Ragismic microtemperaments: Difference between revisions

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[[toc|flat]]

The ragisma is 4375/4374, the smallest 7-limit superparticular ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low cmplexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

=Ennealimmal=
Ennealimmal temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the ennealimma comma, |1 -27 18>, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is <<18 27 18 1 -22 -34||.

Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 equal, though its hardly likely anyone could tell the difference. 

If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example.) In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS. 

Commas: 2401/2400, 4375/4374

POTE generators: 36/35: 49.0205; 10/9: 182.354; 6/5: 315.687; 49/40: 350.980

Map: [<9 1 1 2|, <0 2 3 2|]
Wedgie: <<18 27 18 1 -22 -34||
EDOs: 27, 45, 72, 99, 171, 270, 441, 612
Badness: 0.00361

==11 limit hemiennealimmal==
Commas: 2401/2400, 4375/4374, 3025/3024

POTE generator: 99/98: 17.6219 or 6/5: 315.7114

Map: [<18 0 -1 22 48|, <0 2 3 2 1|]
EDOs: 72, 198, 270, 342, 612, 954, 1566
Badness: 0.00628

==13 limit hemiennealimmal==
Commas: 676/675, 1001/1000, 1716/1715, 3025/3024

POTE generator ~99/98 = 17.7504

Map: [<18 0 -1 22 48 -19|, <0 2 3 2 1 6|]
EDOs: 72, 198, 270
Badness: 0.0125

=Gamera=
Commas: 4375/4374, 589824/588245

POTE generator ~8/7 = 230.336

Map: [<1 6 10 3|, <0 -23 -40 -1|]
EDOs: 26, 73, 99, 224, 323, 422, 735
Badness: 0.0376

=Supermajor=
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of <<37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.

Commas: 4375/4374, 52734375/52706752

POTE generator: ~9/7 = 435.082

Map: [<1 15 19 30|, <0 -37 -46 -75|]
EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214
Badness: 0.0108

=Enneadecal=
Enndedecal temperament tempers out the enneadeca, |-14 -19 19>, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5 or 7 limits, and [[494edo]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.

Commas: 4375/4374, 703125/702464

POTE generator: ~3/2 = 701.880

Map: [<19 0 14 -37|, <0 1 1 3|]
Generators: 28/27, 3
EDOs: 19, 152, 171, 665, 836, 1007, 2185
Badness: 0.0110

=Abigail=
Commas: 4375/4374, 2147483648/2144153025

[[POTE tuning|POTE generator]]: 208.899

Map: [<2 7 13 -1|, <0 -11 -24 19|]
Wedgie: <<22 48 -38 25 -122 -223||
EDOs: 46, 132, 178, 224, 270, 494, 764, 1034, 1798
Badness: 0.0370

==11-limit==
Comma: 3025/3024, 4375/4374, 20614528/20588575

[[POTE tuning|POTE generator]]: 208.901

Map: [<2 7 13 -1 1|, <0 -11 -24 19 17|]
EDOs: 46, 132, 178, 224, 270, 494, 764
Badness: 0.0129

==13-limit==
Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095

[[POTE tuning|POTE generator]]: 208.903

Map: [<2 7 13 -1 1 -2|, <0 -11 -24 19 17 27|]
EDOs: 46, 178, 224, 270, 494, 764, 1258
Badness: 0.00886

=Nearly Micro=

=Octoid=
Commas: 4375/4374, 16875/16807

POTE generator: ~7/5 = 583.940

Map: [<8 1 3 3|, <0 3 4 5|]
Generators: 49/45, 7/5
EDOs: 72, 152, 224
Badness: 0.0427

==11-limit==
Commas: 540/539, 1375/1372, 4000/3993

POTE generator: ~7/5 = 583.692

Map: Map: [<8 1 3 3 16|, <0 3 4 5 3|]
EDOs: 72, 152, 224
Badness: 0.0141

==13-limit==
Commas: 540/539, 1375/1372, 4000/3993, 625/624

POTE generator: ~7/5 = 583.905

Map: Map: [<8 1 3 3 16 -21|, <0 3 4 5 3 13|]
EDOs: 72, 224
Badness: 0.0153

=Amity=
The generator for amity temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&53 temperament, or by its wedgie, <<5 13 -17 9 -41 -76||. [[99edo]] is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.

