Vector โŸจ...] [math]\displaystyle{ {\large ๐“}\hspace{2mu}_i }[/math]
[math]\displaystyle{ 1200ร—\slant{\mathbf{1}}LG_{\text{j}}M_{\text{j}} }[/math]

[math]\displaystyle{ 1200ร—\slant{\mathbf{1}}L }[/math]
[math]\displaystyle{ ๐’ˆ_{\text{j}}M_{\text{j}} }[/math]

[math]\displaystyle{ ๐’‹ }[/math] Just(-prime) tuning map [math]\displaystyle{ \scriptsize \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} \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} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \\ \scriptsize \quad \begin{array} {c} G_{\text{j}} \\[-2pt] \cancel{๐—ฝ} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} M_{\text{j}} \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Cents per prime [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} 1200 \\[-3pt] \left(1, \cancel{1}\right) \end{array} \! \! \begin{array} {c} \slant{\mathbf{1}} \\[-3pt] \left(\cancel{1}, \cancel{d}\right) \end{array} \! \! \begin{array} {c} L \\[-3pt] \left(\cancel{d}, \cancel{d}\right) \end{array} \\ \scriptsize \quad \! \! \begin{array} {c} G_{\text{j}} \\[-3pt] \left(\cancel{d}, \cancel{r}\right) \end{array} \! \! \begin{array} {c} M_{\text{j}} \\[-3pt] \left(\cancel{r}, d\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize \left(1, d_{\text{p}}\right) }[/math] Real Vector โŸจ...] [math]\displaystyle{ j_i }[/math]
[math]\displaystyle{ 1200ร—\slant{\mathbf{1}}LG }[/math] [math]\displaystyle{ ๐’ˆ }[/math] Generator tuning map [math]\displaystyle{ \scriptsize \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} \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} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \\ \scriptsize \quad \begin{array} {c} G \\[-2pt] \cancel{๐—ฝ} \hspace{-2mu} / \hspace{-2mu} ๐—ด \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ด }[/math] Cents per generator [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} 1200 \\[-3pt] (1, \cancel{1}) \end{array} \! \! \begin{array} {c} \slant{\mathbf{1}} \\[-3pt] (\cancel{1}, \cancel{d}) \end{array} \! \! \begin{array} {c} L \\[-3pt] (\cancel{d}, \cancel{d}) \end{array} \\ \scriptsize \quad \! \! \begin{array} {c} G \\[-3pt] (\cancel{d}, r) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, r) }[/math] Real Vector {...] [math]\displaystyle{ g_i }[/math]
[math]\displaystyle{ 1200ร—\slant{\mathbf{1}}LGM }[/math]

[math]\displaystyle{ 1200ร—\slant{\mathbf{1}}LP }[/math]
[math]\displaystyle{ ๐’ˆM }[/math]

[math]\displaystyle{ ๐’• }[/math] (Tempered-prime) tuning map [math]\displaystyle{ \scriptsize \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} \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} L \\[-2pt] \cancel{\mathsf{oct}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \\ \scriptsize \quad \begin{array} {c} G \\[-2pt] \cancel{๐—ฝ} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} M \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Cents per prime [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} 1200 \\[-3pt] \left(1, \cancel{1}\right) \end{array} \! \! \begin{array} {c} \slant{\mathbf{1}} \\[-3pt] \left(\cancel{1}, \cancel{d}\right) \end{array} \! \! \begin{array} {c} L \\[-3pt] \left(\cancel{d}, \cancel{d}\right) \end{array} \\ \scriptsize \quad \! \! \begin{array} {c} G \\[-3pt] \left(\cancel{d}, \cancel{r}\right) \end{array} \! \! \begin{array} {c} M \\[-3pt] \left(\cancel{r}, d\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, d) }[/math] Real Vector โŸจ...] [math]\displaystyle{ t_i }[/math]
[math]\displaystyle{ ๐’• - ๐’‹ }[/math]

[math]\displaystyle{ 1200ร—\slant{\mathbf{1}}L(P - I) }[/math]

[math]\displaystyle{ ๐’“ }[/math] Retuning (or mistuning) map [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Cents per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Real Vector โŸจ...] [math]\displaystyle{ r_i }[/math] Previous name: prime error map
[math]\displaystyle{ ๐’‹\textbf{i} }[/math] [math]\displaystyle{ \mathrm{o} }[/math] (Just) (interval) size [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’‹ \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar Mnemonic: [math]\displaystyle{ \mathrm{o} }[/math]riginal size
[math]\displaystyle{ ๐’ˆM\textbf{i} }[/math]

[math]\displaystyle{ ๐’•\textbf{i} }[/math]

[math]\displaystyle{ \mathrm{a} }[/math] Tempered (interval) size [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’• \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar Mnemonic: [math]\displaystyle{ \mathrm{a} }[/math]ltered size
[math]\displaystyle{ ๐’•\textbf{i} - ๐’‹\textbf{i} }[/math]

[math]\displaystyle{ a - o }[/math]
[math]\displaystyle{ ๐’“\textbf{i} }[/math]

