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| <h2>IMPORTED REVISION FROM WIKISPACES</h2> | | <span style="display: block; text-align: right;">[[一貫性|日本語]]</span> |
| 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:TallKite|TallKite]] and made on <tt>2017-01-12 16:51:59 UTC</tt>.<br>
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| : The original revision id was <tt>603954266</tt>.<br>
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| : The revision comment was: <tt>I cleaned up the examples section, which confused odd limit with prime limit. I also added some links.</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>
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| <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"><span style="display: block; text-align: right;">[[一貫性|日本語]]
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| If N-edo is an [[edo|equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is //consistent// with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[Odd limit|q odd limit intervals]], consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be //q limit consistent//. If each interval in the q-limit is mapped to a unique value by N, then it said to be //uniquely q limit consistent//.
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| See also [[Minimal consistent EDOs|this list]] of odd limits, with the smallest edo that is consistent or uniquely consistent in that odd limit. And [[Consistency levels of small EDOs|this list]] of edos, with the largest odd limit that this edo is consistent or uniquely consistent in.
| | If N-edo is an [[EDO|equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is ''consistent'' with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[Odd_limit|q odd limit intervals]], consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''. |
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| ==Examples==
| | See also [[Minimal_consistent_EDOs|this list]] of odd limits, with the smallest edo that is consistent or uniquely consistent in that odd limit. And [[Consistency_levels_of_small_EDOs|this list]] of edos, with the largest odd limit that this edo is consistent or uniquely consistent in. |
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| An example for a system that is //not// consistent in a particular odd limit is [[25edo]]:
| | ==Examples== |
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| The best approximation for the interval of [[7_6|7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the [[3_2|perfect fifth 3/2]] is 15 steps. Adding the two just intervals gives 3/2 * 7/6 = [[7_4|7/4]], the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7 odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.
| | An example for a system that is ''not'' consistent in a particular odd limit is [[25edo|25edo]]: |
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| An example for a system that //is// consistent in the 7 odd-limit is [[12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. 12edo is also consistent in the 9 odd-limit, but not in the 11 odd-limit.
| | The best approximation for the interval of [[7/6|7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the [[3/2|perfect fifth 3/2]] is 15 steps. Adding the two just intervals gives 3/2 * 7/6 = [[7/4|7/4]], the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7 odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped. |
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| One notable example: [[xenharmonic/46edo|46edo]] is not consistent in the 15 odd limit. The 15:13 interval is very closer slightly closer to 9 degrees of 46edo than to 10 degrees, but the //functional// 15:13 (the difference between 46edo's versions of 15:8 and 13:8) is 10 degrees. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-//integer//-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series.
| | An example for a system that ''is'' consistent in the 7 odd-limit is [[12edo|12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. 12edo is also consistent in the 9 odd-limit, but not in the 11 odd-limit. |
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| ==Generalization to non-octave scales== | | One notable example: [[46edo|46edo]] is not consistent in the 15 odd limit. The 15:13 interval is very closer slightly closer to 9 degrees of 46edo than to 10 degrees, but the ''functional'' 15:13 (the difference between 46edo's versions of 15:8 and 13:8) is 10 degrees. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series. |
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| | ==Generalization to non-octave scales== |
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| It is possible to generalize the concept of consistency to non-edo equal temperaments. Because octaves are no longer equivalent, instead of an odd limit we must use an integer limit, and the term 2^n in the above equation is no longer present. Instead, the set S consists of all intervals u/v where u <= q >= v. | | It is possible to generalize the concept of consistency to non-edo equal temperaments. Because octaves are no longer equivalent, instead of an odd limit we must use an integer limit, and the term 2^n in the above equation is no longer present. Instead, the set S consists of all intervals u/v where u <= q >= v. |
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| This also means that the concept of octave inversion no longer applies: in this example, 13:9 is in S, but 18:13 is not. | | This also means that the concept of octave inversion no longer applies: in this example, 13:9 is in S, but 18:13 is not. |
| ==Links== | | |
| [[http://www.tonalsoft.com/enc/c/consistent.aspx|consistent (TonalSoft encyclopedia)]]</pre></div>
| | ==Links== |
| <h4>Original HTML content:</h4>
| | [http://www.tonalsoft.com/enc/c/consistent.aspx consistent (TonalSoft encyclopedia)] [[Category:edo]] |
