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Sycamore family: Difference between revisions

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The head of the '''sycamore family''' is [[5-limit]] sycamore, which tempers out (25/24)<sup>6</sup>/(5/4) = {{monzo| -16 -6 11 }} = 48828125/47775744. The dual of the [[monzo]] is the [[wedgie]], &lt;&lt;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&amp;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.
The head of the '''sycamore family''' is [[5-limit]] sycamore, which tempers out (25/24)<sup>6</sup>/(5/4) = {{monzo| -16 -6 11 }} = 48828125/47775744. The dual of the [[monzo]] is the [[wedgie]], {{multival| 11 6 -16 }}, which tells us that six classic 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&amp;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.
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 =
= Sycamore =
Comma: 48828125/47775744
[[Comma list]]: 48828125/47775744
 
[[Mapping]]: [{{val| 1 1 2 }}, {{val| 0 11 6 }}]


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


Map: [&lt;1 1 2|, &lt;0 11 6|]
{{Val list|legend=1| 18, 19, 56, 75, 94, 207c, 301c }}
 
[[Badness]]: 0.2100
 
= 7-limit =
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&amp;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.


{{EDOs|legend=1| 18, 19, 56, 75, 94, 207c, 301c }}
Subgroup: 2.3.5.7


Badness: 0.2100
[[Comma list]]: 686/675, 875/864


== 7-limit ==
[[Mapping]]: [{{val| 1 1 2 2 }}, {{val| 0 11 6 15 }}]
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 &lt;&lt;11 6 15 -16 -7 18|| for a wedgie, and may also be called the 19&amp;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
{{Multival|legend=1| 11 6 15 -16 -7 18 }}


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


Map: [&lt;1 1 2 2|, &lt;0 11 6 15|]
{{Val list|legend=1| 18, 19, 56, 75d }}
 
[[Badness]]: 0.0620


{{EDOs|legend=1| 18, 19, 56, 75d }}
== 11-limit ==


Badness: 0.0620
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 100/99, 385/384, 686/675
Commas: 100/99, 385/384, 686/675


[[POTE generator]]: ~25/24 = 64.268
Mapping: [{{val| 1 1 2 2 4 }}, {{val| 0 11 6 15 -10 }}]


Map: [&lt;1 1 2 2 4|, &lt;0 11 6 15 -10|]
POTE generator: ~25/24 = 64.268


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


Badness: 0.0559
Badness: 0.0559


== 13-limit ==
== 13-limit ==
Commas: 91/90, 100/99, 169/168, 385/384


POTE generator: ~25/24 = 64.296
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 2 2 4 3|, &lt;0 11 6 15 -10 13|]
Comma list: 91/90, 100/99, 169/168, 385/384


{{EDOs|legend=1| 18, 19, 37, 56 }}
Mapping: [{{val| 1 1 2 2 4 3 }}, {{val| 0 11 6 15 -10 13 }}]
 
POTE generator: ~26/25 = 64.296
 
{{Val list|legend=1| 18, 19, 37, 56 }}


Badness: 0.0343
Badness: 0.0343


= Betic =
= 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. 94et) or exactly those of Carlos Beta, we get the 19&amp;94 temperament, betic, for the 7-limit. This adds [[225/224]] to the sycamore comma, and has &lt;&lt;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 &lt;&lt;11 6 34 -29 ...||.
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&amp;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 {{multival| 11 6 34 -29 … }}.


Commas: 225/224, 1071875/1062882
Subgroup: 2.3.5.7


[[POTE generator]]: 63.701
[[Comma list]]: 225/224, 1071875/1062882


Map: [&lt;1 1 2 1|, &lt;0 11 6 34|]
[[Mapping]]: [{{val| 1 1 2 1 }}, {{val| 0 11 6 34 }}]


{{EDOs|legend=1| 19, 75, 94, 113, 320c, 433cd }}
{{Multival|legend=1| 11 6 34 -16 23 62 }}
 
[[POTE generator]]: ~28/27 = 63.776
 
{{Val list|legend=1| 19, 75, 94, 113, 320c, 433cd }}


== 11-limit ==
== 11-limit ==


Commas: 225/224, 385/384, 218750/216513
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 218750/216513


[[POTE generator]]: 63.776
Mapping: [{{val| 1 1 2 1 5 }}, {{val| 0 11 6 34 -29 }}]


Map: [&lt;1 1 2 1 5|, &lt;0 11 6 34 -29|]
POTE generator: ~28/27 = 63.776


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


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Sycamore]]
[[Category:Sycamore]]
[[Category:Rank 2]]
[[Category:Rank 2]]
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