Deneb: Difference between revisions
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{{Todo|inline=1|complete page|comment=Add | {{Infobox regtemp | ||
| Title = Deneb | |||
| Subgroups = 3.5.11 | |||
| Comma basis = [[6655/6561]] | |||
| Edo join 1 = b28 | Edo join 2 = b39 | |||
| Mapping = 1; -3 1 | |||
| Generators = 11/9 | |||
| Generators tuning = 340.052 | |||
| Optimization method = CWE | |||
| MOS scales = {{mos scalesig|1L 4s<3/1>|link=1}}, {{mos scalesig|5L 1s<3/1>|link=1}}, {{mos scalesig|6L 5s<3/1>|link=1}}, {{mos scalesig|11L 6s<3/1>|link=1}} | |||
}} | |||
{{Todo|inline=1|complete page|comment=Add minimax error and target scale size. }} | |||
'''Deneb''' is a [[nonoctave]], [[tritave]]-repeating [[rank-2 temperament]] that tempers out the comma [[6655/6561]] in the 3.5.11 subgroup. It is generated by a neutral third that represents a flat [[11/9]], and three generators stacked represent [[9/5]] (and due to tritave-equivalence, -3 generators stacked represents [[5/3]]). It is one of the simplest temperaments in the 3.5.11 subgroup with decent accuracy. | '''Deneb''' is a [[nonoctave]], [[tritave]]-repeating [[rank-2 temperament]] that tempers out the comma [[6655/6561]] in the 3.5.11 subgroup. It is generated by a neutral third that represents a flat [[11/9]], and three generators stacked represent [[9/5]] (and due to tritave-equivalence, -3 generators stacked represents [[5/3]]). It is one of the simplest temperaments in the 3.5.11 subgroup with decent accuracy. | ||
| Line 60: | Line 72: | ||
| 478.409 | | 478.409 | ||
| [[33/25]] | | [[33/25]] | ||
| [[72/55]] | | [[72/55]], [[320/243]] | ||
|- | |- | ||
| 8 | | 8 | ||
| Line 158: | Line 170: | ||
| 1423.546 | | 1423.546 | ||
| [[25/11]] | | [[25/11]] | ||
| [[55/24]] | | [[55/24]], [[729/320]] | ||
|- | |- | ||
| -8 | | -8 | ||
| Line 223: | Line 235: | ||
[[Fomalhaut]] is a [[weak extension]] of Deneb to higher limits that splits the interval of [[11/3]] in three. Fomalhaut was considered in the no-2s no-7s 23-limit (3.5.11.13.17.19.23) from the start. It is considerably more complex than Deneb, and requires at least an 18-note MOS to properly use (technically, all prime harmonics 3-23 are accessible in the 13-note MOS, but the 13-note MOS is very [[hard]], and said prime harmonics are in different directions). | [[Fomalhaut]] is a [[weak extension]] of Deneb to higher limits that splits the interval of [[11/3]] in three. Fomalhaut was considered in the no-2s no-7s 23-limit (3.5.11.13.17.19.23) from the start. It is considerably more complex than Deneb, and requires at least an 18-note MOS to properly use (technically, all prime harmonics 3-23 are accessible in the 13-note MOS, but the 13-note MOS is very [[hard]], and said prime harmonics are in different directions). | ||
Another possible | Another possible extension of Deneb to 3.5.11.13 is [[No-twos subgroup temperaments#Threneb|Threneb]], which temper out the threedie, [[2197/2187]]. This splits the tritave into 3 [[13/9]]s, and 13/9 becomes the new period while the generator remains the same. The EDT join of this extension is b39&b45 (only EDTs that are a multiple of 3 are supported). Unfortunately, the smaller MOS scales generated by this are once again very [[hard]]. | ||
Also, the 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament. The way it is a weak extension of Deneb is that [[27/11]] (the tritave-complement of 11/9) is split into two [[11/7]]s. Mintra can then be further extended to 3.5.7.11.13. | Also, the 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament. The way it is a weak extension of Deneb is that [[27/11]] (the tritave-complement of 11/9) is split into two [[11/7]]s. Mintra can then be further extended to 3.5.7.11.13. | ||