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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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 the biome comma [[91/90]], and '''biosphere temperament''', its rank-5 full 13-limit extension. The term "biome" loosely means "ecosystem" or "climate."
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 20:51:28 UTC</tt>.<br>
: The original revision id was <tt>222577872</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 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 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 to those not used to [[9-odd-limit]] harmony. 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.


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 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). 46EDO is a great tuning for biome, giving nearly-pure harmonies all around, somewhat analogous to the accuracy of 34EDO or 53EDO 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.
This lattice can also be extended to deal with "higher primes", as can 5-limit JI. However, we instead expand 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**=  
== Parent Temperaments ==
Comma: 91/90
=== Biome ===
Subgroup: 2.3.7.13/5
 
Comma list: 91/90
 
Mapping:
 
{{val| 1 0 0 1 }}<br>
{{val| 0 1 0 2 }}<br>
{{val| 0 0 1 -1 }}


Map
{{Optimal ET sequence|legend=1| 5, 9, 14, 17, 22, 27, 32, 46 }}
I have no idea


EDOs: 46 and some other stuff
=== Biosphere ===
Subgroup: 2.3.5.7.11.13


=[[#Rank two temperaments]]Rank two temperaments=
Comma list: 91/90
=[[#Rank two temperaments-Decitonic]]&lt;span style="color: #000000;"&gt;Oceanfront&lt;/span&gt;=
 
Subgroup: 2.3.7.13/5
Mapping:  
Commas: 91/90, 64/63


[[POTE tuning|POTE generator]]: ~4/3 = 486.090 (I think)
{{val| 1 0 0 0 0 1 }}<br>
{{val| 0 1 0 0 0 2 }}<br>
{{val| 0 0 1 0 0 1 }}<br>
{{val| 0 0 0 1 0 -1 }}<br>
{{val| 0 0 0 0 1 0 }}


Map: [&lt;1 2 2 3|, &lt; 0 -1 2 -4|]
{{Optimal ET sequence|legend=1| 8d, 9, 10, 14cf, 15, 17c, 19, 22, 27e, 29, 31f, 37, 38df, 46 }}
EDOs: 27,32
Badness: I have no idea


== Rank two temperaments ==
=== Oceanfront ===
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.
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.
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
Subgroup: 2.3.7.13/5
13-limit: TBD
 
[[Comma list]]: 64/63, 91/90
 
[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -1 2 -4 }}]
 
[[POTE generator]]: ~4/3 = 486.090
 
{{Optimal ET sequence|legend=1| 27, 32 }}
 
Scales: [[Oceanfront scales]]
 
==== Superpyth ====
{{see also| Archytas clan #Superpyth }}
 
Extends 11-limit superpyth as 22&amp;49.
 
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 64/63, 78/77, 91/90, 100/99
 
[[Mapping]]: [{{val| 1 2 6 2 10 9 }}, {{val| 0 -1 -9 2 -16 -13 }}]
 
[[POTE generator]]: ~4/3 = 489.521
 
{{Optimal ET sequence|legend=1| 22, 27e, 49, 76bcde }}
 
[[Badness]]: 0.024673
 
==== Quasisupra ====
{{see also| Archytas clan #Quasisuper }}
 
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 64/63, 78/77, 91/90, 121/120
 
[[Mapping]]: [{{val| 1 2 -3 2 1 0 }}, {{val| 0 -1 13 2 6 9 }}]
 
[[POTE generator]]: ~4/3 = 491.996
 
{{Optimal ET sequence|legend=1| 17c, 22, 39d, 61df, 100bcdf }}
 
[[Badness]]: 0.030219
 
==== Ultrapyth ====
{{see also| Archytas clan #Ultrapyth }}


==**Oceanfront Children**==
===[[#Rank two temperaments-Decitonic]]Ultrapyth===
Subgroup: 2.3.5.7.13
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 = ?
[[Comma list]]: 64/63, 91/90, 4394/4375