[math]\displaystyle{ \mathrm{e} }[/math] (Interval) error [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’“ \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
Optimization
[math]\displaystyle{ p }[/math] Optimization power [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \llangle\,ยท\,\rrangle_p }[/math] Power mean ([math]\displaystyle{ p }[/math]-mean) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
Damage
[math]\displaystyle{ c }[/math] complexity (See complexities section of complexities and simplicities table) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \dfrac1c }[/math] [math]\displaystyle{ s }[/math] Simplicity (See simplicities section of complexities and simplicities table) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ c }[/math] or [math]\displaystyle{ s }[/math] [math]\displaystyle{ w }[/math] Weight (See complexities and simplicities table) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \abs{\mathrm{e}} w }[/math] [math]\displaystyle{ \mathrm{d} }[/math] Damage (See damages table) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
Target-intervals
[math]\displaystyle{ \mathrm{T} }[/math] Target-interval list [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, k) }[/math] Integer Matrix [[...โŸฉ ...] [math]\displaystyle{ \textbf{t}_i }[/math] [math]\displaystyle{ \mathrm{t}_{ij} }[/math]
[math]\displaystyle{ M\mathrm{T} }[/math] [math]\displaystyle{ \mathrm{Y} }[/math] Mapped target-interval list [math]\displaystyle{ \scriptsize \begin{array} {c} M \\[-2pt] ๐—ด \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \small ๐—ด }[/math] Generators [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} M \\[-3pt] \left(r, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (r, k) }[/math] Integer Matrix [[...} ...] [math]\displaystyle{ \textbf{y}_i }[/math] [math]\displaystyle{ \mathrm{y}_{ij} }[/math] Mnemonic: looks like bent-up 'T', or cross between 'M' and 'T'
[math]\displaystyle{ ๐’‹\mathrm{T} }[/math] [math]\displaystyle{ \textbf{o} }[/math] Target-interval (just) size list [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’‹ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’‹ \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, k) }[/math] Real List [...] [math]\displaystyle{ \mathrm{o}_i }[/math] Mnemonic: [math]\displaystyle{ \textbf{o} }[/math]riginal size list
[math]\displaystyle{ ๐’•\mathrm{T} }[/math] [math]\displaystyle{ \textbf{a} }[/math] Tempered target-interval size list [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’• \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’• \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, k) }[/math] Real List [...] [math]\displaystyle{ \mathrm{a}_i }[/math] Mnemonic: [math]\displaystyle{ \textbf{a} }[/math]ltered size list
[math]\displaystyle{ ๐’•\mathrm{T} - ๐’‹\mathrm{T} }[/math]

[math]\displaystyle{ ๐’“\mathrm{T} }[/math]
[math]\displaystyle{ \textbf{a} - \textbf{o} }[/math]

[math]\displaystyle{ \textbf{e} }[/math] target-interval error list [math]\displaystyle{ \scriptsize \begin{array} {c} ๐’“ \\[-2pt] {\small\mathsf{ยข}} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \mathrm{T} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \mathsf{ยข} }[/math] Cents [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} ๐’• \\[-3pt] \left(1, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \mathrm{T} \\[-3pt] \left(\cancel{d}, k\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (1, k) }[/math] Real List [...] [math]\displaystyle{ \mathrm{e}_i }[/math]
[math]\displaystyle{ C }[/math] or [math]\displaystyle{ S }[/math] [math]\displaystyle{ W }[/math] Target-interval weight matrix (See complexities and simplicities table) [math]\displaystyle{ \scriptsize (k, k) }[/math] Real Matrix [[...] ...] [math]\displaystyle{ ๐’˜ }[/math] [math]\displaystyle{ w_i }[/math] or [math]\displaystyle{ w_{ij} }[/math]
[math]\displaystyle{ C }[/math] Target-interval complexity weight matrix (See complexities section of complexities and simplicities table) [math]\displaystyle{ \scriptsize (k, k) }[/math] Real Matrix [[...] ...] [math]\displaystyle{ ๐’„ }[/math] [math]\displaystyle{ c_i }[/math]
[math]\displaystyle{ \dfrac1C }[/math] [math]\displaystyle{ S }[/math] Target-interval simplicity weight matrix (See simplicities section of complexities and simplicities table) [math]\displaystyle{ \scriptsize (k, k) }[/math] Real Matrix [[...] ...] [math]\displaystyle{ ๐’” }[/math] [math]\displaystyle{ s_i }[/math] Entry-wise reciprocal of [math]\displaystyle{ C }[/math]
[math]\displaystyle{ \abs{\textbf{e}} W }[/math]

[math]\displaystyle{ 1200ร—\slant{\mathbf{1}}L\abs{P - I} \mathrm{T}W }[/math]

