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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | The '''height''' is a mathematical tool to measure the [[complexity]] of [[JI]] [[interval]]s. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:guest|guest]] and made on <tt>2012-09-06 21:05:16 UTC</tt>.<br>
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| : The original revision id was <tt>362695182</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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>
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| <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">=Definition:=
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| A **height** is a function on an abelian group which maps elements to real numbers, yielding a type of complexity measurement. Since the (non-zero) rationals form an abelian group under multiplication, we can assign each element a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition which is practical for musical purposes.
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| A height function H(q) on the rationals q should fulfill the following criteria: | | == Definition == |
| # Given any constant C, there are finitely many elements q such that H(q) <= C.
| | A '''height''' is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement (see [[Wikipedia: Height function]]). For example we can assign each element of the positive rational numbers a height, and hence a complexity. While there is no consensus on the restrictions of a height, we will attempt to create a definition for positive rational numbers which is practical for musical purposes. |
| # There is a unique constant K such that H(q) >= K, for all q.
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| # H(q) = H(1/q)
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| Since any rational q can be rewritten as a fraction n/d, we may sub this into the above equation to get H(n/d) = H(d/n). This relation is extremely useful - it tells us that we can switch n and d without any consequences on the outcome of the height.
| | A height function H(''q'') on the positive rationals ''q'' should fulfill the following criteria: |
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| If we have a function F(x) which is strictly increasing on the positive reals, then F(H(q)) will rank elements in the same order as H(q). We can therefore establish the following equivalence relation:
| | # Given any constant ''C'', there are finitely many elements ''q'' such that H(''q'') ≤ ''C''. |
| [[math]]
| | # H(''q'') is bounded below by H(1), so that H(''q'') ≥ H(1) for all q. |
| H(q) \equiv F(H(q)) | | # H(''q'') = H(1) iff ''q'' = 1. |
| [[math]]
| | # H(''q'') = H(1/''q'') |
| | # H(''q''<sup>''n''</sup>) ≥ H(''q'') for any non-negative integer ''n''. |
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| =Examples:= | | If we have a function F which is strictly increasing on the positive reals, then F(H(''q'')) will rank elements in the same order as H(''q''). We can therefore establish the following equivalence relation: |
| || Name: || H(n/d) || H(q) || H(q) simplified by equivalence relation || | | |
| || [[Benedetti Height|Benedetti height]]
| | <math>\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math> |
| (or [[Tenney Height]]) || [[math]] | | |
| nd
| | Exponentiation and logarithm are such functions commonly used for converting a height between arithmetic and logarithmic scales. |
| [[math]] || [[math]] | | |
| 2^{T1(q)} | | A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''<sub>1</sub> and ''q''<sub>2</sub> are considered equivalent if they differ only by factors of 2. |
| [[math]] || [[math]]
| | We can also consider other equivalences. For example, we can assume tritave equivalence by ignoring factors of 3. |
| T1(q)
| | |
| [[math]] || | | == Height versus norm == |
| || Weil Height || [[math]] | | Height functions are applied to ratios, whereas norms are measurements on interval lattices [[wikipedia: embedding|embedded]] in [[wikipedia: Normed vector space|normed vector spaces]]. Some height functions are essentially norms, and they are numerically equal. For example, the [[Tenney height]] is also the Tenney norm. |
| max(n,d) | | |
| [[math]] || [[math]]
| | However, not all height functions are norms, and not all norms are height functions. The [[Benedetti height]] is not a norm, since it does not satisfy the condition of absolute homogeneity. The [[taxicab distance]] is not a height, since there can be infinitely many intervals below a given bound. |
| 2^{(T1(q)+|log_2(q)|)/2} | | |
| [[math]] || [[math]]
| | == Examples of height functions == |
| T1(q)+|log_2(q)| | | {| class="wikitable" |
| [[math]] ||
| | ! Name |
| || Arithmetic Height || [[math]] | | ! Type |
| n+d | | ! H(''n''/''d'') |
| [[math]] || [[math]]
| | ! H(''q'') |
| 2^{T1(q)/2} (q+1)/q^{1/2}
| | ! H(''q'') simplified by equivalence relation |
| [[math]] || [[math]]
| | |- |
| T1(q)+2log_2(q+1)-log_2(q) | | | [[Benedetti height]] <br> (or [[Tenney height]]) |
| [[math]] ||
| | | Height |
| || [[Kees Height]] || [[math]]
| | | <math>nd</math> |
| max(2^{-v_2(n)}n, | | | <math>2^{\large{\|q\|_{T1}}}</math> |
| 2^{-v_2(d)}d) | | | <math>\|q\|_{T1}</math> |
| [[math]] || [[math]]
| | |- |
| 2^{(T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|)/2} | | | [[Wilson height]] |
| [[math]] || [[math]]
| | | Height |
| T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|
| | | <math>\text{sopfr}(n d)</math> |
| [[math]] ||
| | | <math>2^{\large{\text{sopfr}(q)}}</math> |
| || || || || || | | | <math>\text{sopfr}(q)</math> |
| Where T1(q) is the [[xenharmonic/Generalized Tenney Norms and Tp Interval Space#The%20Tenney%20Norm%20(T1%20norm)|tenney norm]] of q in monzo form, and vp(x) is the [[http://en.wikipedia.org/wiki/P-adic_order|p-adic valuation]] of x.
