Equal-step tuning: Difference between revisions

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As you can see, some patterns appear:

* For each pair of superparticular ratios <math>{s1}/{s2}</math> and <math>{s2}/{s3}</math>, there exists a ratio <math>{s1}/{s3}</math> such that <math>{s1}/{s2}</math> and <math>{s2}/{s3}</math> are <math>{s1}/{s3}</math> complementary; although <math>~~s1−s3~~=2</math>, when simplified as a coprime ratio with <math>{s1}/{s3} = {a}/{b}</math>, 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.

* For each pair of superparticular ratios <math>\frac{s+1}{s}</math> and <math>\frac{s}{s-1}</math>, there exists a ratio <math>\frac{s+1}{s-1}</math> such that <math>\frac{s+1}{s}</math> and <math>\frac{s}{s-1}</math> are <math>\frac{s+1}{s-1}</math> complementary; although <math>(s+1)-(s-1)=2</math>, when simplified as a coprime ratio with <math>\frac{s+1}{s-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{s+1}{s}</math> and <math>\frac{s}{s-1}</math> that are <math>\frac{a}{b}</math>-complementary.

* We can observe a converging sequence and pattern for divisions of the ratio <math>{a}/{b}</math> where low errors appears for <math>{s1}/{s2}</math> and <math>{s2}/{s3}</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 divisions of the ratio <math>\frac{s+1}{s-1}</math> where low errors appears for <math>\frac{s+1}{s}</math> and <math>\frac{s}{s-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. -

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 As you can see, some patterns appear:
-* For each pair of superparticular ratios <math>{s1}/{s2}</math> and <math>{s2}/{s3}</math>, there exists a ratio <math>{s1}/{s3}</math> such that <math>{s1}/{s2}</math> and <math>{s2}/{s3}</math> are <math>{s1}/{s3}</math> complementary; although <math>s1−s3=2</math>, when simplified as a coprime ratio with <math>{s1}/{s3} = {a}/{b}</math>, 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.
+* For each pair of superparticular ratios <math>\frac{s+1}{s}</math> and <math>\frac{s}{s-1}</math>, there exists a ratio <math>\frac{s+1}{s-1}</math> such that <math>\frac{s+1}{s}</math> and <math>\frac{s}{s-1}</math> are <math>\frac{s+1}{s-1}</math> complementary; although <math>(s+1)-(s-1)=2</math>, when simplified as a coprime ratio with <math>\frac{s+1}{s-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{s+1}{s}</math> and <math>\frac{s}{s-1}</math> that are <math>\frac{a}{b}</math>-complementary.
-* We can observe a converging sequence and pattern for divisions of the ratio <math>{a}/{b}</math> where low errors appears for <math>{s1}/{s2}</math> and <math>{s2}/{s3}</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 divisions of the ratio <math>\frac{s+1}{s-1}</math> where low errors appears for <math>\frac{s+1}{s}</math> and <math>\frac{s}{s-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. -
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