Ploidacot/Alpha-tricot: Difference between revisions
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{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=1|Cots=3|Pergen=[P8, P11/3]|Forms=5, 7, 12|Title=Alpha-tricot}}'''Alpha-tricot''' is a temperament archetype where the generator is a wide tritone of about | {{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=1|Cots=3|Pergen=[P8, P11/3]|Forms=5, 7, 12|Title=Alpha-tricot}} | ||
'''Alpha-tricot''' is a temperament archetype where the generator is a wide tritone of about 625–640{{cent}}, 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 == | == 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. | 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. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Alpha-tricot intervals (assuming pure fifth and octave) | |+ style="font-size: 105%;" | Alpha-tricot intervals (assuming pure fifth and octave) | ||
!# | |- | ||
!Cents | ! # | ||
!Notation | ! Cents | ||
!Name (generator = fifth) | ! Notation | ||
!Notation | ! Name (generator = fifth) | ||
!Name (generator = fourth) | ! Notation | ||
! Name (generator = fourth) | |||
|- | |- | ||
| | | −9 | ||
|294.14 | | 294.14 | ||
|Eb | | Eb | ||
|minor third | | minor third | ||
|Eb | | Eb | ||
|minor third | | minor third | ||
|- | |- | ||
| | | −8 | ||
|928.12 | | 928.12 | ||
|^Bbb | | ^Bbb | ||
|supradiminished seventh | | supradiminished seventh | ||
|^A | | ^A | ||
|supermajor sixth | | supermajor sixth | ||
|- | |- | ||
| | | −7 | ||
|362.11 | | 362.11 | ||
|vE | | vE | ||
|submajor third | | submajor third | ||
|vFb | | vFb | ||
|subdiminished fourth | | subdiminished fourth | ||
|- | |- | ||
| | | −6 | ||
|996.09 | | 996.09 | ||
|Bb | | Bb | ||
|minor seventh | | minor seventh | ||
|Bb | | Bb | ||
|minor seventh | | minor seventh | ||
|- | |- | ||
| | | −5 | ||
|430.08 | | 430.08 | ||
|^Fb | | ^Fb | ||
|supradiminished fourth | | supradiminished fourth | ||
|^E | | ^E | ||
|supermajor third | | supermajor third | ||
|- | |- | ||
| | | −4 | ||
|1064.06 | | 1064.06 | ||
|vB | | vB | ||
|submajor seventh | | submajor seventh | ||
|vCb | | vCb | ||
|subdiminished octave | | subdiminished octave | ||
|- | |- | ||
| | | −3 | ||
|498.05 | | 498.05 | ||
|F | | F | ||
|perfect fourth | | perfect fourth | ||
|F | | F | ||
|perfect fourth | | perfect fourth | ||
|- | |- | ||
| | | −2 | ||
|1132.03 | | 1132.03 | ||
|^Cb | | ^Cb | ||
|supradiminished octave | | supradiminished octave | ||
|^B | | ^B | ||
|supermajor seventh | | supermajor seventh | ||
|- | |- | ||
| | | −1 | ||
|566.02 | | 566.02 | ||
|vF# | | vF# | ||
|subaugmented fourth | | subaugmented fourth | ||
|vGb | | vGb | ||
|subdiminished fifth | | subdiminished fifth | ||
|- | |- | ||
|0 | | 0 | ||
|0 | | 0 | ||
|C | | C | ||
|perfect unison | | perfect unison | ||
|C | | C | ||
|perfect unison | | perfect unison | ||
|- | |- | ||
|1 | | 1 | ||
|633.99 | | 633.99 | ||
|^Gb | | ^Gb | ||
|supradiminished fifth | | supradiminished fifth | ||
|^F# | | ^F# | ||
|superaugmented fourth | | superaugmented fourth | ||
|- | |- | ||
|2 | | 2 | ||
|67.97 | | 67.97 | ||
|vC# | | vC# | ||
|subaugmented unison | | subaugmented unison | ||
|vDb | | vDb | ||
|subminor second | | subminor second | ||
|- | |- | ||
|3 | | 3 | ||
|701.96 | | 701.96 | ||
|G | | G | ||
|perfect fifth | | perfect fifth | ||
|G | | G | ||
|perfect fifth | | perfect fifth | ||
|- | |- | ||
|4 | | 4 | ||
|135.94 | | 135.94 | ||
|^Db | | ^Db | ||
|supraminor second | | supraminor second | ||
|^C# | | ^C# | ||
|superaugmented unison | | superaugmented unison | ||
|- | |- | ||
|5 | | 5 | ||
|769.93 | | 769.93 | ||
|vG# | | vG# | ||
|subaugmented fifth | | subaugmented fifth | ||
|vAb | | vAb | ||
|subminor sixth | | subminor sixth | ||
|- | |- | ||
|6 | | 6 | ||
|203.91 | | 203.91 | ||
|D | | D | ||
|major second | | major second | ||
|D | | D | ||
|major second | | major second | ||
|- | |- | ||
|7 | | 7 | ||
|837.9 | | 837.9 | ||
|^Ab | | ^Ab | ||
|supraminor sixth | | supraminor sixth | ||
|^G# | | ^G# | ||
|superaugmented fifth | | superaugmented fifth | ||
|- | |- | ||
|8 | | 8 | ||
|271.88 | | 271.88 | ||
|vD# | | vD# | ||
|subaugmented second | | subaugmented second | ||
|vEb | | vEb | ||
|subminor third | | subminor third | ||
|- | |- | ||
|9 | | 9 | ||
|905.87 | | 905.87 | ||
|A | | A | ||
|major sixth | | major sixth | ||
|A | | A | ||
|major sixth | | major sixth | ||
|} | |} | ||
| Line 150: | Line 153: | ||
=== Threedic === | === 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 ( | 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. | ||
=== [[Alphatricot]] === | === [[Alphatricot]] === | ||
| Line 162: | Line 165: | ||
=== Paralimmal === | === Paralimmal === | ||
[[Paralimmal]] maps the generator to [[16/11]], meaning it is best tuned with a sharpened fifth (around | [[Paralimmal]] maps the generator to [[16/11]], meaning it is best tuned with a sharpened fifth (around 715–720{{c}} or so). | ||
{{Todo| unify precision }} | {{Todo| unify precision }} | ||