The Biosphere: Difference between revisions

<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 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.

Comma: 91/90

=[[#Rank two temperaments]]Rank two temperaments=  

=[[#Rank two temperaments-Decitonic]]&lt;span style="color: #000000;"&gt;Oceanfront&lt;/span&gt;=  

Subgroup: 2.3.7.13/5

Commas: 91/90, 64/63

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

Map: [&lt;1 2 2 3|, &lt; 0 -1 2 -4|]

EDOs: 27,32

11-limit: TBD

13-limit: TBD

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

Map: 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

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

Map: [&lt;1 2 3 2 -1 1|, &lt;0 -3 -5 6 33 20|]

EDOs: 37, 59

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.

[[POTE tuning|POTE generator]]: ~4/3 = 251.507 (I think)

Map: [&lt;1 2 3 2|, &lt;0 -2 -1 -3|]

EDOs: 19, 24

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.

11-limit: TBD

==Tropic Children==

=[[#Rank two temperaments-Decitonic]]Pandora=  

Subgroup: 2.3.7.13/5

Commas: 91/90, ???

[[POTE tuning|POTE generator]]: TBD

Map: TBD

- decently pure harmonies (around what 46-equal offers)

- slightly sharp fifths, between those of 46-equal and 17-equal

- ideally not too complex, but I'll take what I can get

11-limit: TBD

<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;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;

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;

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;!-- 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;

Map: [&amp;lt;1 2 2 3|, &amp;lt; 0 -1 2 -4|]&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;

Map: [&amp;lt;1 2 3 2 -1 1|, &amp;lt;0 -3 -5 6 33 20|]&lt;br /&gt;

Map: [&amp;lt;1 2 3 2|, &amp;lt;0 -2 -1 -3|]&lt;br /&gt;