| <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>consistent</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="/%E4%B8%80%E8%B2%AB%E6%80%A7">日本語</a><br />
| | [[Category:temperament]] |
| </span><br />
| | [[Category:term]] |
| If N-edo is an <a class="wiki_link" href="/edo">equal division of the octave</a>, and if for any interval r, N(r) is the best N-edo approximation to r, then N is <em>consistent</em> with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of <a class="wiki_link" href="/Odd%20limit">q odd limit intervals</a>, consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be <em>q limit consistent</em>. If each interval in the q-limit is mapped to a unique value by N, then it said to be <em>uniquely q limit consistent</em>.<br />
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| | [[Category:todo:reduce_mathslang]] |
| See also <a class="wiki_link" href="/Minimal%20consistent%20EDOs">this list</a> of odd limits, with the smallest edo that is consistent or uniquely consistent in that odd limit. And <a class="wiki_link" href="/Consistency%20levels%20of%20small%20EDOs">this list</a> of edos, with the largest odd limit that this edo is consistent or uniquely consistent in.<br />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h2>
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| An example for a system that is <em>not</em> consistent in a particular odd limit is <a class="wiki_link" href="/25edo">25edo</a>:<br />
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| The best approximation for the interval of <a class="wiki_link" href="/7_6">7/6</a> (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the <a class="wiki_link" href="/3_2">perfect fifth 3/2</a> is 15 steps. Adding the two just intervals gives 3/2 * 7/6 = <a class="wiki_link" href="/7_4">7/4</a>, the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7 odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.<br />
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| An example for a system that <em>is</em> consistent in the 7 odd-limit is <a class="wiki_link" href="/12edo">12edo</a>: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. 12edo is also consistent in the 9 odd-limit, but not in the 11 odd-limit.<br />
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| One notable example: <a class="wiki_link" href="http://xenharmonic.wikispaces.com/46edo">46edo</a> is not consistent in the 15 odd limit. The 15:13 interval is very closer slightly closer to 9 degrees of 46edo than to 10 degrees, but the <em>functional</em> 15:13 (the difference between 46edo's versions of 15:8 and 13:8) is 10 degrees. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-<em>integer</em>-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series.<br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Generalization to non-octave scales"></a><!-- ws:end:WikiTextHeadingRule:2 -->Generalization to non-octave scales</h2>
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| It is possible to generalize the concept of consistency to non-edo equal temperaments. Because octaves are no longer equivalent, instead of an odd limit we must use an integer limit, and the term 2^n in the above equation is no longer present. Instead, the set S consists of all intervals u/v where u &lt;= q &gt;= v.<br />
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| This also means that the concept of octave inversion no longer applies: in this example, 13:9 is in S, but 18:13 is not.<br />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Links"></a><!-- ws:end:WikiTextHeadingRule:4 -->Links</h2>
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| <a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/c/consistent.aspx" rel="nofollow">consistent (TonalSoft encyclopedia)</a></body></html></pre></div>
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日本語
If N-edo is an equal division of the octave, and if for any interval r, N(r) is the best N-edo approximation to r, then N is consistent with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of q odd limit intervals, consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be q limit consistent. If each interval in the q-limit is mapped to a unique value by N, then it said to be uniquely q limit consistent.
See also this list of odd limits, with the smallest edo that is consistent or uniquely consistent in that odd limit. And this list of edos, with the largest odd limit that this edo is consistent or uniquely consistent in.
Examples
An example for a system that is not consistent in a particular odd limit is 25edo:
The best approximation for the interval of 7/6 (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the perfect fifth 3/2 is 15 steps. Adding the two just intervals gives 3/2 * 7/6 = 7/4, the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7 odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.
An example for a system that is consistent in the 7 odd-limit is 12edo: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. 12edo is also consistent in the 9 odd-limit, but not in the 11 odd-limit.
One notable example: 46edo is not consistent in the 15 odd limit. The 15:13 interval is very closer slightly closer to 9 degrees of 46edo than to 10 degrees, but the functional 15:13 (the difference between 46edo's versions of 15:8 and 13:8) is 10 degrees. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-integer-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series.
Generalization to non-octave scales
It is possible to generalize the concept of consistency to non-edo equal temperaments. Because octaves are no longer equivalent, instead of an odd limit we must use an integer limit, and the term 2^n in the above equation is no longer present. Instead, the set S consists of all intervals u/v where u <= q >= v.
This also means that the concept of octave inversion no longer applies: in this example, 13:9 is in S, but 18:13 is not.
Links
consistent (TonalSoft encyclopedia)