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'''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.
{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=1|Cots=3|Pergen=[P8, P11/3]|Forms=15, 17, 19, 36|Title=Alpha-tricot|Wedgie=3}}
'''Alpha-tricot''' is a temperament archetype where the generator is a wide tritone of about 630–635{{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
!Name (generator = fifth)
!Notation
!Name (generator = fourth)
|-
|-
| -9
! rowspan="2" | #
|294.135
! rowspan="2" | Cents
|Eb
! colspan="2" | Wide generator = fifth
|minor third
! colspan="2" | Wide generator = fourth
|Eb
|minor third
|-
|-
| -8
! Notation
|928.12
! Name
|^Bbb
! Notation
|supradiminished seventh
! Name
|^A
|supermajor sixth
|-
|-
| -7
| −9
|362.105
| 294.13
|vE
| Eb
|submajor third
| minor third
|vFb
| Eb
|subdiminished fourth
| minor third
|-
|-
| -6
| −8
|996.09
| 928.12
|Bb
| ^Bbb
|minor seventh
| supradiminished seventh
|Bb
| ^A
|minor seventh
| supermajor sixth
|-
|-
| -5
| −7
|430.075
| 362.10
|^Fb
| vE
|supradiminished fourth
| submajor third
|^E
| vFb
|supermajor third
| subdiminished fourth
|-
|-
| -4
| −6
|1064.06
| 996.09
|vB
| Bb
|submajor seventh
| minor seventh
|vCb
| Bb
|subdiminished octave
| minor seventh
|-
|-
| -3
| −5
|498.045
| 430.07
|F
| ^Fb
|perfect fourth
| supradiminished fourth
|F
| ^E
|perfect fourth
| supermajor third
|-
|-
| -2
| −4
|1132.03
| 1064.06
|^Cb
| vB
|supradiminished octave
| submajor seventh
|^B
| vCb
|supermajor seventh
| subdiminished octave
|-
|-
| -1
| −3
|566.015
| 498.04
|vF#
| F
|subaugmented fourth
| perfect fourth
|vGb
| F
|subdiminished fifth
| perfect fourth
|-
|-
|0
| −2
|0
| 1132.03
|C
| ^Cb
|perfect unison
| supradiminished octave
|C
| ^B
|perfect unison
| supermajor seventh
|-
|-
|1
| −1
|633.985
| 566.01
|^Gb
| vF#
|supradiminished fifth
| subaugmented fourth
|^F#
| vGb
|superaugmented fourth
| subdiminished fifth
|-
|-
|2
| 0
|67.97
| 0
|vC#
| C
|subaugmented unison
| perfect unison
|vDb
| C
|subminor second
| perfect unison
|-
|-
|3
| 1
|701.955
| 633.99
|G
| ^Gb
|perfect fifth
| supradiminished fifth
|G
| ^F#
|perfect fifth
| superaugmented fourth
|-
|-
|4
| 2
|135.94
| 67.97
|^Db
| vC#
|supraminor second
| subaugmented unison
|^C#
| vDb
|superaugmented unison
| subminor second
|-
|-
|5
| 3
|769.925
| 701.96
|vG#
| G
|subaugmented fifth
| perfect fifth
|vAb
| G
|subminor sixth
| perfect fifth
|-
|-
|6
| 4
|203.91
| 135.94
|D
| ^Db
|major second
| supraminor second
|D
| ^C#
|major second
| superaugmented unison
|-
|-
|7
| 5
|837.895
| 769.93
|^Ab
| vG#
|supraminor sixth
| subaugmented fifth
|^G#
| vAb
|superaugmented fifth
| subminor sixth
|-
|-
|8
| 6
|271.88
| 203.91
|vD#
| D
|subaugmented second
| major second
|vEb
| D
|subminor third
| major second
|-
|-
|9
| 7
|905.865
| 837.90
|A
| ^Ab
|major sixth
| supraminor sixth
|A
| ^G#
|major sixth
| superaugmented fifth
|-
| 8
| 271.88
| vD#
| subaugmented second
| vEb
| subminor third
|-
| 9
| 905.87
| A
| major sixth
| A
| major sixth
|}
|}


== Temperament interpretations ==
== Temperament interpretations ==
Any valid alpha-tricot temperament assigns a just interpretation to the individual generator.
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{{c}}) sharp of just.


=== [[Alphatricot]] ===
=== [[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.
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.


=== Liese ===
=== 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 ===
=== 2.3.23 subgroup temperament ===
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]], 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 ===
[[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{{c}} or so).
 
[[Category:Ploidacots|Alpha-tricot]]
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