Sycamore family: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
The | The '''sycamore family''' of [[regular temperament|temperaments]] tempers out the [[sycamore comma]] ({{monzo|legend=1| -16 -6 11 }}, [[ratio]]: 48828125/47775744). | ||
== Sycamore == | |||
The head of this family is [[5-limit]] sycamore. 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. | 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. | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
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{{Mapping|legend=1| 1 1 2 | 0 11 6 }} | {{Mapping|legend=1| 1 1 2 | 0 11 6 }} | ||
: mapping generators: ~2, ~25/24 | : mapping generators: ~2, ~25/24 | ||
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{{Optimal ET sequence|legend=1| 18, 19, 56, 75, 94, 207c, 301c }} | {{Optimal ET sequence|legend=1| 18, 19, 56, 75, 94, 207c, 301c }} | ||
[[Badness]] (Sintel): 4. | [[Badness]] (Sintel): 4.93 | ||
== Septimal sycamore == | == Septimal sycamore == | ||
The second element of the [[ | The second element of the [[normal forms #Normal forms for commas|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 undecimal 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 | [[Subgroup]]: 2.3.5.7 | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.7208, ~25/24 = 64.0334 | * [[WE]]: ~2 = 1200.7208{{c}}, ~25/24 = 64.0334{{c}} | ||
* [[CWE]]: ~2 = 1200.0000, ~25/24 = 64.0496 | : [[error map]]: {{val| +0.721 +3.133 -0.672 -6.884 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/24 = 64.0496{{c}} | |||
: error map: {{val| 0.000 +2.591 -2.016 -8.082 }} | |||
{{Optimal ET sequence|legend=1| 18, 19, 56, 75d }} | {{Optimal ET sequence|legend=1| 18, 19, 56, 75d }} | ||
[[Badness]] (Sintel): 1. | [[Badness]] (Sintel): 1.57 | ||
=== 11-limit === | === 11-limit === | ||
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Comma list: 100/99, 385/384, 686/675 | Comma list: 100/99, 385/384, 686/675 | ||
{{Mapping|legend=0| 1 1 2 2 4 | 0 11 6 15 -10 }} | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199.4126, ~25/24 = 64.2363 | * WE: ~2 = 1199.4126{{c}}, ~25/24 = 64.2363{{c}} | ||
* CWE: ~2 = 1200.0000, ~25/24 = 64.2505 | * CWE: ~2 = 1200.0000{{c}}, ~25/24 = 64.2505{{c}} | ||
{{Optimal ET sequence|legend=0| 18, 19, 37, 56 }} | {{Optimal ET sequence|legend=0| 18, 19, 37, 56 }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.85 | ||
=== 13-limit === | === 13-limit === | ||
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Comma list: 91/90, 100/99, 169/168, 385/384 | Comma list: 91/90, 100/99, 169/168, 385/384 | ||
{{Mapping|legend=0| 1 1 2 2 4 3 | 0 11 6 15 -10 13 }} | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199.6597, ~25/24 = 64.2778 | * WE: ~2 = 1199.6597{{c}}, ~25/24 = 64.2778{{c}} | ||
* CWE: ~2 = 1200.0000, ~25/24 = 64.2853 | * CWE: ~2 = 1200.0000{{c}}, ~25/24 = 64.2853{{c}} | ||
{{Optimal ET sequence|legend=0| 18, 19, 37, 56 }} | {{Optimal ET sequence|legend=0| 18, 19, 37, 56 }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.42 | ||
== Betic == | == Betic == | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.6891, ~25/24 = 63.7773 | * [[WE]]: ~2 = 1200.6891{{c}}, ~25/24 = 63.7773{{c}} | ||
* [[CWE]]: ~2 = 1200.0000, ~25/24 = 63.7683 | : [[error map]]: {{val| +0.689 +0.284 -2.272 +0.291 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/24 = 63.7683{{c}} | |||
: error map: {{val| 0.000 -0.504 -3.704 -0.703 }} | |||
{{Optimal ET sequence|legend=1| 19, 56d, 75, 94, 113, 320cc, 433ccd }} | {{Optimal ET sequence|legend=1| 19, 56d, 75, 94, 113, 320cc, 433ccd }} | ||
[[Badness]] (Sintel): 1. | [[Badness]] (Sintel): 1.77 | ||
=== 11-limit === | === 11-limit === | ||
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Comma list: 225/224, 385/384, 218750/216513 | Comma list: 225/224, 385/384, 218750/216513 | ||
{{Mapping|legend=0| 1 1 2 1 5 | 0 11 6 34 -29 }} | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200.4466, ~25/24 = 63.7993 | * WE: ~2 = 1200.4466{{c}}, ~25/24 = 63.7993{{c}} | ||
* CWE: ~2 = 1200.0000, ~25/24 = 63.7796 | * CWE: ~2 = 1200.0000{{c}}, ~25/24 = 63.7796{{c}} | ||
{{Optimal ET sequence|legend=0| 19, 75, 94, 207c }} | {{Optimal ET sequence|legend=0| 19, 75, 94, 207c }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.88 | ||
=== 13-limit === | === 13-limit === | ||
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Comma list: 225/224, 325/324, 385/384, 1875/1859 | Comma list: 225/224, 325/324, 385/384, 1875/1859 | ||
{{Mapping|legend=0| 1 1 2 1 5 2 | 0 11 6 34 -29 32 }} | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200.3946, ~25/24 = 63.7867 | * WE: ~2 = 1200.3946{{c}}, ~25/24 = 63.7867{{c}} | ||
* CWE: ~2 = 1200.0000, ~25/24 = 63.7702 | * CWE: ~2 = 1200.0000{{c}}, ~25/24 = 63.7702{{c}} | ||
{{Optimal ET sequence|legend=0| 19, 75, 94, 113, 207c }} | {{Optimal ET sequence|legend=0| 19, 75, 94, 113, 207c }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.34 | ||
[[Category:Sycamore family ]] <!-- main article --> | [[Category:Sycamore family ]] <!-- main article --> | ||