Ploidacot/Alpha-tricot: Difference between revisions
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{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=1|Cots=3|Pergen=[P8, P11/3]|Forms=15, 17, 19, 36|Title=Alpha-tricot|Wedgie=3}} | {{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 | '''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 == | ||
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|+ style="font-size: 105%;" | Alpha-tricot intervals (assuming pure fifth and octave) | |+ style="font-size: 105%;" | Alpha-tricot intervals (assuming pure fifth and octave) | ||
|- | |- | ||
! # | ! rowspan="2" | # | ||
! Cents | ! rowspan="2" | Cents | ||
! colspan="2" | Wide generator = fifth | |||
! colspan="2" | Wide generator = fourth | |||
|- | |||
! Notation | ! Notation | ||
! Name | ! Name | ||
! Notation | ! Notation | ||
! Name | ! Name | ||
|- | |- | ||
| −9 | | −9 | ||
| 294. | | 294.13 | ||
| Eb | | Eb | ||
| minor third | | minor third | ||
| Line 30: | Line 33: | ||
|- | |- | ||
| −7 | | −7 | ||
| 362. | | 362.10 | ||
| vE | | vE | ||
| submajor third | | submajor third | ||
| Line 44: | Line 47: | ||
|- | |- | ||
| −5 | | −5 | ||
| 430. | | 430.07 | ||
| ^Fb | | ^Fb | ||
| supradiminished fourth | | supradiminished fourth | ||
| Line 58: | Line 61: | ||
|- | |- | ||
| −3 | | −3 | ||
| 498. | | 498.04 | ||
| F | | F | ||
| perfect fourth | | perfect fourth | ||
| Line 72: | Line 75: | ||
|- | |- | ||
| −1 | | −1 | ||
| 566. | | 566.01 | ||
| vF# | | vF# | ||
| subaugmented fourth | | subaugmented fourth | ||
| Line 128: | Line 131: | ||
|- | |- | ||
| 7 | | 7 | ||
| 837. | | 837.90 | ||
| ^Ab | | ^Ab | ||
| supraminor sixth | | supraminor sixth | ||
| Line 161: | Line 164: | ||
[[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 715–720{{c}} 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]] | [[Category:Ploidacots|Alpha-tricot]] | ||