Xenharmonic Wiki

Douglas Blumeyer's RTT How-To: Difference between revisions

← Older editNewer edit →
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,160 edits
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,160 edits
add units and variables table
Line 1,158: Line 1,158:
|monzo
|monzo
|}
|}
The following table gives the names, units, dimensions, recommended single-letter variable names, and other helpful information about the objects that come together in RTT take us from intervals to tuned pitches in the temperament.
Note that units are given here in a plain font, scalars are italic, vectors are bold, and matrices are capital italic, as per convention.
{| class="wikitable"
|+ '''Table 3.''' RTT units, dimensions, variables
! colspan="3" |variable
! rowspan="2" |name
! colspan="3" |units
! colspan="2" |dimensions
! rowspan="2" |type
! rowspan="2" |[[variance|<math>v</math>]]
|-
!unreduced
!red.
![[RTT library in Wolfram Language|Wolfram library]]
!unreduced
!red.
!read as
!unreduced
!red.
|-
|
|<math>\textbf{i}</math>
|<code>i</code>
|[[Prime count vector|interval]]
|
|<math>\text{p}</math>
|primes
|
|<math>d\!×\!1</math>
|vector
|contra
|-
|
|<math>M</math>
|<code>m</code>
|[[Mapping|(temperament) mapping]]
|
|<math>\large{{}^\text{g}{\mskip -5mu/\mskip -3mu}_\text{p}}</math>
|generators per prime
|
|<math>r\!×\!d</math>
|matrix
|co
|-
|<math>M.\textbf{i}</math>
|<math>M\textbf{i}</math>
|<code>mi</code>
|[[Tmonzos_and_tvals|mapped interval]]
|<math>\large{}{}^\text{g}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{p}}}·\!\normalsize{}\cancel{\text{p}}</math>
|<math>\text{g}</math>
|generators
|<math>\small{}(r\!×\!\!\cancel{d}\!)(\!\cancel{d}\!\!×\!1)</math>
|<math>r\!×\!1</math>
|vector
|contra
|-
|
|<math>G</math>
|<code>g</code>
|[[Tenney-Euclidean_tuning#Computing_TE_tuning_using_pseudoinverse|generators (matrix)]]
|
|<math>\large{}{}^\text{p}{\mskip -5mu/\mskip -3mu}_\text{g}</math>
|primes per generator
|
|<math>d\!×\!r</math>
|matrix
|contra
|-
|<math>G.M</math>
|<math>P</math>
|<code>p</code>
|[[Projection matrix|(tuning) projection (matrix)]]
|<math>\large{}{}^{\text{p}}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{g}}}\!·\!{}^{\cancel{\text{g}}\!}{\mskip -5mu/\mskip -3mu}_{\text{p}}</math>
|<math>\large{}{}^\text{p}{\mskip -5mu/\mskip -3mu}_\text{p}</math>
|primes per prime
|<math>\small{}(d\!×\!\!\cancel{r}\!)(\!\cancel{r}\!\!×\!d)</math>
|<math>d\!×\!d</math>
|matrix
|
|-
|<math>G.M.\textbf{i}</math><br><math>P.\textbf{i}</math>
|<math>P\textbf{i}</math>
|<code>pi</code>
|projected interval
|<math>\large{}{}^{\text{p}}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{g}}}\!·\!{}^{\cancel{\text{g}}\!}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{p}}}\!·\!\normalsize{}\cancel{\text{p}}</math>
|<math>\text{p}</math>
|primes
|<math>\small{}(d\!×\!\!\cancel{r}\!)(\!\cancel{r}\!\!×\!\!\cancel{d}\!)(\!\cancel{d}\!\!×\!1)</math>
|<math>d\!×\!1</math>
|vector
|contra
|-
|
|<math>\textbf{p}</math>
|<code>ptm</code>
|[[JIP|prime tuning map]]
|
|<math>\large{}{}^\text{oct}{\mskip -5mu/\mskip -3mu}_\text{p}</math>
|octaves per prime
|
|<math>1\!×\!d</math>
|vector
|co
|-
|<math>\textbf{p}.G</math>
|<math>\textbf{g}</math>
|<code>gtm</code>
|[[generator tuning map]]
|<math>\large{}{}^\text{oct}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{p}}}\!·\!{}^{\cancel{\text{p}}\!}{\mskip -5mu/\mskip -3mu}_\text{g}</math>
|<math>\large{}{}^\text{oct}{\mskip -5mu/\mskip -3mu}_\text{g}</math>
|octaves per generator
|<math>\small{}(1\!×\!\!\cancel{d}\!)(\!\cancel{d}\!\!×\!r)</math>
|<math>1\!×\!r</math>
|vector
|co
|-
|<math>\textbf{p}.G.M</math><br><math>\textbf{p}.P</math><br><math>\textbf{g}.M</math>
|<math>\textbf{t}</math>
|<code>tm</code>
|[[tuning map|(temperament) tuning map]]
|<math>\large{}{}^\text{oct}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{p}}}\!·\!{}^{\cancel{\text{p}}\!}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{g}}}\!·\!{}^{\cancel{\text{g}}\!}{\mskip -5mu/\mskip -3mu}_\text{p}</math>
|<math>\large{}{}^\text{oct}{\mskip -5mu/\mskip -3mu}_\text{p}</math>
|octaves per prime
|<math>\small{}(1\!×\!\!\cancel{d}\!)(\!\cancel{d}\!\!×\!\!\cancel{r}\!)(\!\cancel{r}\!\!×\!d)</math>
