Ploidacot/Alpha-tricot: Difference between revisions
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{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=1|Cots=3|Pergen=[P8, P11/3]|Forms= | {{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 | ! rowspan="2" | # | ||
!Notation | ! rowspan="2" | Cents | ||
!Name | ! colspan="2" | Wide generator = fifth | ||
!Notation | ! colspan="2" | Wide generator = fourth | ||
!Name | |- | ||
! Notation | |||
! Name | |||
! Notation | |||
! Name | |||
|- | |- | ||
| | | −9 | ||
|294. | | 294.13 | ||
|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. | | 362.10 | ||
|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. | | 430.07 | ||
|^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. | | 498.04 | ||
|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. | | 566.01 | ||
|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. | | 837.90 | ||
|^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 156: | ||
=== 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]] === | ||
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 | 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. | ||
=== | === 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 | [[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]] | |||
Latest revision as of 12:03, 26 January 2026
| Pergen | [P8, P11/3] |
| Numeral form | 1-sheared 3-cot |
| Pure generator size | 566.01 ¢ |
| Pure period size | 1200 ¢ |
| Forms | 15, 17, 19, 36 |
| Characteristic multival entry | 3 |
Alpha-tricot is a temperament archetype where the generator is a wide tritone of about 630–635 ¢, 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 | Wide generator = fifth | Wide generator = fourth | ||
|---|---|---|---|---|---|
| Notation | Name | Notation | Name | ||
| −9 | 294.13 | Eb | minor third | Eb | minor third |
| −8 | 928.12 | ^Bbb | supradiminished seventh | ^A | supermajor sixth |
| −7 | 362.10 | vE | submajor third | vFb | subdiminished fourth |
| −6 | 996.09 | Bb | minor seventh | Bb | minor seventh |
| −5 | 430.07 | ^Fb | supradiminished fourth | ^E | supermajor third |
| −4 | 1064.06 | vB | submajor seventh | vCb | subdiminished octave |
| −3 | 498.04 | F | perfect fourth | F | perfect fourth |
| −2 | 1132.03 | ^Cb | supradiminished octave | ^B | supermajor seventh |
| −1 | 566.01 | vF# | subaugmented fourth | vGb | subdiminished fifth |
| 0 | 0 | C | perfect unison | C | perfect unison |
| 1 | 633.99 | ^Gb | supradiminished fifth | ^F# | superaugmented fourth |
| 2 | 67.97 | vC# | subaugmented unison | vDb | subminor second |
| 3 | 701.96 | G | perfect fifth | G | perfect fifth |
| 4 | 135.94 | ^Db | supraminor second | ^C# | superaugmented unison |
| 5 | 769.93 | vG# | subaugmented fifth | vAb | subminor sixth |
| 6 | 203.91 | D | major second | D | major second |
| 7 | 837.90 | ^Ab | supraminor sixth | ^G# | superaugmented fifth |
| 8 | 271.88 | vD# | subaugmented second | vEb | subminor third |
| 9 | 905.87 | A | major sixth | A | major sixth |
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 6/5s), and is best tuned with a fifth slightly (0–4 ¢) sharp of just.
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 (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
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 ¢ or so).