User:Contribution/Successive superparticular complementary pair - Revision history

Contribution at 00:21, 28 October 2025

2025-10-28T00:21:57Z

← Older revision		Revision as of 00:21, 28 October 2025
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			== Context ==

			Read this first: [[Equal-step_tuning#Alpha-beta-gamma_family_of_equal_divisions]]

	== The Alpha-Beta-Gamma family ==		== The Alpha-Beta-Gamma family ==

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	== The fact ==		== The converging Alpha-Beta-Gamma sequence ==

			As a fact, for each <math>n\ge 2</math>, equal divisions of <math>R_n=\dfrac{n+1}{n-1}</math> where low errors appear for <math>S_n=\dfrac{n+1}{n}</math> and <math>B_n=\dfrac{n}{n-1}</math> forms a converging sequence and pattern, with the happy equal divisions of <math>R_n</math> being:
			* '''Alpha:''' <math>k_\alpha=2n-1</math>
			* '''Beta:''' <math>k_\beta=2n+1</math>
			* '''Gamma:''' <math>k_\gamma=4n=k_\alpha+k_\beta</math>

	~~As a fact~~, ~~all Alpha scales~~ are ~~(s1 + s2)ED(a / b), all Beta scales are (s2 + s3)ED(a / b),~~ and ~~all Gamma scales are (s1 + s2 + s2 + s3)ED(a / b)~~.		In this sequence, the errors are lower and lower.

	{{todo\|Why this pattern\|inline=1\|comment=Explain why low errors make this pattern appears.}}		{{todo\|Why this pattern\|inline=1\|comment=Explain why divisions of ratios where low errors appear for successive superparticular complementary pair make this pattern appears.}}

	{\| class="wikitable sortable right-1 left-2 right-3 left-4 right-5 left-6 right-7 left-8 right-9 left-10 right-11 left-12 right-13 left-14 right-15 left-16 right-17 left-18 right-19 left-20"		{\| class="wikitable sortable right-1 left-2 right-3 left-4 right-5 left-6 right-7 left-8 right-9 left-10 right-11 left-12 right-13 left-14 right-15 left-16 right-17 left-18 right-19 left-20"

Contribution at 00:03, 28 October 2025

2025-10-28T00:03:44Z

← Older revision		Revision as of 00:03, 28 October 2025
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	~~{{todo\|Finish the article and move it\|inline~~=~~1\|comment~~=~~When the article is finished and the table explained, move it to the main root}}~~		== The Alpha-Beta-Gamma family ==

	For each pair of superparticular ratios <math>{s1}/{s2}</math> and <math>{s2}/{s3}</math>, there exists a ratio <math>{a}/{b}</math> such that <math>{s1}/{s2}</math> and <math>{s2}/{s3}</math> are <math>{a}/{b}</math> complementary; it is observed that <math>a−b=1</math> or <math>a−b=2</math>.
	In other words, for each ratio <math>a/b</math> where <math>a−b=1</math> or <math>a−b=2</math>, there exists a pair of superparticular ratios <math>{s1}/{s2}</math> and <math>{s2}/{s3}</math> that are <math>{a}/{b}</math> complementary.

	Bellow is a table that show for equal divisions of <math>a/b</math> the cent error in the mapping of superparticular ratios <math>{s1}/{s2}</math> and <math>{s2}/{s3}</math> that are <math>a/b</math> complementary.

	We can observe a converging sequence and pattern for low errors: 5, 7, 12; then 7, 9, 16; then 9, 11, 20; then 11, 13, 24; then 13, 15, 28; then 15, 17, 32; then 17, 19, 36; then 19, 21, 40; then 21, 23, 44; etc. --

	~~{{todo\|Tables\|inline=1\|comment=Explain the tables.}}~~

	~~{{todo\|Pattern\|inline=1\|comment=Clarify the observed pattern and create a descriptive name for it, such as the "~~Alpha-Beta-Gamma pattern" or the "Alpha-Beta-Gamma class" when referring to the group of scales. Assign distinct names to each scale within this class. For instance, 5edo might be called "2/1 Alpha", 7edo could be "2/1 Beta", and 12edo could be "2/1 Gamma". Additionally, compute the Dave Benson optimization for each scale as an alternative tuning. Note: 23edo with stretched octave is "7/5 Alpha".
	Consider this second version for naming: "Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5." Consistency and Clarity: The second version ("Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5") places the descriptive name ("Alpha," "Beta," "Gamma") before the ratio. This makes it clear that "Alpha," "Beta," "Gamma," and so on are categories or types, while "2/1" and "7/5" are specific tunings or ratios within those categories. This ordering helps maintain a logical structure that is easier to follow.}}

