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Minimal consistent EDOs: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Idiosyncratic terms}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
An [[edo]] ''N'' is ''[[consistent]]'' with respect to the [[Odd limit|''q''-odd-limit]] if the closest approximations of the odd harmonics of the q-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics. It is ''[[distinctly consistent]]'' if every one of those closest approximations is a distinct value, and ''purely consistent''{{idiosyncratic}} if its [[relative interval error|relative errors]] on odd harmonics up to and including ''q'' never exceed 25%. Below is a table of the smallest consistent, and the smallest distinctly consistent, edo for every odd number up to 135. Odd limits of {{nowrap|2<sup>''n''</sup> &minus; 1}} are '''highlighted'''.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2017-01-06 12:32:26 UTC</tt>.<br>
: The original revision id was <tt>603198706</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">An e is //consistent// with respect to a set of rational numbers s if the [[patent val]] mapping of every element of s is the nearest N-edo approximation. It is //uniquely consistent// if every element of s is mapped to a unique value. If the set s is the q [[odd limit]], we say N is q-limit consistent and q-limit uniquely consistent, respectively. Below is a table of the least constent, and least uniquely consistent, edo for every odd number up to 135.


|| Odd limit || Smallest consistent || Smallest uniquely consistent ||
<onlyinclude>{| class="wikitable center-all"
|| 1 || 1 || 1 ||
|+ style="font-size: 105%;" | Smallest consistent EDOs per odd limit
|| 3 || 1 || 3 ||
|-
|| 5 || 3 || 9 ||
! Odd<br>limit !! Smallest<br>consistent edo* !! Smallest distinctly<br>consistent edo !! Smallest purely<br>consistent edo* !! Smallest edo<br>consistent to<br>[[Consistency #Generalization|distance 2]]* !! Smallest edo<br>distinctly consistent<br>to distance 2
|| 7 || 4 || 27 ||
|- style="font-weight: bold; background-color: #dddddd;"
|| 9 || 5 || 41 ||
| 1 || 1 || 1 || 1 || 1 || 1
|| 11 || 22 || 58 ||
|- style="font-weight: bold; background-color: #dddddd;"
|| 13 || 26 || 87 ||
| 3 || 1 || 3 || 2 || 2 || 3
|| 15 || 29 || 111 ||
|-
|| 17 || 58 || 149 ||
| 5 || 3 || 9 || 3 || 3 || 12
|| 19 || 80 || 217 ||
|- style="font-weight: bold; background-color: #dddddd;"
|| 21 || 94 || 282 ||
| 7 || 4 || 27 || 10 || 31 || 31
|| 23 || 94 || 282 ||
|-
|| 25 || 282 || 388 ||
| 9 || 5 || 41 || 41 || 41 || 41
|| 27 || 282 || 388 ||
|-
|| 29 || 282 || 1323 ||
| 11 || 22 || 58 || 41 || 72 || 72
|| 31 || 311 || 1600 ||
|-
|| 33 || 311 || 1600 ||
| 13 || 26 || 87 || 46 || 270 || 270
|| 35 || 311 || 1600 ||
|- style="font-weight: bold; background-color: #dddddd;"
|| 37 || 311 || 1600 ||
| 15 || 29 || 111 || 87 || 494 || 494
|| 39 || 311 || 2554 ||
|-
|| 41 || 311 || 2554 ||
| 17 || 58 || 149 || 311 || 3395 || 3395
|| 43 ||17461 || 17461 ||
|-
|| 45 || 17461 || 17461 ||
| 19 || 80 || 217 || 311 || 8539 || 8539
|| 47 || 20567 || 20567 ||
|-
|| 49 || 20567 || 20567 ||
| 21 || 94 || 282 || 311 || 8539 || 8539
|| 51 || 20567 || 20567 ||
|-
|| 53 || 20567 || 20567 ||
| 23 || 94 || 282 || 311 || 16808 || 16808
|| 55 || 20567 || 20567 ||
|-
|| 57 || 20567 || 20567 ||
| 25 || 282 || 388 || 311 || 16808 || 16808
|| 59 || 253389 || 253389 ||
|-
|| 61 || 625534 || 625534 ||
| 27 || 282 || 388 || 311 || 16808 || 16808
|| 63 || 625534 || 625534 ||
|-
|| 65 || 625534 || 625534 ||
| 29 || 282 || 1323 || 311 || 16808 || 16808
|| 67 || 625534 || 625534 ||
|- style="font-weight: bold; background-color: #dddddd;"
|| 69 || 759630 || 759630 ||
| 31 || 311 || 1600 || 311 || 16808 || 16808
|| 71 || 759630 || 759630 ||
|-
|| 73 || 759630 || 759630 ||
| 33 || 311 || 1600 || 311 || 16808 || 16808
|| 75 || 2157429 || 2157429 ||
|-
|| 77 || 2157429 || 2157429 ||
| 35 || 311 || 1600 || 311 || 16808 || 16808
|| 79 || 2901533 || 2901533 ||
|-
|| 81 || 2901533 || 2901533 ||
| 37 || 311 || 1600 || 311 || 324296 || 324296
|| 83 || 2901533 || 2901533 ||
|-
|| 85 || 2901533 || 2901533 ||
| 39 || 311 || 2554 || 311 || 2398629 || 2398629