In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds.

==5-limit==
Comma: 1600000/1594323

POTE generator: ~243/200 = 339.519

Map: [<1 3 6|, <0 -5 -13|]
EDOs: 7, 39, 46, 53, 152, 205, 463, 668, 873
Badness: 0.0220

==7-limit==
Commas: 4375/4374, 5120/5103

POTE generator: ~243/200 = 339.432

Map: [<1 3 6 -2|, <0 -5 -13 17|]
EDOs: 7, 39, 46, 53, 99, 251, 350
Badness: 0.0236

==Hitchcock==
Commas: 121/120, 176/175, 2200/2187

POTE generator: ~11/9 = 339.340

Map: [<1 3 6 -2 6|, <0 -5 -13 17 -9|]
EDOs: 7, 39, 46, 53, 99
Badness: 0.0352

==Hemiamity==
Commas: 4375/4374, 5120/5103, 3025/3024

POTE generator: ~ 243/200 = 339.493

Map: [<2 1 -1 13 13|, <0 5 13 -17 -14|]
EDOs: 14, 46, 106, 152, 350

=Parakleismic=
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13>, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being <<13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie <<13 14 35 -36 ...|| adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118. 

Comma: 124440064/1220703125

POTE generator: ~6/5 = 315.240

Map: [<1 5 6|, <0 -13 -14|]
EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496
Badness: 0.0433

==7-limit==
Commas: 3136/3125, 5475/4374

POTE generator: ~6/5 = 315.181

Map: [<1 5 6 12|, <0 -13 -14 -35|]
EDOs: 19, 80, 99, 217, 316, 415
Badness: 0.0274

==11-limit==
Commas: 385/384, 3136/3125, 4375/4374

POTE generator: ~6/5 = 315.251

Map: [<1 5 6 12 -6|, <0 -13 -14 -35 36|]
EDOs: 19, 99, 118
Badness: 0.0497

Original HTML content:

