User:FloraC/Critique on D&D's terminology

This page is a work in progress.

This is a critique on the term choices in Dave Keenan & Douglas Blumeyer's guide to RTT and related articles. The main purpose of this essay is to raise awareness of good communication tools and warn against an unnecessary community fragmentation.

"Domain basis"

Domain and domain basis are D&D's replacements for subgroup and subgroup basis, respectively. They laid out a number of reasons. I will discuss them one by one.

Consistency

First, they argue subgroup and basis are inconsistent terms since they are a mix from different mathematical fields.

The term "subgroup basis" mixes mathematical terminology from different mathematical fields: "subgroup" comes from group theory, while "basis" comes from linear algebra. The equivalent term for "subgroup" in linear algebra is "subspace", and the equivalent term for "basis" in group theory is "minimal generating set".

That is not correct. The basis is a concept that is used across linear algebra and a particular sector of group theory: the study of free abelian groups. If you look up the definition of the free abelian group, it is simple: a free abelian group is an abelian group that has a basis. Indeed, in RTT, JI is modeled as a free abelian group, rather than a group or module in general. That is why bases are used.

Since a free abelian group is a group, it naturally has subgroups. It is not a vector space, so we do not speak of subspaces – but that has to do with their next point below.

Simplicity

They say they prefer linear algebra, the less advanced math theory, to group theory, the more advanced, since they believe the difference between the vector space and the free abelian group is negligible for practical purposes.

This is because group theory is a relatively obscure and advanced field of mathematics, and this article prefers to leverage terminology from the more well-known and basic field of linear algebra whenever possible. […] some argue that RTT cannot be sufficiently described using only linear algebra. This article, however, prioritizes pedagogy of the basics over any potential considerations arising from such advanced RTT problems.

Here they admit that subgroup is technically more correct than subspace. It should be pointed out that their reason seems to be one for taking the correct terms, not against. Specifically, they draw the comparison between the two math disciplines, and it appears that RTT would be more difficult by taking the correct terms. The problem in their logic is, if "advanced RTT problems" only arise occasionally, the actual difficulty levels should be similar.

The story behind it is that RTT does not make use of the full power of group theory. RTT only concerns free abelian groups, which as they appropriately noted, are similar to vector spaces.

Specificity

At this point, they further argue for domain basis than subspace basis: they prefer more specialized musical terms to general mathematical terms.

This article prefers to use specialized terminology for objects in our RTT application, so that we can clearly discuss them independently from the mathematical structures that represent them.

To understand this point of theirs, we must look back at time before they changed interval basis to domain basis. Back then, interval basis was indeed more specialized for music, though not more specific in its form. Yet that is no longer the case. Domain is by no means more specialized or specific than subgroup, if not less. So the reason is obsolete.

Inclusivity

Finally, they argue that subgroup is often assumed to be nonstandard subgroups, and to exclude the standard type, while domain is designed to include both standard and nonstandard types.

"Subgroup" in many typical RTT usages is apparently intended to exclude the standard prime-limit subgroups. This makes it more difficult than necessary to communicate about the standard prime-limit basis, which are still very much subgroups — of the entire space of primes, for one example. So we think this is unnecessary complexity with no clear benefit.

There is no explicit, clear-cut exclusion of standard subgroups in the definition. Technically, a full prime-limit JI is a subgroup of a larger prime-limit JI. It happens that we have distinct vocabulary for standard subgroups, so the word only tends to appear at the nonstandard type. They also say the term "has taken on a specialized meaning in RTT", which I am again not sure about. For one thing, it has never gained a distinct definition. Still, according to my observation, whether it is meant to include the standard type is up to the context.

The hope for a term that invariably includes standard subgroup does not hold itself because language does not work like that. So long as prime limits continue to be used, chances are the actual usage of domain will turn out the same as subgroup.

My suggestion

My suggestion: use subgroup and subgroup basis. These are technically correct, consistent, and as clear as they can get.