Talk:Superparticular ratio: Difference between revisions
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= Generalizing the notion of splitting a ratio into superparticular ratios = | == Generalizing the notion of splitting a ratio into superparticular ratios == | ||
"Every superparticular ratio can be split into the product of two superparticular ratios." Take 5/4, double everything to get 10/8, then convert that to 10/9 x 9/8. You can likewise triple 5/4 to get 15/12 which is 15/14 x 14/13 x 13/12. And so forth. So in fact every superparticular ratio can be split into the product of N superparticular ratios for any natural number N. | "Every superparticular ratio can be split into the product of two superparticular ratios." Take 5/4, double everything to get 10/8, then convert that to 10/9 x 9/8. You can likewise triple 5/4 to get 15/12 which is 15/14 x 14/13 x 13/12. And so forth. So in fact every superparticular ratio can be split into the product of N superparticular ratios for any natural number N. | ||
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Likewise every delta-3 ratio can be split into the product of N superparticular ratios, where N is threeven. And so forth. -[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 07:21, 27 February 2023 (UTC) | Likewise every delta-3 ratio can be split into the product of N superparticular ratios, where N is threeven. And so forth. -[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 07:21, 27 February 2023 (UTC) | ||
== Redundant statement == | |||
In the Properties section it says "The logarithmic sum (i.e. product) of two successive superparticular ratios is either a superparticular ratio or a superpartient ratio." And yet every ratio is either a superparticular ratio or a superpartient ratio. -[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 07:53, 1 March 2023 (UTC) | |||
: As I discussed on Discord, I left it there when I cleaned up the page recently, because I think it highlights how the property for the logarithmic difference (i.e. quotient) does not extend to the logarithmic sum (i.e. product). I guess it could be rewritten as "is not necessarily superparticular", from that perspective. Technically, the property guarantees that the product is greater than 1, because the initial ratios were taken as greater than 1, although that isn't very surprising in itself. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 01:23, 2 March 2023 (UTC) | |||
Revision as of 01:23, 2 March 2023
Generalizing the notion of splitting a ratio into superparticular ratios
"Every superparticular ratio can be split into the product of two superparticular ratios." Take 5/4, double everything to get 10/8, then convert that to 10/9 x 9/8. You can likewise triple 5/4 to get 15/12 which is 15/14 x 14/13 x 13/12. And so forth. So in fact every superparticular ratio can be split into the product of N superparticular ratios for any natural number N.
Every delta-2 ratio can be split into the product of two superparticular ratios, because (N+2)/N = (N+2)/(N+1) x (N+1)/N. For example, 5/3 = 5/4 x 4/3. (So in my earlier example where 5/4 becomes 10/8, I simply converted a reduced delta-1 ratio into an unreduced delta-2 ratio.) You can likewise double everything in a delta-2 ratio, converting it into a delta-4 ratio, e.g. 5/3 becomes 10/6. Obviously every delta-4 ratio can be split into 4 delta-1 ratios. Likewise a delta-2 ratio can be tripled to a delta-6 ratio, and split into 6 parts. So in fact every delta-2 ratio can be split into the product of N superparticular ratios, where N is even.
Likewise every delta-3 ratio can be split into the product of N superparticular ratios, where N is threeven. And so forth. -TallKite (talk) 07:21, 27 February 2023 (UTC)
Redundant statement
In the Properties section it says "The logarithmic sum (i.e. product) of two successive superparticular ratios is either a superparticular ratio or a superpartient ratio." And yet every ratio is either a superparticular ratio or a superpartient ratio. -TallKite (talk) 07:53, 1 March 2023 (UTC)
- As I discussed on Discord, I left it there when I cleaned up the page recently, because I think it highlights how the property for the logarithmic difference (i.e. quotient) does not extend to the logarithmic sum (i.e. product). I guess it could be rewritten as "is not necessarily superparticular", from that perspective. Technically, the property guarantees that the product is greater than 1, because the initial ratios were taken as greater than 1, although that isn't very surprising in itself. --Fredg999 (talk) 01:23, 2 March 2023 (UTC)