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Lists of edo-distinct temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
For any [[equal temperament]], there are technically an unlimited number of different [[abstract regular temperaments]] that it "[[support]]s" in the sense of tempering out all the same commas, thus allowing music in the non-equal temperament to be faithfully and mechanically played in the equal temperament.
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-06-17 21:21:54 UTC</tt>.<br>
: The original revision id was <tt>514264974</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>
<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">For any [[equal]] temperament, there are technically an unlimited number of different [[abstract regular temperament|abstract regular temperaments]] that it "supports" in the sense of tempering out all the same commas, thus allowing music in the non-equal temperament to be faithfully and mechanically played in the equal temperament.


However, some of these non-equal regular temperaments become exactly the same as each other when tuned to the equal temperament. For example, in [[12edo]], [[pajara]] and [[injera]] become the same, even though in anything other than 12 they are distinct temperaments.
However, some of these non-equal regular temperaments become exactly the same as each other when tuned to the equal temperament. For example, in [[12edo]], [[pajara]] and [[injera]] become the same, even though in anything other than 12 they are distinct temperaments.


Two temperaments are called "edo-distinct" if they remain distinct even when tuned to the same equal temperament. For rank-2, it turns out that the number of edo-distinct rank-2 temperaments in N-edo is floor(N/2); that is, N/2 if N is even and (N-1)/2 if N is odd.
Two temperaments are called '''edo-distinct''' if they remain distinct even when tuned to the same equal temperament. For rank-2, it turns out that the number of edo-distinct rank-2 temperaments in N-edo is floor(N/2); that is, N/2 if N is even and (N-1)/2 if N is odd.


=Rank-2 lists=  
== Rank-2 lists ==
|| [[List of edo-distinct 12et rank two temperaments|12]] || [[List of edo-distinct 12f rank two temperaments|12f]] ||
{| class="wikitable"
|| [[List of edo-distinct 15et rank two temperaments|15]] ||   ||
|-
|| [[List of edo-distinct 16et rank two temperaments|16]] ||   ||
| | [[List_of_edo-distinct_12et_rank_two_temperaments|12]]
|| [[List of edo-distinct 17et rank two temperaments|17]] || [[List of edo-distinct 17c rank two temperaments|17c]] ||
| | [[List_of_edo-distinct_12f_rank_two_temperaments|12f]]
|| [[List of edo-distinct 19et rank two temperaments|19]] ||   ||
|-
|| [[List of edo-distinct 22et rank two temperaments|22]] ||   ||
| | [[List_of_edo-distinct_15et_rank_two_temperaments|15]]
|| [[List of edo-distinct 24et rank two temperaments|24]] ||   ||
| |  
|| [[List of edo-distinct 26et rank two temperaments|26]] ||   ||
|-
|| [[List of edo-distinct 27et rank two temperaments|27]] || [[List of edo-distinct 27e rank two temperaments|27e]] ||
| | [[List_of_edo-distinct_16et_rank_two_temperaments|16]]
|| [[List of edo-distinct 29et rank two temperaments|29]] ||   ||
| |  
|| [[List of edo-distinct 31et rank two temperaments|31]] ||   ||
|-
|| [[List of edo-distinct 34et rank two temperaments|34]] || [[List of edo-distinct 34d rank two temperaments|34d]] ||
| | [[List_of_edo-distinct_17et_rank_two_temperaments|17]]
|| [[List of edo-distinct 41et rank two temperaments|41]] ||   ||
| | [[List_of_edo-distinct_17c_rank_two_temperaments|17c]]
|| [[List of edo-distinct 43et rank two temperaments|43]] ||   ||
|-
|| [[List of edo-distinct 46et rank two temperaments|46]] ||   ||
| | [[List_of_edo-distinct_19et_rank_two_temperaments|19]]
|| [[List of edo-distinct 50et rank two temperaments|50]] ||   ||
| |  
|| [[List of edo-distinct 53et rank two temperaments|53]] ||   ||
|-
|| [[List of edo-distinct 58et rank two temperaments|53]] ||   ||
| | [[List_of_edo-distinct_22et_rank_two_temperaments|22]]
|| [[List of edo-distinct 72et rank two temperaments|72]] ||   ||
| |  
|-
| | [[List_of_edo-distinct_24et_rank_two_temperaments|24]]
| |  
|-
| | [[List_of_edo-distinct_26et_rank_two_temperaments|26]]
| |  
|-
| | [[List_of_edo-distinct_27et_rank_two_temperaments|27]]
| | [[List_of_edo-distinct_27e_rank_two_temperaments|27e]]
|-
| | [[List_of_edo-distinct_29et_rank_two_temperaments|29]]
| |  
|-
| | [[List_of_edo-distinct_31et_rank_two_temperaments|31]]
| |  
|-
| | [[List_of_edo-distinct_34et_rank_two_temperaments|34]]
| | [[List_of_edo-distinct_34d_rank_two_temperaments|34d]]
|-
| | [[List_of_edo-distinct_41et_rank_two_temperaments|41]]
| |  
|-
| | [[List_of_edo-distinct_43et_rank_two_temperaments|43]]
| |  
|-
| | [[List_of_edo-distinct_46et_rank_two_temperaments|46]]
| |  
|-
| | [[List_of_edo-distinct_50et_rank_two_temperaments|50]]
| |  
|-
| | [[List_of_edo-distinct_53et_rank_two_temperaments|53]]
| |  
|-
| | [[List_of_edo-distinct_58et_rank_two_temperaments|58]]
| |  
|-
| | [[List_of_edo-distinct_72et_rank_two_temperaments|72]]
| |  
|}