<html><head><title>Ragismic microtemperaments</title></head><body><!-- ws:start:WikiTextTocRule:42:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><a href="#Ennealimmal">Ennealimmal</a><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --> | <a href="#Gamera">Gamera</a><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --> | <a href="#Supermajor">Supermajor</a><!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --> | <a href="#Enneadecal">Enneadecal</a><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --> | <a href="#Abigail">Abigail</a><!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --> | <a href="#Nearly Micro">Nearly Micro</a><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --> | <a href="#Octoid">Octoid</a><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --> | <a href="#Amity">Amity</a><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --> | <a href="#Parakleismic">Parakleismic</a><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: -->
<!-- ws:end:WikiTextTocRule:64 --><br />
The ragisma is 4375/4374, the smallest 7-limit superparticular ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word &quot;relatively&quot; should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low cmplexity, with the same caveat about &quot;relatively&quot;; however 27/25 is the period for ennealimmal.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Ennealimmal"></a><!-- ws:end:WikiTextHeadingRule:0 -->Ennealimmal</h1>
Ennealimmal temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the ennealimma comma, |1 -27 18&gt;, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is &lt;&lt;18 27 18 1 -22 -34||.<br />
<br />
Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 equal, though its hardly likely anyone could tell the difference. <br />
<br />
If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example.) In particular, people fond of the idea of &quot;tritaves&quot; as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a &quot;tritave&quot;. Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS. <br />
<br />
Commas: 2401/2400, 4375/4374<br />
<br />
POTE generators: 36/35: 49.0205; 10/9: 182.354; 6/5: 315.687; 49/40: 350.980<br />
<br />
Map: [&lt;9 1 1 2|, &lt;0 2 3 2|]<br />
Wedgie: &lt;&lt;18 27 18 1 -22 -34||<br />
EDOs: 27, 45, 72, 99, 171, 270, 441, 612<br />
Badness: 0.00361<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Ennealimmal-11 limit hemiennealimmal"></a><!-- ws:end:WikiTextHeadingRule:2 -->11 limit hemiennealimmal</h2>
Commas: 2401/2400, 4375/4374, 3025/3024<br />
<br />
POTE generator: 99/98: 17.6219 or 6/5: 315.7114<br />
<br />
Map: [&lt;18 0 -1 22 48|, &lt;0 2 3 2 1|]<br />
EDOs: 72, 198, 270, 342, 612, 954, 1566<br />
Badness: 0.00628<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Ennealimmal-13 limit hemiennealimmal"></a><!-- ws:end:WikiTextHeadingRule:4 -->13 limit hemiennealimmal</h2>
Commas: 676/675, 1001/1000, 1716/1715, 3025/3024<br />
<br />
POTE generator ~99/98 = 17.7504<br />
<br />
Map: [&lt;18 0 -1 22 48 -19|, &lt;0 2 3 2 1 6|]<br />
EDOs: 72, 198, 270<br />
Badness: 0.0125<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Gamera"></a><!-- ws:end:WikiTextHeadingRule:6 -->Gamera</h1>
Commas: 4375/4374, 589824/588245<br />
<br />
POTE generator ~8/7 = 230.336<br />
<br />
Map: [&lt;1 6 10 3|, &lt;0 -23 -40 -1|]<br />
EDOs: 26, 73, 99, 224, 323, 422, 735<br />
Badness: 0.0376<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Supermajor"></a><!-- ws:end:WikiTextHeadingRule:8 -->Supermajor</h1>
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of &lt;&lt;37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.<br />
<br />
Commas: 4375/4374, 52734375/52706752<br />
<br />
POTE generator: ~9/7 = 435.082<br />
<br />
Map: [&lt;1 15 19 30|, &lt;0 -37 -46 -75|]<br />
EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214<br />
Badness: 0.0108<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Enneadecal"></a><!-- ws:end:WikiTextHeadingRule:10 -->Enneadecal</h1>
Enndedecal temperament tempers out the enneadeca, |-14 -19 19&gt;, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of <a class="wiki_link" href="/19edo">19edo</a> up to just ones. <a class="wiki_link" href="/171edo">171edo</a> is a good tuning for either the 5 or 7 limits, and <a class="wiki_link" href="/494edo">494edo</a> shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use <a class="wiki_link" href="/665edo">665edo</a> for a tuning.<br />
<br />
Commas: 4375/4374, 703125/702464<br />
<br />
POTE generator: ~3/2 = 701.880<br />
<br />
Map: [&lt;19 0 14 -37|, &lt;0 1 1 3|]<br />
Generators: 28/27, 3<br />
EDOs: 19, 152, 171, 665, 836, 1007, 2185<br />
Badness: 0.0110<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Abigail"></a><!-- ws:end:WikiTextHeadingRule:12 -->Abigail</h1>
Commas: 4375/4374, 2147483648/2144153025<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 208.899<br />
<br />
Map: [&lt;2 7 13 -1|, &lt;0 -11 -24 19|]<br />