|<math>1\!×\!d</math>
|vector
|co
|-
|<math>\textbf{p}.G.M.\textbf{i}</math><br><math>\textbf{p}.P.\textbf{i}</math><br><math>\textbf{g}.M.\textbf{i}</math>
|<math>\textbf{t}\textbf{i}</math>
|<code>ti</code>
|tuned interval
|<math>\large{}{}^\text{oct}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{p}}}\!·\!{}^{\cancel{\text{p}}\!}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{g}}}\!·\!{}^{\cancel{\text{g}}\!}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{p}}}\!·\!\normalsize{}\cancel{\text{p}}</math>
|<math>\text{oct}</math>
|octaves
|<math>\small{}(1\!×\!\!\cancel{d}\!)(\!\cancel{d}\!\!×\!\!\cancel{r}\!)(\!\cancel{r}\!\!×\!\!\cancel{d}\!)(\!\cancel{d}\!\!×\!1)</math>
|<math>1\!×\!1</math>
|vector
|
|-
|
|<math>C</math>
|<code>c</code>
|[[comma basis]]
|
|<math>\text{p}</math>
|primes
|
|<math>d\!×\!n</math>
|matrix
|contra
|-
|
|<math>1200</math>
|
|octaves-to-cents conversion
|
|<math>\large{}{}^\text{c}{\mskip -5mu/\mskip -3mu}_\text{oct}</math>
|cents per octave
|
|<math>1\!×\!1</math>
|scalar
|
|-
|
|<math>T</math>
|<code>t</code>
|[[regular temperament|temperament]]
|
|
|
|
|
|
|
|-
|
|<math>d</math>
|<code>d</code>
|[[dimensionality]]
|
|
|
|
|<math>1\!×\!1</math>
|scalar
|
|-
|
|<math>r</math>
|<code>r</code>
|[[rank]]
|
|
|
|
|<math>1\!×\!1</math>
|scalar
|
|-
|
|<math>n</math>
|<code>n</code>
|[[nullity]]
|
|
|
|
|<math>1\!×\!1</math>
|scalar
|
|-
|
|<math>v</math>
|<code>v</code>
|[[variance]]
|
|
|
|
|
|
|
|-
|
|<math>\text{g}</math>
|
|generators
|
|
|
|
|
|units
|
|-
|
|<math>\text{p}</math>
|
|primes
|
|
|
|
|
|units
|
|-
|
|<math>\text{oct}</math>
|
|octaves
|
|
|
|
|
|units
|
|-
|
|<math>\text{c}</math>
|
|cents
|
|
|
|
|
|units
|
|}
== Example ==
So, let's look at the example of ¼-comma meantone it is. Our four atomic objects are:
* example interval, <math>\textbf{i}</math>: {{vector|-1 1 0}}
* mapping, <math>M</math>: {{ket|{{map|1 1 0}} {{map|0 1 4}}}}
* generators, <math>G</math>: {{bra|{{vector|1 0 0}} {{vector|0 0 ¼}}}}
* prime tuning map, <math>\textbf{p}</math>: 1200{{map|log₂2 log₂3 log₂5}} = {{map|1200.000 1901.955 2786.314}}
And so the various possible compositions of these objects are:
* generator tuning map, <math>\textbf{p}.G = \textbf{g}</math>: {{map|1200.000 696.578}}
* mapped interval, <math>M\textbf{i}</math>: [0 1}
* tuning map, <math>\textbf{p}.G.M = \textbf{t}</math>: {{map|1200.000 1896.58 2786.31}}
* projection matrix, <math>G.M = P</math>: [[1 1 0] [0 0 0] [0 ¼ 1]]
* projected interval: <math>G.M.\textbf{i} = P\textbf{i}</math>: {{vector|0 0 ¼}}
And the ultimate output is <math>\textbf{t}\textbf{i}</math>, which is 0.580. That multiplied by 1200 gives 696.578, the cents of ¼-comma meantone's fifth, which was the example interval we chose.
= Outro =


You’ve made it to the end. This is pretty much everything that I understand about RTT at this point (May 2021). This took me a little over a month of full-time funemployed exploration to learn and put together; at the onset, while having considered myself a xenharmonicist for 15 years, I only understood a few scattered things about RTT, such as ET maps and commas and the overlapping MOS concepts. I hope that equipped with my results here it shouldn’t take a newcomer nearly that much time to get going. And of course I hope to continue learning more myself and perhaps even contributing to the theory one day, since there’s still plenty of frontier here.
You’ve made it to the end. This is pretty much everything that I understand about RTT at this point (May 2021). This took me a little over a month of full-time funemployed exploration to learn and put together; at the onset, while having considered myself a xenharmonicist for 15 years, I only understood a few scattered things about RTT, such as ET maps and commas and the overlapping MOS concepts. I hope that equipped with my results here it shouldn’t take a newcomer nearly that much time to get going. And of course I hope to continue learning more myself and perhaps even contributing to the theory one day, since there’s still plenty of frontier here.
Retrieved from "https://en.xen.wiki/w/Douglas_Blumeyer%27s_RTT_How-To"