The Biosphere: Difference between revisions

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The biosphere is the name given to the collection of temperaments that are children of or related to **//biome temperament//**, the rank 3 2.3.7.13/5 subgroup temperament eliminating 91/90. The term "biome" loosely means "ecosystem" or "climate." This temperament is so named because temperaments that arise from eliminating 91/90 can evoke synesthetic associations of different "natural" settings, some very familiar and some much less so.

The next low-numbered triad after 4:5:6 with a 3/2 on the outside is 6:7:9, but its inversion, 14:18:21, can sound extremely dissonant. On the other hand, you also have 10:13:15, which is another standout triad of low complexity with a fifth on the outside, but its inversion, 26:30:39, is also relatively complex. Tempering out 91/90 makes both of these problems disappear by connecting the two together, such that the utonal inverse of 6:7:9 becomes 10:13:15. Hence, you end up with a tonal system that relates and connects two of the most xenharmonic triads in existence (at least those with 3/2 on the outside). 91/90 tempering thus enriches septimal harmony in this way.

The rank-3 biome temperament is of particular theoretical interest because it generates a rank-3 lattice that is analogous to the 5-limit JI lattice. As 5-limit JI is the basis for which all 5-limit linear temperaments are derived, the rank-3 biome temperament can serve as a basis to derive useful 2.3.7.13/5 linear temperaments. Instead of our base triads being 4:5:6 and its utonal inversion 10:12:15, we instead treat 6:7:9 and its utonal inversion 10:13:15 as fundamental to the system. The three dimensions of the system can be thought of as 2/1, 3/2, and 7/6 (or 9/7, or 13/10). 46-EDO is a great tuning for biome, giving nearly-pure harmonies all around, somewhat analogous to the accuracy of 34-EDO or 53-EDO in approximating 5-limit JI.

This lattice can also be extended to deal with "higher primes," as can 5-limit JI, but instead by expanding the subgroup outward from the center, so that the "higher primes" we look at are things like like 5, 11, and 13. However, it may prove more useful at first to think purely within the 2.3.7.13/5 subgroup, so as to first come to understand the xenharmonic possibilities of the system.

=**Biome Temperament**= 
Comma: 91/90

Map
I have no idea

EDOs: 46 and some other stuff

=[[#Rank two temperaments]]Rank two temperaments= 
==[[#Rank two temperaments-Decitonic]]//Oceanfront//== 
Subgroup: 2.3.7.13/5
Commas: 91/90, 64/63

[[POTE tuning|POTE generator]]: ~4/3 = 486.090 (I think)

Map: [<1 2 2 3|, < 0 -1 2 -4|]
EDOs: 27,32
Badness: I have no idea

Oceanfront is very similar to the familiar 7-limit superpyth temperament, in which 16/9 is equated with 7/4, 32/27 equated with 7/6, and 81/64 with 9/7. Oceanfront aims to equate 81/64 with 13/10 instead, however, so the fifths are even sharper than those of superpyth - 713.910 cents is the optimal POTE generator. The general structure of this scale is similar to that of meantone[7], except that the "major" triads in this scale are 10:13:15, and the minor triads are 6:7:9.

The sharp fifths of this scale can be a little more dissonant than meantone ears are used to, as can the flat fifths of something like mavila. This scale is very much like a brighter cousin of mavila in that regard.

11-limit: TBD
13-limit: TBD

===[[#Rank two temperaments-Decitonic]]Ultrapyth=== 
Subgroup: 2.3.5.7.13
Commas: 91/90, 64/63, ???? (insert best 5-limit comma here to create an analogous system to superpyth)

[[POTE tuning|POTE generator]]: ~4/3 = ?

Map: TBD
EDOs: TBD
Badness: TBD

This is a placeholder for the future "Ultrapyth" temperament, which will extend superpyth as you'd expect. If the best way to do this is the same as "porcupinefish" below, then we'll come up with something else.

===[[#Rank two temperaments-Decitonic]]Porcupinefish=== 
Subgroup: 13-limit
Commas: 91/90, 64/63, 250/243, 121/120

[[POTE tuning|POTE generator]]: ~10/9 = 162.474 (I think)

Map: [<1 2 3 2 -1 1|, <0 -3 -5 6 33 20|]
EDOs: 37, 59
Badness: I have no idea

Porcupinefish is the 13-limit extension of porcupine that you get by adding 91/90 to the usual mix of porcupine temperaments. Its name is derived from that it is a combination of the porcupine and oceanfront temperaments.


==[[#Rank two temperaments-Decitonic]]//Oceanfront//== 
Subgroup: 2.3.7.13/5
Commas: 91/90, 64/63

[[POTE tuning|POTE generator]]: ~4/3 = 486.090 (I think)

Map: [<1 2 2 3|, < 0 -1 2 -4|]
EDOs: 27,32
Badness: I have no idea

Oceanfront is very similar to the familiar 7-limit superpyth temperament, in which 16/9 is equated with 7/4, 32/27 equated with 7/6, and 81/64 with 9/7. Oceanfront aims to equate 81/64 with 13/10 instead, however, so the fifths are even sharper than those of superpyth - 713.910 cents is the optimal POTE generator. The general structure of this scale is similar to that of meantone[7], except that the "major" triads in this scale are 10:13:15, and the minor triads are 6:7:9.

The sharp fifths of this scale can be a little more dissonant than meantone ears are used to, as can the flat fifths of something like mavila. This scale is very much like a brighter cousin of mavila in that regard.

11-limit: TBD
13-limit: TBD

Original HTML content:

<html><head><title>The Biosphere</title></head><body>The biosphere is the name given to the collection of temperaments that are children of or related to <strong><em>biome temperament</em></strong>, the rank 3 2.3.7.13/5 subgroup temperament eliminating 91/90. The term &quot;biome&quot; loosely means &quot;ecosystem&quot; or &quot;climate.&quot; This temperament is so named because temperaments that arise from eliminating 91/90 can evoke synesthetic associations of different &quot;natural&quot; settings, some very familiar and some much less so.<br />
<br />
The next low-numbered triad after 4:5:6 with a 3/2 on the outside is 6:7:9, but its inversion, 14:18:21, can sound extremely dissonant. On the other hand, you also have 10:13:15, which is another standout triad of low complexity with a fifth on the outside, but its inversion, 26:30:39, is also relatively complex. Tempering out 91/90 makes both of these problems disappear by connecting the two together, such that the utonal inverse of 6:7:9 becomes 10:13:15. Hence, you end up with a tonal system that relates and connects two of the most xenharmonic triads in existence (at least those with 3/2 on the outside). 91/90 tempering thus enriches septimal harmony in this way.<br />
<br />
The rank-3 biome temperament is of particular theoretical interest because it generates a rank-3 lattice that is analogous to the 5-limit JI lattice. As 5-limit JI is the basis for which all 5-limit linear temperaments are derived, the rank-3 biome temperament can serve as a basis to derive useful 2.3.7.13/5 linear temperaments. Instead of our base triads being 4:5:6 and its utonal inversion 10:12:15, we instead treat 6:7:9 and its utonal inversion 10:13:15 as fundamental to the system. The three dimensions of the system can be thought of as 2/1, 3/2, and 7/6 (or 9/7, or 13/10). 46-EDO is a great tuning for biome, giving nearly-pure harmonies all around, somewhat analogous to the accuracy of 34-EDO or 53-EDO in approximating 5-limit JI.<br />
<br />
This lattice can also be extended to deal with &quot;higher primes,&quot; as can 5-limit JI, but instead by expanding the subgroup outward from the center, so that the &quot;higher primes&quot; we look at are things like like 5, 11, and 13. However, it may prove more useful at first to think purely within the 2.3.7.13/5 subgroup, so as to first come to understand the xenharmonic possibilities of the system.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Biome Temperament"></a><!-- ws:end:WikiTextHeadingRule:0 --><strong>Biome Temperament</strong></h1>
 Comma: 91/90<br />
<br />
Map<br />
I have no idea<br />
<br />
EDOs: 46 and some other stuff<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 --><!-- ws:start:WikiTextAnchorRule:12:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Rank two temperaments&quot; title=&quot;Anchor: Rank two temperaments&quot;/&gt; --><a name="Rank two temperaments"></a><!-- ws:end:WikiTextAnchorRule:12 -->Rank two temperaments</h1>
 <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Rank two temperaments-Oceanfront"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule:13:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Rank two temperaments-Decitonic&quot; title=&quot;Anchor: Rank two temperaments-Decitonic&quot;/&gt; --><a name="Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextAnchorRule:13 --><em>Oceanfront</em></h2>
 Subgroup: 2.3.7.13/5<br />
Commas: 91/90, 64/63<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~4/3 = 486.090 (I think)<br />
<br />
Map: [&lt;1 2 2 3|, &lt; 0 -1 2 -4|]<br />
EDOs: 27,32<br />
Badness: I have no idea<br />
<br />
Oceanfront is very similar to the familiar 7-limit superpyth temperament, in which 16/9 is equated with 7/4, 32/27 equated with 7/6, and 81/64 with 9/7. Oceanfront aims to equate 81/64 with 13/10 instead, however, so the fifths are even sharper than those of superpyth - 713.910 cents is the optimal POTE generator. The general structure of this scale is similar to that of meantone[7], except that the &quot;major&quot; triads in this scale are 10:13:15, and the minor triads are 6:7:9.<br />
<br />
The sharp fifths of this scale can be a little more dissonant than meantone ears are used to, as can the flat fifths of something like mavila. This scale is very much like a brighter cousin of mavila in that regard.<br />
<br />
11-limit: TBD<br />
13-limit: TBD<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Rank two temperaments-Oceanfront-Ultrapyth"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:14:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Rank two temperaments-Decitonic&quot; title=&quot;Anchor: Rank two temperaments-Decitonic&quot;/&gt; --><a name="Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextAnchorRule:14 -->Ultrapyth</h3>
 Subgroup: 2.3.5.7.13<br />
Commas: 91/90, 64/63, ???? (insert best 5-limit comma here to create an analogous system to superpyth)<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~4/3 = ?<br />
<br />
Map: TBD<br />
EDOs: TBD<br />
Badness: TBD<br />
<br />
This is a placeholder for the future &quot;Ultrapyth&quot; temperament, which will extend superpyth as you'd expect. If the best way to do this is the same as &quot;porcupinefish&quot; below, then we'll come up with something else.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Rank two temperaments-Oceanfront-Porcupinefish"></a><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextAnchorRule:15:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Rank two temperaments-Decitonic&quot; title=&quot;Anchor: Rank two temperaments-Decitonic&quot;/&gt; --><a name="Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextAnchorRule:15 -->Porcupinefish</h3>
 Subgroup: 13-limit<br />
Commas: 91/90, 64/63, 250/243, 121/120<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~10/9 = 162.474 (I think)<br />
<br />
Map: [&lt;1 2 3 2 -1 1|, &lt;0 -3 -5 6 33 20|]<br />
EDOs: 37, 59<br />
Badness: I have no idea<br />
<br />
Porcupinefish is the 13-limit extension of porcupine that you get by adding 91/90 to the usual mix of porcupine temperaments. Its name is derived from that it is a combination of the porcupine and oceanfront temperaments.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Rank two temperaments-Oceanfront"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- ws:start:WikiTextAnchorRule:16:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Rank two temperaments-Decitonic&quot; title=&quot;Anchor: Rank two temperaments-Decitonic&quot;/&gt; --><a name="Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextAnchorRule:16 --><em>Oceanfront</em></h2>
 Subgroup: 2.3.7.13/5<br />
Commas: 91/90, 64/63<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~4/3 = 486.090 (I think)<br />
<br />
Map: [&lt;1 2 2 3|, &lt; 0 -1 2 -4|]<br />
EDOs: 27,32<br />
Badness: I have no idea<br />
<br />
Oceanfront is very similar to the familiar 7-limit superpyth temperament, in which 16/9 is equated with 7/4, 32/27 equated with 7/6, and 81/64 with 9/7. Oceanfront aims to equate 81/64 with 13/10 instead, however, so the fifths are even sharper than those of superpyth - 713.910 cents is the optimal POTE generator. The general structure of this scale is similar to that of meantone[7], except that the &quot;major&quot; triads in this scale are 10:13:15, and the minor triads are 6:7:9.<br />
<br />
The sharp fifths of this scale can be a little more dissonant than meantone ears are used to, as can the flat fifths of something like mavila. This scale is very much like a brighter cousin of mavila in that regard.<br />
<br />
11-limit: TBD<br />
13-limit: TBD</body></html>