[math]\displaystyle{ \textbf{d} }[/math] Target-interval damage list (See damages table) [math]\displaystyle{ \scriptsize (1, k) }[/math] Real List [...] [math]\displaystyle{ \mathrm{d}_i }[/math]
[math]\displaystyle{ k }[/math] Target-interval count [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar Mnemonic: [math]\displaystyle{ k }[/math]ount
Held-intervals
[math]\displaystyle{ \mathrm{H} }[/math] Held-interval basis [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, h) }[/math] Matrix [[...โŸฉ ...] [math]\displaystyle{ \textbf{h}_i }[/math] [math]\displaystyle{ \mathrm{h}_{ij} }[/math]
[math]\displaystyle{ h }[/math] Held-interval count [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
Exploring temperaments
[math]\displaystyle{ \mathrm{C} }[/math] Comma basis [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, n) }[/math] Integer Matrix [[...โŸฉ ...] [math]\displaystyle{ \textbf{c}_i }[/math] [math]\displaystyle{ \mathrm{c}_{ij} }[/math] Jargon name: monzo list
[math]\displaystyle{ \textbf{c} }[/math] Comma [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, 1) }[/math] Integer Vector [...โŸฉ [math]\displaystyle{ \mathrm{c}_i }[/math] Specific type: vector (prime-count vector or PC-vector)
Computation
ꗨ⁠ ⁠ꗨ[math]\displaystyle{ \,ยท\, }[/math]ꗨ⁠ ⁠ꗨ[math]\displaystyle{ _p }[/math] Power sum ([math]\displaystyle{ p }[/math]-sum) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
All-interval tuning schemes
[math]\displaystyle{ \mathrm{I} }[/math] [math]\displaystyle{ \mathrm{T}_{\text{p}} }[/math] Prime proxy target-interval list [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, d) }[/math] Integer Matrix โŸจ[...โŸฉ ...] [math]\displaystyle{ \slant{\mathbf{1}} }[/math]
[math]\displaystyle{ X }[/math] Complexity pretransformer [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)} }[/math] or [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(} }[/math]<alt>-[math]\displaystyle{ \scriptsize\mathsf{C)} }[/math][note 6] [math]\displaystyle{ \small\mathsf{(C)} }[/math] or [math]\displaystyle{ \small\mathsf{(} }[/math]<alt>-[math]\displaystyle{ \small\mathsf{C)} }[/math] Complexity weight or <alternative>-complexity weight [math]\displaystyle{ \scriptsize (d, d) }[/math] or [math]\displaystyle{ \scriptsize (d+1, d+1) }[/math] Real Matrix [โŸจ...] ...โŸฉ [math]\displaystyle{ ๐’™_i }[/math] [math]\displaystyle{ ๐’™ }[/math] [math]\displaystyle{ x_i }[/math] or [math]\displaystyle{ x_{ij} }[/math]
[math]\displaystyle{ \text{diag}({\large\textbf{๐“}}\hspace{2mu}) }[/math] [math]\displaystyle{ L }[/math] Log-prime matrix [math]\displaystyle{ \small\mathsf{oct} }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Octaves per prime [math]\displaystyle{ \scriptsize (d, d) }[/math] Real Matrix [โŸจ...] ...โŸฉ โŸจ[...โŸฉ ...] [math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu}_i }[/math] [math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu} }[/math] [math]\displaystyle{ {\large ๐“}\hspace{2mu}_{ij} }[/math]
[math]\displaystyle{ q }[/math] Interval complexity norm power [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \norm{ยท}_q }[/math] Power norm ([math]\displaystyle{ p }[/math]-norm) [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \dfrac1{1-\frac1q} }[/math] [math]\displaystyle{ \text{dual}(q) }[/math] Dual norm power [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \norm{X\mathbf{i}}_q }[/math] Interval complexity [math]\displaystyle{ \small\mathsf{(C)} }[/math] or [math]\displaystyle{ \small\mathsf{(} }[/math]<alt>-[math]\displaystyle{ \small\mathsf{C)} }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
[math]\displaystyle{ \norm{๐’“X^{-1}}_{\text{dual}(q)} }[/math] Retuning magnitude [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(C^{-1})} }[/math] or [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(} }[/math]<alt>-[math]\displaystyle{ \small\mathsf{C^{-1})} }[/math] [math]\displaystyle{ \scriptsize (1, 1) }[/math] Real Scalar
Alternative complexities
[math]\displaystyle{ ๐’‘ }[/math] Prime list[note 7] [math]\displaystyle{ \scriptsize (1, d) }[/math] Integer List [...] [math]\displaystyle{ p_i }[/math]
[math]\displaystyle{ \slant{\mathbf{1}} }[/math] Summation map [math]\displaystyle{ \scriptsize (1, d) }[/math] Integer Vector โŸจ...] [math]\displaystyle{ 1 }[/math]
[math]\displaystyle{ 1200 }[/math] Octaves-to-cents conversion ยข/oct Cents per octave [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
[math]\displaystyle{ Z }[/math] Size-sensitizing matrix [math]\displaystyle{ \scriptsize (d+1, d) }[/math] Real Matrix [โŸจโ€ฆ]...] [math]\displaystyle{ ๐’›_i }[/math] [math]\displaystyle{ z_{ij} }[/math]
Non-standard domain bases
[math]\displaystyle{ B_s }[/math] (Domain) basis (change) matrix [math]\displaystyle{ \small ๐—ฝ }[/math]/[math]\displaystyle{ \small ๐—ฏ }[/math] Primes per nonprime basis elements [math]\displaystyle{ \scriptsize (d_p, d_b) }[/math] Integer Matrix [[...] ...] [[...] ...] [math]\displaystyle{ b_i }[/math] [math]\displaystyle{ b_{ij} }[/math]
[math]\displaystyle{ B_{Ls} }[/math] [math]\displaystyle{ \small ๐—• }[/math]/[math]\displaystyle{ \small ๐—ฏ }[/math] Superspace basis elements per (subspace) basis elements [math]\displaystyle{ \scriptsize (d_L, d_s) }[/math]
Embedding and projection
[math]\displaystyle{ G }[/math] Generator embedding (matrix) [math]\displaystyle{ \small ๐—ฝ }[/math]/[math]\displaystyle{ \small ๐—ด }[/math] Primes per generator [math]\displaystyle{ \scriptsize (d, r) }[/math] Real Matrix [{...] ...โŸฉ {[...โŸฉ ...] [math]\displaystyle{ ๐’ˆ_i }[/math] [math]\displaystyle{ g_{ij} }[/math]
[math]\displaystyle{ G_cF^{-1}FM_c }[/math]

[math]\displaystyle{ \mathrm{V}\textit{ฮ›}\mathrm{V}^{-1} }[/math]

