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| |monzo | | |monzo |
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| 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.
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| Note that units are given here in a plain font, scalars are italic, vectors are bold, and matrices are capital italic, as per convention.
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| {| class="wikitable"
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| |+ '''Table 3.''' RTT units, dimensions, variables
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| ! colspan="3" |variable
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| ! rowspan="2" |name
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| ! colspan="3" |units
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| ! colspan="2" |dimensions
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| ! rowspan="2" |type
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| ! rowspan="2" |[[variance|<math>v</math>]]
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| |-
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| !unreduced
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| !red.
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| ![[RTT library in Wolfram Language|Wolfram library]]
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| !unreduced
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| !red.
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| !read as
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| !unreduced
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| !red.
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| |-
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| |<math>\textbf{i}</math>
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| |<code>i</code>
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| |[[Prime-count vector|interval]]
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| |<math>\text{p}</math>
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| |primes
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| |<math>d\!×\!1</math>
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| |vector
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| |contra
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| |-
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| |<math>M</math>
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| |<code>m</code>
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| |[[Mapping|(temperament) mapping]]
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| |<math>\large{{}^\text{g}{\mskip -5mu/\mskip -3mu}_\text{p}}</math>
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| |generators per prime
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| |<math>r\!×\!d</math>
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| |matrix
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| |co
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| |-
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| |<math>M.\textbf{i}</math>
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| |<math>M\textbf{i}</math>
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| |<code>mi</code>
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| |[[Tmonzos_and_tvals|mapped interval]]
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| |<math>\large{}{}^\text{g}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{p}}}·\!\normalsize{}\cancel{\text{p}}</math>
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| |<math>\text{g}</math>
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| |generators
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| |<math>\small{}(r\!×\!\!\cancel{d}\!)(\!\cancel{d}\!\!×\!1)</math>
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| |<math>r\!×\!1</math>
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| |vector
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| |contra
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| |-
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| |<math>G</math>
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| |<code>g</code>
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| |[[Tenney-Euclidean_tuning#Computing_TE_tuning_using_pseudoinverse|generators (matrix)]]
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| |<math>\large{}{}^\text{p}{\mskip -5mu/\mskip -3mu}_\text{g}</math>
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| |primes per generator
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| |<math>d\!×\!r</math>
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| |matrix
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| |contra
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| |-
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| |<math>G.M</math>
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| |<math>P</math>
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| |<code>p</code>
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| |[[Projection matrix|(tuning) projection (matrix)]]
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| |<math>\large{}{}^{\text{p}}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{g}}}\!·\!{}^{\cancel{\text{g}}\!}{\mskip -5mu/\mskip -3mu}_{\text{p}}</math>
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| |<math>\large{}{}^\text{p}{\mskip -5mu/\mskip -3mu}_\text{p}</math>
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| |primes per prime
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| |<math>\small{}(d\!×\!\!\cancel{r}\!)(\!\cancel{r}\!\!×\!d)</math>
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| |<math>d\!×\!d</math>
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| |matrix
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| |-
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| |<math>G.M.\textbf{i}</math><br><math>P.\textbf{i}</math>
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| |<math>P\textbf{i}</math>
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| |<code>pi</code>
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| |projected interval
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| |<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>
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| |<math>\text{p}</math>
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| |primes
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| |<math>\small{}(d\!×\!\!\cancel{r}\!)(\!\cancel{r}\!\!×\!\!\cancel{d}\!)(\!\cancel{d}\!\!×\!1)</math>
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| |<math>d\!×\!1</math>
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| |vector
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| |contra
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| |-
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| |<math>\textbf{p}</math>
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| |<code>ptm</code>
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| |[[JIP|prime tuning map]]
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| |<math>\large{}{}^\text{oct}{\mskip -5mu/\mskip -3mu}_\text{p}</math>
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| |octaves per prime
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| |<math>1\!×\!d</math>
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| |vector
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| |co
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| |<math>\textbf{p}.G</math>
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| |<math>\textbf{g}</math>
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| |<code>gtm</code>
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| |[[generator tuning map]]
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| |<math>\large{}{}^\text{oct}{\mskip -5mu/\mskip -3mu}_{\!\!\cancel{\text{p}}}\!·\!{}^{\cancel{\text{p}}\!}{\mskip -5mu/\mskip -3mu}_\text{g}</math>
