Ploidacot/Alpha-tricot: Difference between revisions

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== Temperament interpretations ==

Any valid alpha-tricot temperament assigns a just interpretation to the individual generator.

=== Threedic ===

Probably the most sensical RTT interpretation of the generator is as [[13/9]], tempering out the comma [[2197/2187]], the threedie, in the 2.3.13 subgroup. This temperament is in fact every other step of [[kleismic]] (which splits 13/9 into two [[6/5]]s), and is best tuned with a fifth slightly (0-4 cents) sharp of just.

=== [[Alphatricot]] ===

Formerly inaccurately just called "tricot", this is a 5-limit microtemperament with the alpha-tricot structure (hence its name). ~~In the 2.3.5.13 interpretations, the generator represents either 13/9 or 75/52, and~~ 5/4 is found at ''29'' generators up, the submajor third in the "superaugmented fourth" notation. As a microtemperament, it is tuned best when the fifth is around just.

Formerly inaccurately just called "tricot", this is a 5-limit microtemperament with the alpha-tricot structure (hence its name). 5/4 is found at ''29'' generators up, the submajor third in the "superaugmented fourth" notation, and the generator can be held to represent either 13/9 (threedic, above), or [[75/52]] ([[140625/140608|catasmic]], which is far more accurate) in the 2.3.5.13 subgroup. As a microtemperament, it is tuned best when the fifth is around just.

=== Liese ===

[[Liese]] identifies the generator with 10/7. To extend to the full 7-limit, the meantone mapping of 5 is used (12 generators or 4 fifths up), which has the benefit of flattening the fifth to make the 10/7 more accurate.

[[Liese]] identifies the generator with [[10/7]]. To extend to the full 7-limit, the [[meantone]] mapping of 5 is used (12 generators or 4 fifths up), which has the benefit of flattening the fifth to make the 10/7 more accurate.

=== 12288/12167 ===

An obvious mapping for the generator is 23/16. This temperament is best tuned by flattening the generator slightly to compromise between the tunings of 23/16 and 3/2. At this point, meantone is a reasonable choice for 5, as it is also tuned with a flattened fifth.

An obvious mapping for the generator is [[23/16]]. This temperament is best tuned by flattening the generator slightly to compromise between the tunings of 23/16 and 3/2. At this point, meantone is a reasonable choice for 5, as it is also tuned with a flattened fifth.

=== Paralimmal ===

[[Paralimmal]] maps the generator to 16/11, meaning it is best tuned with a sharpened fifth (around 715-720 cents or so).

[[Paralimmal]] maps the generator to [[16/11]], meaning it is best tuned with a sharpened fifth (around 715-720 cents or so).

@@ Line 148: / Line 148: @@
 == Temperament interpretations ==
 Any valid alpha-tricot temperament assigns a just interpretation to the individual generator.
+=== Threedic ===
+Probably the most sensical RTT interpretation of the generator is as [[13/9]], tempering out the comma [[2197/2187]], the threedie, in the 2.3.13 subgroup. This temperament is in fact every other step of [[kleismic]] (which splits 13/9 into two [[6/5]]s), and is best tuned with a fifth slightly (0-4 cents) sharp of just.
 === [[Alphatricot]] ===
-Formerly inaccurately just called "tricot", this is a 5-limit microtemperament with the alpha-tricot structure (hence its name). In the 2.3.5.13 interpretations, the generator represents either 13/9 or 75/52, and 5/4 is found at ''29'' generators up, the submajor third in the "superaugmented fourth" notation. As a microtemperament, it is tuned best when the fifth is around just.
+Formerly inaccurately just called "tricot", this is a 5-limit microtemperament with the alpha-tricot structure (hence its name). 5/4 is found at ''29'' generators up, the submajor third in the "superaugmented fourth" notation, and the generator can be held to represent either 13/9 (threedic, above), or [[75/52]] ([[140625/140608|catasmic]], which is far more accurate) in the 2.3.5.13 subgroup. As a microtemperament, it is tuned best when the fifth is around just.
 === Liese ===
-[[Liese]] identifies the generator with 10/7. To extend to the full 7-limit, the meantone mapping of 5 is used (12 generators or 4 fifths up), which has the benefit of flattening the fifth to make the 10/7 more accurate.
+[[Liese]] identifies the generator with [[10/7]]. To extend to the full 7-limit, the [[meantone]] mapping of 5 is used (12 generators or 4 fifths up), which has the benefit of flattening the fifth to make the 10/7 more accurate.
 === 12288/12167 ===
-An obvious mapping for the generator is 23/16. This temperament is best tuned by flattening the generator slightly to compromise between the tunings of 23/16 and 3/2. At this point, meantone is a reasonable choice for 5, as it is also tuned with a flattened fifth.
+An obvious mapping for the generator is [[23/16]]. This temperament is best tuned by flattening the generator slightly to compromise between the tunings of 23/16 and 3/2. At this point, meantone is a reasonable choice for 5, as it is also tuned with a flattened fifth.
 === Paralimmal ===
-[[Paralimmal]] maps the generator to 16/11, meaning it is best tuned with a sharpened fifth (around 715-720 cents or so).
+[[Paralimmal]] maps the generator to [[16/11]], meaning it is best tuned with a sharpened fifth (around 715-720 cents or so).
 {{Todo| unify precision }}