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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 [[sycamore comma]]. 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]] [[support]]s 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.
{{Technical data page}}
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 [[sycamore comma]]. 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 &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 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.
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: mapping generators: ~2, ~25/24
: mapping generators: ~2, ~25/24


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 63.779
[[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]]: 0.209966
[[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 &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.
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.


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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 }}


{{Multival|legend=1| 11 6 15 -16 -7 18 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.7208, ~25/24 = 64.0334
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 63.995
* [[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]]: 0.062018
[[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 tuning (POTE): ~2 = 1\1, ~25/24 = 64.268
Optimal tunings:
* WE: ~2 = 1199.4126, ~25/24 = 64.2363
* CWE: ~2 = 1200.0000, ~25/24 = 64.2505


{{Optimal ET sequence|legend=1| 18, 19, 37, 56 }}
{{Optimal ET sequence|legend=0| 18, 19, 37, 56 }}


Badness: 0.055940
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 tuning (POTE): ~2 = 1\1, ~25/24 = 64.296
Optimal tunings:
* WE: ~2 = 1199.6597, ~25/24 = 64.2778
* CWE: ~2 = 1200.0000, ~25/24 = 64.2853


{{Optimal ET sequence|legend=1| 18, 19, 37, 56 }}
{{Optimal ET sequence|legend=0| 18, 19, 37, 56 }}


Badness: 0.034295
Badness (Sintel): 1.417


== 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. 94edo) 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 … }}.
{{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 &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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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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 }}


{{Multival|legend=1| 11 6 34 -16 23 62 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.6891, ~25/24 = 63.7773
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 63.741
* [[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]]: 0.069748
[[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 tuning (POTE): ~2 = 1\1, ~28/27 = 63.776
Optimal tunings:
* WE: ~2 = 1200.4466, ~25/24 = 63.7993
* CWE: ~2 = 1200.0000, ~25/24 = 63.7796


{{Optimal ET sequence|legend=1| 19, 75, 94, 207c }}
{{Optimal ET sequence|legend=0| 19, 75, 94, 207c }}


Badness: 0.056874
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 tuning (POTE): ~2 = 1\1, ~28/27 = 63.766
Optimal tunings:
* WE: ~2 = 1200.3946, ~25/24 = 63.7867
* CWE: ~2 = 1200.0000, ~25/24 = 63.7702


{{Optimal ET sequence|legend=1| 19, 75, 94, 113, 207c }}
{{Optimal ET sequence|legend=0| 19, 75, 94, 113, 207c }}


Badness: 0.032475
Badness (Sintel): 1.342


[[Category:Temperament families]]
[[Category:Temperament families]]
Retrieved from "https://en.xen.wiki/w/Sycamore_family"