Map: TBD
[[Mapping]]: [{{val|1 2 8 2 11}}, {{val|0 -1 -14 2 -18}}]
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.
[[POTE generator]]: ~4/3 = 486.255


===[[#Rank two temperaments-Decitonic]]Porcupinefish===
{{Optimal ET sequence|legend=1| 5, 32, 37 }}
Subgroup: 13-limit
Commas: 91/90, 64/63, 250/243, 121/120


[[POTE tuning|POTE generator]]: ~10/9 = 162.474 (I think)
===== Full 13-limit ultrapyth =====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 3 2 -1 1|, &lt;0 -3 -5 6 33 20|]
[[Comma list]]: 55/54, 64/63, 91/90, 1573/1568
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.
[[Mapping]]: [{{val| 1 2 8 2 -1 11 }}, {{val| 0 -1 -14 2 11 -18 }}]


[[POTE generator]]: ~4/3 = 486.500
{{Optimal ET sequence|legend=1| 5, 32, 37 }}
[[Badness]]: 0.049172
===== Ultramarine =====
Subgroup: 2.3.5.7.11.13
[[Comma list]]: 64/63, 91/90, 100/99, 847/845
[[Mapping]]: [{{val| 1 2 8 2 14 11 }}, {{val| 0 -1 -14 2 -26 -18 }}]
[[POTE generator]]: ~4/3 = 486.189
{{Optimal ET sequence|legend=1| 5e, 32e, 37, 79bcef, 116bbcef }}
[[Badness]]: 0.045653
==== Porcupinefish ====
{{see also| Porcupine family #Porcupinefish }}
Porcupinefish is the 13-limit extension of [[Porcupine|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: 2.3.5.7.11.13
[[Comma list]]: 55/54, 64/63, 91/90, 100/99
[[Mapping]]: [{{val| 1 2 3 2 4 6 }}, {{val| 0 -3 -5 6 -4 -17 }}]
[[POTE generator]]: ~10/9 = 162.277
{{Optimal ET sequence|legend=1| 15, 22, 37, 59 }}
[[Badness]]: 0.025314
=== Tropic ===
Tropic is the merger of the biosphere and the [[The Archipelago|archipelago]]. It is also a subgroup relative of semaphore temperament, since [[49/48]] vanishes. Of note is that [[676/675]] vanishes, so that two 7/6's (or 15/13)'s is equated with 4/3. While this temperament doesn't take advantage of the nearly pure harmonies that biome tempering can offer, particularly where 7/4 is involved, it still has some use, particularly for those who don't mind a bit more error in their tunings.


=[[#Rank two temperaments-Decitonic]]Tropic=
Subgroup: 2.3.7.13/5
Subgroup: 2.3.7.13/5
Commas: 91/90, 49/48


[[POTE tuning|POTE generator]]: ~4/3 = 251.507 (I think)
[[Comma list]]: 49/48, 91/90
 
[[Mapping]]: [{{val| 1 2 3 2 }}, {{val| 0 -2 -1 -3 }}]
 
[[POTE generator]]: ~7/6 = 251.507
 
{{Optimal ET sequence|legend=1| 19, 24 }}
 
==== Godzilla ====
{{see also| Meantone family #Godzilla }}
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 49/48, 81/80, 91/90
 
[[Mapping]]: [{{val|1 0 -4 2 -5}}, {{val|0 2 8 1 11}}]
 
[[POTE generator]]: ~7/6 = 252.429
 
{{Optimal ET sequence|legend=1| 5, 14cf, 19 }}
 
===== Full 13-limit godzilla =====
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 45/44, 49/48, 78/77, 81/80
 
[[Mapping]]: [{{val|1 0 -4 2 -6 -5}}, {{val|0 2 8 1 12 11}}]
 