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| |<math>\large{}{}^\text{oct}{\mskip -5mu/\mskip -3mu}_\text{g}</math>
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| |octaves per generator
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| |<math>\small{}(1\!×\!\!\cancel{d}\!)(\!\cancel{d}\!\!×\!r)</math>
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| |<math>1\!×\!r</math>
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| |vector
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| |co
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| |-
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| |<math>\textbf{p}.G.M</math><br><math>\textbf{p}.P</math><br><math>\textbf{g}.M</math>
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| |<math>\textbf{t}</math>
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| |<code>tm</code>
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| |[[tuning map|(temperament) tuning map]]
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| |<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>
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| |<math>\large{}{}^\text{oct}{\mskip -5mu/\mskip -3mu}_\text{p}</math>
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| |octaves per prime
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| |<math>\small{}(1\!×\!\!\cancel{d}\!)(\!\cancel{d}\!\!×\!\!\cancel{r}\!)(\!\cancel{r}\!\!×\!d)</math>
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| |<math>1\!×\!d</math>
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| |vector
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| |co
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| |-
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| |<math>\textbf{p}.G.M.\textbf{i}</math><br><math>\textbf{p}.P.\textbf{i}</math><br><math>\textbf{g}.M.\textbf{i}</math>
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| |<math>\textbf{t}\textbf{i}</math>
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| |<code>ti</code>
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| |tuned interval
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| |<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>
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| |<math>\text{oct}</math>
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| |octaves
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| |<math>\small{}(1\!×\!\!\cancel{d}\!)(\!\cancel{d}\!\!×\!\!\cancel{r}\!)(\!\cancel{r}\!\!×\!\!\cancel{d}\!)(\!\cancel{d}\!\!×\!1)</math>
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| |<math>1\!×\!1</math>
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| |vector
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| |<math>C</math>
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| |<code>c</code>
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| |[[comma basis]]
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| |<math>\text{p}</math>
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| |primes
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| |<math>d\!×\!n</math>
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| |matrix
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| |contra
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| |-
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| |<math>1200</math>
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| |octaves-to-cents conversion
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| |<math>\large{}{}^\text{c}{\mskip -5mu/\mskip -3mu}_\text{oct}</math>
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| |cents per octave
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| |<math>1\!×\!1</math>
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| |scalar
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| |<math>T</math>
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| |<code>t</code>
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| |[[regular temperament|temperament]]
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| |-
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| |<math>d</math>
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| |<code>d</code>
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| |[[dimensionality]]
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| |<math>1\!×\!1</math>
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| |scalar
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| |<math>r</math>
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| |<code>r</code>
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| |[[rank]]
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| |<math>1\!×\!1</math>
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| |scalar
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| |<math>n</math>
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| |<code>n</code>
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| |[[nullity]]
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| |<math>1\!×\!1</math>
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| |scalar
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| |-
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| |<math>v</math>
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| |<code>v</code>
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| |[[variance]]
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| |-
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| |<math>\text{g}</math>
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| |generators
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| |units
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| |<math>\text{p}</math>
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| |primes
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| |units
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| |<math>\text{oct}</math>
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| |octaves
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| |units
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| |-
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| |<math>\text{c}</math>
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| |cents
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| |units
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| |}
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| == Example ==
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| So, let's look at the example of ¼-comma meantone it is. Our four atomic objects are:
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| * example interval, <math>\textbf{i}</math>: {{vector|-1 1 0}}
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| * mapping, <math>M</math>: {{ket|{{map|1 1 0}} {{map|0 1 4}}}}
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| * generators, <math>G</math>: {{bra|{{vector|1 0 0}} {{vector|0 0 ¼}}}}
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| * prime tuning map, <math>\textbf{p}</math>: 1200{{map|log₂2 log₂3 log₂5}} = {{map|1200.000 1901.955 2786.314}}
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| And so the various possible compositions of these objects are:
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| * generator tuning map, <math>\textbf{p}.G = \textbf{g}</math>: {{map|1200.000 696.578}}
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| * mapped interval, <math>M\textbf{i}</math>: [0 1}
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| * tuning map, <math>\textbf{p}.G.M = \textbf{t}</math>: {{map|1200.000 1896.58 2786.31}}
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| * projection matrix, <math>G.M = P</math>: [[1 1 0] [0 0 0] [0 ¼ 1]]
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| * projected interval: <math>G.M.\textbf{i} = P\textbf{i}</math>: {{vector|0 0 ¼}}
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| 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.
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| = Outro = | | = Outro = |