Talk:847/845

As (125/121)/(175/169)?

What's important about this? What's the point of introducing intervals with 5³ and 5² when the comma only contains one instance of 5? —FloraC (talk) 16:43, 6 May 2026 (UTC)

It's important because tempering out 847/845 equates two of the simplest very small intervals in the 5.7.11.13 subgroup. That alone is enough justification. You are considering one thing (out of many) that can make a description of a comma useful, and taking its absence to mean that the description is useless. Consider the page for 243/242; only two of the many equivalent expressions there don't use any other factors at all (2187/2048/(33/32)² is a particularly "bad" example) yet it would be hard to argue that they aren't useful or worth including. Squib (talk) 22:20, 6 May 2026 (UTC)

I did some number crunching; turns out that 125/121 and 175/169 are actually the simplest two intervals (by factor count) smaller than 100 cents, and they differ by 847/845. The next two (aside from 847/845 itself) ALSO differ by 847/845. Squib (talk) 00:20, 7 May 2026 (UTC)

125/121 and 175/169 are quartertone-sized yet they should be treated as commas by default? I don't see why. And considering them in the 5.7.11.13 subgroup should imply this subgroup is of some significance in the first place, which is yet to be proven. The rastma is described in terms of significant intervals that belong to significant subgroups. It's the plain opposite with what we have here. Given the current state I think it's at best a trivia for the curious and at worst actively unhelpful cuz users are unlikely to be familiar with those intervals. The best way to get to 847/845 is already shown: (7/5)/(13/11)², and no alternatives can be as clear as that. —FloraC (talk) 08:51, 7 May 2026 (UTC)

The 5.7.11.13 subgroup is interesting mostly because it lacks primes 2 and 3, but also because of its structure. 847/845 is by far the smallest comma of any reasonable complexity, and it's extremely natural to temper it out, which makes it very important in this subgroup. I'm not treating 125/121 and 175/169 like commas, but they're right next to each other and it's relevant that they're separated by 847/845. It's not simpler than (7/5)/(13/11)², but it's still an important result of tempering it out in its subgroup. Squib (talk) 02:26, 20 May 2026 (UTC)

> The 5.7.11.13 subgroup is interesting mostly because it lacks primes 2 and 3, but also because of its structure.

> it's relevant that [125/121 and 175/169] are separated by 847/845.

> it's still an important result of tempering it out in its subgroup.

Maybe stop begging the question all the time. Provide substances to your arguments.

—FloraC (talk) 08:23, 22 May 2026 (UTC)

I think I understand what you mean. Here's my attempt to formulate my logic more precisely: The 5.7.11.13 subgroup is musically significant; 847/845 plays an important role in that subgroup; this interval relationship is a relevant description of that role.

The significance of the 5.7.11.13 subgroup is primarily due to its lack of primes 2 and 3, which play unique roles in traditional music and much of microtonal music. This forces the exploration of new harmony. (It is also significant in part because it strongly suggests tempering one comma above all others, much more than most subgroups do.)

I believe I have already explained 847/845's importance in the 5.7.11.13 subgroup, but for what it's worth, sintel's temperament search is more quantitative. Searching for 5.7.11.13 rank-3 temperaments gives 847/845 with a badness of 0.58, and the second result has a badness of 1.25, more than double the first's. (For comparison, I did the same search in about 20% of the 125 other four-prime 23-limit subgroups, and found only one other similarly extreme comma: 325/323 in the 5.13.17.19 subgroup.) 847/845 is a very significant interval in a significant subgroup, which merits a description of its role in that subgroup.

The small intervals present in a subgroup are relevant to the structure and use of that subgroup. In the 5.7.11.13 subgroup, there are four simple ones: 125/121 and 175/169; and 637/605 and 343/325. 637/625 (33¢) separates the pairs from each other, and 847/845 (4¢) separates the intervals within each pair. This is an important part of 847/845's role in the 5.7.11.13 subgroup, so it belongs on this page.

Squib (talk) 02:14, 23 May 2026 (UTC)

> The significance of the 5.7.11.13 subgroup is primarily due to its lack of primes 2 and 3, which play unique roles in traditional music and much of microtonal music.

Why does the lack of primes 2 and 3 makes the subgroup significant rather than insignificant?

> This forces the exploration of new harmony.

What new harmony? The utility of the concept is unproven without a constructive compositional theory. Do you have such a theory?

—FloraC (talk) 07:46, 23 May 2026 (UTC)

You're putting the cart before the horse. The value of the 5.7.11.13 subgroup is precisely that it is incompatible with existing compositional theories; the whole point of XA is to explore unfamiliar harmony. We don't need a fully realized compositional theory to document a subgroup's musical potential. I'm trying to build one, and documenting the intervals that are available once the familiar ones are gone is a necessary step. Squib (talk) 10:55, 23 May 2026 (UTC)

At this point you're essentially waving the interesting number paradox as a magic wand. Plz don't do that; read our Notability Guidelines.

> the whole point of XA is to explore unfamiliar harmony.

Not really…

> I'm trying to build one.

I'll patiently wait.

—FloraC (talk) 11:58, 23 May 2026 (UTC) (updated FloraC (talk) 12:15, 23 May 2026 (UTC))

This isn't the interesting number paradox. The roles of 2 and 3 aren't being taken up by the next "uninteresting" primes; I'm trying to abandon those roles entirely. That's what makes this subgroup notable. Squib (talk) 13:33, 23 May 2026 (UTC)

> I'll patiently wait.

I see your point. I haven't documented this system yet, so readers don't have the context to understand what makes the interval relationship meaningful. Documenting it all from scratch is daunting, though... Squib (talk) 16:45, 23 May 2026 (UTC)