@@ Line 1: / Line 1: @@
 <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:mbattaglia1|mbattaglia1]] and made on <tt>2011-04-24 19:40:59 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-04-24 20:38:58 UTC</tt>.<br>
-: The original revision id was <tt>222566636</tt>.<br>
+: The original revision id was <tt>222575758</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">&lt;span style="display: block; display: none;"&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;white-space: pre-wrap ! important" class="old-revision-html">The biosphere is the name given to the collection of temperaments that are children of or related to **//biome temperament//**, the rank 3 2.3.7.13/5 subgroup temperament eliminating 91/90. The term "biome" loosely means "ecosystem" or "climate." This temperament is so named because temperaments that arise from eliminating 91/90 can evoke synesthetic associations of different "natural" settings, some very familiar and some much less so.
-||~ Details || last edit by &lt;span style="outline: medium none;"&gt;[[image:http://www.wikispaces.com/user/pic/1202793136/genewardsmith-sm.jpg width="16" height="16" caption="genewardsmith" link="http://www.wikispaces.com/user/view/genewardsmith"]]&lt;/span&gt; &lt;span style="outline: medium none;"&gt;[[http://www.wikispaces.com/user/view/genewardsmith|genewardsmith]]&lt;/span&gt; [[page/diff/The Archipelago/219253274|Apr 11, 2011 3:26 pm]] - [[page/history/the Archipelago|36 revisions]] || [[image:http://www.wikispaces.com/i/w/W_close.gif caption="hide details" link="the Archipelago#"]] ||
-||~ Tags || * [[tag/view/listen|listen]]
-* [[tag/view/theory|theory]]
-[[the Archipelago#|edit]]
-listentheory
-Save[[the Archipelago#|Cancel]] ||
-&lt;/span&gt;
-The biosphere is the name given to the collection of temperaments that are children of or related to **//biome temperament//**, the rank 3 2.3.7.13/10 subgroup temperament eliminating 91/90. The term "biome" loosely means "ecosystem" or "climate." This temperament is so named because temperaments that arise from eliminating 91/90 can evoke synesthetic associations of different "natural" settings, some very familiar and some much less so.
-This temperament makes the utonal inverse of 6:7:9 out to be 10:13:15, whereas it is normally the much more complex (and arguably more dissonant) 14:18:21. Eliminating 91/90 thus enriches septimal harmony by increasing the concordance of "utonal" septimal triads such as 6:7:9 or tetrads such as 4:6:7:9.
+The next low-numbered triad after 4:5:6 with a 3/2 on the outside is 6:7:9, but its inversion, 14:18:21, can sound extremely dissonant. On the other hand, you also have 10:13:15, which is another standout triad of low complexity with a fifth on the outside, but its inversion, 26:30:39, is also relatively complex. Tempering out 91/90 makes both of these problems disappear by connecting the two together, such that the utonal inverse of 6:7:9 becomes 10:13:15. Hence, you end up with a tonal system that relates and connects two of the most xenharmonic triads in existence (at least those with 3/2 on the outside). 91/90 tempering thus enriches septimal harmony in this way.
-Biome temperament, from which is of particular theoretical interest because it generates a rank-3 lattice that is analogous to the 5-limit JI lattice.
+The rank-3 biome temperament is of particular theoretical interest because it generates a rank-3 lattice that is analogous to the 5-limit JI lattice. As 5-limit JI is the basis for which all 5-limit linear temperaments are derived, the rank-3 biome temperament can serve as a basis to derive useful 2.3.7.13/5 linear temperaments. Instead of our base triads being 4:5:6 and its utonal inversion 10:12:15, we instead treat 6:7:9 and its utonal inversion 10:13:15 as fundamental to the system. The three dimensions of the system can be thought of as 2/1, 3/2, and 7/6 (or 9/7, or 13/10). 46-EDO is a great tuning for biome, giving nearly-pure harmonies all around, somewhat analogous to the accuracy of 34-EDO or 53-EDO in approximating 5-limit JI.
-The barbados triad is of particular theoretical interest because, when reduced to lowest terms, it is the 10:13:15 triad. Thus, this triad is only slightly higher in complexity than the 5-limit 10:12:15 minor triad, which means it may be of distinct value as a relatively unexplored musical consonance. It is one of only a few low-complexity triads with a 3/2 on the outer dyad, some others being 4:5:6, 6:7:9, and 10:12:15. It works out to 0-454-702 cents, which means that it is an //ultramajor// triad, with a third sharper even than the 9/7 supermajor third.
+This lattice can also be extended to deal with "higher primes," as can 5-limit JI, but instead by expanding the subgroup outward from the center, so that the "higher primes" we look at are things like like 5, 11, and 13. However, it may prove more useful at first to think purely within the 2.3.7.13/5 subgroup, so as to first come to understand the xenharmonic possibilities of the system.
-Compared to the 7-limit 14:18:21 supermajor triad, 10:13:15 is lower in triadic complexity (10:13:15 vs 14:18:21), but contains dyads that are on average higher in complexity (9/7 vs 13/10 and 7/6 vs 15/13). Its inverse, however, is the ultraminor 26:30:39, which is far more complex than the 7-limit subminor 6:7:9. Temperaments in which 91/90 vanishes equate the two types of triads.
+=**Biome Temperament**=
+Comma: 91/90
-[[24edo]] approximates this triad to within an error of four cents, and [[29edo]] does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below.
-Comma: 676/675
 Map
-&lt;1 0 0 0 0 -1|
+I have no idea
-&lt;0 2 0 0 0 3|
-&lt;0 0 1 0 0 1|
-&lt;0 0 0 1 0 0|
-&lt;0 0 0 0 1 0|
-EDOs: 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940
-[[Optimal patent val]]: [[940edo]]
-=[[#Rank four temperaments]]Rank four temperaments=
-==[[#Rank four temperaments-1001/1000]]1001/1000==
-Commas: 676/675, 1001/1000
-EDOs: 15, 19, 29, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940
-[[Optimal patent val]]: [[940edo]]
-==[[#Rank four temperaments-49/48]]49/48==
-Commas: 49/48, 91/90
-==[[#Rank four temperaments-1716/1715]]1716/1715==
-Commas: 676/675, 1716/1715
-==[[#Rank four temperaments-364/363]]364/363==
-Commas: 364/363, 676/675
-===[[#Rank four temperaments-364/363-351/350]]351/350===
-Commas: 351/350, 676/675
-=[[#Rank three temperaments]]Rank three temperaments=
-==[[#Rank three temperaments-Greenland]][[Breed family|Greenland]]==
-Commas: 676/675, 1001/1000, 1716/1715
-Map: [&lt;2 0 1 3 7 -1|, &lt;0 2 1 1 -2 4|, &lt;0 0 2 1 3 2|]
-Edos: 58, 72, 130, 198, 270, 940
-[[Optimal patent val]]: [[940edo]]
-Badness: 0.000433
-[[Spectrum of a temperament|Spectrum]]: 15/13, 7/5, 8/7, 7/6, 4/3, 15/14, 5/4, 18/13, 13/12, 14/13, 13/10, 6/5, 16/15, 11/10, 9/7, 9/8, 16/13, 10/9, 14/11, 11/8, 15/11, 12/11, 13/11, 11/9
-==[[#Rank three temperaments-History]][[Werckismic temperaments|History]]==
-Commas: 364/363, 441/440, 1001/1000
-EDOs: 15, 29, 43, 58, 72, 87, 130, 217, 289
-[[Optimal patent val]]: [[289edo]]
-Badness: 0.000540
-Spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7
-==[[#Rank three temperaments-Borneo]]Borneo==
-Commas: 676/675, 1001/1000, 3025/3024
-Map: [&lt;3 0 0 4 8 -3|, &lt;0 2 0 -4 1 3|, &lt;0 0 1 2 0 1|]
-EDOs: 15, 72, 87, 111, 159, 183, 198, 270