[math]\displaystyle{ P }[/math] Projection (matrix) [math]\displaystyle{ \scriptsize \begin{array} {c} G \\[-2pt] ๐—ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} M \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} ๐—ฝ \end{array} }[/math] [math]\displaystyle{ \small ๐—ฝ }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Primes per prime [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} G \\[-3pt] \left(d, \cancel{r}\right) \end{array} \! \! \begin{array} {c} M \\[-3pt] \left(\cancel{r}, d\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (d, d) }[/math] Real Matrix [โŸจ...] ...โŸฉ โŸจ[...โŸฉ ...] [math]\displaystyle{ ๐’‘_i }[/math] [math]\displaystyle{ p_i }[/math]
[math]\displaystyle{ GM\textbf{i} }[/math] [math]\displaystyle{ P\textbf{i} }[/math] Projected interval [math]\displaystyle{ \scriptsize \begin{array} {c} G \\[-2pt] ๐—ฝ \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ด} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} M \\[-2pt] \cancel{๐—ด} \hspace{-2mu} / \hspace{-2mu} \cancel{๐—ฝ} \end{array} \begin{array} {c} \\[-2pt] ยท \end{array} \begin{array} {c} \textbf{i} \\[-2pt] \cancel{๐—ฝ} \end{array} }[/math] [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize \! \! \begin{array} {c} G \\[-3pt] \left(d, \cancel{r}\right) \end{array} \! \! \begin{array} {c} M \\[-3pt] \left(\cancel{r}, \cancel{d}\right) \end{array} \! \! \begin{array} {c} \textbf{i} \\[-3pt] \left(\cancel{d}, 1\right) \end{array} \! \! }[/math] [math]\displaystyle{ \scriptsize (d, 1) }[/math] Real Vector [...โŸฉ Specific type: vector (prime-count vector or PC-vector)
[math]\displaystyle{ \mathrm{U} }[/math] Unchanged-interval basis [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, r) }[/math] Matrix [[...โŸฉ ...] [math]\displaystyle{ \textbf{u}_i }[/math] [math]\displaystyle{ \mathrm{u}_{ij} }[/math] Jargon name: eigenmonzo list
[math]\displaystyle{ \textit{ฮ›} }[/math] Scaling factor (eigenvalue) matrix [math]\displaystyle{ \scriptsize (d, d) }[/math] Matrix [โŸจโ€ฆ] โ€ฆโŸฉ โŸจ[โ€ฆโŸฉ โ€ฆ] [math]\displaystyle{ ๐€ }[/math] [math]\displaystyle{ ฮป_i }[/math] Mnemonic: [math]\displaystyle{ \mathrm{V} }[/math] is mirrored of [math]\displaystyle{ \textit{ฮ›} }[/math] which it combines with to create the projection matrix; previous name: eigenvalue matrix
[math]\displaystyle{ \mathrm{V} }[/math] Unrotated vector (eigenvector) list [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (d, d) }[/math] Matrix โŸจ[...โŸฉ ...] [math]\displaystyle{ \textbf{v}_i }[/math] [math]\displaystyle{ \mathrm{v}_{ij} }[/math] Mnemonic: [math]\displaystyle{ \mathrm{V} }[/math] is mirrored of [math]\displaystyle{ \textit{ฮ›} }[/math] which it combines with to create the projection matrix; jargon name: eigenmonzo and comma list
[math]\displaystyle{ F }[/math] Generator form matrix [math]\displaystyle{ \scriptsize (r, r) }[/math] Matrix [{...] โ€ฆ} [math]\displaystyle{ ๐’‡_i }[/math] [math]\displaystyle{ f_{ij} }[/math]
[math]\displaystyle{ I }[/math] [math]\displaystyle{ M_{\text{j}} }[/math] JI mapping (matrix) [math]\displaystyle{ \small ๐—ด }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Generators per prime [math]\displaystyle{ \scriptsize (d, d) }[/math] Integer Matrix [โŸจ...] ...} โŸจ[...} ...] [math]\displaystyle{ \slant{\mathbf{1}} }[/math]
[math]\displaystyle{ I }[/math] [math]\displaystyle{ G_{\text{j}} }[/math] JI generator embedding (matrix) [math]\displaystyle{ \small ๐—ฝ }[/math]/[math]\displaystyle{ \small ๐—ด }[/math] Primes per generator [math]\displaystyle{ \scriptsize (d, d) }[/math] Integer Matrix [{...] ...โŸฉ {[...โŸฉ ...] [math]\displaystyle{ \slant{\mathbf{1}} }[/math]
[math]\displaystyle{ K }[/math] Constraint (matrix) [math]\displaystyle{ \scriptsize (k, r) }[/math] [math]\displaystyle{ \scriptsize \{0, +1, -1\} }[/math] Matrix [[...] ...] [math]\displaystyle{ ๐’Œ_i }[/math] [math]\displaystyle{ k_{ij} }[/math] Mnemonic: [math]\displaystyle{ K }[/math]onstraint
