Sycamore family: Difference between revisions
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: mapping generators: ~2, ~25/24 | : mapping generators: ~2, ~25/24 | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.6031{{c}}, ~25/24 = 63.8108{{c}} | |||
: [[error map]]: {{val| +0.603 +0.567 -2.242 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/24 = 63.8234{{c}} | |||
: error map: {{val| 0.000 +0.103 -3.373 }} | |||
{{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]]: | [[Badness]] (Sintel): 4.925 | ||
== Septimal sycamore == | == Septimal sycamore == | ||
{{main| Sycamore and betic }} | |||
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]] is an excellent tuning for 7-limit sycamore, and [[56edo]] for the 11-limit version. | ||
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{{Mapping|legend=1| 1 1 2 2 | 0 11 6 15 }} | {{Mapping|legend=1| 1 1 2 2 | 0 11 6 15 }} | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.7208, ~25/24 = 64.0334 | |||
* [[CWE]]: ~2 = 1200.0000, ~25/24 = 64.0496 | |||
{{Optimal ET sequence|legend=1| 18, 19, 56, 75d }} | {{Optimal ET sequence|legend=1| 18, 19, 56, 75d }} | ||
[[Badness]]: | [[Badness]] (Sintel): 1.569 | ||
=== 11-limit === | === 11-limit === | ||
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Mapping: {{mapping| 1 1 2 2 4 | 0 11 6 15 -10 }} | Mapping: {{mapping| 1 1 2 2 4 | 0 11 6 15 -10 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.4126, ~25/24 = 64.2363 | |||
* CWE: ~2 = 1200.0000, ~25/24 = 64.2505 | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 18, 19, 37, 56 }} | ||
Badness: | Badness (Sintel): 1.849 | ||
=== 13-limit === | === 13-limit === | ||
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Mapping: {{mapping| 1 1 2 2 4 3 | 0 11 6 15 -10 13 }} | Mapping: {{mapping| 1 1 2 2 4 3 | 0 11 6 15 -10 13 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.6597, ~25/24 = 64.2778 | |||
* CWE: ~2 = 1200.0000, ~25/24 = 64.2853 | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 18, 19, 37, 56 }} | ||
Badness: | Badness (Sintel): 1.417 | ||
== Betic == | == Betic == | ||
{{main| Sycamore and 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. 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. | 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. | ||
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{{Mapping|legend=1| 1 1 2 1 | 0 11 6 34 }} | {{Mapping|legend=1| 1 1 2 1 | 0 11 6 34 }} | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.6891, ~25/24 = 63.7773 | |||
* [[CWE]]: ~2 = 1200.0000, ~25/24 = 63.7683 | |||
{{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]]: | [[Badness]] (Sintel): 1.765 | ||
=== 11-limit === | === 11-limit === | ||
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Mapping: {{mapping| 1 1 2 1 5 | 0 11 6 34 -29 }} | Mapping: {{mapping| 1 1 2 1 5 | 0 11 6 34 -29 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.4466, ~25/24 = 63.7993 | |||
* CWE: ~2 = 1200.0000, ~25/24 = 63.7796 | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 75, 94, 207c }} | ||
Badness: | Badness (Sintel): 1.880 | ||
=== 13-limit === | === 13-limit === | ||
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Mapping: {{mapping| 1 1 2 1 5 2 | 0 11 6 34 -29 32 }} | Mapping: {{mapping| 1 1 2 1 5 2 | 0 11 6 34 -29 32 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.3946, ~25/24 = 63.7867 | |||
* CWE: ~2 = 1200.0000, ~25/24 = 63.7702 | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 75, 94, 113, 207c }} | ||
Badness: | Badness (Sintel): 1.342 | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||