Wedgie: &lt;&lt;22 48 -38 25 -122 -223||<br />
EDOs: 46, 132, 178, 224, 270, 494, 764, 1034, 1798<br />
Badness: 0.0370<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Abigail-11-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->11-limit</h2>
Comma: 3025/3024, 4375/4374, 20614528/20588575<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 208.901<br />
<br />
Map: [&lt;2 7 13 -1 1|, &lt;0 -11 -24 19 17|]<br />
EDOs: 46, 132, 178, 224, 270, 494, 764<br />
Badness: 0.0129<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Abigail-13-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->13-limit</h2>
Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 208.903<br />
<br />
Map: [&lt;2 7 13 -1 1 -2|, &lt;0 -11 -24 19 17 27|]<br />
EDOs: 46, 178, 224, 270, 494, 764, 1258<br />
Badness: 0.00886<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="Nearly Micro"></a><!-- ws:end:WikiTextHeadingRule:18 -->Nearly Micro</h1>
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Octoid"></a><!-- ws:end:WikiTextHeadingRule:20 -->Octoid</h1>
Commas: 4375/4374, 16875/16807<br />
<br />
POTE generator: ~7/5 = 583.940<br />
<br />
Map: [&lt;8 1 3 3|, &lt;0 3 4 5|]<br />
Generators: 49/45, 7/5<br />
EDOs: 72, 152, 224<br />
Badness: 0.0427<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Octoid-11-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->11-limit</h2>
Commas: 540/539, 1375/1372, 4000/3993<br />
<br />
POTE generator: ~7/5 = 583.692<br />
<br />
Map: Map: [&lt;8 1 3 3 16|, &lt;0 3 4 5 3|]<br />
EDOs: 72, 152, 224<br />
Badness: 0.0141<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Octoid-13-limit"></a><!-- ws:end:WikiTextHeadingRule:24 -->13-limit</h2>
Commas: 540/539, 1375/1372, 4000/3993, 625/624<br />
<br />
POTE generator: ~7/5 = 583.905<br />
<br />
Map: Map: [&lt;8 1 3 3 16 -21|, &lt;0 3 4 5 3 13|]<br />
EDOs: 72, 224<br />
Badness: 0.0153<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Amity"></a><!-- ws:end:WikiTextHeadingRule:26 -->Amity</h1>
The generator for amity temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&amp;53 temperament, or by its wedgie, &lt;&lt;5 13 -17 9 -41 -76||. <a class="wiki_link" href="/99edo">99edo</a> is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.<br />
<br />
In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Amity-5-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->5-limit</h2>
Comma: 1600000/1594323<br />
<br />
POTE generator: ~243/200 = 339.519<br />
<br />
Map: [&lt;1 3 6|, &lt;0 -5 -13|]<br />
EDOs: 7, 39, 46, 53, 152, 205, 463, 668, 873<br />
Badness: 0.0220<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Amity-7-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->7-limit</h2>
Commas: 4375/4374, 5120/5103<br />
<br />
POTE generator: ~243/200 = 339.432<br />
<br />
Map: [&lt;1 3 6 -2|, &lt;0 -5 -13 17|]<br />
EDOs: 7, 39, 46, 53, 99, 251, 350<br />
Badness: 0.0236<br />
<br />
<!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Amity-Hitchcock"></a><!-- ws:end:WikiTextHeadingRule:32 -->Hitchcock</h2>
Commas: 121/120, 176/175, 2200/2187<br />
<br />
POTE generator: ~11/9 = 339.340<br />
<br />
Map: [&lt;1 3 6 -2 6|, &lt;0 -5 -13 17 -9|]<br />
EDOs: 7, 39, 46, 53, 99<br />
Badness: 0.0352<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Amity-Hemiamity"></a><!-- ws:end:WikiTextHeadingRule:34 -->Hemiamity</h2>
Commas: 4375/4374, 5120/5103, 3025/3024<br />
<br />
POTE generator: ~ 243/200 = 339.493<br />
<br />
Map: [&lt;2 1 -1 13 13|, &lt;0 5 13 -17 -14|]<br />
EDOs: 14, 46, 106, 152, 350<br />
<br />
<!-- ws:start:WikiTextHeadingRule:36:&lt;h1&gt; --><h1 id="toc18"><a name="Parakleismic"></a><!-- ws:end:WikiTextHeadingRule:36 -->Parakleismic</h1>
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13&gt;, with the <a class="wiki_link" href="/118edo">118edo</a> tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being &lt;&lt;13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie &lt;&lt;13 14 35 -36 ...|| adding 385/384. For the 7-limit <a class="wiki_link" href="/99edo">99edo</a> may be preferred, but in the 11-limit it is best to stick with 118. <br />
<br />
Comma: 124440064/1220703125<br />
<br />
POTE generator: ~6/5 = 315.240<br />
<br />
Map: [&lt;1 5 6|, &lt;0 -13 -14|]<br />
EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496<br />
Badness: 0.0433<br />
<br />
<!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Parakleismic-7-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->7-limit</h2>
Commas: 3136/3125, 5475/4374<br />
<br />
POTE generator: ~6/5 = 315.181<br />
<br />
Map: [&lt;1 5 6 12|, &lt;0 -13 -14 -35|]<br />
EDOs: 19, 80, 99, 217, 316, 415<br />
Badness: 0.0274<br />
<br />
<!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Parakleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->11-limit</h2>
Commas: 385/384, 3136/3125, 4375/4374<br />
<br />
POTE generator: ~6/5 = 315.251<br />
<br />
Map: [&lt;1 5 6 12 -6|, &lt;0 -13 -14 -35 36|]<br />
EDOs: 19, 99, 118<br />
Badness: 0.0497</body></html>

