Sycamore family: Difference between revisions

The head of the sycamore family is [[5-limit|5-limit]] sycamore, which tempers out (25/24)^6/(5/4) = |-16 -6 11> = 48828125/47775744. The dual of the [[monzo|monzo]] is the [[wedgie|wedgie]], <<11 6 -16||, which tells us that six chromatic semitone [[generator|generator]]s give 5/4 (and hence five 6/5) and eleven give 3/2. [[94edo|94edo]] supports sycamore, and 5\94 is reommendable 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|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|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_tuning|POTE generator]]: ~25/24 = 63.779

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

EDOs: [[18edo|18]], [[19edo|19]], [[56edo|56]], [[75edo|75]], [[94edo|94]], [[207edo|207c]], [[301edo|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|75edo]] is an excellent tuning for 7-limit sycamore, and [[56edo|56edo]] for the 11-limit version.

Commas: 686/675, 875/864

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

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

EDOs: 18, 19, 56, 75d

Badness: 0.0620

==11-limit==

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

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

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

EDOs: 18, 19, [[37edo|37]], [[56edo|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: 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 (eg 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_tuning|POTE generator]]: 63.701

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

EDOs: 19, 75, 94, [[113edo|113]], [[133edo|133]], [[320edo|320c]], [[433edo|433cd]]

11-limit

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

[[POTE_tuning|POTE generator]]: 63.776

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

EDOs: 19, 75, 94, 207c

[[Category:Theory]]

[[Category:Temperament family]]

[[Category:Sycamore]]