[[POTE generator]]: ~7/6 = 253.603
 
{{Optimal ET sequence|legend=1| 5e, 14cf, 19, 33cdff, 52cdff }}
 
[[Badness]]: 0.022503
 
===== Varan =====
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 49/48, 66/65, 77/75, 81/80
 
[[Mapping]]: [{{val|1 0 -4 2 -10 -5}}, {{val|0 2 8 1 17 11}}]
 
[[POTE generator]]: ~7/6 = 251.165
 
{{Optimal ET sequence|legend=1| 19e, 24, 43de }}
 
[[Badness]]: 0.025676
 
===== Baragon =====
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 49/48, 56/55, 81/80, 91/90
 
[[Mapping]]: [{{val|1 0 -4 2 9 -5}}, {{val|0 2 8 1 -7 11}}]
 
[[POTE generator]]: ~7/6 = 251.198
 
{{Optimal ET sequence|legend=1| 5, 14cef, 19, 24, 43d }}
 
[[Badness]]: 0.026703
 
==== Anguirus ====
{{see also| Diaschismic family #Anguirus }}
 
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 49/48, 56/55, 91/90, 352/351
 
[[Mapping]]: [{{val| 2 4 3 6 9 7 }}, {{val| 0 -2 4 -1 -5 1 }}]
 
[[POTE generator]]: ~8/7 = 247.691
 
{{Optimal ET sequence|legend=1| 10, 24, 34, 58d, 92def }}
 
[[Badness]]: 0.030829
 
=== Echidnic ===
{{see also| Diaschismic family #Echidnic }}
 
13-limit echidnic temperament, the 10&amp;46 temperament, is about as accurate as a biosphere temperament can get.
 
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 91/90, 169/168, 385/384, 441/440
 
[[Mapping]]: [{{val| 2 2 7 6 3 7 }}, {{val| 0 3 -6 -1 10 1 }}]


Map: [&lt;1 2 3 2|, &lt;0 -2 -1 -3|]
[[POTE generator]]: ~8/7 = 235.088
EDOs: 19, 24
Badness: I have no idea


Tropic is the merger of the biosphere and the archipelago. It is also a subgroup relative of semaphore temperament, since 49/48 vanishes. Of note is that 676/675 vanishes, so that two 7/6's (or 15/13)'s is equated with 4/3. While this temperament doesn't take advantage of the nearly pure harmonies that biome tempering can offer, particularly where 7/4 is involved, it still has some use, particularly for those who don't mind a bit more error in their tunings.
{{Optimal ET sequence|legend=1| 10, 46, 102, 148f, 194bcdf }}


