Dave Keenan & Douglas Blumeyer's guide to RTT/Conventions for names, variables, units, and notations: Difference between revisions
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This is an appendix to [[Dave Keenan]] & [[Douglas Blumeyer]]'s guide to RTT, or "[[D&D's guide]]" for short. The tables in this article present our recommendations for communicating about [[regular temperament theory]] (RTT), in particular the names and notations for temperament matrices, tuning schemes, interval complexities, and measurement units. | This is an appendix to [[Dave Keenan]] & [[Douglas Blumeyer]]'s guide to RTT, or "[[D&D's guide]]" for short. The tables in this article present our recommendations for communicating about [[regular temperament theory]] (RTT), in particular the names and notations for temperament matrices, tuning schemes, interval complexities, and measurement units. | ||
Our recommendations are designed to make this topic easy to learn for musicians who do not have technical backgrounds, though we have generally deferred to established mathematical, scientific, and engineering conventions for the benefit of those who do. | Our recommendations are designed to make this topic easy to learn for musicians who do not have technical backgrounds, though we have generally deferred to established mathematical, scientific, and engineering conventions for the benefit of those who do. | ||
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]]. | |||
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 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: | ||
| Line 10: | Line 11: | ||
|- | |- | ||
! | ! | ||
| | | Units → | ||
! rowspan="2" | | ! rowspan="2" | | ||
| | | Simple units | ||
| | | Compound or no units | ||
|- | |- | ||
| ↓ | | ↓ Order | ||
| ↓ | | ↓ Style → | ||
| | | Upright | ||
| '' | | ''Italic'' | ||
|- | |- | ||
! scope="col" height="8px" ! colspan="2" | | ! scope="col" height="8px" ! colspan="2" | | ||
| Line 26: | Line 26: | ||
|- | |- | ||
| 0 | | 0 | ||
| | | Plain | ||
! rowspan="3" | | ! rowspan="3" | | ||
| | | Scalar with simple unit | ||
| '' | | ''Scalar'' with no unit | ||
|- | |- | ||
| 1 | | 1 | ||
| ''' | | '''Bold''' | ||
| ''' | | '''Vector''' | ||
| ''''' | | '''''Map''''' (row vector) | ||
|- | |- | ||
| 2 | | 2 | ||
| UPPERCASE | | UPPERCASE | ||
| LIST or BASIS | | LIST or BASIS | ||
| | | True ''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. | ||
== Basic == | == Basic == | ||
| Line 327: | Line 327: | ||
𝒕\textbf{i}</math> | 𝒕\textbf{i}</math> | ||
| <math>\mathrm{a}</math> | | <math>\mathrm{a}</math> | ||
| | | {{subpage|tuning_fundamentals|uprev|s=Example 3|text=tempered (interval) size}} | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 425: | Line 425: | ||
| | | | ||
| <math>c</math> | | <math>c</math> | ||
| | | {{subpage|tuning_fundamentals|uprev|s=complexity}} | ||
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math><ref>For educational purposes, we use the 𝟙 symbol here to represent the implicit [[Wikipedia:Dimensionless_quantity| dimensionless unit]] that the weighting annotation "(C)" is attached to. But this symbol should not be shown in the reduced result. Another way to understand how we arrive at a bare annotation for the units of this quantity is to consider that ''w'' = ''d'' / | ''e''| whose units are ¢(W) / ¢ and the cents cancel.</ref> | | <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math><ref group="note">For educational purposes, we use the 𝟙 symbol here to represent the implicit [[Wikipedia:Dimensionless_quantity| dimensionless unit]] that the weighting annotation "(C)" is attached to. But this symbol should not be shown in the reduced result. Another way to understand how we arrive at a bare annotation for the units of this quantity is to consider that ''w'' = ''d'' / | ''e''| whose units are ¢(W) / ¢ and the cents cancel.</ref> | ||
