Ploidacot/Alpha-tricot
Alpha-tricot is a temperament archetype where the generator is a wide tritone of about 625-640 cents, three of which stack to form a perfect twelfth of 3/1, and the period is a 2/1 octave. Alpha-tricot temperaments generate the 2L 5s, 2L 7s, and 2L 9s MOS structures. Alpha-tricot temperaments split the diatonic whole tone into three equal parts, producing both supermajor/subminor and supraminor/submajor intervals.
Intervals and notation
There is no agreed-upon notation for alpha-tricot, and constructing one by extending Pythagorean notation is complicated due to the fact that it does not split the chromatic or diatonic semitone, but rather their sum. Thus, there are two main options, based on interpreting the generator as a supradiminished fifth or a superaugmented fourth. The former is more melodically intuitive, but the latter adheres to the structure of certain temperaments better.
| # | Cents | Notation | Name (generator = fifth) | Notation | Name (generator = fourth) |
|---|---|---|---|---|---|
| -9 | 294.135 | Eb | minor third | Eb | minor third |
| -8 | 928.12 | ^Bbb | supradiminished seventh | ^A | supermajor sixth |
| -7 | 362.105 | vE | submajor third | vFb | subdiminished fourth |
| -6 | 996.09 | Bb | minor seventh | Bb | minor seventh |
| -5 | 430.075 | ^Fb | supradiminished fourth | ^E | supermajor third |
| -4 | 1064.06 | vB | submajor seventh | vCb | subdiminished octave |
| -3 | 498.045 | F | perfect fourth | F | perfect fourth |
| -2 | 1132.03 | ^Cb | supradiminished octave | ^B | supermajor seventh |
| -1 | 566.015 | vF# | subaugmented fourth | vGb | subdiminished fifth |
| 0 | 0 | C | perfect unison | C | perfect unison |
| 1 | 633.985 | ^Gb | supradiminished fifth | ^F# | superaugmented fourth |
| 2 | 67.97 | vC# | subaugmented unison | vDb | subminor second |
| 3 | 701.955 | G | perfect fifth | G | perfect fifth |
| 4 | 135.94 | ^Db | supraminor second | ^C# | superaugmented unison |
| 5 | 769.925 | vG# | subaugmented fifth | vAb | subminor sixth |
| 6 | 203.91 | D | major second | D | major second |
| 7 | 837.895 | ^Ab | supraminor sixth | ^G# | superaugmented fifth |
| 8 | 271.88 | vD# | subaugmented second | vEb | subminor third |
| 9 | 905.865 | A | major sixth | A | major sixth |
Temperament interpretations
Any valid alpha-tricot temperament assigns a just interpretation to the individual generator.
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 interpretation, the generator is 13/9, 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.
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.
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.
Paralimmal
Paralimmal maps the generator to 16/11, meaning it is best tuned with a sharpened fifth (around 715-720 cents or so).