Sycamore family - Revision history

Xenllium at 22:49, 22 May 2026

2026-05-22T22:49:52Z

← Older revision		Revision as of 22:49, 22 May 2026
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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 & 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]] ([[~~POTE~~]]): ~2 = ~~1\1~~, ~25/24 = 63.~~995~~		[[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]]: 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 ==
			{{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]] ([[~~POTE~~]]): ~2 = ~~1\1~~, ~28/27 = 63.~~741~~		[[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]]: 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]]

FloraC: Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

2025-11-27T12:27:42Z

Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

← Older revision		Revision as of 12:27, 27 November 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Sycamore family ]] <!-- main article -->		[[Category:Sycamore family ]] <!-- main article -->
	[[Category:Sycamore\| ]] <!-- key article -->		[[Category:Sycamore\| ]] <!-- key article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

Lériendil: Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

2025-06-24T00:35:22Z

Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

← Older revision		Revision as of 00:35, 24 June 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
			[[Category:Pages with mostly numerical content]]
	[[Category:Sycamore family ]] <!-- main article -->		[[Category:Sycamore family ]] <!-- main article -->
	[[Category:Sycamore\| ]] <!-- key article -->		[[Category:Sycamore\| ]] <!-- key article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

VectorGraphics: Phasing out wedgies

2025-06-12T04:11:12Z

Phasing out wedgies

← Older revision		Revision as of 04:11, 12 June 2025
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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]] ([[POTE]]): ~2 = 1\1, ~25/24 = 63.995		[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 63.995
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	== 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 & 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 & 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]] ([[POTE]]): ~2 = 1\1, ~28/27 = 63.741		[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 63.741

FloraC: Text replacement - "{{Technical data page}}

" to "{{Technical data page}}"

2025-04-13T13:45:25Z

Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"

← Older revision		Revision as of 13:45, 13 April 2025
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	{{Technical data page}}~~<br><br>~~		{{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 & 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.		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 & 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.

BudjarnLambeth: {{Technical data page}}

2025-04-13T06:52:43Z

{{Technical data page}}<br><br>

← Older revision		Revision as of 06:52, 13 April 2025
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			{{Technical data page}}<br><br>
	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 & 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.		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 & 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.

Sintel: Don't mention wedgies in lede

2024-12-21T19:22:26Z

Don't mention wedgies in lede

← Older revision		Revision as of 19:22, 21 December 2024
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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 & 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.		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 & 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.

Zhenlige at 15:51, 26 April 2024

2024-04-26T15:51:30Z

← Older revision		Revision as of 15:51, 26 April 2024
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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 & 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.		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 & 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 ==

FloraC: Update keys

2023-09-03T09:27:37Z

Update keys

← Older revision		Revision as of 09:27, 3 September 2023
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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~~]] [[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.		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 & 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 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 ==		== 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

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

	{{Optimal ET sequence\|legend=1\| 18, 19, 56, 75, 94, 207c, 301c }}		{{Optimal ET sequence\|legend=1\| 18, 19, 56, 75, 94, 207c, 301c }}
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	== 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&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 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.

	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 }}		{{Multival\|legend=1\| 11 6 15 -16 -7 18 }}

	[[POTE ~~generator~~]]: ~25/24 = 63.995		[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 63.995

	{{Optimal ET sequence\|legend=1\| 18, 19, 56, 75d }}		{{Optimal ET sequence\|legend=1\| 18, 19, 56, 75d }}
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	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 tuning (POTE): ~2 = 1\1, ~25/24 = 64.268

	{{Optimal ET sequence\|legend=1\| 18, 19, 37, 56 }}		{{Optimal ET sequence\|legend=1\| 18, 19, 37, 56 }}
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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: [{{~~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 tuning (POTE): ~2 = 1\1, ~25/24 = 64.296

	{{Optimal ET sequence\|legend=1\| 18, 19, 37, 56 }}		{{Optimal ET sequence\|legend=1\| 18, 19, 37, 56 }}
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	== 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&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 & 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 … }}.

	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 }}		{{Multival\|legend=1\| 11 6 34 -16 23 62 }}

	[[POTE ~~generator~~]]: ~28/27 = 63.741		[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 63.741

	{{Optimal ET sequence\|legend=1\| 19, 56d, 75, 94, 113, 320cc, 433ccd }}		{{Optimal ET sequence\|legend=1\| 19, 56d, 75, 94, 113, 320cc, 433ccd }}
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	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 tuning (POTE): ~2 = 1\1, ~28/27 = 63.776

	{{Optimal ET sequence\|legend=1\| 19, 75, 94, 207c }}		{{Optimal ET sequence\|legend=1\| 19, 75, 94, 207c }}
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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: [{{~~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 tuning (POTE): ~2 = 1\1, ~28/27 = 63.766

	{{Optimal ET sequence\|legend=1\| 19, 75, 94, 113, 207c }}		{{Optimal ET sequence\|legend=1\| 19, 75, 94, 113, 207c }}
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Sycamore]]		[[Category:Sycamore family ]] <!-- main article -->
			[[Category:Sycamore\| ]] <!-- key article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

Cmloegcmluin: "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

2023-05-07T16:52:11Z

"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

← Older revision		Revision as of 16:52, 7 May 2023
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	[[POTE generator]]: ~25/24 = 63.779		[[POTE generator]]: ~25/24 = 63.779

	{{~~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]]: 0.209966
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	[[POTE generator]]: ~25/24 = 63.995		[[POTE generator]]: ~25/24 = 63.995

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

	[[Badness]]: 0.062018		[[Badness]]: 0.062018
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	POTE generator: ~25/24 = 64.268		POTE generator: ~25/24 = 64.268

	Optimal ~~GPV~~ sequence~~: {{Val list~~\| 18, 19, 37, 56 }}		{{Optimal ET sequence\|legend=1\| 18, 19, 37, 56 }}

	Badness: 0.055940		Badness: 0.055940
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	POTE generator: ~25/24 = 64.296		POTE generator: ~25/24 = 64.296

	Optimal ~~GPV~~ sequence~~: {{Val list~~\| 18, 19, 37, 56 }}		{{Optimal ET sequence\|legend=1\| 18, 19, 37, 56 }}

	Badness: 0.034295		Badness: 0.034295
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	[[POTE generator]]: ~28/27 = 63.741		[[POTE generator]]: ~28/27 = 63.741

	{{~~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]]: 0.069748
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	POTE generator: ~28/27 = 63.776		POTE generator: ~28/27 = 63.776

	Optimal ~~GPV~~ sequence~~: {{Val list~~\| 19, 75, 94, 207c }}		{{Optimal ET sequence\|legend=1\| 19, 75, 94, 207c }}

	Badness: 0.056874		Badness: 0.056874
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	POTE generator: ~28/27 = 63.766		POTE generator: ~28/27 = 63.766

	Optimal ~~GPV~~ sequence~~: {{Val list~~\| 19, 75, 94, 113, 207c }}		{{Optimal ET sequence\|legend=1\| 19, 75, 94, 113, 207c }}

	Badness: 0.032475		Badness: 0.032475