Sycamore family

The head of the sycamore family is 5-limit sycamore, which tempers out (25/24)⁶/(5/4) = [-16 -6 11⟩ = 48828125/47775744, the sycamore comma. The dual of the monzo is the wedgie, ⟨⟨11 6 -16]], which tells us that six classic chromatic semitone generators 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.

Sycamore

Subgroup: 2.3.5

Comma list: 48828125/47775744

Mapping: [⟨1 1 2], ⟨0 11 6]]

mapping generators: ~2, ~25/24

Optimal tuning (POTE): ~2 = 1\1, ~25/24 = 63.779

Optimal ET sequence: 18, 19, 56, 75, 94, 207c, 301c

Badness: 0.209966

Septimal sycamore

The second element of the 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 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.

Subgroup: 2.3.5.7

Comma list: 686/675, 875/864

Mapping: [⟨1 1 2 2], ⟨0 11 6 15]]

Wedgie: ⟨⟨11 6 15 -16 -7 18]]

Optimal tuning (POTE): ~2 = 1\1, ~25/24 = 63.995

Optimal ET sequence: 18, 19, 56, 75d

Badness: 0.062018

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 686/675

Mapping: [⟨1 1 2 2 4], ⟨0 11 6 15 -10]]

Optimal tuning (POTE): ~2 = 1\1, ~25/24 = 64.268

Optimal ET sequence: 18, 19, 37, 56

Badness: 0.055940

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 100/99, 169/168, 385/384

Mapping: [⟨1 1 2 2 4 3], ⟨0 11 6 15 -10 13]]

Optimal tuning (POTE): ~2 = 1\1, ~25/24 = 64.296

Optimal ET sequence: 18, 19, 37, 56

Badness: 0.034295

Betic

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. 94edo) 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. 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 …]].

Subgroup: 2.3.5.7

Comma list: 225/224, 1071875/1062882

Mapping: [⟨1 1 2 1], ⟨0 11 6 34]]

Wedgie: ⟨⟨11 6 34 -16 23 62]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 63.741

Optimal ET sequence: 19, 56d, 75, 94, 113, 320cc, 433ccd

Badness: 0.069748

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 218750/216513

Mapping: [⟨1 1 2 1 5], ⟨0 11 6 34 -29]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 63.776

Optimal ET sequence: 19, 75, 94, 207c

Badness: 0.056874

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 1875/1859

Mapping: [⟨1 1 2 1 5 2], ⟨0 11 6 34 -29 32]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 63.766

Optimal ET sequence: 19, 75, 94, 113, 207c

Badness: 0.032475