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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&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 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 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.


== Sycamore ==
== Sycamore ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 48828125/47775744
[[Comma list]]: 48828125/47775744


[[Mapping]]: [{{val| 1 1 2 }}, {{val| 0 11 6 }}]
{{Mapping|legend=1| 1 1 2 | 0 11 6 }}


[[POTE generator]]: ~25/24 = 63.779
: mapping generators: ~2, ~25/24


{{Val list|legend=1| 18, 19, 56, 75, 94, 207c, 301c }}
[[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 }}


[[Badness]]: 0.209966
{{Optimal ET sequence|legend=1| 18, 19, 56, 75, 94, 207c, 301c }}
 
[[Badness]] (Sintel): 4.925


== Septimal sycamore ==
== 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&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|75EDO]] is an excellent tuning for 7-limit sycamore, and [[56edo|56EDO]] for the 11-limit version.
{{main| Sycamore and betic }}


Subgroup: 2.3.5.7
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.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 686/675, 875/864
[[Comma list]]: 686/675, 875/864


[[Mapping]]: [{{val| 1 1 2 2 }}, {{val| 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
* [[CWE]]: ~2 = 1200.0000, ~25/24 = 64.0496


[[POTE generator]]: ~25/24 = 63.995
{{Optimal ET sequence|legend=1| 18, 19, 56, 75d }}


{{Val list|legend=1| 18, 19, 56, 75d }}
[[Badness]] (Sintel): 1.569
 
[[Badness]]: 0.062018


=== 11-limit ===
=== 11-limit ===
Line 38: Line 47:
Comma list: 100/99, 385/384, 686/675
Comma list: 100/99, 385/384, 686/675


Mapping: [{{val| 1 1 2 2 4 }}, {{val| 0 11 6 15 -10 }}]
Mapping: {{mapping| 1 1 2 2 4 | 0 11 6 15 -10 }}


POTE generator: ~25/24 = 64.268
Optimal tunings:  
* WE: ~2 = 1199.4126, ~25/24 = 64.2363
* CWE: ~2 = 1200.0000, ~25/24 = 64.2505


Vals: {{Val list| 18, 19, 37, 56 }}
{{Optimal ET sequence|legend=0| 18, 19, 37, 56 }}


Badness: 0.055940
Badness (Sintel): 1.849


=== 13-limit ===
=== 13-limit ===
Line 51: Line 62:
Comma list: 91/90, 100/99, 169/168, 385/384
Comma list: 91/90, 100/99, 169/168, 385/384


Mapping: [{{val| 1 1 2 2 4 3 }}, {{val| 0 11 6 15 -10 13 }}]
Mapping: {{mapping| 1 1 2 2 4 3 | 0 11 6 15 -10 13 }}


POTE generator: ~25/24 = 64.296
Optimal tunings:  
* WE: ~2 = 1199.6597, ~25/24 = 64.2778
* CWE: ~2 = 1200.0000, ~25/24 = 64.2853


Vals: {{Val list| 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 }}


Subgroup: 2.3.5.7
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


[[Comma list]]: 225/224, 1071875/1062882
[[Comma list]]: 225/224, 1071875/1062882


[[Mapping]]: [{{val| 1 1 2 1 }}, {{val| 0 11 6 34 }}]
{{Mapping|legend=1| 1 1 2 1 | 0 11 6 34 }}
 
{{Multival|legend=1| 11 6 34 -16 23 62 }}


[[POTE generator]]: ~28/27 = 63.741
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.6891, ~25/24 = 63.7773
* [[CWE]]: ~2 = 1200.0000, ~25/24 = 63.7683


{{Val list|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 ===
Line 81: Line 96:
Comma list: 225/224, 385/384, 218750/216513
Comma list: 225/224, 385/384, 218750/216513


Mapping: [{{val| 1 1 2 1 5 }}, {{val| 0 11 6 34 -29 }}]
Mapping: {{mapping| 1 1 2 1 5 | 0 11 6 34 -29 }}


POTE generator: ~28/27 = 63.776
Optimal tunings:  
* WE: ~2 = 1200.4466, ~25/24 = 63.7993
* CWE: ~2 = 1200.0000, ~25/24 = 63.7796


Vals: {{Val list| 19, 75, 94, 207c }}
{{Optimal ET sequence|legend=0| 19, 75, 94, 207c }}


Badness: 0.056874
Badness (Sintel): 1.880


=== 13-limit ===
=== 13-limit ===
Line 94: Line 111:
Comma list: 225/224, 325/324, 385/384, 1875/1859
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 }}]
Mapping: {{mapping| 1 1 2 1 5 2 | 0 11 6 34 -29 32 }}


POTE generator: ~28/27 = 63.766
Optimal tunings:  
* WE: ~2 = 1200.3946, ~25/24 = 63.7867
* CWE: ~2 = 1200.0000, ~25/24 = 63.7702


Vals: {{Val list| 19, 75, 94, 113, 207c }}
{{Optimal ET sequence|legend=0| 19, 75, 94, 113, 207c }}


Badness: 0.032475
Badness (Sintel): 1.342


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Sycamore family ]] <!-- main article -->
[[Category:Sycamore]]
[[Category:Sycamore| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Sycamore_family"