Subgroup monzos and vals: Difference between revisions

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== Conversion ==

If '''m'''G is a smonzo of the subgroup G, and if ''S'' is a [[subgroup basis matrix]] whose columns form a basis for the subgroup G, then the corresponding monzo '''m''' is given by

If '''m'''''G'' is a smonzo of the subgroup ''G'', and if ''S'' is a [[subgroup basis matrix]] whose columns form a basis for the subgroup ''G'', then the corresponding monzo '''m''' is given by

~~<math>\displaystyle~~ \vec m = S\vec ~~m_{\rm G}</math>~~

$$\vec m = S\vec m_G$$

Conversely, if '''m''' is a monzo for an interval in the subgroup, then we can take the [[pseudoinverse]] of ''S'', ''S''+, and the corresponding smonzo is given by

~~<math>\displaystyle~~ \vec ~~m_{\rm G}~~ = S^+ \vec m~~</math>~~

$$\vec m_G = S^+ \vec m$$

For example, consider the subgroup generated by the [[The Archipelago|barbados triad]], ~~1-13~~/~~10-3~~/~~2-2~~. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix ''S'' with columns consisting of the monzos for 2, 3, and 13/10. Now the monzo for 676/675 is {{monzo| 2 -3 2 0 0 2 }}, and left-multiplying this by ''S''+ gives the smonzo {{monzo| 2 -3 2 }}. We may check this is the correct smonzo from 22 3-3 (13/5)2, which is 676/675 as desired.

For example, consider the subgroup generated by the [[The Archipelago|barbados triad]], 1–13/10–3/2–2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix ''S'' with columns consisting of the monzos for 2, 3, and 13/10. Now the monzo for 676/675 is {{monzo| 2 -3 2 0 0 2 }}, and left-multiplying this by ''S''+ gives the smonzo {{monzo| 2 -3 2 }}. We may check this is the correct smonzo from 22 3-3 (13/5)2, which is 676/675 as desired.

[[Category:Subgroup]]

[[Category:Val]]

[[Category:Monzo]]

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 == Conversion ==
-If '''m'''<sub>G</sub> is a smonzo of the subgroup G, and if ''S'' is a [[subgroup basis matrix]] whose columns form a basis for the subgroup G, then the corresponding monzo '''m''' is given by
+If '''m'''<sub>''G''</sub> is a smonzo of the subgroup ''G'', and if ''S'' is a [[subgroup basis matrix]] whose columns form a basis for the subgroup ''G'', then the corresponding monzo '''m''' is given by
-<math>\displaystyle \vec m = S\vec m_{\rm G}</math>
+$$\vec m = S\vec m_G$$
 Conversely, if '''m''' is a monzo for an interval in the subgroup, then we can take the [[pseudoinverse]] of ''S'', ''S''<sup>+</sup>, and the corresponding smonzo is given by
-<math>\displaystyle \vec m_{\rm G} = S^+ \vec m</math>
+$$\vec m_G = S^+ \vec m$$
-For example, consider the subgroup generated by the [[The Archipelago|barbados triad]], 1-13/10-3/2-2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix ''S'' with columns consisting of the monzos for 2, 3, and 13/10. Now the monzo for 676/675 is {{monzo| 2 -3 2 0 0 2 }}, and left-multiplying this by ''S''<sup>+</sup> gives the smonzo {{monzo| 2 -3 2 }}. We may check this is the correct smonzo from 2<sup>2</sup> 3<sup>-3</sup> (13/5)<sup>2</sup>, which is 676/675 as desired.
+For example, consider the subgroup generated by the [[The Archipelago|barbados triad]], 1–13/10–3/2–2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix ''S'' with columns consisting of the monzos for 2, 3, and 13/10. Now the monzo for 676/675 is {{monzo| 2 -3 2 0 0 2 }}, and left-multiplying this by ''S''<sup>+</sup> gives the smonzo {{monzo| 2 -3 2 }}. We may check this is the correct smonzo from 2<sup>2</sup> 3<sup>-3</sup> (13/5)<sup>2</sup>, which is 676/675 as desired.
 [[Category:Subgroup]]
 [[Category:Val]]
 [[Category:Monzo]]