	~~{{todo\|Add data\|inline~~=~~1\|comment~~=Include columns for the following optimizations: Pure Equave, Dave Benson, TE, POTE, and CTE. To prevent the table from becoming overcrowded, choose between using cents or EDO. Cents are preferable since these are EDONOI. Retain the EDO column for pure equaves to provide a sense of scale, but place it outside the optimization subgroup column.}}

	{\| class="wikitable"		{\| class="wikitable"
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	\| 0, -0.003, 0.003		\| 0, -0.003, 0.003
	\|}		\|}
	~~{{todo\|Temperaments\|inline=1\|comment=Compute the temperaments associated to each Alpha-Beta-Gamma scales.}}~~

	== The fact ==		== The fact ==

	As a fact, all Alpha scales are (s1 + s2)ED(a / b), all Beta scales are (s2 + s3)ED(a / b), and all Gamma scales are (s1 + s2 + s2 + s3)ED(a / b).		As a fact, all Alpha scales are (s1 + s2)ED(a / b), all Beta scales are (s2 + s3)ED(a / b), and all Gamma scales are (s1 + s2 + s2 + s3)ED(a / b).

			{{todo\|Why this pattern\|inline=1\|comment=Explain why low errors make this pattern appears.}}

	{\| class="wikitable sortable right-1 left-2 right-3 left-4 right-5 left-6 right-7 left-8 right-9 left-10 right-11 left-12 right-13 left-14 right-15 left-16 right-17 left-18 right-19 left-20"		{\| class="wikitable sortable right-1 left-2 right-3 left-4 right-5 left-6 right-7 left-8 right-9 left-10 right-11 left-12 right-13 left-14 right-15 left-16 right-17 left-18 right-19 left-20"

Contribution at 23:55, 27 October 2025

2025-10-27T23:55:13Z

Show changes

Contribution at 16:17, 25 June 2025

2025-06-25T16:17:59Z

← Older revision		Revision as of 16:17, 25 June 2025
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	{{todo\|Temperaments\|inline=1\|comment=Compute the temperaments associated to each Alpha-Beta-Gamma scales.}}		{{todo\|Temperaments\|inline=1\|comment=Compute the temperaments associated to each Alpha-Beta-Gamma scales.}}

	== ~~Coincidence?~~ ==		== The fact ==

	As a ~~coincidence (?)~~, all Alpha scales are (s1 + s2)ED(a / b), all Beta scales are (s2 + s3)ED(a / b), and all Gamma scales are (s1 + s2 + s2 + s3)ED(a / b).		As a fact, all Alpha scales are (s1 + s2)ED(a / b), all Beta scales are (s2 + s3)ED(a / b), and all Gamma scales are (s1 + s2 + s2 + s3)ED(a / b).

	{\| class="wikitable sortable right-1 left-2 right-3 left-4 right-5 left-6 right-7 left-8 right-9 left-10 right-11 left-12 right-13 left-14 right-15 left-16 right-17 left-18 right-19 left-20"		{\| class="wikitable sortable right-1 left-2 right-3 left-4 right-5 left-6 right-7 left-8 right-9 left-10 right-11 left-12 right-13 left-14 right-15 left-16 right-17 left-18 right-19 left-20"

Contribution at 15:09, 5 September 2024

2024-09-05T15:09:58Z

← Older revision		Revision as of 15:09, 5 September 2024
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	Consider this second version for naming: "Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5." Consistency and Clarity: The second version ("Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5") places the descriptive name ("Alpha," "Beta," "Gamma") before the ratio. This makes it clear that "Alpha," "Beta," "Gamma," and so on are categories or types, while "2/1" and "7/5" are specific tunings or ratios within those categories. This ordering helps maintain a logical structure that is easier to follow.}}		Consider this second version for naming: "Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5." Consistency and Clarity: The second version ("Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5") places the descriptive name ("Alpha," "Beta," "Gamma") before the ratio. This makes it clear that "Alpha," "Beta," "Gamma," and so on are categories or types, while "2/1" and "7/5" are specific tunings or ratios within those categories. This ordering helps maintain a logical structure that is easier to follow.}}

	{{todo\|Add data\|inline=1\|comment=Include columns for the following optimizations: Pure Equave, Dave Benson, TE, POTE, and CTE. To prevent the table from becoming overcrowded, choose between using cents or EDO. Cents are preferable since these are EDONOI. Retain the EDO column for pure equaves to provide a sense of scale, but place it outside the optimization subgroup column~~. --- These are non-octave scales. Octave-based optimizations such as POTE, CTE or CTWE don't really make sense. However, TE is still relevant~~.}}		{{todo\|Add data\|inline=1\|comment=Include columns for the following optimizations: Pure Equave, Dave Benson, TE, POTE, and CTE. To prevent the table from becoming overcrowded, choose between using cents or EDO. Cents are preferable since these are EDONOI. Retain the EDO column for pure equaves to provide a sense of scale, but place it outside the optimization subgroup column.}}