| | |- |
| | | [[Weil height]] |
| | | Height |
| | | <math>\max \left( {n , d} \right)</math> |
| | | <math>2^{\large{\frac{1}{2}(\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid)}}</math> |
| | | <math>\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid</math> |
| | |- |
| | | Arithmetic height |
| | | Height |
| | | <math>n + d</math> |
| | | <math>\dfrac {\left( {q + 1} \right)} {\sqrt{q}} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}</math> |
| | | <math>\|q\|_{T1} + 2 \log_2 \left( {q + 1} \right) - \log_2 \left( {q} \right)</math> |
| | |- |
| | | Harmonic semi-height |
| | | Semi-Height |
| | | <math>\dfrac {n d} {n + d}</math> |
| | | <math>\dfrac {\sqrt{q}} {\left( {q + 1} \right)} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}</math> |
| | | <math>\|q\|_{T1} - 2 \log_2 \left( {q + 1} \right) + \log_2 \left( {q} \right)</math> |
| | |- |
| | | [[Kees semi-height]] |
| | | Semi-Height |
| | | <math>\max \left( {2^{-v_2 \left( {n} \right)} n, 2^{-v_2 \left( {d} \right)} d} \right)</math> |
| | | <math>2^{\large{\left(\frac{1}{2}\left(\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(q) - v_2(q) \mid \right)\right)}}</math> |
| | | <math>\|{2^{-v_2 \left( {q} \right)} q}\|_{T1} + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |</math> |
| | |} |
| | |
| | Where ||''q''||<sub>T1</sub> is the [[Generalized Tenney norms and Tp interval space #The Tenney Norm (T1 norm)|tenney norm]] of ''q'' in monzo form, and v<sub>''p''</sub>(''q'') is the [[Wikipedia: P-adic order|''p''-adic valuation]] of ''q''. |
| | |
| | The function sopfr (''nd'') is the [https://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of ''n''·''d''. Equivalently, this is the L<sub>1</sub> norm on monzos, but where each prime is weighted by ''p'' rather than log (''p''). This is called "Wilson's Complexity" in [[John Chalmers]]'s ''Divisions of the Tetrachord''<ref>[http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf ''Division of the Tetrachord''], page 55. John Chalmers. </ref>. |
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| Some useful identities: | | Some useful identities: |
| [[math]]
| | * <math>n = 2^{\large{\frac{1}{2}(\|q\|_{T1} + \log_2(q))}}</math> |
| n=2^{(T1(q)\pm|log_2(q)|)/2} | | * <math>d = 2^{\large{\frac{1}{2}(\|q\|_{T1} - \log_2(q))}}</math> |
| [[math]]
| | * <math>n d = 2^{\|q\|_{T1}}</math> |
| [[math]]
| | |
| d=2^{(T1(q)\mp|log_2(q)|)/2} | | Height functions can also be put on the points of [http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html projective varieties]. Since [[abstract regular temperament]]s can be identified with rational points on [[Wikipedia: Grassmannian|Grassmann varieties]], complexity measures of regular temperaments are also height functions. |
| [[math]]
| | |
| [[math]]
| | See [[Dave Keenan & Douglas Blumeyer's guide to RTT/Alternative complexities]] for an extensive discussion of heights and semi-heights used in regular temperament theory. |
| nd=2^{T1(q)}
| | |
| [[math]]</pre></div>
| | == History == |
| <h4>Original HTML content:</h4>
| | The concept of height was introduced to xenharmonics by [[Gene Ward Smith]] in 2001<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_31418#31488 Yahoo! Tuning Group | ''Super Particular Stepsize'']</ref>; it comes from the mathematical field of number theory (for more information, see [[Wikipedia: Height function]]). It is not to be confused with the musical notion of [[Wikipedia: Pitch (music) #Theories of pitch perception|''pitch height'' (as opposed to ''pitch chroma'')]]<ref>Though it has also been used to refer to the size of an interval in cents. On page 23 of [https://www.plainsound.org/pdfs/JC&ToH.pdf ''John Cage and the Theor of Harmony''], Tenney writes: "The one-dimensional continuum of pitch-height (i.e. 'pitch' as ordinarily defined)", and graphs it ''as opposed to'' his concept of "harmonic distance", which was ironically the first measurement named by Gene Ward Smith as a "height": "Tenney height".</ref>. |
| <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>Height</title></head><body><!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:16 -->Definition:</h1>
| | |
| A <strong>height</strong> is a function on an abelian group which maps elements to real numbers, yielding a type of complexity measurement. Since the (non-zero) rationals form an abelian group under multiplication, we can assign each element a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition which is practical for musical purposes.<br />
| | == See also == |
| <br />
| | * [[Commas by taxicab distance]] |
| A height function H(q) on the rationals q should fulfill the following criteria:<br />
| | * [[Harmonic entropy]] |
| <ol><li>Given any constant C, there are finitely many elements q such that H(q) &lt;= C.</li><li>There is a unique constant K such that H(q) &gt;= K, for all q.</li><li>H(q) = H(1/q)</li></ol><br />
| | |
| Since any rational q can be rewritten as a fraction n/d, we may sub this into the above equation to get H(n/d) = H(d/n). This relation is extremely useful - it tells us that we can switch n and d without any consequences on the outcome of the height.<br />
| | == References == |
| <br />
| |
| If we have a function F(x) which is strictly increasing on the positive reals, then F(H(q)) will rank elements in the same order as H(q). We can therefore establish the following equivalence relation:<br />
