Sycamore family
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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. Its generator is a classic chromatic semitone, and stacking six of these gives 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 near pure 3/2 at 702.162258 cents 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
- WE: ~2 = 1200.6031 ¢, ~25/24 = 63.8108 ¢
- error map: ⟨+0.603 +0.567 -2.242]
- CWE: ~2 = 1200.0000 ¢, ~25/24 = 63.8234 ¢
- error map: ⟨0.000 +0.103 -3.373]
Optimal ET sequence: 18, 19, 56, 75, 94, 207c, 301c
Badness (Sintel): 4.925
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]]
Optimal ET sequence: 18, 19, 56, 75d
Badness (Sintel): 1.569
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 tunings:
- WE: ~2 = 1199.4126, ~25/24 = 64.2363
- CWE: ~2 = 1200.0000, ~25/24 = 64.2505
Optimal ET sequence: 18, 19, 37, 56
Badness (Sintel): 1.849
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 tunings:
- WE: ~2 = 1199.6597, ~25/24 = 64.2778
- CWE: ~2 = 1200.0000, ~25/24 = 64.2853
Optimal ET sequence: 18, 19, 37, 56
Badness (Sintel): 1.417
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.
Subgroup: 2.3.5.7
Comma list: 225/224, 1071875/1062882
Mapping: [⟨1 1 2 1], ⟨0 11 6 34]]
Optimal ET sequence: 19, 56d, 75, 94, 113, 320cc, 433ccd
Badness (Sintel): 1.765
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 tunings:
- WE: ~2 = 1200.4466, ~25/24 = 63.7993
- CWE: ~2 = 1200.0000, ~25/24 = 63.7796
Optimal ET sequence: 19, 75, 94, 207c
Badness (Sintel): 1.880
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 tunings:
- WE: ~2 = 1200.3946, ~25/24 = 63.7867
- CWE: ~2 = 1200.0000, ~25/24 = 63.7702
Optimal ET sequence: 19, 75, 94, 113, 207c
Badness (Sintel): 1.342