| <math>\small\mathsf{(C)}</math> | | <math>\small\mathsf{(C)}</math> | ||
| complexity weight | | complexity weight | ||
| Line 555: | Line 555: | ||
| <math>𝒋\mathrm{T}</math> | | <math>𝒋\mathrm{T}</math> | ||
| <math>\textbf{o}</math> | | <math>\textbf{o}</math> | ||
| | | {{subpage|tuning_fundamentals|uprev|s=primes|text=target-interval (just) size list}} | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 661: | Line 661: | ||
| | | | ||
| <math>C</math> | | <math>C</math> | ||
| | | {{subpage|tuning_fundamentals|uprev|s=complexity-weight_damage|text=target-interval complexity weight matrix}} | ||
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> | | <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> | ||
| <math>\small\mathsf{(C)}</math> | | <math>\small\mathsf{(C)}</math> | ||
| Line 679: | Line 679: | ||
| <math>\dfrac1C</math> | | <math>\dfrac1C</math> | ||
| <math>S</math> | | <math>S</math> | ||
| | | {{subpage|tuning fundamentals|uprev|s=complexity-weight_damage|text=target-interval simplicity weight matrix}} | ||
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math> | | <math>\small\mathsf{𝟙}\scriptsize\mathsf{(S)}</math> | ||
| <math>\small\mathsf{(S)}</math> | | <math>\small\mathsf{(S)}</math> | ||
| Line 693: | Line 693: | ||
| <math>𝒔</math> | | <math>𝒔</math> | ||
| <math>s_i</math> | | <math>s_i</math> | ||
| | | entry-wise reciprocal of <math>C</math> | ||
|- | |- | ||
| <math>| \textbf{e}| W</math> | | <math>| \textbf{e}| W</math> | ||
| <math>\textbf{d}</math> | | <math>\textbf{d}</math> | ||
| [[target-interval damage list]]<ref>You may sometimes see annotated units without parentheses, such as "dBA", but this is not compliant with SI standards, so we always keep the parentheses.</ref> | | [[target-interval damage list]]<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 | | <math>\scriptsize | ||
\begin{array} {c} | \textbf{e}| \\[-2pt] {\small\mathsf{¢}} \end{array} | \begin{array} {c} | \textbf{e}| \\[-2pt] {\small\mathsf{¢}} \end{array} | ||
| Line 819: | Line 819: | ||
=== Units === | === Units === | ||
We recommend using a narrow no-break space (U+202F) between quantities and their units.<ref>Per https://physics.nist.gov/cuu/Units/checklist.html and https://academia.stackexchange.com/questions/54885/should-there-be-a-space-between-a-value-and-the-units-used | 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 | ||
.</ref> For how to type this, see the [[#WinCompose| WinCompose]] section below. | .</ref> For how to type this, see the [[#WinCompose| WinCompose]] section below. | ||
| Line 836: | Line 836: | ||
| yes | | yes | ||
|- | |- | ||
| <math>\mathsf{¢}</math><ref>It seems there is no standard symbol for a musical cent, except the word spelled in full (see https://en.wikipedia.org/wiki/Cent_(music)). But it seems unlikely anyone will interpret the cent currency symbol "¢" following a number in a musical context as anything other than musical cents.</ref> | | <math>\mathsf{¢}</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 "¢" following a number in a musical context as anything other than musical cents.</ref> | ||
| cents | | cents | ||
| | | | ||
| Line 866: | Line 866: | ||
=== Tuning schemes === | === Tuning schemes === | ||
Copied from | Copied from {{subpage|tuning fundamentals|uprev|s=Systematic tuning scheme names}}. | ||
