Consistency: Difference between revisions
Wikispaces>MasonGreen1 **Imported revision 570782573 - Original comment: ** |
Wikispaces>TallKite **Imported revision 603954266 - Original comment: I cleaned up the examples section, which confused odd limit with prime limit. I also added some links.** |
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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> | ||
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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//. | 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//. | ||
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. | |||
==Examples== | ==Examples== | ||
An example for a system that is //not// consistent in a particular odd limit is [[25edo]]: | 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|7/6]] (the septimal subminor third) in 25edo is 6 steps, the best approximation for the [[3_2|perfect fifth 3/2]] is 15 steps. | 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. | ||
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 | |||
An example for a system that //is// consistent in the | 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. | ||
==Generalization== | 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. | ||
==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. | 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. | 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== | ==Links== | ||
[[http://www.tonalsoft.com/enc/c/consistent.aspx|consistent (TonalSoft encyclopedia)]]</pre></div> | [[http://www.tonalsoft.com/enc/c/consistent.aspx|consistent (TonalSoft encyclopedia)]]</pre></div> | ||
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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 /> | 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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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> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h2> | ||
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 /> | <br /> | ||
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 /> | |||
<br /> | <br /> | ||
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, the best approximation for the <a class="wiki_link" href="/3_2">perfect fifth 3/2</a> is 15 steps. | 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 /> | ||
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 | |||
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An example for a system that <em>is</em> consistent in the | 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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<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Generalization"></a><!-- ws:end:WikiTextHeadingRule:2 -->Generalization</h2> | 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 /> | 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 /> | ||
<br /> | <br /> | ||
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 /> | 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 /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Links"></a><!-- ws:end:WikiTextHeadingRule:4 -->Links</h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Links"></a><!-- ws:end:WikiTextHeadingRule:4 -->Links</h2> | ||
<a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/c/consistent.aspx" rel="nofollow">consistent (TonalSoft encyclopedia)</a></body></html></pre></div> | <a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/c/consistent.aspx" rel="nofollow">consistent (TonalSoft encyclopedia)</a></body></html></pre></div> | ||