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Wikispaces>mbattaglia1 **Imported revision 222628542 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 224001534 - Original comment: ** |
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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- | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-04-29 02:27:18 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>224001534</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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Revision as of 02:27, 29 April 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author mbattaglia1 and made on 2011-04-29 02:27:18 UTC.
- The original revision id was 224001534.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
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, and **//biosphere temperament//**, its rank five full 13-limit extension. 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 others fresh and exotic. 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 by increasing the concordance of its utonalities. 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. 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: 91/90 [[POTE tuning|POTE generator]]s: TBD Map: <1 0 0 1| <0 1 0 2| <0 0 1 -1| EDOs: 46 and some other stuff =**Biosphere**= Subgroup: Full 13-limit Comma: 91/90 Map: <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| EDOs: 46 and some other stuff =[[#Rank two temperaments]]Rank two temperaments= =[[#Rank two temperaments-Decitonic]]<span style="color: #000000;">Oceanfront</span>= 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 ==**Oceanfront Children**== ===[[#Rank two temperaments-Decitonic]]Ultrapyth=== Subgroup: 2.3.5.7.13 Commas: 91/90, 64/63, 245/243 [[POTE tuning|POTE generator]]: ~4/3 = 490.304 Map: [< 1 2 6 2 7|, <0 -1 -9 2 -8|] EDOs: 32 Badness: How would I know This temperament extends superpyth as you'd expect. Full 13-limit: TBD ===[[#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]]Tropic= Subgroup: 2.3.7.13/5 Commas: 91/90, 49/48 [[POTE tuning|POTE generator]]: ~4/3 = 251.507 (I think) Map: [<1 2 3 2|, <0 -2 -1 -3|] 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. 11-limit: TBD 13-limit: TBD ==Tropic Children== TBD =[[#Rank two temperaments-Decitonic]]Pandora= Subgroup: 2.3.7.13/5 Commas: 91/90, ??? [[POTE tuning|POTE generator]]: TBD Map: TBD EDOs: TBD Badness: TBD Pandora is the name of a future temperament that meets all of the following criteria: - 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 - also ideally generating a decent albitonic scale 11-limit: TBD 13-limit: TBD ==Pandora Children== 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, and <strong><em>biosphere temperament</em></strong>, its rank five full 13-limit extension. 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 others fresh and exotic.<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 by increasing the concordance of its utonalities.<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 "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.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Parent Temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Parent Temperaments</h1> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Biome"></a><!-- ws:end:WikiTextHeadingRule:2 --><strong>Biome</strong></h1> Subgroup: 2.3.7.13/5<br /> Comma: 91/90<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>s: TBD<br /> <br /> Map:<br /> <1 0 0 1|<br /> <0 1 0 2|<br /> <0 0 1 -1|<br /> <br /> EDOs: 46 and some other stuff<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Biosphere"></a><!-- ws:end:WikiTextHeadingRule:4 --><strong>Biosphere</strong></h1> Subgroup: Full 13-limit<br /> Comma: 91/90<br /> <br /> Map:<br /> <1 0 0 0 0 1|<br /> <0 1 0 0 0 2|<br /> <0 0 1 0 0 1|<br /> <0 0 0 1 0 -1|<br /> <0 0 0 0 1 0|<br /> <br /> EDOs: 46 and some other stuff<br /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:26:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank two temperaments" title="Anchor: Rank two temperaments"/> --><a name="Rank two temperaments"></a><!-- ws:end:WikiTextAnchorRule:26 -->Rank two temperaments</h1> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Oceanfront"></a><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextAnchorRule:27:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank two temperaments-Decitonic" title="Anchor: Rank two temperaments-Decitonic"/> --><a name="Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextAnchorRule:27 --><span style="color: #000000;">Oceanfront</span></h1> 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: [<1 2 2 3|, < 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 "major" 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:10:<h2> --><h2 id="toc5"><a name="Oceanfront-Oceanfront Children"></a><!-- ws:end:WikiTextHeadingRule:10 --><strong>Oceanfront Children</strong></h2> <!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="Oceanfront-Oceanfront Children-Ultrapyth"></a><!-- ws:end:WikiTextHeadingRule:12 --><!-- ws:start:WikiTextAnchorRule:28:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank two temperaments-Decitonic" title="Anchor: Rank two temperaments-Decitonic"/> --><a name="Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextAnchorRule:28 -->Ultrapyth</h3> Subgroup: 2.3.5.7.13<br /> Commas: 91/90, 64/63, 245/243<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~4/3 = 490.304<br /> <br /> Map: [< 1 2 6 2 7|, <0 -1 -9 2 -8|]<br /> EDOs: 32<br /> Badness: How would I know<br /> <br /> This temperament extends superpyth as you'd expect.<br /> <br /> Full 13-limit: TBD<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h3> --><h3 id="toc7"><a name="Oceanfront-Oceanfront Children-Porcupinefish"></a><!-- ws:end:WikiTextHeadingRule:14 --><!-- ws:start:WikiTextAnchorRule:29:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank two temperaments-Decitonic" title="Anchor: Rank two temperaments-Decitonic"/> --><a name="Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextAnchorRule:29 -->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: [<1 2 3 2 -1 1|, <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:16:<h1> --><h1 id="toc8"><a name="Tropic"></a><!-- ws:end:WikiTextHeadingRule:16 --><!-- ws:start:WikiTextAnchorRule:30:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank two temperaments-Decitonic" title="Anchor: Rank two temperaments-Decitonic"/> --><a name="Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextAnchorRule:30 -->Tropic</h1> Subgroup: 2.3.7.13/5<br /> Commas: 91/90, 49/48<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~4/3 = 251.507 (I think)<br /> <br /> Map: [<1 2 3 2|, <0 -2 -1 -3|]<br /> EDOs: 19, 24<br /> Badness: I have no idea<br /> <br /> 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.<br /> <br /> 11-limit: TBD<br /> 13-limit: TBD<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Tropic-Tropic Children"></a><!-- ws:end:WikiTextHeadingRule:18 -->Tropic Children</h2> TBD<br /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h1> --><h1 id="toc10"><a name="Pandora"></a><!-- ws:end:WikiTextHeadingRule:20 --><!-- ws:start:WikiTextAnchorRule:31:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank two temperaments-Decitonic" title="Anchor: Rank two temperaments-Decitonic"/> --><a name="Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextAnchorRule:31 -->Pandora</h1> Subgroup: 2.3.7.13/5<br /> Commas: 91/90, ???<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: TBD<br /> <br /> Map: TBD<br /> EDOs: TBD<br /> Badness: TBD<br /> <br /> Pandora is the name of a future temperament that meets all of the following criteria:<br /> - decently pure harmonies (around what 46-equal offers)<br /> - slightly sharp fifths, between those of 46-equal and 17-equal<br /> - ideally not too complex, but I'll take what I can get<br /> - also ideally generating a decent albitonic scale<br /> <br /> 11-limit: TBD<br /> 13-limit: TBD<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h2> --><h2 id="toc11"><a name="Pandora-Pandora Children"></a><!-- ws:end:WikiTextHeadingRule:22 -->Pandora Children</h2> TBD<br /> <!-- ws:start:WikiTextHeadingRule:24:<h1> --><h1 id="toc12"><!-- ws:end:WikiTextHeadingRule:24 --> </h1> </body></html>