-[[Optimal patent val]]: [[270edo]]
-Badness: 0.000549
-Spectrum: 12/11, 15/13, 11/8, 4/3, 11/10, 18/13, 6/5, 5/4, 13/12, 15/11, 11/9, 13/10, 10/9, 7/5, 16/15, 13/11, 9/8, 16/13, 8/7, 14/11, 15/14, 7/6, 14/13, 9/7
-==[[#Rank three temperaments-Madagascar]][[Cataharry family|Madagascar]]==
-Commas: 351/350, 540/539, 676/675
-EDOs: 19, 53, 58, 72, 111, 130, 183, 313
-[[Optimal patent val]]: [[313edo]]
-Badness: 0.000560
-Spectrum: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11
-[[madagascar19]]
-==[[#Rank three temperaments-Baffin]]Baffin==
-Commas: 676/675, 1001/1000, 4225/4224
-Map: [&lt;1 0 0 13 -9 1|, &lt;0 2 0 -7 4 3|, &lt;0 0 1 -2 4 1|]
-EDOs: 34, 43, 53, 87, 130, 183, 217, 270, 940
-[[Optimal patent val]]: [[940edo]]
-Badness: 0.000604
-Spectrum: 15/13, 16/15, 13/12, 4/3, 16/13, 5/4, 18/13, 13/10, 6/5, 9/8, 11/10, 8/7, 7/5, 15/11, 10/9, 13/11, 15/14, 11/8, 7/6, 14/13, 12/11, 9/7, 11/9, 14/11
+EDOs: 46 and some other stuff
 =[[#Rank two temperaments]]Rank two temperaments=
-Rank two temperaments tempering out 676/675 include the 13-limit versions of [[Ragismic microtemperaments|hemiennealimmal]], [[Breedsmic temperaments|harry]], [[Kleismic family|tritikleismic]], [[Kleismic family|catakleimsic]], [[Marvel temperaments|negri]], [[Hemifamity temperaments|mystery]], [[Hemifamity temperaments|buzzard]], [[Kleismic family|quadritikleismic]].
+==[[#Rank two temperaments-Decitonic]]//Oceanfront//==
+Subgroup: 2.3.7.13/5
+Commas: 91/90, 64/63
-It is interesting to note the Graham complexity of 15/13 in these temperaments. This is 18 in hemiennealimmal, 6 in harry, 9 in tritikleismic, 3 in catakleismic, 2 in negri, 2 in buzzard, 12 in quadritikleismic. Catakleismic and buzzard are particularly interesting from an archipelago point of view. Mystery is special case, since the 15/13 part of it belongs to [[29edo]] alone.
+[[POTE tuning|POTE generator]]: ~4/3 = 486.090 (I think)
-==[[#Rank two temperaments-Decitonic]]Decitonic==
+Map: [&lt;1 2 2 3|, &lt; 0 -1 2 -4|]
-Commas: 676/675, 1001/1000, 1716/1715, 4225/4224
+EDOs: 27,32
+Badness: I have no idea
-[[POTE tuning|POTE generator]]: ~15/13 = 248.917
+Oceanfront is very similar to the familiar 7-limit superpyth temperament, in which 16/9 is equated with 7/4, 32/27 equated with 7/6, and 81/64 with 9/7. Oceanfront aims to equate 81/64 with 13/10 instead, however, so the fifths are even sharper than those of superpyth - 713.910 cents is the optimal POTE generator. The general structure of this scale is similar to that of meantone[7], except that the "major" triads in this scale are 10:13:15, and the minor triads are 6:7:9.
-Map: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]
+The sharp fifths of this scale can be a little more dissonant than meantone ears are used to, as can the flat fifths of something like mavila. This scale is very much like a brighter cousin of mavila in that regard.
-EDOs: 130, 270, 940, 1480
-Badness: 0.0135
-==[[#Rank two temperaments-Avicenna]]Avicenna==
+-limit: TBD
-Commas: 676/675, 1001/1000, 3025/3024, 4096/4095
+-limit: TBD
-[[POTE tuning|POTE generator]]: ~13/12 = 137.777
+===[[#Rank two temperaments-Decitonic]]Ultrapyth===
+Subgroup: 2.3.5.7.13
+Commas: 91/90, 64/63, ???? (insert best 5-limit comma here to create an analogous system to superpyth)
-Map: [&lt;3 2 8 16 9 8|, &lt;0 8 -3 -22 4 9|]
+[[POTE tuning|POTE generator]]: ~4/3 = ?
-EDOs: 87, 183, 270
-Badness: 0.0156
-=[[#Subgroup temperaments]]Subgroup temperaments=
+Map: TBD
+EDOs: TBD
+Badness: TBD
-==[[#Subgroup temperaments-Barbados]]Barbados==
+This is a placeholder for the future "Ultrapyth" temperament, which will extend superpyth as you'd expect. If the best way to do this is the same as "porcupinefish" below, then we'll come up with something else.
-Subgroup: 2.3.13/5
-Commas: 676/675
-Perhaps the simplest method of making use of the barbados triad and other characteristic island harmonies is to strip things down to essentials by tempering the 2.3.13/5 [[Just intonation subgroups|just intontation subgroup]]. The minimax tuning for this makes the generator 2/sqrt(3), or 249.0225 cents. EDOs which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with MOS of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
+===[[#Rank two temperaments-Decitonic]]Porcupinefish===
+Subgroup: 13-limit
+Commas: 91/90, 64/63, 250/243, 121/120
-[[POTE tuning|POTE generator]]: ~15/13 = 248.621
+[[POTE tuning|POTE generator]]: ~10/9 = 162.474 (I think)
-[[Smonzos and Svals|Sval map]]: [&lt;1 0 -1|, &lt;0 2 3|]
+Map: [&lt;1 2 3 2 -1 1|, &lt;0 -3 -5 6 33 20|]
-EDOs: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362
+EDOs: 37, 59
-Badness: 0.002335
+Badness: I have no idea
-==[[#Subgroup temperaments-Trinidad]]Trinidad==
+Porcupinefish is the 13-limit extension of porcupine that you get by adding 91/90 to the usual mix of porcupine temperaments. Its name is derived from that it is a combination of the porcupine and oceanfront temperaments.
-Subgroup: 2.3.5.13
-Commas: 325/324, 625/624
-Trinidad may be viewed as the reduction of [[Kleismic family|catakleismic temperament]] to the 2.3.5.13 subgroup. Another way to put it is that it is the rank two 2.3.5.13 subgroup temperament tempering out 325/324, 625/624 and hence also 676/675.
-[[POTE tuning|POTE generator]]: 317.076
+==[[#Rank two temperaments-Decitonic]]//Oceanfront//==
+Subgroup: 2.3.7.13/5
+Commas: 91/90, 64/63
-[[Smonzos and Svals|Sval map]]: [&lt;1 0 1 0 |, &lt;0 6 5 14|]
+[[POTE tuning|POTE generator]]: ~4/3 = 486.090 (I think)
-EDOs: 15, 19, 34, 53, 87, 140, 193, 246
-==[[#Subgroup temperaments-Parizekmic]]Parizekmic==
+Map: [&lt;1 2 2 3|, &lt; 0 -1 2 -4|]
-Subgroup: 2.3.5.13
+EDOs: 27,32
-Commas: 676/675
+Badness: I have no idea
-Closely related to barbados temperament is parizekmic, the rank three 2.3.5.13 subgroup temperament tempering out 676/675. This is generated by 2, 5, and 15/13, where the minimax tuning makes 2 and 5 pure, and 15/13 sharp by sqrt(676/675), or 1.28145 cents. This is, in other words, the same sqrt(4/3) generator as the minimax tuning for barbados, and it gives parizekmic a just 5-limit, with barbados triads where the 13/10 is a cent flat.
+Oceanfront is very similar to the familiar 7-limit superpyth temperament, in which 16/9 is equated with 7/4, 32/27 equated with 7/6, and 81/64 with 9/7. Oceanfront aims to equate 81/64 with 13/10 instead, however, so the fifths are even sharper than those of superpyth - 713.910 cents is the optimal POTE generator. The general structure of this scale is similar to that of meantone[7], except that the "major" triads in this scale are 10:13:15, and the minor triads are 6:7:9.
-[[Smonzos and Svals|Sval map]]
+The sharp fifths of this scale can be a little more dissonant than meantone ears are used to, as can the flat fifths of something like mavila. This scale is very much like a brighter cousin of mavila in that regard.
-&lt;1 0 0 -1|
-&lt;0 2 0 3|
-&lt;0 0 1 1|
-===[[#Subgroup temperaments-Parizekmic-Music]]Music===
+-limit: TBD
-[[http://micro.soonlabel.com/petr_parizek/pp_pump_675.mp3|Petr's Pump]], a comma pump based ditty in Pariekmic temperament.
+-limit: TBD</pre></div>
-EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270 </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;The Biosphere&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; display: none;"&gt;&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;The Biosphere&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The biosphere is the name given to the collection of temperaments that are children of or related to &lt;strong&gt;&lt;em&gt;biome temperament&lt;/em&gt;&lt;/strong&gt;, the rank 3 2.3.7.13/5 subgroup temperament eliminating 91/90. The term &amp;quot;biome&amp;quot; loosely means &amp;quot;ecosystem&amp;quot; or &amp;quot;climate.&amp;quot; This temperament is so named because temperaments that arise from eliminating 91/90 can evoke synesthetic associations of different &amp;quot;natural&amp;quot; settings, some very familiar and some much less so.&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;th&gt;Details&lt;br /&gt;
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-        &lt;td&gt;last edit by &lt;span style="outline: medium none;"&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:83:&amp;lt;a href=&amp;quot;http://www.wikispaces.com/user/view/genewardsmith&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://www.wikispaces.com/user/pic/1202793136/genewardsmith-sm.jpg&amp;quot; alt=&amp;quot;genewardsmith&amp;quot; title=&amp;quot;genewardsmith&amp;quot; style=&amp;quot;height: 16px; width: 16px;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;table class="captionBox"&gt;&lt;tr&gt;&lt;td class="captionedImage"&gt;&lt;a href="http://www.wikispaces.com/user/view/genewardsmith" rel="nofollow"&gt;&lt;img src="http://www.wikispaces.com/user/pic/1202793136/genewardsmith-sm.jpg" alt="genewardsmith" title="genewardsmith" style="height: 16px; width: 16px;" /&gt;&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td class="imageCaption"&gt;genewardsmith&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:83 --&gt;&lt;/span&gt; &lt;span style="outline: medium none;"&gt;&lt;a class="wiki_link_ext" href="http://www.wikispaces.com/user/view/genewardsmith" rel="nofollow"&gt;genewardsmith&lt;/a&gt;&lt;/span&gt; &lt;a class="wiki_link" href="http://page.wikispaces.com/-/diff/The%20Archipelago/219253274"&gt;Apr 11, 2011 3:26 pm&lt;/a&gt; - [[page/history/the Archipelago|36 revisions]]&lt;br /&gt;
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-        &lt;th&gt;Tags&lt;br /&gt;
-&lt;/th&gt;
-        &lt;td&gt;&lt;ul&gt;&lt;li&gt;[[tag/view/listen|listen]]&lt;/li&gt;&lt;li&gt;[[tag/view/theory|theory]]&lt;/li&gt;&lt;/ul&gt;[[the Archipelago#|edit]]&lt;br /&gt;
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-The biosphere is the name given to the collection of temperaments that are children of or related to &lt;strong&gt;&lt;em&gt;biome temperament&lt;/em&gt;&lt;/strong&gt;, the rank 3 2.3.7.13/10 subgroup temperament eliminating 91/90. The term &amp;quot;biome&amp;quot; loosely means &amp;quot;ecosystem&amp;quot; or &amp;quot;climate.&amp;quot; This temperament is so named because temperaments that arise from eliminating 91/90 can evoke synesthetic associations of different &amp;quot;natural&amp;quot; settings, some very familiar and some much less so.&lt;br /&gt;
 &lt;br /&gt;
-This temperament makes the utonal inverse of 6:7:9 out to be 10:13:15, whereas it is normally the much more complex (and arguably more dissonant) 14:18:21. Eliminating 91/90 thus enriches septimal harmony by increasing the concordance of &amp;quot;utonal&amp;quot; septimal triads such as 6:7:9 or tetrads such as 4:6:7:9.&lt;br /&gt;
+The next low-numbered triad after 4:5:6 with a 3/2 on the outside is 6:7:9, but its inversion, 14:18:21, can sound extremely dissonant. On the other hand, you also have 10:13:15, which is another standout triad of low complexity with a fifth on the outside, but its inversion, 26:30:39, is also relatively complex. Tempering out 91/90 makes both of these problems disappear by connecting the two together, such that the utonal inverse of 6:7:9 becomes 10:13:15. Hence, you end up with a tonal system that relates and connects two of the most xenharmonic triads in existence (at least those with 3/2 on the outside). 91/90 tempering thus enriches septimal harmony in this way.&lt;br /&gt;
 &lt;br /&gt;
-Biome temperament, from which is of particular theoretical interest because it generates a rank-3 lattice that is analogous to the 5-limit JI lattice.&lt;br /&gt;
+The rank-3 biome temperament is of particular theoretical interest because it generates a rank-3 lattice that is analogous to the 5-limit JI lattice. As 5-limit JI is the basis for which all 5-limit linear temperaments are derived, the rank-3 biome temperament can serve as a basis to derive useful 2.3.7.13/5 linear temperaments. Instead of our base triads being 4:5:6 and its utonal inversion 10:12:15, we instead treat 6:7:9 and its utonal inversion 10:13:15 as fundamental to the system. The three dimensions of the system can be thought of as 2/1, 3/2, and 7/6 (or 9/7, or 13/10). 46-EDO is a great tuning for biome, giving nearly-pure harmonies all around, somewhat analogous to the accuracy of 34-EDO or 53-EDO in approximating 5-limit JI.&lt;br /&gt;
 &lt;br /&gt;
-The barbados triad is of particular theoretical interest because, when reduced to lowest terms, it is the 10:13:15 triad. Thus, this triad is only slightly higher in complexity than the 5-limit 10:12:15 minor triad, which means it may be of distinct value as a relatively unexplored musical consonance. It is one of only a few low-complexity triads with a 3/2 on the outer dyad, some others being 4:5:6, 6:7:9, and 10:12:15. It works out to 0-454-702 cents, which means that it is an &lt;em&gt;ultramajor&lt;/em&gt; triad, with a third sharper even than the 9/7 supermajor third.&lt;br /&gt;
+This lattice can also be extended to deal with &amp;quot;higher primes,&amp;quot; as can 5-limit JI, but instead by expanding the subgroup outward from the center, so that the &amp;quot;higher primes&amp;quot; we look at are things like like 5, 11, and 13. However, it may prove more useful at first to think purely within the 2.3.7.13/5 subgroup, so as to first come to understand the xenharmonic possibilities of the system.&lt;br /&gt;
 &lt;br /&gt;
-Compared to the 7-limit 14:18:21 supermajor triad, 10:13:15 is lower in triadic complexity (10:13:15 vs 14:18:21), but contains dyads that are on average higher in complexity (9/7 vs 13/10 and 7/6 vs 15/13). Its inverse, however, is the ultraminor 26:30:39, which is far more complex than the 7-limit subminor 6:7:9. Temperaments in which 91/90 vanishes equate the two types of triads.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Biome Temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;strong&gt;Biome Temperament&lt;/strong&gt;&lt;/h1&gt;
-&lt;br /&gt;
+ Comma: 91/90&lt;br /&gt;
-&lt;a class="wiki_link" href="/24edo"&gt;24edo&lt;/a&gt; approximates this triad to within an error of four cents, and &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt; does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below.&lt;br /&gt;
-&lt;br /&gt;
-Comma: 676/675&lt;br /&gt;
 &lt;br /&gt;
 Map&lt;br /&gt;
-&amp;lt;1 0 0 0 0 -1|&lt;br /&gt;
+I have no idea&lt;br /&gt;
-&amp;lt;0 2 0 0 0 3|&lt;br /&gt;
-&amp;lt;0 0 1 0 0 1|&lt;br /&gt;
-&amp;lt;0 0 0 1 0 0|&lt;br /&gt;
-&amp;lt;0 0 0 0 1 0|&lt;br /&gt;
-EDOs: 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940&lt;br /&gt;