| |
| <!-- ws:start:WikiTextMathRule:0:
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| [[math]]&lt;br/&gt; | |
| H(q) \equiv F(H(q))&lt;br/&gt;[[math]]
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| --><script type="math/tex">H(q) \equiv F(H(q))</script><!-- ws:end:WikiTextMathRule:0 --><br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc1"><a name="Examples:"></a><!-- ws:end:WikiTextHeadingRule:18 -->Examples:</h1>
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| <table class="wiki_table">
| | <references/> |
| <tr>
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| <td>Name:<br />
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| </td>
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| <td>H(n/d)<br />
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| </td>
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| <td>H(q)<br />
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| </td>
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| <td>H(q) simplified by equivalence relation<br />
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| </td>
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| </tr>
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| <tr>
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| <td><a class="wiki_link" href="/Benedetti%20Height">Benedetti height</a><br />
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| (or <a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)<br />
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| </td>
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| <td><!-- ws:start:WikiTextMathRule:1:
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| [[math]]&lt;br/&gt;
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| nd&lt;br/&gt;[[math]]
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| --><script type="math/tex">nd</script><!-- ws:end:WikiTextMathRule:1 --><br />
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| </td>
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| <td><!-- ws:start:WikiTextMathRule:2:
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| [[math]]&lt;br/&gt;
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| 2^{T1(q)}&lt;br/&gt;[[math]]
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| --><script type="math/tex">2^{T1(q)}</script><!-- ws:end:WikiTextMathRule:2 --><br />
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| </td>
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| <td><!-- ws:start:WikiTextMathRule:3:
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| [[math]]&lt;br/&gt;
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| T1(q)&lt;br/&gt;[[math]]
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| --><script type="math/tex">T1(q)</script><!-- ws:end:WikiTextMathRule:3 --><br />
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| </td>
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| </tr>
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| <tr>
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| <td>Weil Height<br />
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| </td>
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| <td><!-- ws:start:WikiTextMathRule:4:
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| [[math]]&lt;br/&gt;
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| max(n,d)&lt;br/&gt;[[math]]
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| --><script type="math/tex">max(n,d)</script><!-- ws:end:WikiTextMathRule:4 --><br />
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| </td>
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| <td><!-- ws:start:WikiTextMathRule:5:
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| [[math]]&lt;br/&gt;
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| 2^{(T1(q)+|log_2(q)|)/2}&lt;br/&gt;[[math]]
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| --><script type="math/tex">2^{(T1(q)+|log_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:5 --><br />
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| </td>
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| <td><!-- ws:start:WikiTextMathRule:6:
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| [[math]]&lt;br/&gt;
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| T1(q)+|log_2(q)|&lt;br/&gt;[[math]]
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| --><script type="math/tex">T1(q)+|log_2(q)|</script><!-- ws:end:WikiTextMathRule:6 --><br />
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| </td>
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| </tr>
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| <tr>
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| <td>Arithmetic Height<br />
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| </td>
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| <td><!-- ws:start:WikiTextMathRule:7:
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| [[math]]&lt;br/&gt;
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| n+d&lt;br/&gt;[[math]]
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| --><script type="math/tex">n+d</script><!-- ws:end:WikiTextMathRule:7 --><br />
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| </td>
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| <td><!-- ws:start:WikiTextMathRule:8:
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| [[math]]&lt;br/&gt;