[math]\displaystyle{ ๐’ƒ }[/math] (Generator tuning map) blend map [math]\displaystyle{ \scriptsize (1, ฯ„-1) }[/math] Real Vector [...] [math]\displaystyle{ b_i }[/math]
[math]\displaystyle{ B }[/math] (Generator tuning map) blend matrix [math]\displaystyle{ \scriptsize (d, ฯ„-1) }[/math] Real Matrix [[...โŸฉ...] [math]\displaystyle{ ๐’ƒ_{i} }[/math] [math]\displaystyle{ b_{ij} }[/math]
[math]\displaystyle{ D }[/math] (Generator tuning map) deltas matrix [math]\displaystyle{ \mathsf{ยข} }[/math]/[math]\displaystyle{ \small ๐—ด }[/math] Cents per generator [math]\displaystyle{ \scriptsize (ฯ„-1,r) }[/math] Real Matrix [{...] ...] [math]\displaystyle{ ๐œน_i }[/math] [math]\displaystyle{ ๐›ฟ_{ij} }[/math]
[math]\displaystyle{ ฯ„ }[/math] Tied basic minimax tuning count Integer Scalar
Exterior algebra
[math]\displaystyle{ ๐•ž }[/math] Multimap [math]\displaystyle{ \small ๐—ด }[/math]/[math]\displaystyle{ \small ๐—ฝ }[/math] Generators per prime [math]\displaystyle{ \scriptsize (1, d) }[/math] Integer Multivector โŸจ...] or โŸจโŸจ...]] or โŸจโŸจโŸจ...]]] ... [math]\displaystyle{ ๐•ž_i }[/math]
[math]\displaystyle{ ๐•” }[/math] Multicomma [math]\displaystyle{ \small ๐—ฝ }[/math] Primes [math]\displaystyle{ \scriptsize (1, n) }[/math] Integer Multivector [...โŸฉ or [[...โŸฉโŸฉ or [[[...โŸฉโŸฉโŸฉ ... [math]\displaystyle{ ๐•”_i }[/math]
[math]\displaystyle{ ๐•ง }[/math] (Generic temperament multivector) [math]\displaystyle{ \scriptsize (1, {{d}\choose{r}}) }[/math] or [math]\displaystyle{ \scriptsize (1, {{d}\choose{n}}) }[/math] Integer Multivector โŸจ...] or โŸจโŸจ...]] or โŸจโŸจโŸจ...]]] ... [...โŸฉ or [[...โŸฉโŸฉ or [[[...โŸฉโŸฉโŸฉ ... [math]\displaystyle{ ๐•ง_i }[/math]
[math]\displaystyle{ A }[/math] (Generic temperament matrix) [math]\displaystyle{ \scriptsize (g, d) }[/math] or [math]\displaystyle{ \scriptsize (d, g) }[/math] Integer Matrix [โŸจ...] ...} โŸจ[...} ...] or [[...โŸฉ ...] [math]\displaystyle{ ๐’‚_i }[/math] [math]\displaystyle{ ๐’‚_i }[/math] [math]\displaystyle{ ๐’‚ }[/math] [math]\displaystyle{ a_{ij} }[/math]
[math]\displaystyle{ v }[/math] Variance
[math]\displaystyle{ g }[/math] Grade [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
Temperament addition
[math]\displaystyle{ \min(r, n) }[/math] [math]\displaystyle{ g_\text{min} }[/math] Min-grade [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
[math]\displaystyle{ \max(r, n) }[/math] [math]\displaystyle{ g_\text{max} }[/math] Max-grade [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
[math]\displaystyle{ L_\text{dep} }[/math] Linear-dependence basis [math]\displaystyle{ \scriptsize \left(l_\text{dep}, d\right) }[/math] or [math]\displaystyle{ \scriptsize \left(d, l_\text{dep}\right) }[/math] Integer Matrix [โŸจ...]] or [[...] ...โŸฉ โŸจ[...]] or [[...โŸฉ ...] [math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu}_{\text{dep}i} }[/math] [math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu}_{\text{dep}i} }[/math] [math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu}_\text{dep} }[/math] [math]\displaystyle{ {\large ๐“}\hspace{2mu}_{\text{dep}ij} }[/math]
[math]\displaystyle{ L_\text{ind} }[/math] Linear-independence basis [math]\displaystyle{ \scriptsize \left(l_\text{ind}, d\right) }[/math] or [math]\displaystyle{ \scriptsize \left(d, l_\text{ind}\right) }[/math] Integer Matrix [โŸจ...]] or [[...] ...โŸฉ โŸจ[...]] or [[...โŸฉ ...] [math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu}_{\text{ind}i} }[/math] [math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu}_{\text{ind}i} }[/math] [math]\displaystyle{ {\large\textbf{๐“}}\hspace{2mu}_\text{ind} }[/math] [math]\displaystyle{ {\large ๐“}\hspace{2mu}_{\text{ind}ij} }[/math]
[math]\displaystyle{ \dim(L_\text{dep}) }[/math] [math]\displaystyle{ l_\text{dep} }[/math] Linear-dependence [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar
[math]\displaystyle{ \dim(L_\text{ind}) }[/math] [math]\displaystyle{ l_\text{ind} }[/math] Linear-independence [math]\displaystyle{ \scriptsize (1, 1) }[/math] Integer Scalar

