Nobody's "perfect"

In my previous blog post, What is a "perfect" interval, really?, I described the traditional "interval classes" as:
- Perfect: diminished/ perfect /augmented
- Imperfect: diminished/ minor/major /augmented

Then I asked, "is this distinction meaningful? And if so, how?"

The short answer is "no, it's not meaningful."

The long answer, provided by Andy Milne, is that all Moment of Symmetry (MOS) scales (also called well-formed scales), including the diatonic scale, have two interval sizes. The wider interval could always be called the "major" interval, and the narrower one, the "minor" interval. That's the interval-naming rule associated with the "imperfect" interval class.

In short, the traditional distinction between "perfect" vs. "imperfect" intervals is a distinction without a difference. One could classify all intervals as being "imperfect," in any MOS scale, without loss of information. Because all tonal intervals would therefore fall into a single class, the need for any such classification would be removed.

An Alternative to Perfection
For example, the diatonic scale includes two fourths, one smaller (e.g., F to B) and one larger (e.g., C to F). Traditionally, fourths are members of the "perfect" interval class, with the narrower interval being named the perfect fourth and the wider interval being named the augmented fourth. However, these intervals could instead be named the minor fourth and major fourth, respectively.

Likewise, the diatonic scale includes two sizes of fifths, one smaller (e.g., B to F) and one larger (e.g., C to G). Traditionally, fifths are members of the perfect class, with the narrower interval being named the diminished fifth and the wider interval being named the perfect fifth. However, these intervals could instead be named the minor fifth and major fifth, respectively.

Using only the imperfect interval class's interval-naming rule would result in the following interval names (with changed names in italics):

There are two reasons why using two naming rules, when one will suffice, is bad.

Error of the First Type
The first reason why its bad to make a distinction between perfect and imperfect intervals is that this distinction implies that there is a difference between them, when no such difference exists. The diligent student, on learning of the distinction, will naturally attempt to discover what the difference is. When the student fails to find such a difference (because there is no difference to find), this failure will tend to reinforce the student's feeling that music is Kafkaesque — that is, "marked by a senseless, disorienting, often menacing complexity." Alternatively, the student may internalize the failure, concluding that "I'm just too stupid to understand music."

Either way, this is bad pedagogy.

There's a simple rule to follow, while designing systems, that can help one avoid Kafkaesquery: Occam's Razor, which states that "entities should not be multiplied unnecessarily." In this case, the interval-naming rules of the perfect and imperfect interval classes are the "entities." If only one rule is needed, then Occam's razor says, don't create more rules.

In short, the first reason why it is bad to have two interval classes, when one will suffice, is that having two classes implies the existence of meaningful patterns that do not, in fact, exist. Exposing meaningless patterns is a false positive error (also known as a "type I error").

Error of the Second Type
The second reason why it is bad to have two interval classes, when one will suffice, is that having two classes hides the existence of meaningful patterns. Hiding meaningful patterns is a false negative error (also known as a "type II error").

Consider the relationships of interval-names to modes. Using only the one "imperfect" interval class, the relationship between interval-names and modes reveals the major-minor axis through the modes. The table below shows the diatonic modes in circle-of-fifths order, from Fa to Ti (using movable Do with a La-based minor, of course):

This table (and hence the one-class interval-naming system) exposes a number of clear patterns.
- Fa-mode (Lydian) is "the most major" mode; all of its intervals are major.
- Ti-mode (Locrian) is "the most minor" mode; all of its intervals are minor.
- As one moves down the table from Fa-mode to Ti mode (and hence around the circle of fifths), each mode introduces a new minor interval, making it "more minor" than the previous mode.
- The "new minor interval" of each mode is always the interval from that mode's tonic to Fa.

It's not that these patterns were impossible to detect when using two interval-naming rules; it's just that using a single rule makes these patterns easier to detect. Increased ease-of-detection reduces the chance of false negative errors. Alternatively put, it increases the chance that students will detect (and therefore have a chance to understand) these meaningful patterns.

Fossilized Tradition
The music establishment uses this overly-complex interval-naming system for one reason, and one reason only: because the music establishment uses this overly-complex interval-naming system. It's done because it's done. A few centuries ago, some intervals were renamed in accordance with the major/minor system, but some of the old names stuck.

Consonance
I am aware that the "perfect" moniker is applied only to "the most consonant" intervals, i.e., the unison, octave, wide fifth, and narrow fourth. At most, that's a reason
- to name the narrow fourth "perfect" while calling the wide fourth "major," and
- to name the wide fifth "perfect" while calling the narrow fifth "minor."

Diatonic and Chromatic Intervals
Another meaningful distinction that is blurred by the traditional interval-naming scheme is the difference between diatonic intervals and chromatic intervals. By calling the FaTi interval an "augmented fourth" and the TiFa interval a "diminished fifth," no distinction is made, in the naming of intervals, as to whether they are diatonic or chromatic.

Using only the "imperfect" interval-naming rule makes a clear distinction between these two different kinds of intervals: minor and major intervals occur in the diatonic scale, whereas diminished and augmented intervals do not. This is a meaningful difference, easily detected.

Communication Compatibility
Inspection of the table of interval names above shows that the likelihood of confusion, between people using the old names and those using the new, is quite low.

It's OK to have many different names that refer to the same thing, as in "Bob," "my brother Bob," "my father Bob," "Mr. Bob Johnson," etc. That's a many:1 mapping. No problem. So when a traditionalist hears someone say "minor fifth" for the first time, he'll need to learn a new name, but there's no ambiguity. Four of the five changed names fall into this many:1 category (minor fourth, major fourth, minor fifth, major fifth).

Two changed names, however, use existing names differently that they were used before (augmented fourth and diminished fifth). When a traditionalist hears someone say "diminished fifth," he'll have to wonder...is that an "old style" diminished fifth, or a "new style" diminished fifth?

That would be a problem...if it were to ever come up. However, it won't, because the "doubly altered" intervals with which new-style diminished fifth and augmented fourth may be confused are exceedingly rare. That is, someone using the new-style interval names would almost never say "diminished fifth" or "augmented fourth;" it simply wouldn't come up.

Likewise, if a person familiar with the new-style names heard a traditionalist describe an interval as a "diminished fifth," the new-stylist would say, "WTF? Why would you use a chromatic fifth there, instead of a diatonic fifth?" and after just one such interaction, the name-mapping would be clear to both parties in a very memorable way.

Hence, the ease-of-learning benefits of the proposed new names are very likely to outweigh the compatibility cost of the change.