|| 87 || 2901533 || 2901533 ||
|-
|| 91 || 2901533 || 2901533 ||
| 41 || 311 || 2554 || 311 || 19164767 || 19164767
|| 93 || 2901533 || 2901533 ||
|-
|| 95 || 2901533 || 2901533 ||
| 43 || 17461 || 17461 || 20567 || 19735901 || 19735901
|| 97 || 2901533 || 2901533 ||
|-
|| 99 || 2901533 || 2901533 ||
| 45 || 17461 || 17461 || 20567 || 19735901 || 19735901
|| 101 || 2901533 || 2901533 ||
|-
|| 103 || 2901533 || 2901533 ||
| 47 || 20567 || 20567 || 20567 || 152797015 || 152797015
|| 105 || 2901533 || 2901533 ||
|-
|| 107 || 2901533 || 2901533 ||
| 49 || 20567 || 20567 || 459944 ||  ||  
|| 109 || 2901533 || 2901533 ||
|-
|| 111 || 2901533 || 2901533 ||
| 51 || 20567 || 20567 || 459944 ||  ||  
|| 113 || 2901533 || 2901533 ||
|-
|| 115 || 2901533 || 2901533 ||
| 53 || 20567 || 20567 || 1705229 ||  ||  
|| 117 || 2901533 || 2901533 ||
|-
|| 119 || 2901533 || 2901533 ||
| 55 || 20567 || 20567 || 1705229 ||  ||  
|| 121 || 2901533 || 2901533 ||
|-
|| 123 || 2901533 || 2901533 ||
| 57 || 20567 || 20567 || 1705229 ||  ||  
|| 125 || 2901533 || 2901533 ||
|-
|| 127 || 2901533 || 2901533 ||
| 59 || 253389 || 253389 || 3159811 ||  ||  
|| 129 || 2901533 || 2901533 ||
|-
|| 131 || 2901533 || 2901533 ||
| 61 || 625534 || 625534 || 3159811 ||  ||  
|| 133 || 70910024.|| 70910024.||
|- style="font-weight: bold; background-color: #dddddd;"
|| 135 || 70910024.|| 70910024.||
| 63 || 625534 || 625534 || 3159811 ||  ||  
</pre></div>
|-
<h4>Original HTML content:</h4>
| 65 || 625534 || 625534 || 3159811 ||  ||  
<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;Minimal consistent EDOs&lt;/title&gt;&lt;/head&gt;&lt;body&gt;An e is &lt;em&gt;consistent&lt;/em&gt; with respect to a set of rational numbers s if the &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; mapping of every element of s is the nearest N-edo approximation. It is &lt;em&gt;uniquely consistent&lt;/em&gt; if every element of s is mapped to a unique value. If the set s is the q &lt;a class="wiki_link" href="/odd%20limit"&gt;odd limit&lt;/a&gt;, we say N is q-limit consistent and q-limit uniquely consistent, respectively. Below is a table of the least constent,  and least uniquely consistent, edo for every odd number up to 135.&lt;br /&gt;
|-
&lt;br /&gt;
| 67 || 625534 || 625534 || 7317929 ||  ||  
|-
| 69 || 759630 || 759630 || 8595351 ||  ||  
|-
| 71 || 759630 || 759630 || 8595351 ||  ||  
|-
| 73 || 759630 || 759630 || 27783092 ||  ||  
|-
| 75 || 2157429 || 2157429 || 34531581 ||  ||  
|-
| 77 || 2157429 || 2157429 || 34531581 ||  ||  
|-
| 79 || 2901533 || 2901533 || 50203972 ||  ||  
|-
| 81 || 2901533 || 2901533 || 50203972 ||  ||  
|-
| 83 || 2901533 || 2901533 || 50203972 ||  ||  
|-
| 85 || 2901533 || 2901533 || 50203972 ||  ||  
|-
| 87 || 2901533 || 2901533 || 50203972 ||  ||
|-
| 89 || 2901533 || 2901533 || 50203972 ||  ||
|-
| 91 || 2901533 || 2901533 || 50203972 ||  ||  
|-
| 93 || 2901533 || 2901533 || 50203972 ||  ||  
|-
| 95 || 2901533 || 2901533 || 50203972 ||  ||  
|-
| 97 || 2901533 || 2901533 || 1297643131 ||  ||  
|-
| 99 || 2901533 || 2901533 || 1297643131 ||  ||  
|-
| 101 || 2901533 || 2901533 || 3888109922 ||  ||  
|-
| 103 || 2901533 || 2901533 || 3888109922 ||  ||  
|-
| 105 || 2901533 || 2901533 || 3888109922 ||  ||  
|-
| 107 || 2901533 || 2901533 || 13805152233 ||  ||  
|-
| 109 || 2901533 || 2901533 || 27218556026 ||  ||  
|-
| 111 || 2901533 || 2901533 || 27218556026 ||  ||  
|-
| 113 || 2901533 || 2901533 || 27218556026 ||  ||  
|-
| 115 || 2901533 || 2901533 || 27218556026 ||  ||  
|-
| 117 || 2901533 || 2901533 || 27218556026 ||  ||  
|-
| 119 || 2901533 || 2901533 || 42586208631 ||  ||  
|-
| 121 || 2901533 || 2901533 || 42586208631 ||  ||  
|-
| 123 || 2901533 || 2901533 || 42586208631 ||  ||  
|-
| 125 || 2901533 || 2901533 || 42586208631 ||  ||  
|- style="font-weight: bold; background-color: #dddddd;"
| 127 || 2901533 || 2901533 || 42586208631 ||  ||  
|-
| 129 || 2901533 || 2901533 || 42586208631 ||  ||  
|-
| 131 || 2901533 || 2901533 || 93678217813** ||  ||  
|-
| 133 || 70910024 || 70910024 || 93678217813 ||  ||  
|-
| 135 || 70910024 || 70910024 || 93678217813 ||  ||
|}
<nowiki />* Apart from 0edo


