Monzos and interval space: Difference between revisions
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<span style="display: block; text-align: right;">[[de:Intervallraum]]</span> | <span style="display: block; text-align: right;">[[de:Intervallraum]]</span> | ||
This page gives the formal mathematical definition of a monzo. For a simpler explanation with examples, visit the [[monzos]] page. | This page gives the formal mathematical definition of a '''monzo''' and shows its relation to '''interval space'''. For a simpler explanation with examples, visit the [[monzos]] page. | ||
=Definition= | == Definition == | ||
A [[Harmonic_Limit|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving | A [[Harmonic_Limit|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving | ||
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<math>\| |e_2 \, e_3 \dotso e_p \rangle \| = |e_2| + |e_3| \log_2 3 + \dotsb + |e_p| \log_2 p</math> | <math>\| |e_2 \, e_3 \dotso e_p \rangle \| = |e_2| + |e_3| \log_2 3 + \dotsb + |e_p| \log_2 p</math> | ||
which is a [http://en.wikipedia.org/wiki/Normed_vector_space vector space norm]; hence we may [http://en.wikipedia.org/wiki/Embedding embed] the p-limit monzos into a normed vector I space of dimension n = π(p) via a map M:monzos ⟶ I. The monzos under this embedding now define a [http://en.wikipedia.org/wiki/Lattice_%28group%29 lattice], which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard [http://mathworld.wolfram.com/L1-Norm.html L1 norm]. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while {{ | which is a [http://en.wikipedia.org/wiki/Normed_vector_space vector space norm]; hence we may [http://en.wikipedia.org/wiki/Embedding embed] the p-limit monzos into a normed vector I space of dimension n = π(p) via a map M:monzos ⟶ I. The monzos under this embedding now define a [http://en.wikipedia.org/wiki/Lattice_%28group%29 lattice], which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard [http://mathworld.wolfram.com/L1-Norm.html L1 norm]. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while {{monzo|1 0}} represents 2, so does {{monzo|0 log3(2)}}. | ||
Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have [[Tenney-Euclidean_metrics|Tenney-Euclidean interval space]] instead of Tenney interval space. Explicitly, if we take the monzo {{ | Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have [[Tenney-Euclidean_metrics|Tenney-Euclidean interval space]] instead of Tenney interval space. Explicitly, if we take the monzo {{monzo|e2 e3 ... ep}} then the Tenney-Euclidean norm, or TE norm, of it is | ||
<math>\sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2}</math> | <math>\sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2}</math> | ||
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and if the coordinates are the weighted interval space coordinates, then the TE norm is the [http://mathworld.wolfram.com/L2-Norm.html standard Euclidean, or L2, norm]. | and if the coordinates are the weighted interval space coordinates, then the TE norm is the [http://mathworld.wolfram.com/L2-Norm.html standard Euclidean, or L2, norm]. | ||
=Alternate Definition= | == Alternate Definition == | ||
Given a rational number q, we can rewrite it in monzo form by the following definition: | Given a rational number q, we can rewrite it in monzo form by the following definition: | ||
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Where vp(q) is the [http://en.wikipedia.org/wiki/P-adic_order p-adic valuation] of q. | Where vp(q) is the [http://en.wikipedia.org/wiki/P-adic_order p-adic valuation] of q. | ||
=Example= | == Example == | ||
The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of {{ | The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of {{monzo|4 -1 -1}}. In weighted coordinates, that becomes {{monzo|4 -log2(3) -log2(5)}}, approximately {{monzo|4 -1.585 -2.322}}. | ||
The TE norm is therefore | The TE norm is therefore | ||
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</math> | </math> | ||
=See also= | == See also == | ||
* [[Fractional monzos]] | |||
* [[Vals and tuning space]] | |||
[[Category: | [[Category:Interval measure]] | ||
[[Category: | [[Category:Interval space]] | ||
[[Category: | [[Category:Math]] | ||
[[Category: | [[Category:Monzo]] | ||
[[Category: | [[Category:Theory]] | ||