11-limit: TBD
[[Badness]]: 0.028874
13-limit: TBD


==****Tropic Children****==
[[Category:Regular temperament theory]]
TBD</pre></div>
[[Category:Commatic realms]]
<h4>Original HTML content:</h4>
[[Category:Biome]]
<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;
[[Category:Biosphere]]
&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;
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;
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;
&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;
Comma: 91/90&lt;br /&gt;
&lt;br /&gt;
Map&lt;br /&gt;
I have no idea&lt;br /&gt;
&lt;br /&gt;
EDOs: 46 and some other stuff&lt;br /&gt;
&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: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&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:16 --&gt;Rank two temperaments&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Oceanfront"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:17:&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:17 --&gt;&lt;span style="color: #000000;"&gt;Oceanfront&lt;/span&gt;&lt;/h1&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="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~4/3 = 486.090 (I think)&lt;br /&gt;
&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;
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;
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;
11-limit: TBD&lt;br /&gt;
13-limit: TBD&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="Oceanfront-Oceanfront Children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;strong&gt;Oceanfront Children&lt;/strong&gt;&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="Oceanfront-Oceanfront Children-Ultrapyth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:18:&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:18 --&gt;Ultrapyth&lt;/h3&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;: ~4/3 = ?&lt;br /&gt;
&lt;br /&gt;
Map: TBD&lt;br /&gt;
EDOs: TBD&lt;br /&gt;
Badness: TBD&lt;br /&gt;
&lt;br /&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:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Oceanfront-Oceanfront Children-Porcupinefish"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:19:&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:19 --&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;: ~10/9 = 162.474 (I think)&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 3 2 -1 1|, &amp;lt;0 -3 -5 6 33 20|]&lt;br /&gt;
EDOs: 37, 59&lt;br /&gt;
Badness: I have no idea&lt;br /&gt;
&lt;br /&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;
&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="Tropic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:20:&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:20 --&gt;Tropic&lt;/h1&gt;
Subgroup: 2.3.7.13/5&lt;br /&gt;
Commas: 91/90, 49/48&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~4/3 = 251.507 (I think)&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 3 2|, &amp;lt;0 -2 -1 -3|]&lt;br /&gt;
EDOs: 19, 24&lt;br /&gt;
Badness: I have no idea&lt;br /&gt;
&lt;br /&gt;
Tropic is the merger of the biosphere and the archipelago. It is also a subgroup relative of semaphore temperament, since 49/48 vanishes. Of note is that 676/675 vanishes, so that two 7/6's (or 15/13)'s is equated with 4/3. While this temperament doesn't take advantage of the nearly pure harmonies that biome tempering can offer, particularly where 7/4 is involved, it still has some use, particularly for those who don't mind a bit more error in their tunings.&lt;br /&gt;
&lt;br /&gt;
11-limit: TBD&lt;br /&gt;
13-limit: TBD&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Tropic-**Tropic Children**"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Tropic Children&lt;/h2&gt;
TBD&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 01:19, 13 December 2025

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 the biome comma 91/90, and biosphere temperament, its rank-5 full 13-limit extension. The term "biome" loosely means "ecosystem" or "climate."

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 to those not used to 9-odd-limit harmony. 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.

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). 46EDO is a great tuning for biome, giving nearly-pure harmonies all around, somewhat analogous to the accuracy of 34EDO or 53EDO in approximating 5-limit JI.

This lattice can also be extended to deal with "higher primes", as can 5-limit JI. However, we instead expand 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.

Parent Temperaments

Biome

Subgroup: 2.3.7.13/5

Comma list: 91/90

Mapping:

⟨1 0 0 1]
⟨0 1 0 2]
⟨0 0 1 -1]

Optimal ET sequence: 5, 9, 14, 17, 22, 27, 32, 46

Biosphere

Subgroup: 2.3.5.7.11.13

Comma list: 91/90

Mapping:

⟨1 0 0 0 0 1]
⟨0 1 0 0 0 2]
⟨0 0 1 0 0 1]
⟨0 0 0 1 0 -1]
⟨0 0 0 0 1 0]

Optimal ET sequence: 8d, 9, 10, 14cf, 15, 17c, 19, 22, 27e, 29, 31f, 37, 38df, 46

Rank two temperaments

Oceanfront

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.

Subgroup: 2.3.7.13/5

Comma list: 64/63, 91/90

Mapping: [⟨1 2 2 3], ⟨0 -1 2 -4]]

POTE generator: ~4/3 = 486.090

Optimal ET sequence: 27, 32

Scales: Oceanfront scales

Superpyth

See also: Archytas clan § Superpyth

Extends 11-limit superpyth as 22&49.

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 100/99

Mapping: [⟨1 2 6 2 10 9], ⟨0 -1 -9 2 -16 -13]]

POTE generator: ~4/3 = 489.521

Optimal ET sequence: 22, 27e, 49, 76bcde

Badness: 0.024673

Quasisupra

See also: Archytas clan § Quasisuper

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 121/120

Mapping: [⟨1 2 -3 2 1 0], ⟨0 -1 13 2 6 9]]

POTE generator: ~4/3 = 491.996

Optimal ET sequence: 17c, 22, 39d, 61df, 100bcdf

Badness: 0.030219

Ultrapyth

See also: Archytas clan § Ultrapyth

Subgroup: 2.3.5.7.13

Comma list: 64/63, 91/90, 4394/4375

Mapping: [⟨1 2 8 2 11], ⟨0 -1 -14 2 -18]]

