S-expression: Difference between revisions
m →Mathematical derivation: style consistency correction |
m →Mathematical derivation: alternative statement of conclusion about semiparticulars |
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In conclusion: S''k''/S(''k'' + 2) is superparticular for k ≠ 3 (mod 4) and is odd-particular when k = 3 (mod 4). | In conclusion: S''k''/S(''k'' + 2) is superparticular for k ≠ 3 (mod 4) and is odd-particular when k = 3 (mod 4). | ||
Alternatively stated: S(''k'' - 1)/S(''k'' + 1) is superparticular for ''k'' ≠ 0 (mod 4) and is odd-particular when k = 0 (mod 4). This alternative statement highlights an interesting fact that the four harmonics related by tempering S(''k'' - 1)/S(''k'' + 1) are (''k'' - 2):(''k'' - 1):(''k'' + 1):(''k'' + 2) through tempering ((''k''+2)/(''k''-2)) / ((''k''+1)/(''k''-1))<sup>2</sup> meaning the ''k''th harmonic is the only one not included and therefore a semiparticular is odd-particular if the excluded "harmonic in the middle" (around which the two on each side are symmetric in terms of placement) is a multiple of 4 and is superparticular otherwise. | |||
[[Category:Elementary math]] | [[Category:Elementary math]] | ||