==No fives==
=== No-fives ===
|| [[List of edo-distinct 17et no-fives rank two temperaments|17]] ||   ||</pre></div>
{| class="wikitable"
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Lists of edo-distinct temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;For any &lt;a class="wiki_link" href="/equal"&gt;equal&lt;/a&gt; temperament, there are technically an unlimited number of different &lt;a class="wiki_link" href="/abstract%20regular%20temperament"&gt;abstract regular temperaments&lt;/a&gt; that it &amp;quot;supports&amp;quot; in the sense of tempering out all the same commas, thus allowing music in the non-equal temperament to be faithfully and mechanically played in the equal temperament.&lt;br /&gt;
| | [[List_of_edo-distinct_17et_no-fives_rank_two_temperaments|17]]
&lt;br /&gt;
| |  
However, some of these non-equal regular temperaments become exactly the same as each other when tuned to the equal temperament. For example, in &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/pajara"&gt;pajara&lt;/a&gt; and &lt;a class="wiki_link" href="/injera"&gt;injera&lt;/a&gt; become the same, even though in anything other than 12 they are distinct temperaments.&lt;br /&gt;
|}
&lt;br /&gt;
Two temperaments are called &amp;quot;edo-distinct&amp;quot; if they remain distinct even when tuned to the same equal temperament. For rank-2, it turns out that the number of edo-distinct rank-2 temperaments in N-edo is floor(N/2); that is, N/2 if N is even and (N-1)/2 if N is odd.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Rank-2 lists"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Rank-2 lists&lt;/h1&gt;


&lt;table class="wiki_table"&gt;
[[Category:Lists of edo-distinct temperaments| ]] <!-- main article -->
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2012et%20rank%20two%20temperaments"&gt;12&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2012f%20rank%20two%20temperaments"&gt;12f&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2015et%20rank%20two%20temperaments"&gt;15&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2016et%20rank%20two%20temperaments"&gt;16&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2017et%20rank%20two%20temperaments"&gt;17&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2017c%20rank%20two%20temperaments"&gt;17c&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2019et%20rank%20two%20temperaments"&gt;19&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2022et%20rank%20two%20temperaments"&gt;22&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2024et%20rank%20two%20temperaments"&gt;24&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2026et%20rank%20two%20temperaments"&gt;26&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2027et%20rank%20two%20temperaments"&gt;27&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2027e%20rank%20two%20temperaments"&gt;27e&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2029et%20rank%20two%20temperaments"&gt;29&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2031et%20rank%20two%20temperaments"&gt;31&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2034et%20rank%20two%20temperaments"&gt;34&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2034d%20rank%20two%20temperaments"&gt;34d&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2041et%20rank%20two%20temperaments"&gt;41&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2043et%20rank%20two%20temperaments"&gt;43&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2046et%20rank%20two%20temperaments"&gt;46&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2050et%20rank%20two%20temperaments"&gt;50&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2053et%20rank%20two%20temperaments"&gt;53&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2058et%20rank%20two%20temperaments"&gt;53&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2072et%20rank%20two%20temperaments"&gt;72&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Rank-2 lists-No fives"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;No fives&lt;/h2&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2017et%20no-fives%20rank%20two%20temperaments"&gt;17&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;/body&gt;&lt;/html&gt;</pre></div>
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