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 [[sycamore comma]]. Its [[generator]] is a [[25/24 | classic chromatic semitone]], and stacking six of these gives 5/4 (and hence five 6/5) and eleven give 3/2. [[94edo]] [[support]]s 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 [[cent]]s 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|legend=1| 1 1 2 | 0 11 6 }}

: mapping generators: ~2, ~25/24

[[Optimal tuning]]s:  

: error map: {{val| 0.000 +0.103 -3.373 }}

{{Optimal ET sequence|legend=1| 18, 19, 56, 75, 94, 207c, 301c }}

[[Badness]] (Sintel): 4.925

== Septimal sycamore ==

The second element of the [[Normal lists #Normal interval list|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.

[[Comma list]]: 686/675, 875/864

{{Mapping|legend=1| 1 1 2 2 | 0 11 6 15 }}

[[Optimal tuning]]s:  

* [[CWE]]: ~2 = 1200.0000, ~25/24 = 64.0496

{{Optimal ET sequence|legend=1| 18, 19, 56, 75d }}

[[Badness]] (Sintel): 1.569

Subgroup: 2.3.5.7.11

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

Mapping: {{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|legend=0| 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: {{mapping| 1 1 2 2 4 3 | 0 11 6 15 -10 13 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000, ~25/24 = 64.2853

{{Optimal ET sequence|legend=0| 18, 19, 37, 56 }}

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Category:Temperament families]]

[[Category:Sycamore family ]] <!-- main article -->

[[Category:Sycamore| ]] <!-- key article -->