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-03-23 07:36:14 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-27 02:52:36 UTC</tt>.<br>
-: The original revision id was <tt>213134168</tt>.<br>
+: The original revision id was <tt>238903247</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">The ragisma is 4375/4374, the smallest 7-limit superparticular ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low cmplexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
+<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">[[toc|flat]]
-==Ennealimmal==
+The ragisma is 4375/4374, the smallest 7-limit superparticular ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low cmplexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
+=Ennealimmal=
 Ennealimmal temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the ennealimma comma, |1 -27 18&gt;, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is &lt;&lt;18 27 18 1 -22 -34||.
@@ Line 24: / Line 26: @@
 Badness: 0.00361
-===11 limit hemiennealimmal===
+==11 limit hemiennealimmal==
 Commas: 2401/2400, 4375/4374, 3025/3024
@@ Line 42: / Line 44: @@
 Badness: 0.0125
-==Gamera==
+=Gamera=
 Commas: 4375/4374, 589824/588245
@@ Line 51: / Line 53: @@
 Badness: 0.0376
-==Supermajor==
+=Supermajor=
 The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of &lt;&lt;37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.
@@ Line 62: / Line 64: @@
 Badness: 0.0108
-==Enneadecal==
+=Enneadecal=
 Enndedecal temperament tempers out the enneadeca, |-14 -19 19&gt;, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5 or 7 limits, and [[494edo]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
@@ Line 74: / Line 76: @@
 Badness: 0.0110
-==Abigail==
+=Abigail=
 Commas: 4375/4374, 2147483648/2144153025
@@ Line 84: / Line 86: @@
 Badness: 0.0370
-===11-limit===
+==11-limit==
 Comma: 3025/3024, 4375/4374, 20614528/20588575
@@ Line 93: / Line 95: @@
 Badness: 0.0129
-===13-limit===
+==13-limit==
 Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095
@@ Line 104: / Line 106: @@
 =Nearly Micro=
-==Octoid==
+=Octoid=
 Commas: 4375/4374, 16875/16807
@@ Line 114: / Line 116: @@
 Badness: 0.0427
-===11-limit===
+==11-limit==
 Commas: 540/539, 1375/1372, 4000/3993
@@ Line 123: / Line 125: @@
 Badness: 0.0141
-===13-limit===
+==13-limit==
 Commas: 540/539, 1375/1372, 4000/3993, 625/624
@@ Line 132: / Line 134: @@
 Badness: 0.0153
-==Amity==
+=Amity=
 The generator for amity temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&amp;53 temperament, or by its wedgie, &lt;&lt;5 13 -17 9 -41 -76||. [[99edo]] is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.
 In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds.
-===5-limit===
+==5-limit==
 Comma: 1600000/1594323
@@ Line 146: / Line 148: @@
 Badness: 0.0220
-===7-limit===
+==7-limit==
 Commas: 4375/4374, 5120/5103
@@ Line 155: / Line 157: @@
 Badness: 0.0236
-===Hitchcock===
+==Hitchcock==
 Commas: 121/120, 176/175, 2200/2187
@@ Line 164: / Line 166: @@
 Badness: 0.0352
-===Hemiamity===
+==Hemiamity==
 Commas: 4375/4374, 5120/5103, 3025/3024
@@ Line 172: / Line 174: @@
 EDOs: 14, 46, 106, 152, 350
-==Parakleismic==
+=Parakleismic=
 In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13&gt;, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being &lt;&lt;13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie &lt;&lt;13 14 35 -36 ...|| adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.
@@ Line 183: / Line 185: @@
 Badness: 0.0433
-===7-limit===
+==7-limit==
 Commas: 3136/3125, 5475/4374
@@ Line 192: / Line 194: @@
 Badness: 0.0274
-===11-limit===
+==11-limit==
 Commas: 385/384, 3136/3125, 4375/4374
@@ Line 201: / Line 203: @@
 Badness: 0.0497</pre></div>
 <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;Ragismic microtemperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The ragisma is 4375/4374, the smallest 7-limit superparticular ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word &amp;quot;relatively&amp;quot; should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low cmplexity, with the same caveat about &amp;quot;relatively&amp;quot;; however 27/25 is the period for ennealimmal.&lt;br /&gt;
+<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;Ragismic microtemperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:42:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;a href="#Ennealimmal"&gt;Ennealimmal&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt; | &lt;a href="#Gamera"&gt;Gamera&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt; | &lt;a href="#Supermajor"&gt;Supermajor&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt; | &lt;a href="#Enneadecal"&gt;Enneadecal&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt; | &lt;a href="#Abigail"&gt;Abigail&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt; | &lt;a href="#Nearly Micro"&gt;Nearly Micro&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt; | &lt;a href="#Octoid"&gt;Octoid&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt; | &lt;a href="#Amity"&gt;Amity&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt; | &lt;a href="#Parakleismic"&gt;Parakleismic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt;&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt;&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt;
