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&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 '''sycamore family''' of [[regular temperament|temperaments]] tempers out the [[sycamore comma]] ({{monzo|legend=1| -16 -6 11 }}, [[ratio]]: 48828125/47775744).  


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.
== 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.
 
[[Subgroup]]: 2.3.5


== Sycamore  ==
[[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 }}
: mapping generators: ~2, ~25/24


[[POTE generator]]: ~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 }}


{{Val list|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): 4.93


== 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.
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


[[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 }}


[[POTE generator]]: ~25/24 = 63.995
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7208{{c}}, ~25/24 = 64.0334{{c}}
: [[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 }}


{{Val list|legend=1| 18, 19, 56, 75d }}
{{Optimal ET sequence|legend=1| 18, 19, 56, 75d }}


[[Badness]]: 0.062018
[[Badness]] (Sintel): 1.57


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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{{c}}, ~25/24 = 64.2363{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/24 = 64.2505{{c}}


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


Badness: 0.055940
Badness (Sintel): 1.85


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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{{c}}, ~25/24 = 64.2778{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/24 = 64.2853{{c}}


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


Badness: 0.034295
Badness (Sintel): 1.42


== 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. 94et) 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 … }}.
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


[[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{{c}}, ~25/24 = 63.7773{{c}}
: [[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 }}


{{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): 1.77


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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{{c}}, ~25/24 = 63.7993{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/24 = 63.7796{{c}}


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


Badness: 0.056874
Badness (Sintel): 1.88


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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{{c}}, ~25/24 = 63.7867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/24 = 63.7702{{c}}


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


Badness: 0.032475
Badness (Sintel): 1.34


[[Category:Regular temperament theory]]
[[Category:Sycamore family ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Sycamore| ]] <!-- key article -->
[[Category:Sycamore]]
[[Category:Temperament families]]
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 04:53, 25 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The sycamore family of temperaments tempers out the sycamore comma (monzo: [-16 -6 11⟩, ratio: 48828125/47775744).

Sycamore

The head of this family is 5-limit sycamore. Its generator is a classic chromatic semitone, and stacking six of these gives 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 near pure 3/2 at 702.162258 cents 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

Comma list: 48828125/47775744

Mapping: [⟨1 1 2], ⟨0 11 6]]

mapping generators: ~2, ~25/24

Optimal tunings:

  • WE: ~2 = 1200.6031 ¢, ~25/24 = 63.8108 ¢
error map: ⟨+0.603 +0.567 -2.242]
  • CWE: ~2 = 1200.0000 ¢, ~25/24 = 63.8234 ¢
error map: ⟨0.000 +0.103 -3.373]

Optimal ET sequence: 18, 19, 56, 75, 94, 207c, 301c

Badness (Sintel): 4.93

Septimal sycamore

The second element of the 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

Comma list: 686/675, 875/864

Mapping: [⟨1 1 2 2], ⟨0 11 6 15]]

Optimal tunings:

  • WE: ~2 = 1200.7208 ¢, ~25/24 = 64.0334 ¢
error map: ⟨+0.721 +3.133 -0.672 -6.884]
  • CWE: ~2 = 1200.0000 ¢, ~25/24 = 64.0496 ¢
error map: ⟨0.000 +2.591 -2.016 -8.082]

Optimal ET sequence: 18, 19, 56, 75d

Badness (Sintel): 1.57

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 686/675

Mapping: [⟨1 1 2 2 4], ⟨0 11 6 15 -10]]

Optimal tunings:

  • WE: ~2 = 1199.4126 ¢, ~25/24 = 64.2363 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/24 = 64.2505 ¢

Optimal ET sequence: 18, 19, 37, 56

Badness (Sintel): 1.85

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 100/99, 169/168, 385/384

Mapping: [⟨1 1 2 2 4 3], ⟨0 11 6 15 -10 13]]

Optimal tunings:

  • WE: ~2 = 1199.6597 ¢, ~25/24 = 64.2778 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/24 = 64.2853 ¢

Optimal ET sequence: 18, 19, 37, 56

Badness (Sintel): 1.42

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.

Subgroup: 2.3.5.7

Comma list: 225/224, 1071875/1062882

Mapping: [⟨1 1 2 1], ⟨0 11 6 34]]

Optimal tunings:

  • WE: ~2 = 1200.6891 ¢, ~25/24 = 63.7773 ¢
error map: ⟨+0.689 +0.284 -2.272 +0.291]
  • CWE: ~2 = 1200.0000 ¢, ~25/24 = 63.7683 ¢
error map: ⟨0.000 -0.504 -3.704 -0.703]

Optimal ET sequence: 19, 56d, 75, 94, 113, 320cc, 433ccd

Badness (Sintel): 1.77

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 218750/216513

Mapping: [⟨1 1 2 1 5], ⟨0 11 6 34 -29]]

Optimal tunings:

  • WE: ~2 = 1200.4466 ¢, ~25/24 = 63.7993 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/24 = 63.7796 ¢

Optimal ET sequence: 19, 75, 94, 207c

Badness (Sintel): 1.88

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 1875/1859

Mapping: [⟨1 1 2 1 5 2], ⟨0 11 6 34 -29 32]]

Optimal tunings:

  • WE: ~2 = 1200.3946 ¢, ~25/24 = 63.7867 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/24 = 63.7702 ¢

Optimal ET sequence: 19, 75, 94, 113, 207c

Badness (Sintel): 1.34

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