POTE generator: ~4/3 = 486.255

Optimal ET sequence: 5, 32, 37

Full 13-limit ultrapyth

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 1573/1568

Mapping: [⟨1 2 8 2 -1 11], ⟨0 -1 -14 2 11 -18]]

POTE generator: ~4/3 = 486.500

Optimal ET sequence: 5, 32, 37

Badness: 0.049172

Ultramarine

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 91/90, 100/99, 847/845

Mapping: [⟨1 2 8 2 14 11], ⟨0 -1 -14 2 -26 -18]]

POTE generator: ~4/3 = 486.189

Optimal ET sequence: 5e, 32e, 37, 79bcef, 116bbcef

Badness: 0.045653

Porcupinefish

See also: Porcupine family § Porcupinefish

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: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 100/99

Mapping: [⟨1 2 3 2 4 6], ⟨0 -3 -5 6 -4 -17]]

POTE generator: ~10/9 = 162.277

Optimal ET sequence: 15, 22, 37, 59

Badness: 0.025314

Tropic

Tropic is the merger of the biosphere and the archipelago. It is also a subgroup relative of semaphore temperament, since 49/48 vanishes. Of note is that 676/675 vanishes, so that two 7/6's (or 15/13)'s is equated with 4/3. While this temperament doesn't take advantage of the nearly pure harmonies that biome tempering can offer, particularly where 7/4 is involved, it still has some use, particularly for those who don't mind a bit more error in their tunings.

Subgroup: 2.3.7.13/5

Comma list: 49/48, 91/90

Mapping: [⟨1 2 3 2], ⟨0 -2 -1 -3]]

POTE generator: ~7/6 = 251.507

Optimal ET sequence: 19, 24

Godzilla

See also: Meantone family § Godzilla

Subgroup: 2.3.5.7.13

Comma list: 49/48, 81/80, 91/90

Mapping: [⟨1 0 -4 2 -5], ⟨0 2 8 1 11]]

POTE generator: ~7/6 = 252.429

Optimal ET sequence: 5, 14cf, 19

Full 13-limit godzilla

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 78/77, 81/80

Mapping: [⟨1 0 -4 2 -6 -5], ⟨0 2 8 1 12 11]]

POTE generator: ~7/6 = 253.603

Optimal ET sequence: 5e, 14cf, 19, 33cdff, 52cdff

Badness: 0.022503

Varan

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 66/65, 77/75, 81/80

Mapping: [⟨1 0 -4 2 -10 -5], ⟨0 2 8 1 17 11]]

POTE generator: ~7/6 = 251.165

Optimal ET sequence: 19e, 24, 43de

Badness: 0.025676

Baragon

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 81/80, 91/90

Mapping: [⟨1 0 -4 2 9 -5], ⟨0 2 8 1 -7 11]]

POTE generator: ~7/6 = 251.198

Optimal ET sequence: 5, 14cef, 19, 24, 43d

Badness: 0.026703

Anguirus

See also: Diaschismic family § Anguirus

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 352/351

Mapping: [⟨2 4 3 6 9 7], ⟨0 -2 4 -1 -5 1]]

POTE generator: ~8/7 = 247.691

Optimal ET sequence: 10, 24, 34, 58d, 92def

Badness: 0.030829

Echidnic

See also: Diaschismic family § Echidnic

13-limit echidnic temperament, the 10&46 temperament, is about as accurate as a biosphere temperament can get.

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 169/168, 385/384, 441/440

Mapping: [⟨2 2 7 6 3 7], ⟨0 3 -6 -1 10 1]]

POTE generator: ~8/7 = 235.088

Optimal ET sequence: 10, 46, 102, 148f, 194bcdf

Badness: 0.028874

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