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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{interwiki |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | en = Harmonic limit |
| : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-15 13:31:16 UTC</tt>.<br>
| | | de = P-Limit |
| : The original revision id was <tt>602249856</tt>.<br>
| | | ja = リミット |
| : 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>
| | {{Wikipedia|Limit (music)}} |
| <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">A positive rational number q belongs to the **p-limit**, called the **p harmonic** or **prime limit**, for a given [[prime number]] p if and only if it can be factored into primes (with positive or negative integer exponents) of size less than or equal to p. For any prime number p, the set of all rational numbers in the p-limit defines a [[http://en.wikipedia.org/wiki/Free_abelian_group|finitely generated free abelian group]]. The rank of this group is equal to pi(p), the number of prime numbers less than or equal to p. Hence, for example, the rank of the [[7-limit]] is 4, as it is generated by 2, 3, 5 and 7. Another way to express the p-limit is that it consists of the ratios of [[http://en.wikipedia.org/wiki/Smooth_number|p-smooth numbers]], where a p-smooth number is an integer with prime factors no larger than p.
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| ==List of small p-limits==
| | In [[just intonation]], the '''''p''-limit''' (or '''''p''-prime-limit''') is the set of [[frequency ratio]]s that can be expressed using only [[prime numbers]] less than or equal to ''p''. |
| With increasing limits, the tonal space becomes more dense.
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| * [[2-limit]] contains only multiples of the [[octave]] (2/1), see [[1edo]]
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| * [[3-limit]] contains [[3_2|3/2]], the [[just perfect fifth]]
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| * [[5-limit]] contains [[5_4|5/4]], the just major third
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| * [[7-limit]] contains [[7_4|7/4]], the harmonic seventh
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| * [[11-limit]] contains [[11_8|11/8]], the Alphorn-Fa
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| * [[13-limit]]
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| * [[17-limit]]
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| * [[19-limit]]
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| * [[23-limit]]
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| * [[29-limit]]
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| * [[31-limit]]
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| * [[37-limit]]
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| * [[41-limit]]
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| * [[43-limit]]
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| * [[47-limit]]
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| * [[53-limit]]
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| * [[59-limit]]
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| * [[61-limit]]
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| ==See also== | | A frequency ratio belongs to the ''p''-limit if and only if both its numerator and denominator can be [[prime factorization|factored]] completely into prime numbers no larger than ''p'' (with positive or negative integer exponents). In mathematics, such numbers are known as {{w|smooth number|''p''-smooth numbers}}. |
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| | An interval does not need to contain the prime ''p'' itself to be within the ''p''-limit. For example, [[3/2]] belongs to the [[13-limit]] because both 2 and 3 are smaller than 13. Conversely, containing the prime ''p'' does not guarantee membership in the ''p''-limit. For instance, [[23/13]] is not within the 13-limit because 23 is a prime number larger than 13. |
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| | All prime limits contain infinitely many intervals. Even if we [[octave reduction|restrict]] our consideration to intervals within a single octave, all prime limits except the [[2-limit]] still contain infinitely many distinct ratios. |
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| | == Prime limits as subgroups == |
| | Prime limits are essentially [[just intonation subgroups]] that do not skip any primes. For any prime number ''p'', the ''p''-limit creates a well-defined mathematical structure, called ''{{w|free abelian group}}''. This structure has a dimension (or rank) equal to the number of prime numbers less than or equal to ''p''. For example, the [[7-limit]] works with intervals built from the primes 2, 3, 5, and 7, so it has 4 dimensions. |
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| | Often, composers and theorists find it more practical to work with smaller subsets of a prime limit rather than using all possible intervals within that limit. This becomes increasingly important for higher limits, as the number of practical tuning systems that can reasonably approximate the full set of intervals diminishes. For discussing these subsets that exist within a larger prime limit, some theorists use the term "''p''-horizon" to refer to the collection of all possible subsets within a ''p''-limit. |
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| | == Proper harmonic limit == |
| | While harmonic limit encompasses all ratios up to a given prime, '''proper harmonic limit''' classifies JI ratios based only based on the ''highest'' prime they contain in either the numerator or denominator. Equivalently, it is all of the intervals of a prime limit that are not found in a lower prime limit. It has been called '''harmonic class''' or '''HC'''. |
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| | A ratio belongs to the proper ''p''-prime limit if and only if ''p'' is the highest prime number found in its factorization. For example: |
| | * [[7/4]] is proper 7-limit because 7 is the highest prime in its factorization. |
| | * [[5/4]] is proper 5-limit, not proper 7-limit, even though it's within the 7-limit. |
