Sycamore family: Difference between revisions
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The head of the sycamore family is [[ | 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]], <<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&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 [[ | 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: 48828125/47775744 | ||
[[ | [[POTE generator]]: ~25/24 = 63.779 | ||
Map: [<1 1 2|, <0 11 6|] | Map: [<1 1 2|, <0 11 6|] | ||
EDOs | {{EDOs|legend=1| 18, 19, 56, 75, 94, 207c, 301c }} | ||
Badness: 0.2100 | Badness: 0.2100 | ||
= 7-limit = | |||
The second element of the [[ | 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]] is an excellent tuning for 7-limit sycamore, and [[56edo]] for the 11-limit version. | ||
Commas: 686/675, 875/864 | Commas: 686/675, 875/864 | ||
[[ | [[POTE generator]]: ~25/24 = 63.995 | ||
Map: [<1 1 2 2|, <0 11 6 15|] | Map: [<1 1 2 2|, <0 11 6 15|] | ||
EDOs | {{EDOs|legend=1| 18, 19, 56, 75d }} | ||
Badness: 0.0620 | Badness: 0.0620 | ||
==11-limit== | == 11-limit == | ||
Commas: 100/99, 385/384, 686/675 | Commas: 100/99, 385/384, 686/675 | ||
[[ | [[POTE generator]]: ~25/24 = 64.268 | ||
Map: [<1 1 2 2 4|, <0 11 6 15 -10|] | Map: [<1 1 2 2 4|, <0 11 6 15 -10|] | ||
EDOs | {{EDOs|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 | Commas: 91/90, 100/99, 169/168, 385/384 | ||
| Line 46: | Line 46: | ||
Map: [<1 1 2 2 4 3|, <0 11 6 15 -10 13|] | Map: [<1 1 2 2 4 3|, <0 11 6 15 -10 13|] | ||
EDOs | {{EDOs|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 ( | 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&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 | Commas: 225/224, 1071875/1062882 | ||
[[ | [[POTE generator]]: 63.701 | ||
Map: [<1 1 2 1|, <0 11 6 34|] | Map: [<1 1 2 1|, <0 11 6 34|] | ||
EDOs | {{EDOs|legend=1| 19, 75, 94, 113, 320c, 433cd }} | ||
11-limit | == 11-limit == | ||
Commas: 225/224, 385/384, 218750/216513 | Commas: 225/224, 385/384, 218750/216513 | ||
[[ | [[POTE generator]]: 63.776 | ||
Map: [<1 1 2 1 5|, <0 11 6 34 -29|] | Map: [<1 1 2 1 5|, <0 11 6 34 -29|] | ||
EDOs | {{EDOs|legend=1| 19, 75, 94, 207c }} | ||
[[Category:Theory]] | [[Category:Theory]] | ||
[[Category:Temperament family]] | [[Category:Temperament family]] | ||
[[Category:Sycamore]] | [[Category:Sycamore]] | ||
[[Category:Rank 2]] | |||
Revision as of 12:41, 11 January 2021
The head of the sycamore family is 5-limit sycamore, which tempers out (25/24)6/(5/4) = [-16 -6 11⟩ = 48828125/47775744. The dual of the monzo is the wedgie, <<11 6 -16||, which tells us that six 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
Comma: 48828125/47775744
POTE generator: ~25/24 = 63.779
Map: [<1 1 2|, <0 11 6|]
EDOs: 18, 19, 56, 75, 94, 207c, 301c
Badness: 0.2100
7-limit
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 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 is an excellent tuning for 7-limit sycamore, and 56edo for the 11-limit version.
Commas: 686/675, 875/864
POTE generator: ~25/24 = 63.995
Map: [<1 1 2 2|, <0 11 6 15|]
Badness: 0.0620
11-limit
Commas: 100/99, 385/384, 686/675
POTE generator: ~25/24 = 64.268
Map: [<1 1 2 2 4|, <0 11 6 15 -10|]
Badness: 0.0559
13-limit
Commas: 91/90, 100/99, 169/168, 385/384
POTE generator: ~25/24 = 64.296
Map: [<1 1 2 2 4 3|, <0 11 6 15 -10 13|]
Badness: 0.0343
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&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 generator: 63.701
Map: [<1 1 2 1|, <0 11 6 34|]
EDOs: 19, 75, 94, 113, 320c, 433cd
11-limit
Commas: 225/224, 385/384, 218750/216513
POTE generator: 63.776
Map: [<1 1 2 1 5|, <0 11 6 34 -29|]