Units

Symbol Name Vectorized
[math]\displaystyle{ \small ๐—ด }[/math] Generators Yes
[math]\displaystyle{ \small ๐—ฝ }[/math] Primes Yes
[math]\displaystyle{ \small ๐—ฏ }[/math] (Subspace) basis elements Yes
[math]\displaystyle{ \small ๐—• }[/math] Superspace basis elements Yes
[math]\displaystyle{ \mathsf{ยข} }[/math] Cents
[math]\displaystyle{ \mathsf{ยข}\small{(} }[/math]<weight>[math]\displaystyle{ \small\mathsf{)} }[/math] Weighted cents
[math]\displaystyle{ \small\mathsf{oct} }[/math] Octaves

Tuning schemes

Retuning (or mistuning) magnitude Damage Target
intervals 		Systematic name Previously named tuning schemes that are specific types of this tuning scheme Of interest?
Weight Optimization
Interval complexity Slope Initial Name Power
Norm pretransformer Norm power Norm pretransformer Norm power Initial Name Multiplier
Initial Name Multiplier Initial Name Power Initial Name Multiplier Initial Name Power Abbreviated Read ("____ tuning scheme")
<none> <n/a> Maximum ∞ <none> (t) Taxicab 1 S Simplicity-weight 1/Complexity <n/a> Minimax ∞ All Minimax-S Minimax simplicity-weight damage "TOP"/"T1"/"TIPTOP"*, "CTOP", "POTOP"/"POTT"* yes
<various> <various> Minimax-<alt>-S Minimax <alternative>-simplicity-weight damage "BOP", "Weil", "Kees" yes
<none> Euclidean 2 <none> E Euclidean 2 Minimax-ES Minimax Euclideanized-simplicity-weight damage "TE"/"T2"/"TOP-RMS", "CTE", "POTE" yes
<various> <various> Minimax-E-<alt>-S Minimax Euclideanized-<alternative>-simplicity-weight damage "Frobenius", "BE", "WE", "KE" Yes
<n/a> <n/a> U Unity-weight <none> <set> <set> Minimax-U <set> Minimax unity-weight damage "Minimax" yes
<none> (t) Taxicab 1 S Simplicity-weight 1/Complexity <set> Minimax-S <set> Minimax simplicity-weight damage Yes
<various> <set> Minimax-<alt>-S <set> Minimax <alternative>-simplicity-weight damage
<none> E Euclidean 2 <set> Minimax-ES <set> Minimax Euclideanized-simplicity-weight damage
<various> <set> Minimax-E-<alt>-S <set> Minimax Euclideanized-<alternative>-simplicity-weight damage
<none> (t) Taxicab 1 C Complexity-weight Complexity <set> Cinimax-C <set> Cinimax complexity-weight damage Yes
<various> <set> Minimax-<alt>-C <set> Minimax <alternative>-complexity-weight damage
<none> E Euclidean 2 <set> Minimax-EC <set> Minimax Euclideanized-complexity-weight damage
<various> <set> Minimax-E-<alt>-C <set> Minimax Euclideanized-<alternative>-complexity-weight damage
<n/a> U Unity-weight <none> MiniRMS 2 <set> MiniRMS-U <set> MiniRMS unity-weight damage "Least squares" yes
<none> (t) Taxicab 1 S Simplicity-weight 1/Complexity <set> MiniRMS-S <set> MiniRMS simplicity-weight damage Yes
<various> <set> MiniRMS-<alt>-S <set> MiniRMS <alternative>-simplicity-weight damage
<none> E Euclidean 2 <set> MiniRMS-ES <set> MiniRMS Euclideanized-simplicity-weight damage
<various> <set> MiniRMS-E-<alt>-S <set> MiniRMS Euclideanized-<alternative>-simplicity-weight damage
<none> (t) Taxicab 1 C Complexity-weight Complexity <set> MiniRMS-C <set> MiniRMS complexity-weight damage yes
<various> <set> MiniRMS-<alt>-C <set> MiniRMS <alternative>-complexity-weight damage
<none> E Euclidean 2 <set> MiniRMS-EC <set> MiniRMS Euclideanized-complexity-weight damage
<various> <set> MiniRMS-E-<alt>-C <set> MiniRMS Euclideanized-<alternative>-complexity-weight damage
<n/a> U Unity-weight <none> Miniaverage 1 <set> Miniaverage-U <set> Miniaverage unity-weight damage yes
<none> (t) Taxicab 1 S Simplicity-weight 1/Complexity <set> Miniaverage-S <set> Miniaverage simplicity-weight damage Yes
<various> <set> Miniaverage-<alt>-S <set> Miniaverage <alternative>-simplicity-weight damage
<none> E Euclidean 2 <set> Miniaverage-ES <set> Miniaverage Euclideanized-simplicity-weight damage
<various> <set> Miniaverage-E-<alt>-S <set> Miniaverage Euclideanized-<alternative>-simplicity-weight damage
<none> (t) Taxicab 1 C Complexity-weight Complexity <set> Miniaverage-C <set> Miniaverage complexity-weight damage Yes
<various> <set> Miniaverage-<alt>-C <set> Miniaverage <alternative>-complexity-weight damage
<none> E Euclidean 2 <set> Miniaverage-EC <set> Miniaverage Euclideanized-complexity-weight damage
<various> <set> Miniaverage-E-<alt>-C <set> Miniaverage Euclideanized-<alternative>-complexity-weight damage

Damages

Quantity Unit
Abbreviation Name Symbol Name
U-damage Unity-weight damage [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(U)} }[/math] Unity-weighted cents
C-damage Complexity-weight damage [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(C)} }[/math] Complexity-weighted cents
<alt>-C-damage <alternative>-complexity-weight damage [math]\displaystyle{ \mathsf{ยข} }[/math][math]\displaystyle{ \small\mathsf{(} }[/math]<alt>-[math]\displaystyle{ \small\mathsf{C)} }[/math] <alternative>-complexity-weighted cents
EC-damage Euclideanized-complexity-weight damage [math]\displaystyle{ \mathsf{ยข} }[/math][math]\displaystyle{ \small\mathsf{(EC)} }[/math] Euclideanized-complexity-weighted cents
E-<alt>-C-damage Euclideanized-<alternative>-complexity-weight damage [math]\displaystyle{ \mathsf{ยข} }[/math][math]\displaystyle{ \small\mathsf{(E} }[/math]-<alt>-[math]\displaystyle{ \small\mathsf{C)} }[/math] Euclideanized-<alternative>-complexity-weighted cents
S-damage Simplicity-weight damage [math]\displaystyle{ \mathsf{ยข}\small\mathsf{(S)} }[/math] Simplicity-weighted cents
<alt>-S-damage <alternative>-simplicity-weight damage [math]\displaystyle{ \mathsf{ยข} }[/math][math]\displaystyle{ \small\mathsf{(} }[/math]<alt>-[math]\displaystyle{ \small\mathsf{S)} }[/math] <alternative>-simplicity-weighted cents
ES-damage Euclideanized-simplicity-weight damage [math]\displaystyle{ \mathsf{ยข} }[/math][math]\displaystyle{ \small\mathsf{(ES)} }[/math] Euclideanized-simplicity-weighted cents
E-<alt>-S-damage Euclideanized-<alternative>-simplicity-weight damage [math]\displaystyle{ \mathsf{ยข} }[/math][math]\displaystyle{ \small\mathsf{(E} }[/math]-<alt>-[math]\displaystyle{ \small\mathsf{S)} }[/math] Euclideanized-<alternative>-simplicity-weighted cents

Complexity and simplicity

Quantity Unit
Abbreviation Name Unit Name
C Complexity [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(C)} }[/math] = [math]\displaystyle{ \small\mathsf{(C)} }[/math] Complexity weight
<alt>-C <alternative> complexity [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(} }[/math]<alt>-[math]\displaystyle{ \scriptsize\mathsf{C)} }[/math] = [math]\displaystyle{ \small\mathsf{(} }[/math]<alt>-[math]\displaystyle{ \small\mathsf{C)} }[/math] <alternative>-complexity weight
EC Euclideanized complexity [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(EC)} }[/math] = [math]\displaystyle{ \small\mathsf{(EC)} }[/math] Euclideanized-complexity weight
E-<alt>-C Euclideanized-<alternative> complexity [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(E} }[/math]-<alt>-[math]\displaystyle{ \scriptsize\mathsf{C)} }[/math] = [math]\displaystyle{ \small\mathsf{(E} }[/math]-<alt>-[math]\displaystyle{ \small\mathsf{C)} }[/math] Euclideanized-<alternative>-complexity weight
S Simplicity [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(S)} }[/math] = [math]\displaystyle{ \small\mathsf{(S)} }[/math] Simplicity weight
<alt>-S <alternative> simplicity [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(} }[/math]<alt>-[math]\displaystyle{ \scriptsize\mathsf{S)} }[/math] = [math]\displaystyle{ \small\mathsf{(} }[/math]<alt>-[math]\displaystyle{ \small\mathsf{S)} }[/math] <alternative>-simplicity weight
ES Euclideanized simplicity [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(ES)} }[/math] = [math]\displaystyle{ \small\mathsf{(ES)} }[/math] Euclideanized-simplicity weight
E-<alt>-S Euclideanized-<alternative> simplicity [math]\displaystyle{ \small\mathsf{๐Ÿ™}\scriptsize\mathsf{(E} }[/math]-<alt>-[math]\displaystyle{ \scriptsize\mathsf{S)} }[/math] = [math]\displaystyle{ \small\mathsf{(E} }[/math]-<alt>-[math]\displaystyle{ \small\mathsf{S)} }[/math] Euclideanized-<alternative>-simplicity weight