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| 2^{T1(q)/2} (q+1)/q^{1/2}&lt;br/&gt;[[math]]
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| --><script type="math/tex">2^{T1(q)/2} (q+1)/q^{1/2}</script><!-- ws:end:WikiTextMathRule:8 --><br />
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| </td>
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| <td><!-- ws:start:WikiTextMathRule:9:
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| [[math]]&lt;br/&gt;
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| T1(q)+2log_2(q+1)-log_2(q)&lt;br/&gt;[[math]]
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| --><script type="math/tex">T1(q)+2log_2(q+1)-log_2(q)</script><!-- ws:end:WikiTextMathRule:9 --><br />
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| </td>
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| </tr>
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| <tr>
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| <td><a class="wiki_link" href="/Kees%20Height">Kees Height</a><br />
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| </td>
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| <td><!-- ws:start:WikiTextMathRule:10:
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| [[math]]&lt;br/&gt;
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| max(2^{-v_2(n)}n,&lt;br /&gt;
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| 2^{-v_2(d)}d)&lt;br/&gt;[[math]]
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| --><script type="math/tex">max(2^{-v_2(n)}n,
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| 2^{-v_2(d)}d)</script><!-- ws:end:WikiTextMathRule:10 --><br />
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| </td>
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| <td><!-- ws:start:WikiTextMathRule:11:
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| [[math]]&lt;br/&gt;
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| 2^{(T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|)/2}&lt;br/&gt;[[math]]
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| --><script type="math/tex">2^{(T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:11 --><br />
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| </td>
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| <td><!-- ws:start:WikiTextMathRule:12:
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| [[math]]&lt;br/&gt;
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| T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|&lt;br/&gt;[[math]]
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| --><script type="math/tex">T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|</script><!-- ws:end:WikiTextMathRule:12 --><br />
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| </td>
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| </tr>
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| <tr>
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| <td><br />
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| </td>
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| <td><br />
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| </td>
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| <td><br />
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| </td>
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| <td><br />
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| </td>
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| </tr>
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| </table>
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|
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| Where T1(q) is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Generalized%20Tenney%20Norms%20and%20Tp%20Interval%20Space#The%20Tenney%20Norm%20(T1%20norm)">tenney norm</a> of q in monzo form, and vp(x) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/P-adic_order" rel="nofollow">p-adic valuation</a> of x.<br />
| | [[Category:Math]] |
| <br />
| | [[Category:Interval complexity measures]] |
| Some useful identities:<br />
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| <!-- ws:start:WikiTextMathRule:13:
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| [[math]]&lt;br/&gt;
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| n=2^{(T1(q)\pm|log_2(q)|)/2}&lt;br/&gt;[[math]]
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| --><script type="math/tex">n=2^{(T1(q)\pm|log_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:13 --><br />
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| <!-- ws:start:WikiTextMathRule:14:
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| [[math]]&lt;br/&gt;
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| d=2^{(T1(q)\mp|log_2(q)|)/2}&lt;br/&gt;[[math]]
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| --><script type="math/tex">d=2^{(T1(q)\mp|log_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:14 --><br />
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| <!-- ws:start:WikiTextMathRule:15:
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| [[math]]&lt;br/&gt;
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| nd=2^{T1(q)}&lt;br/&gt;[[math]]
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| --><script type="math/tex">nd=2^{T1(q)}</script><!-- ws:end:WikiTextMathRule:15 --></body></html></pre></div>
| |