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]], {{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|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. | |||
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&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. | ||
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{{Val list|legend=1| 18, 19, 56, 75, 94, 207c, 301c }} | {{Val list|legend=1| 18, 19, 56, 75, 94, 207c, 301c }} | ||
[[Badness]]: 0. | [[Badness]]: 0.209966 | ||
= | = 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. | 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|75EDO]] is an excellent tuning for 7-limit sycamore, and [[56edo|56EDO]] for the 11-limit version. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
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{{Val list|legend=1| 18, 19, 56, 75d }} | {{Val list|legend=1| 18, 19, 56, 75d }} | ||
[[Badness]]: 0. | [[Badness]]: 0.062018 | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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POTE generator: ~25/24 = 64.268 | POTE generator: ~25/24 = 64.268 | ||
{{Val list | Vals: {{Val list| 18, 19, 37, 56 }} | ||
Badness: 0. | Badness: 0.055940 | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 54: | Line 51: | ||
Mapping: [{{val| 1 1 2 2 4 3 }}, {{val| 0 11 6 15 -10 13 }}] | Mapping: [{{val| 1 1 2 2 4 3 }}, {{val| 0 11 6 15 -10 13 }}] | ||
POTE generator: ~ | POTE generator: ~25/24 = 64.296 | ||
{{Val list | Vals: {{Val list| 18, 19, 37, 56 }} | ||
Badness: 0. | Badness: 0.034295 | ||
= Betic = | = Betic = | ||
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{{Multival|legend=1| 11 6 34 -16 23 62 }} | {{Multival|legend=1| 11 6 34 -16 23 62 }} | ||
[[POTE generator]]: ~28/27 = 63. | [[POTE generator]]: ~28/27 = 63.741 | ||
{{Val list|legend=1| 19, 75, 94, 113, 320cc, 433ccd }} | {{Val list|legend=1| 19, 75, 94, 113, 320cc, 433ccd }} | ||
[[Badness]]: 0.069748 | |||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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POTE generator: ~28/27 = 63.776 | POTE generator: ~28/27 = 63.776 | ||
{{Val list| | Vals: {{Val list| 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: [{{val| 1 1 2 1 5 2 }}, {{val| 0 11 6 34 -29 32 }}] | |||
POTE generator: ~28/27 = 63.766 | |||
Vals: {{Val list| 19, 75, 94, 113, 207c }} | |||
Badness: 0.032475 | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||