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| As you can see, some patterns appear:
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| * For each pair of superparticular ratios <math>\frac{n+1}{n}</math> and <math>\frac{n}{n-1}</math>, there exists a ratio <math>\frac{n+1}{n-1}</math> such that <math>\frac{n+1}{n}</math> and <math>\frac{n}{n-1}</math> are <math>\frac{n+1}{n-1}</math> complementary; although <math>(n+1)-(n-1)=2</math>, when simplified as a coprime ratio with <math>\frac{n+1}{n-1}=\frac{a}{b}</math>, it is observed that <math>a-b\in{1,2}</math>. In other words, for each ratio <math>\frac{a}{b}</math> where <math>a-b\in{1,2}</math>, there exists a pair of superparticular ratios <math>\frac{n+1}{n}</math> and <math>\frac{n}{n-1}</math> that are <math>\frac{a}{b}</math> complementary.
| | A pair of small and big successive superparticulars |
| | <math>S_n=\dfrac{n+1}{n}</math> and <math>B_n=\dfrac{n}{n-1}</math> |
| | has product |
| | <math>\dfrac{n+1}{n}\cdot\dfrac{n}{n-1}=\dfrac{n+1}{n-1}</math>. |
| | Thus they are complementary in the ratio <math>R_n=\dfrac{n+1}{n-1}</math>. |
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| * We can observe a converging sequence and pattern for divisions of the ratio <math>\frac{n+1}{n-1}</math> where low errors appears for <math>\frac{n+1}{n}</math> and <math>\frac{n}{n-1}</math>: 3, 5, 8; then 5, 7, 12; then 7, 9, 16; then 9, 11, 20; then 11, 13, 24; then 13, 15, 28; etc. -
| | We can observe a converging sequence and pattern for equal divisions of the ratio <math>R_n</math> where low errors appears for <math>S_n</math> and <math>B_n</math>: 3, 5, 8; then 5, 7, 12; then 7, 9, 16; then 9, 11, 20; then 11, 13, 24; then 13, 15, 28; etc. - |
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| * We can describe the pattern as follow: Ratio divided = (n+1) / (n-1); if n%2 = 0 then a-b = 2 else a-b = 1; Small SSCP = (n+1) / n; Big SSCP = n / (n-1); Alpha division = <math>2\times {s2} - 1</math>; Beta division = <math>2\times {s2} + 1</math>; Gamma division = Alpha division + Beta division = 4 x n
| | For each <math>n\ge 2</math> consider the three divisions of <math>R_n</math>: |
| | * '''Alpha:''' <math>k_\alpha=2n-1</math> |
| | * '''Beta:''' <math>k_\beta=2n+1</math> |
| | * '''Gamma:''' <math>k_\gamma=4n=k_\alpha+k_\beta</math> |
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| * <math>\text{Ratio divided}frac{n+1}{n-1}</math>
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| * a - b: <math>n \equiv 0 \pmod{2} \Rightarrow a-b=2</math>; <math>n \equiv 1 \pmod{2} \Rightarrow a-b=1</math>
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| * <math>\text{Small SSCP}=\frac{n+1}{n}</math>
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| * <math>\text{Big SSCP}=frac{n}{n-1}</math>
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| * <math>\text{Alpha}=2n-1</math>
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| * <math>\text{Beta}=2n+1</math>
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| * <math>\text{Alpha}+\text{Beta}=(2n-1)+(2n+1)=4n</math>
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| * Ratio divided : <math>\frac{n+1}{,n-1,}</math>
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| * a-b : <math>a-b=\begin{cases}2,& n\equiv 0\pmod{2}\[4pt]1,& n\equiv 1\pmod{2}\end{cases}</math>
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| * Small SSCP : <math>\frac{n+1}{n}</math>
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| * Big SSCP : <math>\frac{n}{,n-1,}</math>
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| * Alpha division : <math>Alph2n-1</math>
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| * Beta division : <math>2n+1</math>
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| * Gamma division : <math>(2n-1)+(2n+1)=4n</math>
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| * Alpha types flatten the smaller interval and sharpen the larger; Beta types do the reverse; Gamma types again flatten the smaller and sharpen the larger. | | * Alpha types flatten the smaller interval and sharpen the larger; Beta types do the reverse; Gamma types again flatten the smaller and sharpen the larger. |
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| * The Alpha, Beta, and Gamma types bring their interval pairs increasingly close to just intonation. | | * The Alpha, Beta, and Gamma types bring their interval pairs increasingly close to just intonation. |
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