WinCompose

Are you tired of every time web-searching for and copy-pasting special characters that you use over and over in RTT discussions, or would like to use if only it were easy, such as โ™ฏ, โ™ญ, ยข, โˆš, ยฐ, โ‚‚, ร—, โˆ’1, โŸฉ, โˆž, and ฯ•? Well, try 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 โŽ„. So for example, you type โ™ฏ as โŽ„##, โ™ญ as โŽ„bb, ยข as โŽ„c/, โˆš as โŽ„v/, ยฐ as โŽ„00, โ‚‚ as โŽ„-2, ร— as โŽ„xx, โˆ’1 as โŽ„11, โŸฉ as โŽ„>>, โˆž as โŽ„88, and ฯ• as โŽ„8f.

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

Dave Keenan & Douglas Blumeyer's compose-key sequences
Compose-key sequence Resulting text Description
Keyboard key symbols
โŽ„โŽ„โŽ„ โŽ„ Compose key symbol (the right alt key by default)
โŽ„\โฃ โฃ Spacebar symbol
โŽ„\โ–ถ๏ธŽ etc. โ–ถ๏ธŽ etc. Right etc. arrow key symbols
โŽ„\A or โŽ„\O โŒฅ Alt or option key symbol
โŽ„\B โŒซ Backspace key symbol
โŽ„\C โœฒ Control key symbol
โŽ„\D โŒฆ Delete key symbol
โŽ„\E โŽ‹ Escape key symbol
โŽ„\L โ‡ช Caps lock key symbol
โŽ„\R or โŽ„\.E โŽ Return or enter key symbol
โŽ„\S โ‡ง Shift key symbol
โŽ„\T โญพ Tab key symbol
โŽ„() โ—Œ Dotted circle, represents any character (such as the character preceding a combining mark)
Double key sequences
โŽ„โฃโฃ โ€ฏ Narrow no-break space (used between quantities and their units)
โŽ„.. ยท Middle dot (used to multiply units when juxtaposition is ambiguous)
โŽ„:: รท Divide sign
โŽ„;; โ—Œฬฒฬ… Combining overline and low line (undirected value)
โŽ„| | โ€– Power norm bracket
โŽ„<< โŸจ Left angle bracket
โŽ„>> โŸฉ Right angle bracket
โŽ„~~ โ‰ˆ Approximately equal
โŽ„** โ˜… Black star
โŽ„'' โ€ฒ prime mark
โŽ„11 โปยน Power of −1 or inverse
โŽ„22 through โŽ„77 ยฒ ยณ โด โต โถ โท Squared, cubed, fourth through seventh power
โŽ„88 โˆž Infinity
โŽ„00 ยฐ Degree sign
โŽ„nn โฟ Superscript small n
โŽ„-- โ‚‹ Subscript minus sign
โŽ„__ โ—Œฬฒ Combining low line (underline)
โŽ„== โ‰ก Modular congruence
โŽ„// โ„ Fraction slash (use with super and subscripts to create fractions)
โŽ„## โ™ฏ Musical sharp
โŽ„bb โ™ญ Musical flat
โŽ„dd โˆ‚ Partial derivative
โŽ„ff ฯ• Small phi symbol
โŽ„gg ษก Single-storey (opentail) small g
โŽ„ll โ„“ Script small L
โŽ„uu ยต Micro sign
โŽ„xx ร— Multiplication sign
โŽ„DD โˆ† Delta (small difference) operator
โŽ„FF ฮฆ Greek capital phi
โŽ„QQ ฯ˜ Greek capital letter archaic qoppa (small quotient operator)
โŽ„TT แต€ Superscript capital T (matrix transpose)
โŽ„++ โบ Superscript plus sign (matrix pseudoinverse)
โŽ„โ–ถ๏ธŽโ–ถ๏ธŽ etc. โ†’ etc. Right etc. arrows
Multiplication operators
โŽ„xx ร— Multiplication sign
โŽ„Xx or โŽ„xX โจฏ Vector or cross product (barely distinguishable from multiplication sign)
โŽ„XX โœ• Large multiplication sign (a better symbol for cross product)
โŽ„x* โ‹† Star operator (prefix: tensor complement, Hodge)
โŽ„X* โˆ— Asterisk operator (infix: scalar product, Dorst)
โŽ„x. โ‹… Dot (product) operator
โŽ„X. โ€ข Bullet (infix: fat dot product, Dorst)
Other operators
โŽ„v/ โˆš Square root sign
โŽ„3v/ โˆ› Cube root sign
โŽ„4v/ โˆœ Fourth root sign
โŽ„-+ โ‚Š Subscript plus sign
โŽ„-- โ‚‹ Subscript minus sign
โŽ„-= โ‚Œ Subscript equals sign
โŽ„++ โบ Superscript plus sign (matrix pseudoinverse)
โŽ„+- or โŽ„+= ยฑ Plus or minus sign
โŽ„=+ โˆ“ Minus or plus sign
โŽ„=- โˆ’ Minus sign
โŽ„== โ‰ก Modular congruence
โŽ„/\ โˆง Logical AND, wedge product, progressive product
โŽ„\/ โˆจ Logical OR, vee product, regressive product
โŽ„โŽ„/\ โ‹€ Larger logical AND, wedge product, progressive product
โŽ„โŽ„\/ โ‹ Larger logical OR, vee product, regressive product
