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. 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. |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|
| Alpha 3/1 | 3ed3/1 | 1.892789 | 633.985000 | 1.907395926960071 | 629.130000247253548 | 3/1 mapping: 3\3<3/1> | equal division error: 0.000 | optimization error: -14.565 |
| 2/1 mapping: 2\3<3/1> | equal division error: 67.970 | optimization error: 58.260 | ||||||
| 3/2 mapping: 1\3<3/1> | equal division error: -67.970 | optimization error: -72.825 | ||||||
| Beta 3/1 | 5ed3/1 | 3.154649 | 380.391000 | 3.141862316907629 | 381.939079106781893 | 3/1 mapping: 5\5<3/1> | equal division error: 0.000 | optimization error: 7.740 |
| 2/1 mapping: 3\5<3/1> | equal division error: -58.827 | optimization error: -54.183 | ||||||
| 3/2 mapping: 2\5<3/1> | equal division error: 58.827 | optimization error: 61.923 | ||||||
| Gamma 3/1 | 8ed3/1 | 5.047438 | 237.744375 | 5.042556213760587 | 237.974540913461853 | 3/1 mapping: 8\8<3/1> | equal division error: 0.000 | optimization error: 1.841 |
| 2/1 mapping: 5\8<3/1> | equal division error: -11.278 | optimization error: -10.127 | ||||||
| 3/2 mapping: 3\8<3/1> | equal division error: 11.278 | optimization error: 11.969 |
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Todo: Temperaments
Compute the temperaments associated to each Alpha-Beta-Gamma scales. |
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).