{| class="wikitable center-all mw-collapsible" | {| class="wikitable center-all mw-collapsible" | ||
| Line 1,253: | Line 1,253: | ||
𝒕\textbf{i}</math> | 𝒕\textbf{i}</math> | ||
| <math>\mathrm{a}</math> | | <math>\mathrm{a}</math> | ||
| | | {{subpage|tuning fundamentals|uprev|s=Example 3|text=tempered (interval) size}} | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 1,351: | Line 1,351: | ||
| | | | ||
| <math>c</math> | | <math>c</math> | ||
| | | {{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) | ||
| | | | ||
| Line 1,463: | Line 1,463: | ||
| <math>𝒋\mathrm{T}</math> | | <math>𝒋\mathrm{T}</math> | ||
| <math>\textbf{o}</math> | | <math>\textbf{o}</math> | ||
| | | {{subpage|tuning fundamentals|uprev|s=primes|text=target-interval (just) size list}} | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 1,565: | Line 1,565: | ||
| | | | ||
| <math>C</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) | | colspan="3" | (see complexities section of complexities and simplicities table) | ||
| | | | ||
| Line 1,581: | Line 1,581: | ||
| <math>\dfrac1C</math> | | <math>\dfrac1C</math> | ||
| <math>S</math> | | <math>S</math> | ||
| | | {{subpage|tuning_fundamentals|uprev|s=complexity-weight_damage|text=target-interval simplicity weight matrix}} | ||
| colspan="3" | (see simplicities section of complexities and simplicities table) | | colspan="3" | (see simplicities section of complexities and simplicities table) | ||
| | | | ||
| Line 1,593: | Line 1,593: | ||
| <math>𝒔</math> | | <math>𝒔</math> | ||
| <math>s_i</math> | | <math>s_i</math> | ||
| | | entry-wise reciprocal of <math>C</math> | ||
|- | |- | ||
| <math>| \textbf{e}| W</math> | | <math>| \textbf{e}| W</math> | ||
| Line 1,783: | Line 1,783: | ||
| | | | ||
| <math>q</math> | | <math>q</math> | ||
| | | {{subpage|all-interval_tuning_schemes|uprev|s=Dual norms|text=interval complexity norm power}} | ||
| | | | ||
| | | | ||
| Line 1,819: | Line 1,819: | ||
| <math>\dfrac1{1-\frac1q}</math> | | <math>\dfrac1{1-\frac1q}</math> | ||
| <math>\text{dual}(q)</math> | | <math>\text{dual}(q)</math> | ||
| | | {{subpage|all-interval tuning schemes|uprev|s=Dual norms|text=dual norm power}} | ||
| | | | ||
| | | | ||
| Line 2,518: | Line 2,518: | ||
𝒕\textbf{i}</math> | 𝒕\textbf{i}</math> | ||
| <math>\mathrm{a}</math> | | <math>\mathrm{a}</math> | ||
| | | {{subpage|tuning fundamentals|uprev|s=Example 3|text=tempered (interval) size}} | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} 𝒕 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 2,616: | Line 2,616: | ||
| | | | ||
| <math>c</math> | | <math>c</math> | ||
| | | {{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) | ||
| | | | ||
| Line 2,728: | Line 2,728: | ||
| <math>𝒋\mathrm{T}</math> | | <math>𝒋\mathrm{T}</math> | ||
| <math>\textbf{o}</math> | | <math>\textbf{o}</math> | ||
| | | {{subpage|tuning fundamentals|uprev|s=primes|text=target-interval (just) size list}} | ||
| <math>\scriptsize | | <math>\scriptsize | ||
\begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | \begin{array} {c} 𝒋 \\[-2pt] {\small\mathsf{¢}} \hspace{-2mu} / \hspace{-2mu} \cancel{𝗽} \end{array} | ||
| Line 2,830: | Line 2,830: | ||
| | | | ||
| <math>C</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) | | colspan="3" | (see complexities section of complexities and simplicities table) | ||
| | | | ||
| Line 2,846: | Line 2,846: | ||
| <math>\dfrac1C</math> | | <math>\dfrac1C</math> | ||
| <math>S</math> | | <math>S</math> | ||
| | | {{subpage|tuning fundamentals|uprev|s=complexity-weight_damage|text=target-interval simplicity weight matrix}} | ||