-&lt;a class="wiki_link" href="/Optimal%20patent%20val"&gt;Optimal patent val&lt;/a&gt;: &lt;a class="wiki_link" href="/940edo"&gt;940edo&lt;/a&gt;&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 four temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:40:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank four temperaments&amp;quot; title=&amp;quot;Anchor: Rank four temperaments&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank four temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:40 --&gt;Rank four temperaments&lt;/h1&gt;
- &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Rank four temperaments-1001/1000"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:41:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank four temperaments-1001/1000&amp;quot; title=&amp;quot;Anchor: Rank four temperaments-1001/1000&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank four temperaments-1001/1000"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:41 --&gt;1001/1000&lt;/h2&gt;
- Commas: 676/675, 1001/1000&lt;br /&gt;
-&lt;br /&gt;
-EDOs: 15, 19, 29, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940&lt;br /&gt;
-&lt;a class="wiki_link" href="/Optimal%20patent%20val"&gt;Optimal patent val&lt;/a&gt;: &lt;a class="wiki_link" href="/940edo"&gt;940edo&lt;/a&gt;&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="Rank four temperaments-49/48"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:42:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank four temperaments-49/48&amp;quot; title=&amp;quot;Anchor: Rank four temperaments-49/48&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank four temperaments-49/48"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:42 --&gt;49/48&lt;/h2&gt;
- Commas: 49/48, 91/90&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="Rank four temperaments-1716/1715"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:43:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank four temperaments-1716/1715&amp;quot; title=&amp;quot;Anchor: Rank four temperaments-1716/1715&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank four temperaments-1716/1715"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:43 --&gt;1716/1715&lt;/h2&gt;
- Commas: 676/675, 1716/1715&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="Rank four temperaments-364/363"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:44:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank four temperaments-364/363&amp;quot; title=&amp;quot;Anchor: Rank four temperaments-364/363&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank four temperaments-364/363"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:44 --&gt;364/363&lt;/h2&gt;
- Commas: 364/363, 676/675&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Rank four temperaments-364/363-351/350"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:45:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank four temperaments-364/363-351/350&amp;quot; title=&amp;quot;Anchor: Rank four temperaments-364/363-351/350&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank four temperaments-364/363-351/350"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:45 --&gt;351/350&lt;/h3&gt;
- Commas: 351/350, 676/675&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Rank three temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:46:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank three temperaments&amp;quot; title=&amp;quot;Anchor: Rank three temperaments&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank three temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:46 --&gt;Rank three temperaments&lt;/h1&gt;
- &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Rank three temperaments-Greenland"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:47:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank three temperaments-Greenland&amp;quot; title=&amp;quot;Anchor: Rank three temperaments-Greenland&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank three temperaments-Greenland"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:47 --&gt;&lt;a class="wiki_link" href="/Breed%20family"&gt;Greenland&lt;/a&gt;&lt;/h2&gt;
- Commas: 676/675, 1001/1000, 1716/1715&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;2 0 1 3 7 -1|, &amp;lt;0 2 1 1 -2 4|, &amp;lt;0 0 2 1 3 2|]&lt;br /&gt;
-Edos: 58, 72, 130, 198, 270, 940&lt;br /&gt;
-&lt;a class="wiki_link" href="/Optimal%20patent%20val"&gt;Optimal patent val&lt;/a&gt;: &lt;a class="wiki_link" href="/940edo"&gt;940edo&lt;/a&gt;&lt;br /&gt;
-Badness: 0.000433&lt;br /&gt;
-&lt;br /&gt;
-&lt;a class="wiki_link" href="/Spectrum%20of%20a%20temperament"&gt;Spectrum&lt;/a&gt;: 15/13, 7/5, 8/7, 7/6, 4/3, 15/14, 5/4, 18/13, 13/12, 14/13, 13/10, 6/5, 16/15, 11/10, 9/7, 9/8, 16/13, 10/9, 14/11, 11/8, 15/11, 12/11, 13/11, 11/9&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Rank three temperaments-History"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:48:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank three temperaments-History&amp;quot; title=&amp;quot;Anchor: Rank three temperaments-History&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank three temperaments-History"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:48 --&gt;&lt;a class="wiki_link" href="/Werckismic%20temperaments"&gt;History&lt;/a&gt;&lt;/h2&gt;
- Commas: 364/363, 441/440, 1001/1000&lt;br /&gt;
-&lt;br /&gt;
-EDOs: 15, 29, 43, 58, 72, 87, 130, 217, 289&lt;br /&gt;
-&lt;a class="wiki_link" href="/Optimal%20patent%20val"&gt;Optimal patent val&lt;/a&gt;: &lt;a class="wiki_link" href="/289edo"&gt;289edo&lt;/a&gt;&lt;br /&gt;
-Badness: 0.000540&lt;br /&gt;
-&lt;br /&gt;
-Spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Rank three temperaments-Borneo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:49:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank three temperaments-Borneo&amp;quot; title=&amp;quot;Anchor: Rank three temperaments-Borneo&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank three temperaments-Borneo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:49 --&gt;Borneo&lt;/h2&gt;
- Commas: 676/675, 1001/1000, 3025/3024&lt;br /&gt;
-&lt;br /&gt;
-Map: [&amp;lt;3 0 0 4 8 -3|, &amp;lt;0 2 0 -4 1 3|, &amp;lt;0 0 1 2 0 1|]&lt;br /&gt;
-EDOs: 15, 72, 87, 111, 159, 183, 198, 270&lt;br /&gt;
-&lt;a class="wiki_link" href="/Optimal%20patent%20val"&gt;Optimal patent val&lt;/a&gt;: &lt;a class="wiki_link" href="/270edo"&gt;270edo&lt;/a&gt;&lt;br /&gt;
-Badness: 0.000549&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-Spectrum: 12/11, 15/13, 11/8, 4/3, 11/10, 18/13, 6/5, 5/4, 13/12, 15/11, 11/9, 13/10, 10/9, 7/5, 16/15, 13/11, 9/8, 16/13, 8/7, 14/11, 15/14, 7/6, 14/13, 9/7&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Rank three temperaments-Madagascar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:50:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank three temperaments-Madagascar&amp;quot; title=&amp;quot;Anchor: Rank three temperaments-Madagascar&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank three temperaments-Madagascar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:50 --&gt;&lt;a class="wiki_link" href="/Cataharry%20family"&gt;Madagascar&lt;/a&gt;&lt;/h2&gt;
- Commas: 351/350, 540/539, 676/675&lt;br /&gt;
-&lt;br /&gt;
-EDOs: 19, 53, 58, 72, 111, 130, 183, 313&lt;br /&gt;