+&lt;!-- ws:end:WikiTextTocRule:64 --&gt;&lt;br /&gt;
+The ragisma is 4375/4374, the smallest 7-limit superparticular ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word &amp;quot;relatively&amp;quot; should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low cmplexity, with the same caveat about &amp;quot;relatively&amp;quot;; however 27/25 is the period for ennealimmal.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Ennealimmal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Ennealimmal&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Ennealimmal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Ennealimmal&lt;/h1&gt;
 Ennealimmal temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the ennealimma comma, |1 -27 18&amp;gt;, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is &amp;lt;&amp;lt;18 27 18 1 -22 -34||.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 219: / Line 223: @@
 Badness: 0.00361&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Ennealimmal-11 limit hemiennealimmal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11 limit hemiennealimmal&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Ennealimmal-11 limit hemiennealimmal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11 limit hemiennealimmal&lt;/h2&gt;
 Commas: 2401/2400, 4375/4374, 3025/3024&lt;br /&gt;
 &lt;br /&gt;
@@ Line 228: / Line 232: @@
 Badness: 0.00628&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-13 limit hemiennealimmal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13 limit hemiennealimmal&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Ennealimmal-13 limit hemiennealimmal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13 limit hemiennealimmal&lt;/h2&gt;
 Commas: 676/675, 1001/1000, 1716/1715, 3025/3024&lt;br /&gt;
 &lt;br /&gt;
@@ Line 237: / Line 241: @@
 Badness: 0.0125&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Gamera"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Gamera&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Gamera"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Gamera&lt;/h1&gt;
 Commas: 4375/4374, 589824/588245&lt;br /&gt;
 &lt;br /&gt;
@@ Line 246: / Line 250: @@
 Badness: 0.0376&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-Supermajor"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Supermajor&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Supermajor"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Supermajor&lt;/h1&gt;
 The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of &amp;lt;&amp;lt;37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 257: / Line 261: @@
 Badness: 0.0108&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x-Enneadecal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Enneadecal&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Enneadecal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Enneadecal&lt;/h1&gt;
 Enndedecal temperament tempers out the enneadeca, |-14 -19 19&amp;gt;, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt; up to just ones. &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt; is a good tuning for either the 5 or 7 limits, and &lt;a class="wiki_link" href="/494edo"&gt;494edo&lt;/a&gt; shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use &lt;a class="wiki_link" href="/665edo"&gt;665edo&lt;/a&gt; for a tuning.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 269: / Line 273: @@
 Badness: 0.0110&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x-Abigail"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Abigail&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Abigail"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Abigail&lt;/h1&gt;
 Commas: 4375/4374, 2147483648/2144153025&lt;br /&gt;
 &lt;br /&gt;
@@ Line 279: / Line 283: @@
 Badness: 0.0370&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x-Abigail-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;11-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Abigail-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;11-limit&lt;/h2&gt;
 Comma: 3025/3024, 4375/4374, 20614528/20588575&lt;br /&gt;
 &lt;br /&gt;
@@ Line 288: / Line 292: @@
 Badness: 0.0129&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-Abigail-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;13-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Abigail-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;13-limit&lt;/h2&gt;
 Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095&lt;br /&gt;
 &lt;br /&gt;
@@ Line 299: / Line 303: @@
