Sycamore family: Difference between revisions

The head of the '''sycamore family''' is [[5-limit]] sycamore, which tempers out (25/24)⁶/(5/4) = {{monzo| -16 -6 11 }} = 48828125/47775744. The dual of the [[monzo]] is the [[wedgie]], <<11 6 -16||, which tells us that six chromatic semitone [[generator]]s give 5/4 (and hence five 6/5) and eleven give 3/2. [[94edo]] supports sycamore, and 5\94 is recommendable as a generator. It can be described as the 19&94 temperament, and uses a decidedly flat version of the chromatic semitone as a generator. [[MOS]] of 18 or 19 notes to the octave give enough room for sycamore's triads, but 37 notes can be tried by the adventurous.

Another possible tuning uses a generator which is a pure 3/2 divided into 11 parts, and this makes the generator chain of sycamore exactly the same as [[Carlos Beta]]. In fact, Carlos Beta is characterized by Carlos as taking five steps to reach 6/5 and six to reach 5/4, which means it tempers out the sycamore comma. It can be described as the generator chain of sycamore, or sycamore can be called Carlos Beta with octaves.

Comma: 48828125/47775744

[[POTE generator]]: ~25/24 = 63.779

Map: [<1 1 2|, <0 11 6|]

{{EDOs|legend=1| 18, 19, 56, 75, 94, 207c, 301c }}

Badness: 0.2100

== 7-limit ==

The second element of the [[Normal lists|normal comma list]] for septimal sycamore is [[875/864]], the keema, and it also tempers out [[686/675]], the senga, and [[3136/3125]], hemimean. It has <<11 6 15 -16 -7 18|| for a wedgie, and may also be called the 19&56 temperament. This may also be used as the name for the temperament obtained by adding [[100/99]] to sycamore's commas, giving unidecimal sycamore, where 10 generator steps reaches 16/11, 11 reach 3/2, and 15 give 7/4, adding a considerable dose of 11-limit harmonies to the 19-note MOS. [[75edo]] is an excellent tuning for 7-limit sycamore, and [[56edo]] for the 11-limit version.

Commas: 686/675, 875/864

[[POTE generator]]: ~25/24 = 63.995

Map: [<1 1 2 2|, <0 11 6 15|]

{{EDOs|legend=1| 18, 19, 56, 75d }}

Badness: 0.0620

== 11-limit ==

Commas: 100/99, 385/384, 686/675

[[POTE generator]]: ~25/24 = 64.268

Map: [<1 1 2 2 4|, <0 11 6 15 -10|]

{{EDOs|legend=1| 18, 19, 37, 56 }}

Badness: 0.0559

== 13-limit ==

POTE generator: ~25/24 = 64.296

Map: [<1 1 2 2 4 3|, <0 11 6 15 -10 13|]

{{EDOs|legend=1| 18, 19, 37, 56 }}

Badness: 0.0343

Septimal sycamore sharpens the fifth from where it stands in the 5-limit, and lowers accuracy in order to reach 7-limit harmonies. If we retain tunings approximately (e.g. 94et) or exactly those of Carlos Beta, we get the 19&94 temperament, betic, for the 7-limit. This adds [[225/224]] to the sycamore comma, and has <<11 6 34 -16 23 62|| as a wedgie. The Carlos Beta tuning, with pure fifths, is a good tuning choice, but 94 or 113 equal are as well. Betic extends to the 11-limit upon addition of [[385/384]] or [[540/539]] to the list of commas, which means it supports both 7 and 11-limit marvel. The wedgie starts <<11 6 34 -29 ...||.

Commas: 225/224, 1071875/1062882

[[POTE generator]]: 63.701

Map: [<1 1 2 1|, <0 11 6 34|]

{{EDOs|legend=1| 19, 75, 94, 113, 320c, 433cd }}

== 11-limit ==

Commas: 225/224, 385/384, 218750/216513

[[POTE generator]]: 63.776

Map: [<1 1 2 1 5|, <0 11 6 34 -29|]

{{EDOs|legend=1| 19, 75, 94, 207c }}

[[Category:Theory]]

[[Category:Temperament family]]

[[Category:Sycamore]]

[[Category:Rank 2]]