| | * [[9/7]] is proper 7-limit because the highest prime is 7 (since {{nowrap| 9 {{=}} 3<sup>2</sup> }}). |
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| | This distinction helps differentiate between intervals that merely fall within a limit versus those that specifically use a particular prime. Unlike regular harmonic limits, proper harmonic limits are mutually exclusive categories. |
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| | == Alternative classification systems == |
| | Prime limits contain infinitely many intervals, so they cannot truly function as limits on complexity. Rather, they serve as compositional constraints that help manage choices when working with just intonation or regular temperaments. |
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| | Various alternative classification systems exist for characterizing intervals, such as: |
| | * [[Odd-limit]] classifies intervals based on the complexity of the ratio itself. |
| | * [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic interval naming]] characterizes just intervals by their deviation from basic Pythagorean intervals. |
| | * [[Overtone scales]] and [[primodality]] classify scales and chords based on their relative position in the harmonic series. |
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| | == Individual pages of ''p''-limit JI == |
| | {| class="wikitable center-all" |
| | |- |
| | | [[2-limit]] || [[3-limit]] || [[5-limit]] || [[7-limit]] || [[11-limit]] || [[13-limit]] |
| | |- |
| | | [[17-limit]] || [[19-limit]] || [[23-limit]] || [[29-limit]] || [[31-limit]] || [[37-limit]] |
| | |- |
| | | [[41-limit]] || [[43-limit]] || [[47-limit]] || [[53-limit]] || [[59-limit]] || [[61-limit]] |
| | |- |
| | | [[67-limit]] || [[71-limit]] || [[73-limit]] || [[79-limit]] || [[83-limit]] || [[89-limit]] |
| | |} |
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| | == See also == |
| * [[Odd limit]] | | * [[Odd limit]] |
| * [[Harmonic Class (HC)]] | | * [[Cubic and octahedral limits]] |
| * [[consistency]] | | * [[Prime minimum]] |
| * [[http://en.wikipedia.org/wiki/Limit_%28music%29|Limit (music) - Wikipedia]] (covers also the distinction between odd-limit and prime-limit) | | * [[Harmonic class]] |
| * [[http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem|Størmer's theorem - Wikipedia]]</pre></div> | | * [[Wikipedia: Størmer's theorem]] |
| <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Harmonic Limit</title></head><body>A positive rational number q belongs to the <strong>p-limit</strong>, called the <strong>p harmonic</strong> or <strong>prime limit</strong>, for a given <a class="wiki_link" href="/prime%20number">prime number</a> p if and only if it can be factored into primes (with positive or negative integer exponents) of size less than or equal to p. For any prime number p, the set of all rational numbers in the p-limit defines a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">finitely generated free abelian group</a>. The rank of this group is equal to pi(p), the number of prime numbers less than or equal to p. Hence, for example, the rank of the <a class="wiki_link" href="/7-limit">7-limit</a> is 4, as it is generated by 2, 3, 5 and 7. Another way to express the p-limit is that it consists of the ratios of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Smooth_number" rel="nofollow">p-smooth numbers</a>, where a p-smooth number is an integer with prime factors no larger than p.<br />
| | == External links == |
| <br />
| | * [http://tonalsoft.com/enc/l/limit.aspx Tonalsoft Encyclopedia | ''Limit''] |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-List of small p-limits"></a><!-- ws:end:WikiTextHeadingRule:0 -->List of small p-limits</h2>
| | |
| With increasing limits, the tonal space becomes more dense.<br />
| | [[Category:Regular temperament theory]] |
| <ul><li><a class="wiki_link" href="/2-limit">2-limit</a> contains only multiples of the <a class="wiki_link" href="/octave">octave</a> (2/1), see <a class="wiki_link" href="/1edo">1edo</a></li><li><a class="wiki_link" href="/3-limit">3-limit</a> contains <a class="wiki_link" href="/3_2">3/2</a>, the <a class="wiki_link" href="/just%20perfect%20fifth">just perfect fifth</a></li><li><a class="wiki_link" href="/5-limit">5-limit</a> contains <a class="wiki_link" href="/5_4">5/4</a>, the just major third</li><li><a class="wiki_link" href="/7-limit">7-limit</a> contains <a class="wiki_link" href="/7_4">7/4</a>, the harmonic seventh</li><li><a class="wiki_link" href="/11-limit">11-limit</a> contains <a class="wiki_link" href="/11_8">11/8</a>, the Alphorn-Fa</li><li><a class="wiki_link" href="/13-limit">13-limit</a></li><li><a class="wiki_link" href="/17-limit">17-limit</a></li><li><a class="wiki_link" href="/19-limit">19-limit</a></li><li><a class="wiki_link" href="/23-limit">23-limit</a></li><li><a class="wiki_link" href="/29-limit">29-limit</a></li><li><a class="wiki_link" href="/31-limit">31-limit</a></li><li><a class="wiki_link" href="/37-limit">37-limit</a></li><li><a class="wiki_link" href="/41-limit">41-limit</a></li><li><a class="wiki_link" href="/43-limit">43-limit</a></li><li><a class="wiki_link" href="/47-limit">47-limit</a></li><li><a class="wiki_link" href="/53-limit">53-limit</a></li><li><a class="wiki_link" href="/59-limit">59-limit</a></li><li><a class="wiki_link" href="/61-limit">61-limit</a></li></ul><br />
| | [[Category:Prime limit| ]] <!-- main article --> |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-See also"></a><!-- ws:end:WikiTextHeadingRule:2 -->See also</h2>
| | [[Category:Limit]] |
| <ul><li><a class="wiki_link" href="/Odd%20limit">Odd limit</a></li><li><a class="wiki_link" href="/Harmonic%20Class%20%28HC%29">Harmonic Class (HC)</a></li><li><a class="wiki_link" href="/consistency">consistency</a></li><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Limit_%28music%29" rel="nofollow">Limit (music) - Wikipedia</a> (covers also the distinction between odd-limit and prime-limit)</li><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem" rel="nofollow">Størmer's theorem - Wikipedia</a></li></ul></body></html></pre></div>
| | [[Category:Terms]] |