	{\| class="wikitable"		{\| class="wikitable"

Contribution at 14:37, 5 September 2024

2024-09-05T14:37:18Z

← Older revision		Revision as of 14:37, 5 September 2024
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	Consider this second version for naming: "Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5." Consistency and Clarity: The second version ("Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5") places the descriptive name ("Alpha," "Beta," "Gamma") before the ratio. This makes it clear that "Alpha," "Beta," "Gamma," and so on are categories or types, while "2/1" and "7/5" are specific tunings or ratios within those categories. This ordering helps maintain a logical structure that is easier to follow.}}		Consider this second version for naming: "Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5." Consistency and Clarity: The second version ("Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5") places the descriptive name ("Alpha," "Beta," "Gamma") before the ratio. This makes it clear that "Alpha," "Beta," "Gamma," and so on are categories or types, while "2/1" and "7/5" are specific tunings or ratios within those categories. This ordering helps maintain a logical structure that is easier to follow.}}

	{{todo\|Add data\|inline=1\|comment=Include columns for the following optimizations: Pure Equave, Dave Benson, TE, POTE, and CTE. To prevent the table from becoming overcrowded, choose between using cents or EDO. Cents are preferable since these are EDONOI. Retain the EDO column for pure equaves to provide a sense of scale, but place it outside the optimization subgroup column.}}		{{todo\|Add data\|inline=1\|comment=Include columns for the following optimizations: Pure Equave, Dave Benson, TE, POTE, and CTE. To prevent the table from becoming overcrowded, choose between using cents or EDO. Cents are preferable since these are EDONOI. Retain the EDO column for pure equaves to provide a sense of scale, but place it outside the optimization subgroup column. --- These are non-octave scales. Octave-based optimizations such as POTE, CTE or CTWE don't really make sense. However, TE is still relevant.}}

	{\| class="wikitable"		{\| class="wikitable"

Contribution at 00:24, 5 September 2024

2024-09-05T00:24:23Z

← Older revision		Revision as of 00:24, 5 September 2024
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	! Equave		! Equave
	! SSC pair		! SSC pair
	! ~~Degrees~~		! Steps (Equave, SSC pair)
	! Errors (cent)		! Errors (cent)
	! Pure Equave		! Pure Equave

Contribution at 00:20, 5 September 2024

2024-09-05T00:20:57Z

← Older revision		Revision as of 00:20, 5 September 2024
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	! Equave		! Equave
	! SSC pair		! SSC pair
	! ~~Step~~		! Degrees
	! Errors (cent)		! Errors (cent)
	! Pure Equave		! Pure Equave

Contribution at 00:20, 5 September 2024

2024-09-05T00:20:30Z

← Older revision		Revision as of 00:20, 5 September 2024
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	\|+		\|+
	\|-		\|-
	! colspan="3" \| Tuning !! colspan="2" \| Intervals !! colspan="2" \| ~~Intervals Mapping~~ !! colspan="2" \| Various optimizations (~~cents~~)		! colspan="3" \| Tuning !! colspan="2" \| Intervals !! colspan="2" \| Mappings !! colspan="2" \| Various optimizations (cent)
	\|-		\|-
	! Name		! Name
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	! SSC pair		! SSC pair
	! Step		! Step
	! Errors (~~cents~~)		! Errors (cent)
	! Pure Equave		! Pure Equave
	! Dave Benson		! Dave Benson

Contribution at 00:19, 5 September 2024

2024-09-05T00:19:20Z

← Older revision		Revision as of 00:19, 5 September 2024
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	\|+		\|+
	\|-		\|-
	! colspan="3" \| Tuning !! colspan="2" \| Intervals !! colspan="2" \| Mapping !! colspan="2" \| Various optimizations (cents)		! colspan="3" \| Tuning !! colspan="2" \| Intervals !! colspan="2" \| Intervals Mapping !! colspan="2" \| Various optimizations (cents)
	\|-		\|-
	! Name		! Name
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	! Equave		! Equave
	! SSC pair		! SSC pair
	! ~~Mappings (Equave, SSC pair)~~		! Step
	! Errors (~~cent~~)		! Errors (cents)
	! Pure Equave		! Pure Equave
	! Dave Benson		! Dave Benson