Re: Naming intervals -- size matters

Hello, there, Dave Keenan, and thank you for your very readable and
reasonable comments on the major third and neutral third questions,
where our views might actually not be that different, despite some
different stylistic conventions that could reflect musical
expectations in certain tuning systems and genres.

> Hi Margo,

> You (and possibly Paul and Joseph) seem to have made a case for
> calling anything from about 380 cents to about 440 cents an
> (unqualified) major third merely because it is generated by a chain
> of four approximate 2:3s in the particular tuning under
> consideration.

Here you remarks might lead me to qualify that generalization a bit:
my implicit or not-so-implicit qualifications would be that a "major
third" so defined actually fits the musical purposes at hand, and also
that the definition makes sense in a specific universe of discourse.

For example, someone using 46-EDO to approximate 5-limit harmony would
naturally take 15/46 octave rather than 16/46 octave as "the major
third." When I take 16/46 as representing a "usual major third," this
says at least as much about my stylistic conventions as about the
chain of fifths.

> Whereas I would prefer to qualify those more than about 8 cents
> wider than 4:5 with terms like equal, Pythagorean and septimal (as
> Scala does), or in another system: wide, narrow-super, super. I'm
> happy to drop the "wide"s and "narrow"s if a tuning only has one
> kind, but not the "super"s.

First of all, I would glady agree that what I call a "major third" in
a typical neo-medieval tuning system is actually a "supermajor third"
in a usual naming system, and that a reader should feel free to
supply the implicit prefix "_super_major third" at every mention.

While your system of nomenclature is especially well thought out and
crafted, seeking to optimize both descriptiveness and consistency
insofar as possible, I would emphasize that this question goes beyond
specific systems such as yours and Scala's. If we took a poll of a
large number of xenharmonicists, I suspect we'd find that many more
would tend to define "a major third" as around 4:5 rather than 64:81,
11:14, or 7:9, etc., with the latter ratios as "Pythagorean" or
"supermajor."

What my frequent use of "major third" as a synonymous for "supermajor
third" may reflect is the large proportion of time that I spend in
neo-medieval tunings where regular or "usual" thirds may range from
about 408 to 440 cents, or roughly Pythagorean to septimal (excellent
descriptive terms for the regions around 64:81 and 7:9).

Here I "take it as implicit" that a major third is actually
"supermajor" in the more conventional nomenclature, and would have no
problem with a statement like this:

Here we have a minor seventh sonority -- or, in
more familiar terms, a subminor seventh sonority --
quite close to a just 12:14:18:21.

Mainly I want to confirm your point that "supermajor third" would
indeed be a standard description for my "neo-Gothic major third." The
second term is a stylistic norm, while the first is a description as
part of a larger intonational map designed to name intervals in a
range of systems.

By the way, an interesting question: about where in the spectrum does
an interval become "supermajor" or "subminor"? For example, could
Pythagorean be roughly a dividing line between "major/minor" and
"supermajor/subminor" categories? Like the neutral third question you
raise, this could invite some discussion.

> So, Margo, Paul, Joseph or anyone,

> How about neutral thirds? How do you define them? Does their size
> depend on what scale they are in (e.g. how many whatevers generate
> them) or how it is notated?

> I'd say a good definition would be 351 +-8 cents.

While various definitions are possible here, I might say that my own
customary definition is almost identical to yours when translated into
cents: from around 32:39 (about 342.48 cents) to 13:16 (about 359.47
cents). Maybe I'd call this the "classic neutral" zone, leaving open
the qualification that thirds over a considerably wider region can
have "neutral" or "neutral-like" qualities.

For example, I'd consider a regular temperament around 704.61 cents as
having "supraminor/submajor thirds" (taking the Scala categories) at
around 341.46 cents and 363.14 cents. By your definition, or mine,
these are just about on the border of the central "neutral" zone.

Just how the "neutral" and "supraminor/submajor" categories (the
latter represented in Scala terms by 14:17:21) relate or overlap is a
question open to lots of viewpoints and discussions, but maybe this
quick response may give you a first impression as to how I have often
approached the problem.

Also, you raise a noteworthy point: while I use the term "major third"
for intervals ranging from around 5:4 to around 9:7, depending on the
tuning system, the "neutral" or "supraminor/submajor" categories are
considerably more specific.

Again, although I may sometimes rather casually invoke "the chain of
fifths" as an explanation, musical context might be a better approach
in explaning why I'm following the chain of fifths.

Please let me thank you for an opportunity to qualify some of my
previous statements, to compare notes, and also very warmly to invite
more dialogue.

Most appreciatively,

Margo Schulter
mschulter@value.net