Monzos and interval space: Difference between revisions

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This page gives the formal mathematical definition of a '''monzo''' and shows its relation to '''interval space'''.  

A [[Harmonic limit|''p''-limit]] rational number ''q'' can by definition be factored into primes of size less than or equal to ''p'', giving

<math>q = 2^{e_2} \, 3^{e_3} \, 5^{e_5} \dotso p^{e_p}</math>

where the exponents are integers (positive, negative, or zero.) This is often written in [http://mathworld.wolfram.com/Ket.html ket vector] (→ [[Wikipedia: Bra-ket notation]]) notation as

in which case it is called a '''monzo''', where the name refers to the enthusiastic advocacy of [[Joe Monzo]].

The [[Tenney height]] of this monzo is given by

 

<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 [[Wikipedia: Normed vector space|vector space norm]]; hence we may [[Wikipedia: 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 [[Wikipedia: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 {{monzo| ''e''₂ ''e''₃ … ''e''_''p'' }} 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>

 

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].

 

 

<math>q = |v_2 (q) \,v_3 (q) \, v_5 (q) \dotso v_p (q)\rangle</math>

 

 

<math>\| |v_2 (q) \, v_3 (q) \dotso v_p (q) \rangle \| = |v_2 (q)| + |v_3 (q)| \log_2 3 + \dotsb + |v_p (q)| \log_2 p</math>

 

Where ''v''_''p'' (''q'') is the [[Wikipedia: P-adic order|''p''-adic valuation]] of ''q''.

 

== Example ==

The 5-limit interval 16/15 factors as 2⁴ 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 }}.

 

 

<math>\sqrt{(4^2 + \log_2(3)^2 + \log_2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889.

</math>