-&lt;a class="wiki_link" href="/Optimal%20patent%20val"&gt;Optimal patent val&lt;/a&gt;: &lt;a class="wiki_link" href="/313edo"&gt;313edo&lt;/a&gt;&lt;br /&gt;
-Badness: 0.000560&lt;br /&gt;
-&lt;br /&gt;
-Spectrum: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11&lt;br /&gt;
-&lt;a class="wiki_link" href="/madagascar19"&gt;madagascar19&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Rank three temperaments-Baffin"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:51:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank three temperaments-Baffin&amp;quot; title=&amp;quot;Anchor: Rank three temperaments-Baffin&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank three temperaments-Baffin"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:51 --&gt;Baffin&lt;/h2&gt;
- Commas: 676/675, 1001/1000, 4225/4224&lt;br /&gt;
 &lt;br /&gt;
-Map: [&amp;lt;1 0 0 13 -9 1|, &amp;lt;0 2 0 -7 4 3|, &amp;lt;0 0 1 -2 4 1|]&lt;br /&gt;
+EDOs: 46 and some other stuff&lt;br /&gt;
-EDOs: 34, 43, 53, 87, 130, 183, 217, 270, 940&lt;br /&gt;
-&lt;a class="wiki_link" href="/Optimal%20patent%20val"&gt;Optimal patent val&lt;/a&gt;: &lt;a class="wiki_link" href="/940edo"&gt;940edo&lt;/a&gt;&lt;br /&gt;
-Badness: 0.000604&lt;br /&gt;
 &lt;br /&gt;
-Spectrum: 15/13, 16/15, 13/12, 4/3, 16/13, 5/4, 18/13, 13/10, 6/5, 9/8, 11/10, 8/7, 7/5, 15/11, 10/9, 13/11, 15/14, 11/8, 7/6, 14/13, 12/11, 9/7, 11/9, 14/11&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:12:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank two temperaments&amp;quot; title=&amp;quot;Anchor: Rank two temperaments&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:12 --&gt;Rank two temperaments&lt;/h1&gt;
+ &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Rank two temperaments-Oceanfront"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:13:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank two temperaments-Decitonic&amp;quot; title=&amp;quot;Anchor: Rank two temperaments-Decitonic&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank two temperaments-Decitonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:13 --&gt;&lt;em&gt;Oceanfront&lt;/em&gt;&lt;/h2&gt;
+ Subgroup: 2.3.7.13/5&lt;br /&gt;
+Commas: 91/90, 64/63&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:52:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank two temperaments&amp;quot; title=&amp;quot;Anchor: Rank two temperaments&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:52 --&gt;Rank two temperaments&lt;/h1&gt;
+&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~4/3 = 486.090 (I think)&lt;br /&gt;
- Rank two temperaments tempering out 676/675 include the 13-limit versions of &lt;a class="wiki_link" href="/Ragismic%20microtemperaments"&gt;hemiennealimmal&lt;/a&gt;, &lt;a class="wiki_link" href="/Breedsmic%20temperaments"&gt;harry&lt;/a&gt;, &lt;a class="wiki_link" href="/Kleismic%20family"&gt;tritikleismic&lt;/a&gt;, &lt;a class="wiki_link" href="/Kleismic%20family"&gt;catakleimsic&lt;/a&gt;, &lt;a class="wiki_link" href="/Marvel%20temperaments"&gt;negri&lt;/a&gt;, &lt;a class="wiki_link" href="/Hemifamity%20temperaments"&gt;mystery&lt;/a&gt;, &lt;a class="wiki_link" href="/Hemifamity%20temperaments"&gt;buzzard&lt;/a&gt;, &lt;a class="wiki_link" href="/Kleismic%20family"&gt;quadritikleismic&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-It is interesting to note the Graham complexity of 15/13 in these temperaments. This is 18 in hemiennealimmal, 6 in harry, 9 in tritikleismic, 3 in catakleismic, 2 in negri, 2 in buzzard, 12 in quadritikleismic. Catakleismic and buzzard are particularly interesting from an archipelago point of view. Mystery is special case, since the 15/13 part of it belongs to &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt; alone.&lt;br /&gt;
+Map: [&amp;lt;1 2 2 3|, &amp;lt; 0 -1 2 -4|]&lt;br /&gt;
+EDOs: 27,32&lt;br /&gt;
+Badness: I have no idea&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="Rank two temperaments-Decitonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:53:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank two temperaments-Decitonic&amp;quot; title=&amp;quot;Anchor: Rank two temperaments-Decitonic&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank two temperaments-Decitonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:53 --&gt;Decitonic&lt;/h2&gt;
+Oceanfront is very similar to the familiar 7-limit superpyth temperament, in which 16/9 is equated with 7/4, 32/27 equated with 7/6, and 81/64 with 9/7. Oceanfront aims to equate 81/64 with 13/10 instead, however, so the fifths are even sharper than those of superpyth - 713.910 cents is the optimal POTE generator. The general structure of this scale is similar to that of meantone[7], except that the &amp;quot;major&amp;quot; triads in this scale are 10:13:15, and the minor triads are 6:7:9.&lt;br /&gt;
- Commas: 676/675, 1001/1000, 1716/1715, 4225/4224&lt;br /&gt;
 &lt;br /&gt;
-&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/13 = 248.917&lt;br /&gt;
+The sharp fifths of this scale can be a little more dissonant than meantone ears are used to, as can the flat fifths of something like mavila. This scale is very much like a brighter cousin of mavila in that regard.&lt;br /&gt;
 &lt;br /&gt;
-Map: [&amp;lt;10 0 47 36 98 37|, &amp;lt;0 2 -3 -1 -8 0|]&lt;br /&gt;
+-limit: TBD&lt;br /&gt;
-EDOs: 130, 270, 940, 1480&lt;br /&gt;
+-limit: TBD&lt;br /&gt;
-Badness: 0.0135&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Rank two temperaments-Avicenna"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:54:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank two temperaments-Avicenna&amp;quot; title=&amp;quot;Anchor: Rank two temperaments-Avicenna&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank two temperaments-Avicenna"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:54 --&gt;Avicenna&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="Rank two temperaments-Oceanfront-Ultrapyth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:14:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank two temperaments-Decitonic&amp;quot; title=&amp;quot;Anchor: Rank two temperaments-Decitonic&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank two temperaments-Decitonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:14 --&gt;Ultrapyth&lt;/h3&gt;
-  Commas: 676/675, 1001/1000, 3025/3024, 4096/4095&lt;br /&gt;
+  Subgroup: 2.3.5.7.13&lt;br /&gt;
+Commas: 91/90, 64/63, ???? (insert best 5-limit comma here to create an analogous system to superpyth)&lt;br /&gt;
 &lt;br /&gt;
-&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~13/12 = 137.777&lt;br /&gt;
+&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~4/3 = ?&lt;br /&gt;
 &lt;br /&gt;
-Map: [&amp;lt;3 2 8 16 9 8|, &amp;lt;0 8 -3 -22 4 9|]&lt;br /&gt;
+Map: TBD&lt;br /&gt;
-EDOs: 87, 183, 270&lt;br /&gt;
+EDOs: TBD&lt;br /&gt;
-Badness: 0.0156&lt;br /&gt;
+Badness: TBD&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc15"&gt;&lt;a name="Subgroup temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:55:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Subgroup temperaments&amp;quot; title=&amp;quot;Anchor: Subgroup temperaments&amp;quot;/&amp;gt; --&gt;&lt;a name="Subgroup temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:55 --&gt;Subgroup temperaments&lt;/h1&gt;
