Monzos and interval space: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 141273805 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 146156308 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2010-06-01 02:56:18 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>146156308</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A [[Harmonic Limit|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving q = 2^e2 3^e3 ... p^ep, where the exponents are integers (positive, negative, or zero.) This is often written in [[http://mathworld.wolfram.com/Ket.html|ket vector]] notation as |e2 e3 ... ep>, in which case it is called a **monzo**, where the name refers to the enthusiastic advocacy of [[Joe Monzo]]. | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A [[Harmonic Limit|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving q = 2^e2 3^e3 ... p^ep, where the exponents are integers (positive, negative, or zero.) This is often written in [[http://mathworld.wolfram.com/Ket.html|ket vector]]([[http://en.wikipedia.org/wiki/Bra-ket_notation|wp]]) notation as |e2 e3 ... ep>, in which case it is called a **monzo**, where the name refers to the enthusiastic advocacy of [[Joe Monzo]]. | ||
The [[Tenney Height|Tenney height]] of this monzo is given by | The [[Tenney Height|Tenney height]] of this monzo is given by | ||
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and if the coordinates are the standard interval space coordinates, then the Euclidean norm is the [[http://mathworld.wolfram.com/L2-Norm.html|standard Euclidean, or L2, norm]].</pre></div> | and if the coordinates are the standard interval space coordinates, then the Euclidean norm is the [[http://mathworld.wolfram.com/L2-Norm.html|standard Euclidean, or L2, norm]].</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Monzos and Interval Space</title></head><body>A <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> rational number q can by definition be factored into primes of size less than or equal to p, giving q = 2^e2 3^e3 ... p^ep, where the exponents are integers (positive, negative, or zero.) This is often written in <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Ket.html" rel="nofollow">ket vector</a> notation as |e2 e3 ... ep&gt;, in which case it is called a <strong>monzo</strong>, where the name refers to the enthusiastic advocacy of <a class="wiki_link" href="/Joe%20Monzo">Joe Monzo</a>.<br /> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Monzos and Interval Space</title></head><body>A <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> rational number q can by definition be factored into primes of size less than or equal to p, giving q = 2^e2 3^e3 ... p^ep, where the exponents are integers (positive, negative, or zero.) This is often written in <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Ket.html" rel="nofollow">ket vector</a>(<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Bra-ket_notation" rel="nofollow">wp</a>) notation as |e2 e3 ... ep&gt;, in which case it is called a <strong>monzo</strong>, where the name refers to the enthusiastic advocacy of <a class="wiki_link" href="/Joe%20Monzo">Joe Monzo</a>.<br /> | ||
<br /> | <br /> | ||
The <a class="wiki_link" href="/Tenney%20Height">Tenney height</a> of this monzo is given by<br /> | The <a class="wiki_link" href="/Tenney%20Height">Tenney height</a> of this monzo is given by<br /> | ||
Revision as of 02:56, 1 June 2010
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author xenwolf and made on 2010-06-01 02:56:18 UTC.
- The original revision id was 146156308.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
A [[Harmonic Limit|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving q = 2^e2 3^e3 ... p^ep, where the exponents are integers (positive, negative, or zero.) This is often written in [[http://mathworld.wolfram.com/Ket.html|ket vector]]([[http://en.wikipedia.org/wiki/Bra-ket_notation|wp]]) notation as |e2 e3 ... ep>, in which case it is called a **monzo**, where the name refers to the enthusiastic advocacy of [[Joe Monzo]]. The [[Tenney Height|Tenney height]] of this monzo is given by || |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p) |ep| which is a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]]. The monzos with this norm now define a [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]], which is a discrete subgroup spanning a finite dimensional real normed vector space. 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 |1 0> represents 2, so does |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 Euclidean interval space instead of Tenney interval space. Explicitly, if we take the monzo |e2 e3 ... ep> then the Euclidean norm of it is sqrt(e2^2 + (log2(3)e3)^2 ... + (log2(p)ep)^2) and if the coordinates are the standard interval space coordinates, then the Euclidean norm is the [[http://mathworld.wolfram.com/L2-Norm.html|standard Euclidean, or L2, norm]].
Original HTML content:
<html><head><title>Monzos and Interval Space</title></head><body>A <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> rational number q can by definition be factored into primes of size less than or equal to p, giving q = 2^e2 3^e3 ... p^ep, where the exponents are integers (positive, negative, or zero.) This is often written in <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Ket.html" rel="nofollow">ket vector</a>(<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Bra-ket_notation" rel="nofollow">wp</a>) notation as |e2 e3 ... ep>, in which case it is called a <strong>monzo</strong>, where the name refers to the enthusiastic advocacy of <a class="wiki_link" href="/Joe%20Monzo">Joe Monzo</a>.<br /> <br /> The <a class="wiki_link" href="/Tenney%20Height">Tenney height</a> of this monzo is given by<br /> <br /> || |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p) |ep|<br /> <br /> which is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow">vector space norm</a>. The monzos with this norm now define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow">lattice</a>, which is a discrete subgroup spanning a finite dimensional real normed vector space. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow">L1 norm</a>. 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 |1 0> represents 2, so does |0 log3(2)>.<br /> <br /> 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 Euclidean interval space instead of Tenney interval space. Explicitly, if we take the monzo |e2 e3 ... ep> then the Euclidean norm of it is<br /> <br /> sqrt(e2^2 + (log2(3)e3)^2 ... + (log2(p)ep)^2)<br /> <br /> and if the coordinates are the standard interval space coordinates, then the Euclidean norm is the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L2-Norm.html" rel="nofollow">standard Euclidean, or L2, norm</a>.</body></html>