โŽ„| _ โŒŠ Left floor (infix: right contraction, Dorst)
โŽ„_| โŒ‹ Right floor (infix: left contraction, Dorst)
โŽ„| ^ โŒˆ Left ceiling
โŽ„^| โŒ‰ Right ceiling
โŽ„'- โจฝ Righthand interior product
โŽ„-' โจผ (Left-hand) interior product
โŽ„-, ยฌ Not sign (prefix: multivector complement)
โŽ„โŽ„<> โ‹„ Diamond operator (prefix: multivector dual)
โŽ„(.) โจ€ Entry-wise vector multiplication operator
โŽ„(..) โŠ™ Alternative entry-wise vector multiplication operator
โŽ„(/) โŠ˜ Entry-wise vector division operator
Mathematical letter and digit prefixes
โŽ„3โ—Œ ั Cyrillic, โŽ„3q is ya (example)
โŽ„4โ—Œ โ„ต Hebrew, โŽ„4a is aleph (example)
โŽ„5โ—Œ ๐”ž Fraktur, โŽ„5a
โŽ„6โ—Œ แตƒ ยน โ€ฏแชฒ โธ Superscripts, โŽ„6a โŽ„61 โŽ„688 โŽ„68โฃ (not all letters, some only approximate) (same key as ^ but without shift)
โŽ„68โ—Œ แต Superscript greek, โŽ„68b is superscript beta (only a few)
โŽ„7โ—Œ ๐’ถ Script, โŽ„7a
โŽ„8โ—Œ ฮฑ Greek, โŽ„8a is alpha (by sound where possible otherwise letter-shape)
โŽ„8.โ—Œ ฯ‚ Greek variants, โŽ„8.s is final sigma
โŽ„9โ—Œ ๐š ๐Ÿ ๐Ÿ“ ๐Ÿ• ๐Ÿ– ๐ŸŽ Bold, โŽ„9a โŽ„91 โŽ„95โฃ โŽ„97โฃ โŽ„98โฃ โŽ„90โฃ
โŽ„95โ—Œ ๐–† Bold fraktur, โŽ„95a
โŽ„97โ—Œ ๐“ช Bold script, โŽ„97a
โŽ„98โ—Œ ๐›‚ Bold greek, โŽ„98a is bold alpha
โŽ„90โ—Œ ๐’‚ Bold italic, โŽ„90a
โŽ„908โ—Œ ๐œถ Bold italic greek, โŽ„908a is bold italic alpha
โŽ„0โ—Œ ๐‘Ž Italic, โŽ„0a
โŽ„08โ—Œ ๐›ผ Italic greek, โŽ„08a is italic alpha
โŽ„-โ—Œ โ‚ แด€ โ€ฏอš โ‚ˆ Subscripts and small caps, โŽ„-a โŽ„-A โŽ„-88 โŽ„-8โฃ (not all letters, some only approximate) (same key as _ but without shift)
โŽ„-8โ—Œ แตฆ Subscript greek, โŽ„-8b is subscript beta (only a few)
โŽ„{โ—Œ ๐–บ ๐Ÿฃ ๐Ÿซ Sans-serif, โŽ„{a โŽ„{1 โŽ„{9โฃ
โŽ„{9โ—Œ ๐—ฎ ๐Ÿญ Sans-serif bold, โŽ„{9a โŽ„{91
โŽ„}โ—Œ ๐šŠ ๐Ÿท Monospace, โŽ„}a โŽ„}1
โŽ„| โ—Œ ๐•’ ๐Ÿ™ ๐Ÿ  ๐Ÿ˜ Double-struck, โŽ„| a โŽ„| 1 โŽ„| 8โฃ โŽ„| 0โฃ
โŽ„| 8โ—Œ โ„ผ Double-struck greek, โŽ„| 8p (only a few)
โŽ„| 0โ—Œ โ…‡ โ…ˆ Double-struck italic, โŽ„| 0e โŽ„| i (only a few)
Power statistics brackets
โŽ„| | โ€– Power-norm bracket
โŽ„|-1 โ€–โ‚ 1-Norm right bracket
โŽ„|-2 โ€–โ‚‚ 2-Norm right bracket
โŽ„|-8 โ€–โ€ฏอš โˆž-Norm right bracket
โŽ„โŽ„<< โŸช Left power-mean bracket
โŽ„โŽ„>> โŸซ Right power-mean bracket
โŽ„โŽ„{{ โง› Left power-sum bracket (substitute for ꗨ⁠ ⁠ꗨ when HTML is not available)
โŽ„โŽ„}} โงš Right power-sum bracket (substitute for ꗨ⁠ ⁠ꗨ when HTML is not available)
Combining marks
โŽ„\- โ—Œฬถ Combining strike-thru
โŽ„^_ โ—Œฬ… Combining overline
โŽ„__ โ—Œฬฒ Combining low line
โŽ„;; or โŽ„-_ or โŽ„_^ โ—Œฬฒฬ… Combining overline and low line (undirected value)

Keyboard map

Footnotes

  1. โ†‘ 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.
  2. โ†‘ For educational purposes, we use the ๐Ÿ™ symbol here to represent the implicit dimensionless unit that the weighting annotation "(C)" is attached to. But this symbol should not be shown in the reduced result. Another way to understand how we arrive at a bare annotation for the units of this quantity is to consider that w = d/|e| whose units are ยข(W) / ยข and the cents cancel.
  3. โ†‘ 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.
  4. โ†‘ 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 .
  5. โ†‘ 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.
  6. โ†‘ In these tables, "alternative" means any complexity other than the default of log-product complexity, and "alt" stands for its abbreviation.
  7. โ†‘ May be used for a prime-limit or for any prime-only list.
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