User:Contribution/Successive superparticular complementary pair
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Todo: Finish the article and move it
When the article is finished and the table explained, move it to the main root |
For each pair of superparticular ratios [math]\displaystyle{ {s1}/{s2} }[/math] and [math]\displaystyle{ {s2}/{s3} }[/math], there exists a ratio [math]\displaystyle{ {a}/{b} }[/math] such that [math]\displaystyle{ {s1}/{s2} }[/math] and [math]\displaystyle{ {s2}/{s3} }[/math] are [math]\displaystyle{ {a}/{b} }[/math] complementary; it is observed that [math]\displaystyle{ a−b=1 }[/math] or [math]\displaystyle{ a−b=2 }[/math]. In other words, for each ratio [math]\displaystyle{ a/b }[/math] where [math]\displaystyle{ a−b=1 }[/math] or [math]\displaystyle{ a−b=2 }[/math], there exists a pair of superparticular ratios [math]\displaystyle{ {s1}/{s2} }[/math] and [math]\displaystyle{ {s2}/{s3} }[/math] that are [math]\displaystyle{ {a}/{b} }[/math] complementary.
Bellow is a table that show for equal divisions of [math]\displaystyle{ a/b }[/math] the cent error in the mapping of superparticular ratios [math]\displaystyle{ {s1}/{s2} }[/math] and [math]\displaystyle{ {s2}/{s3} }[/math] that are [math]\displaystyle{ 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. --
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Todo: Table
Explain the table. |
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Todo: Pattern
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. |
Coincidence?
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).