Ploidacot/Alpha-tricot: Difference between revisions

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=== 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{{c}}) sharp of just.

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{{c}}) sharp of just.

=== [[Kangaroo]] ===

=== Kangaroo ===

Kangaroo is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with paralimmal (see below). This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-~~not~~ can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.

[[Kangaroo]] is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with paralimmal (see below). This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-note scale can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.

=== [[Alphatricot]] ===

=== Alphatricot ===

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 in the 2.3.5.13 subgroup the generator can be held to represent 13/9 (threedic, above) in alphatricot, or [[75/52]] ([[140625/140608|catasmic]], which is far more accurate) in alphatrillium. As a microtemperament, it is tuned best when the fifth is around just, especially in the case of alphatrillium.

Formerly just called ''tricot'', [[alphatricot]] 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 in the 2.3.5.13 subgroup the generator can be held to represent 13/9 (threedic, above) in alphatricot, or [[75/52]] ([[140625/140608|catasmic]], which is far more accurate) in alphatrillium. As a microtemperament, it is tuned best when the fifth is around just, especially in the case of alphatrillium.

=== 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.

=== 2.3.23 subgroup temperament ===

=== 2.3.23-subgroup temperament ===

An obvious mapping for the generator is [[23/16]], tempering out the comma [[12288/12167]]. 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{{c}} or so).

@@ Line 156: / Line 156: @@
 === 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{{c}}) sharp of just.
+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{{c}}) sharp of just.
-=== [[Kangaroo]] ===
+=== Kangaroo ===
-Kangaroo is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with paralimmal (see below). This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-not can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.
+[[Kangaroo]] is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with paralimmal (see below). This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-note scale can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.
-=== [[Alphatricot]] ===
+=== Alphatricot ===
-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 in the 2.3.5.13 subgroup the generator can be held to represent 13/9 (threedic, above) in alphatricot, or [[75/52]] ([[140625/140608|catasmic]], which is far more accurate) in alphatrillium. As a microtemperament, it is tuned best when the fifth is around just, especially in the case of alphatrillium.
+Formerly just called ''tricot'', [[alphatricot]] 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 in the 2.3.5.13 subgroup the generator can be held to represent 13/9 (threedic, above) in alphatricot, or [[75/52]] ([[140625/140608|catasmic]], which is far more accurate) in alphatrillium. As a microtemperament, it is tuned best when the fifth is around just, especially in the case of alphatrillium.
 === 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.
-=== 2.3.23 subgroup temperament ===
+=== 2.3.23-subgroup temperament ===
 An obvious mapping for the generator is [[23/16]], tempering out the comma [[12288/12167]]. 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{{c}} or so).

Ploidacot/Alpha-tricot: Difference between revisions

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