 &lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Nearly Micro"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Nearly Micro&lt;/h1&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Nearly Micro-Octoid"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Octoid&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Octoid"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Octoid&lt;/h1&gt;
 Commas: 4375/4374, 16875/16807&lt;br /&gt;
 &lt;br /&gt;
@@ Line 309: / Line 313: @@
 Badness: 0.0427&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="Nearly Micro-Octoid-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;11-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Octoid-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;11-limit&lt;/h2&gt;
 Commas: 540/539, 1375/1372, 4000/3993&lt;br /&gt;
 &lt;br /&gt;
@@ Line 318: / Line 322: @@
 Badness: 0.0141&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="Nearly Micro-Octoid-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;13-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Octoid-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;13-limit&lt;/h2&gt;
 Commas: 540/539, 1375/1372, 4000/3993, 625/624&lt;br /&gt;
 &lt;br /&gt;
@@ Line 327: / Line 331: @@
 Badness: 0.0153&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Nearly Micro-Amity"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Amity&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="Amity"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Amity&lt;/h1&gt;
 The generator for amity temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&amp;amp;53 temperament, or by its wedgie, &amp;lt;&amp;lt;5 13 -17 9 -41 -76||. &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt; is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.&lt;br /&gt;
 &lt;br /&gt;
 In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Nearly Micro-Amity-5-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;5-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Amity-5-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;5-limit&lt;/h2&gt;
 Comma: 1600000/1594323&lt;br /&gt;
 &lt;br /&gt;
@@ Line 341: / Line 345: @@
 Badness: 0.0220&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Nearly Micro-Amity-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;7-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Amity-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;7-limit&lt;/h2&gt;
 Commas: 4375/4374, 5120/5103&lt;br /&gt;
 &lt;br /&gt;
@@ Line 350: / Line 354: @@
 Badness: 0.0236&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Nearly Micro-Amity-Hitchcock"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Hitchcock&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Amity-Hitchcock"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Hitchcock&lt;/h2&gt;
 Commas: 121/120, 176/175, 2200/2187&lt;br /&gt;
 &lt;br /&gt;
@@ Line 359: / Line 363: @@
 Badness: 0.0352&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="Nearly Micro-Amity-Hemiamity"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Hemiamity&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Amity-Hemiamity"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Hemiamity&lt;/h2&gt;
 Commas: 4375/4374, 5120/5103, 3025/3024&lt;br /&gt;
 &lt;br /&gt;
@@ Line 367: / Line 371: @@
 EDOs: 14, 46, 106, 152, 350&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Nearly Micro-Parakleismic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Parakleismic&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc18"&gt;&lt;a name="Parakleismic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Parakleismic&lt;/h1&gt;
 In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13&amp;gt;, with the &lt;a class="wiki_link" href="/118edo"&gt;118edo&lt;/a&gt; tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being &amp;lt;&amp;lt;13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie &amp;lt;&amp;lt;13 14 35 -36 ...|| adding 385/384. For the 7-limit &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt; may be preferred, but in the 11-limit it is best to stick with 118. &lt;br /&gt;
 &lt;br /&gt;
@@ Line 378: / Line 382: @@
 Badness: 0.0433&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc19"&gt;&lt;a name="Nearly Micro-Parakleismic-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;7-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Parakleismic-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;7-limit&lt;/h2&gt;
 Commas: 3136/3125, 5475/4374&lt;br /&gt;
 &lt;br /&gt;
@@ Line 387: / Line 391: @@
 Badness: 0.0274&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc20"&gt;&lt;a name="Nearly Micro-Parakleismic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;11-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Parakleismic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;11-limit&lt;/h2&gt;
 Commas: 385/384, 3136/3125, 4375/4374&lt;br /&gt;
 &lt;br /&gt;

Ragismic microtemperaments: Difference between revisions

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