| colspan="3" | (see simplicities section of complexities and simplicities table) | | colspan="3" | (see simplicities section of complexities and simplicities table) | ||
| | | | ||
| Line 2,858: | Line 2,858: | ||
| <math>𝒔</math> | | <math>𝒔</math> | ||
| <math>s_i</math> | | <math>s_i</math> | ||
| | | entry-wise reciprocal of <math>C</math> | ||
|- | |- | ||
| <math>| \textbf{e}| W \\ | | <math>| \textbf{e}| W \\ | ||
| Line 3,013: | Line 3,013: | ||
| | | | ||
| <math>X</math> | | <math>X</math> | ||
| | | {{subpage|alternative complexities|uprev|s=Prescaling_vs._pretransforming|text=complexity pretransformer}} | ||
| <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math><ref>In these tables, "alternative" means any complexity other than the default of log-product complexity, and "alt" stands for its abbreviation.</ref> | | <math>\small\mathsf{𝟙}\scriptsize\mathsf{(C)}</math> or <math>\small\mathsf{𝟙}\scriptsize\mathsf{(}</math><alt>-<math>\scriptsize\mathsf{C)}</math><ref group="note">In these tables, "alternative" means any complexity other than the default of log-product complexity, and "alt" stands for its abbreviation.</ref> | ||
| <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math> | | <math>\small\mathsf{(C)}</math> or <math>\small\mathsf{(}</math><alt>-<math>\small\mathsf{C)}</math> | ||
| complexity weight or <alternative>-complexity weight | | complexity weight or <alternative>-complexity weight | ||
| Line 3,049: | Line 3,049: | ||
| | | | ||
| <math>q</math> | | <math>q</math> | ||
| | | {{subpage|all-interval_tuning_schemes|uprev|s=dual_norms|text=interval complexity norm power}} | ||
| | | | ||
| | | | ||
| Line 3,085: | Line 3,085: | ||
| <math>\dfrac1{1-\frac1q}</math> | | <math>\dfrac1{1-\frac1q}</math> | ||
| <math>\text{dual}(q)</math> | | <math>\text{dual}(q)</math> | ||
| | | {{subpage|all-interval tuning schemes|uprev|s=dual_norms|text=dual norm power}} | ||
| | | | ||
| | | | ||
| Line 3,141: | Line 3,141: | ||
| | | | ||
| <math>𝒑</math> | | <math>𝒑</math> | ||
| | | {{subpage|alternative complexities|uprev|s=formulas|text=prime list}}<ref group="note">May be used for a prime-limit or for any prime-only list.</ref> | ||
| | | | ||
| | | | ||
| Line 3,159: | Line 3,159: | ||
| | | | ||
| <math>\slant{\mathbf{1}}</math> | | <math>\slant{\mathbf{1}}</math> | ||
| | | {{subpage|alternative complexities|uprev|s=proportionality to size|text=summation map}} | ||
| | | | ||
| | | | ||
| Line 3,177: | Line 3,177: | ||
| | | | ||
| <math>1200</math> | | <math>1200</math> | ||
| | | {{subpage|alternative complexities|uprev|s=Proportionality to size|text=octaves-to-cents conversion}} | ||
| | | | ||
| ¢/oct | | ¢/oct | ||
| Line 3,195: | Line 3,195: | ||
| | | | ||
| <math>Z</math> | | <math>Z</math> | ||
| | | {{subpage|alternative complexities|uprev|s=Normifying: size-sensitizing matrix|text=size-sensitizing matrix}} | ||
| | | | ||
| | | | ||
| Line 4,531: | Line 4,531: | ||
| ⎄(.) | | ⎄(.) | ||
| ⨀ | | ⨀ | ||
| | | entry-wise vector multiplication operator | ||
|- | |- | ||
| ⎄(..) | | ⎄(..) | ||
| ⊙ | | ⊙ | ||
| alternative | | alternative entry-wise vector multiplication operator | ||
|- | |- | ||
| ⎄(/) | | ⎄(/) | ||
| ⊘ | | ⊘ | ||
| | | entry-wise vector division operator | ||
|- | |- | ||
! colspan="3" | Mathematical letter and digit prefixes | ! colspan="3" | Mathematical letter and digit prefixes | ||
| Line 4,696: | Line 4,696: | ||
== Footnotes == | == Footnotes == | ||
<references /> | <references group="note" /> | ||
[[Category:Dave Keenan & Douglas Blumeyer's guide to RTT]] | [[Category:Dave Keenan & Douglas Blumeyer's guide to RTT]] | ||
[[Category:Tuning]] | [[Category:Tuning]] | ||