# Talk:Superparticular ratio

## 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)

## Delta-N as a "replacement"

I don't think Kite's use of the word "replacement" in the paragraph about delta-N terminology is appropriate for this article. I think "alternative" would be more suitable, especially since we are talking about terminology (superparticular/epimoric/etc.) that has been in use for many centuries before us, not just in music but in mathematics as well, and that is still actively used today by xenharmonic theorists in expressions like "square superparticular".

I believe that the expression of a single person's preference for a term over another is not something we should find on articles in the main namespace on this wiki; even though we are not literally Wikipedia, because we accept original work and such, I think we should try to follow a few basic guidelines, such as neutral point of view, for articles in the main namespace to ensure that readers can easily tell apart facts from opinions.

Also, the reason I had put this paragraph in the Etymology section at first, instead of the lead section, is that it explains why "delta-1" appears alongside "superparticular" and "epimoric" among the bolded terms at the very beginning (which are very clearly *alternative terms* for the same concept). Since delta-N terminology is a generalization of "superparticular", it might make sense to mention it in the Generalization section, with maybe only a quick overview of the various kinds of generalizations in the lead section. Another option that I have started thinking about (and discussed briefly on Discord) is to merge the pages Superparticular ratio and Superpartient ratio, although I am still hesitant about that because there are several good reasons to keep them separate (different lists of properties, follow the same page structure as Wikipedia, avoid clutter in the lead section, etc.). --Fredg999 (talk) 02:09, 2 March 2023 (UTC)

## Above a part

I'm by no means an expert in Latin, but doesn't *superparticular* mean *"above a part"* or *"the next after a part"*. The *supering* is done by 1 and not by 1/n as in *"above (unity) by one part"*. -Frostburn (talk) 12:17, 23 May 2024 (UTC)

- Yes, the etymology is correct; I know because I'm the one who brought it to the wiki from the work by Thomas Taylor cited in the references. So I don't exactly understand the nature of your confusion, I'm afraid, but perhaps things will be clarified if you review the Generalizations section of the page. The "part"
*is*the 1, which is why a super*bi*particular is of the form (n+2)/n. Happy to try to explain further if it's still unclear, and also work to revise the page to avoid any ambiguity or implicitness. I remember this was really confusing and unintuitive and surprising to me when Dave first tried to explain it to me. --Cmloegcmluin (talk) 15:07, 23 May 2024 (UTC)

## Multiplex ratios

We might add to superparticular (n+1:n) and superpartient ratios mulitplex ratios (n:1). Examples of multiplex ratios would be the 2/1 octave and 3/1 twelfth. Mschulter1325 06:12, 6 June 2024 (UTC)