<nowiki />** Purely consistent to the 137-odd-limit</onlyinclude>


&lt;table class="wiki_table"&gt;
The last entry, 70910024edo, is consistent up to the 135-odd-limit. The next edo is [[5407372813edo|5407372813]], reported to be consistent to the 155-odd-limit.
    &lt;tr&gt;
        &lt;td&gt;Odd limit&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Smallest consistent&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Smallest uniquely consistent&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;41&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;58&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;26&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;87&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;29&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;111&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;58&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;149&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;19&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;80&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;217&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;94&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;282&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;23&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;94&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;282&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;282&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;388&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;282&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;388&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;29&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;282&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1323&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;31&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;311&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1600&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;33&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;311&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1600&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;35&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;311&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1600&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;37&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;311&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1600&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;39&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;311&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2554&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;41&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;311&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2554&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;43&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;17461&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;17461&lt;br /&gt;
&lt;/td&gt;
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    &lt;tr&gt;
        &lt;td&gt;45&lt;br /&gt;
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        &lt;td&gt;17461&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;17461&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;47&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20567&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20567&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;49&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20567&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20567&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;51&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20567&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20567&lt;br /&gt;
&lt;/td&gt;
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    &lt;tr&gt;
        &lt;td&gt;53&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20567&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20567&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;55&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20567&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20567&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;57&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20567&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20567&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;59&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;253389&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;253389&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;61&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;625534&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;625534&lt;br /&gt;
&lt;/td&gt;
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    &lt;tr&gt;
        &lt;td&gt;63&lt;br /&gt;
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        &lt;td&gt;625534&lt;br /&gt;
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        &lt;td&gt;625534&lt;br /&gt;
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        &lt;td&gt;67&lt;br /&gt;
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        &lt;td&gt;625534&lt;br /&gt;
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        &lt;td&gt;625534&lt;br /&gt;
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        &lt;td&gt;69&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;759630&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;759630&lt;br /&gt;
&lt;/td&gt;
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        &lt;td&gt;71&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;759630&lt;br /&gt;
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        &lt;td&gt;759630&lt;br /&gt;
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    &lt;tr&gt;
        &lt;td&gt;73&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;759630&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;759630&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;75&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2157429&lt;br /&gt;
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&lt;/body&gt;&lt;/html&gt;</pre></div>
== OEIS integer sequences links ==
* {{OEIS|A116474|Equal divisions of the octave with progressively increasing consistency levels}}
* {{OEIS|A116475|Equal divisions of the octave with progressively increasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit}}
* {{OEIS|A117577|Equal divisions of the octave with nondecreasing consistency levels.}}
* {{OEIS|A117578|Equal divisions of the octave with nondecreasing consistency limits and distinct approximations for all the ratios in the tonality diamond of that limit}}
 
== See also ==
* [[Consistency limits of small EDOs]]
* {{u|ArrowHead294|Purely consistent EDOs by odd limit}}
 
[[Category:Mapping]]
[[Category:Consistency]]
[[Category:Odd limit]]
Retrieved from "https://en.xen.wiki/w/Minimal_consistent_EDOs"