+This is a placeholder for the future &amp;quot;Ultrapyth&amp;quot; temperament, which will extend superpyth as you'd expect. If the best way to do this is the same as &amp;quot;porcupinefish&amp;quot; below, then we'll come up with something else.&lt;br /&gt;
- &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Subgroup temperaments-Barbados"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:56:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Subgroup temperaments-Barbados&amp;quot; title=&amp;quot;Anchor: Subgroup temperaments-Barbados&amp;quot;/&amp;gt; --&gt;&lt;a name="Subgroup temperaments-Barbados"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:56 --&gt;Barbados&lt;/h2&gt;
- Subgroup: 2.3.13/5&lt;br /&gt;
-Commas: 676/675&lt;br /&gt;
 &lt;br /&gt;
-Perhaps the simplest method of making use of the barbados triad and other characteristic island harmonies is to strip things down to essentials by tempering the 2.3.13/5 &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;just intontation subgroup&lt;/a&gt;. The minimax tuning for this makes the generator 2/sqrt(3), or 249.0225 cents. EDOs which may be used for it are &lt;a class="wiki_link" href="/24edo"&gt;24edo&lt;/a&gt;, &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; and &lt;a class="wiki_link" href="/111edo"&gt;111edo&lt;/a&gt;, with MOS of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="Rank two temperaments-Oceanfront-Porcupinefish"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:15:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank two temperaments-Decitonic&amp;quot; title=&amp;quot;Anchor: Rank two temperaments-Decitonic&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank two temperaments-Decitonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:15 --&gt;Porcupinefish&lt;/h3&gt;
+ Subgroup: 13-limit&lt;br /&gt;
+Commas: 91/90, 64/63, 250/243, 121/120&lt;br /&gt;
 &lt;br /&gt;
-&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/13 = 248.621&lt;br /&gt;
+&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~10/9 = 162.474 (I think)&lt;br /&gt;
 &lt;br /&gt;
-&lt;a class="wiki_link" href="/Smonzos%20and%20Svals"&gt;Sval map&lt;/a&gt;: [&amp;lt;1 0 -1|, &amp;lt;0 2 3|]&lt;br /&gt;
+Map: [&amp;lt;1 2 3 2 -1 1|, &amp;lt;0 -3 -5 6 33 20|]&lt;br /&gt;
-EDOs: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362&lt;br /&gt;
+EDOs: 37, 59&lt;br /&gt;
-Badness: 0.002335&lt;br /&gt;
+Badness: I have no idea&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Subgroup temperaments-Trinidad"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:57:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Subgroup temperaments-Trinidad&amp;quot; title=&amp;quot;Anchor: Subgroup temperaments-Trinidad&amp;quot;/&amp;gt; --&gt;&lt;a name="Subgroup temperaments-Trinidad"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:57 --&gt;Trinidad&lt;/h2&gt;
+Porcupinefish is the 13-limit extension of porcupine that you get by adding 91/90 to the usual mix of porcupine temperaments. Its name is derived from that it is a combination of the porcupine and oceanfront temperaments.&lt;br /&gt;
- Subgroup: 2.3.5.13&lt;br /&gt;
-Commas: 325/324, 625/624&lt;br /&gt;
 &lt;br /&gt;
-Trinidad may be viewed as the reduction of &lt;a class="wiki_link" href="/Kleismic%20family"&gt;catakleismic temperament&lt;/a&gt; to the 2.3.5.13 subgroup. Another way to put it is that it is the rank two 2.3.5.13 subgroup temperament tempering out 325/324, 625/624 and hence also 676/675.&lt;br /&gt;
 &lt;br /&gt;
-&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 317.076&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Rank two temperaments-Oceanfront"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:16:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Rank two temperaments-Decitonic&amp;quot; title=&amp;quot;Anchor: Rank two temperaments-Decitonic&amp;quot;/&amp;gt; --&gt;&lt;a name="Rank two temperaments-Decitonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:16 --&gt;&lt;em&gt;Oceanfront&lt;/em&gt;&lt;/h2&gt;
+ Subgroup: 2.3.7.13/5&lt;br /&gt;
+Commas: 91/90, 64/63&lt;br /&gt;
 &lt;br /&gt;
-&lt;a class="wiki_link" href="/Smonzos%20and%20Svals"&gt;Sval map&lt;/a&gt;: [&amp;lt;1 0 1 0 |, &amp;lt;0 6 5 14|]&lt;br /&gt;
+&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~4/3 = 486.090 (I think)&lt;br /&gt;
-EDOs: 15, 19, 34, 53, 87, 140, 193, 246&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="Subgroup temperaments-Parizekmic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:58:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Subgroup temperaments-Parizekmic&amp;quot; title=&amp;quot;Anchor: Subgroup temperaments-Parizekmic&amp;quot;/&amp;gt; --&gt;&lt;a name="Subgroup temperaments-Parizekmic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:58 --&gt;Parizekmic&lt;/h2&gt;
+Map: [&amp;lt;1 2 2 3|, &amp;lt; 0 -1 2 -4|]&lt;br /&gt;
- Subgroup: 2.3.5.13&lt;br /&gt;
+EDOs: 27,32&lt;br /&gt;
-Commas: 676/675&lt;br /&gt;
+Badness: I have no idea&lt;br /&gt;
 &lt;br /&gt;
-Closely related to barbados temperament is parizekmic, the rank three 2.3.5.13 subgroup temperament tempering out 676/675. This is generated by 2, 5, and 15/13, where the minimax tuning makes 2 and 5 pure, and 15/13 sharp by sqrt(676/675), or 1.28145 cents. This is, in other words, the same sqrt(4/3) generator as the minimax tuning for barbados, and it gives parizekmic a just 5-limit, with barbados triads where the 13/10 is a cent flat.&lt;br /&gt;
+Oceanfront is very similar to the familiar 7-limit superpyth temperament, in which 16/9 is equated with 7/4, 32/27 equated with 7/6, and 81/64 with 9/7. Oceanfront aims to equate 81/64 with 13/10 instead, however, so the fifths are even sharper than those of superpyth - 713.910 cents is the optimal POTE generator. The general structure of this scale is similar to that of meantone[7], except that the &amp;quot;major&amp;quot; triads in this scale are 10:13:15, and the minor triads are 6:7:9.&lt;br /&gt;
 &lt;br /&gt;
-&lt;a class="wiki_link" href="/Smonzos%20and%20Svals"&gt;Sval map&lt;/a&gt;&lt;br /&gt;
+The sharp fifths of this scale can be a little more dissonant than meantone ears are used to, as can the flat fifths of something like mavila. This scale is very much like a brighter cousin of mavila in that regard.&lt;br /&gt;
-&amp;lt;1 0 0 -1|&lt;br /&gt;
-&amp;lt;0 2 0 3|&lt;br /&gt;
-&amp;lt;0 0 1 1|&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="Subgroup temperaments-Parizekmic-Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:59:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Subgroup temperaments-Parizekmic-Music&amp;quot; title=&amp;quot;Anchor: Subgroup temperaments-Parizekmic-Music&amp;quot;/&amp;gt; --&gt;&lt;a name="Subgroup temperaments-Parizekmic-Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:59 --&gt;Music&lt;/h3&gt;
+-limit: TBD&lt;br /&gt;
- &lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/petr_parizek/pp_pump_675.mp3" rel="nofollow"&gt;Petr's Pump&lt;/a&gt;, a comma pump based ditty in Pariekmic temperament.&lt;br /&gt;
+-limit: TBD&lt;/body&gt;&lt;/html&gt;</pre></div>
-EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270&lt;/body&gt;&lt;/html&gt;</pre></div>

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