On the Naming of Intervals

March 21, 2008

I recently read Margo Schulter's interesting and insightful paper "Regions of the Interval Spectrum: Some Concepts and Names". This topic is very dear to me. Obviously it is also a topic fraught with difficulty, perhaps the greatest being psychoacoustic agreement with theoretical propositions. Anyone doing work in this area relies to some extent on speculation and gut instinct, and my work is no different. Schulter wisely restricted her discussion to the realm of theory, making only a few remarks about perception.

In Schulter's paper, assumptions about where interval ordinal categories come from are not stated; however, the categories used in the text and outlined in the conclusion suggest some assumptions. In other words, nowhere is it explicitly stated why any interval is called a "third" of some kind, or why it is "minor". This is also lacking in the sources cited, such as the Scala web page or Dave Keenan's web page. I have outlined my own approach to this on my theory website but this takes many pages to communicate. The next paragraph summarizes my own point of view on diatonic interval names as succinctly as possible. To get straight to the point, please skip the next paragraph.

(Diatonic intervals are derived from a Pythagorean system, but only after seven letter names of seven core tones corresponding to staff positions reflect the scale order of seven diatonic naturals, such that the ordinals are ascribed to the intervals corresponding to raw staff distance, and the interval qualities correspond to the mathematical relations of the tones when taking each tone as an origin from which to measure distances to the other tones. The spelling of an interval is correlated with its notation and hence its ordinal identity and quality, the Perfects ( 1, 4, 5, 8 ) being so named because they are used to construct the system (and they also sound beatless), major and minor ( 2, 3, 6, 7 ) so named because they are incidental to the system (and they cause beating, although the M2 does not actually beat). The traditional 3-Limit system includes 7 letters times five accidentals equaling 35 tones from double-flats to double-sharps, theoretically allowing such intervals as a quintupally augmented fourth from Fbb to Bx and its inversion from Bx to Fbb, the quintupally diminished fifth, which are never used in practice. Thirteen diatonic intervals are understood as basic building blocks; between seventeen and twenty-one intervals can be considered common, including such less commonly used qualities as augmented 6ths and diminished 3rds.)

Now, back to business… Although assumptions such as those described above are not stated in Schulter's paper, at the outset "Pythagorean", "pental" (which I find a much more useful term than "classic", which is asystematic and has no clear meaning), "septimal", et cetera are very clearly defined; however, pairs of terms which feature prominently in the text, "small" and "narrow", and "large" and "wide" are not clearly defined, and they appear to be used interchangeably. These terms also appear to be mixed in with other terms using prefixes like "sub", "super", and "ultra". I feel all such terms should be clearly defined, and their usage should be both systematic and consistent.

An interval such as a "small minor third" is clearly a modified third; that is, considered grammatically, "small" and "minor" are adjectives, and "third" is a noun. This is perfectly clear. Moreover, the adjective "small" tells us something directly about the size of this interval, which is what we are most interested in when it comes to categorizing intervals. On the other hand, a "subminor third" is also clearly a modified third, but it represents a categorically different structure and has a different meaning than a "small minor third"; in this new construction, "third" is the noun, and "subminor" is the adjective, but it is a variation of the known adjective "minor" with a known prefix "sub" which creates what is called a derivational morphological variant, that is here "subminor". A good taxonomic system should use modifiers consistently and should not mix them with morphological variants, but more importantly, the word "sub" also means "below", so it does not describe size but in fact position. This is confusing, and syntactically incorrect. The use of "sub", "super", etc. for the description of intervals should for this reason be discouraged. These terms describe position and should be reserved for single tones only. So, it makes sense to talk about a distance "from A up to sub-C", but not a "subminor third".

When I write "from A up to sub-C", I am describing a distance from a Pythagorean diatonic A to a Pythagorean diatonic C which has been shifted down (sub = below) by one comma, so this interval is a minor third which has been made smaller by one comma, and so should be properly called a "small minor third". In my system, the terms "small" and "large" are used for intervals made smaller or larger by one comma. This is simple and consistent. The terms "narrow" and "wide" are used for intervals made smaller or larger by two commas, also applied consistently. I feel strongly that use of the term "neutral" to describe intervals should be discouraged, as this word comes across as a qualitative or functional assessment or prescription rather than a description of the size of an interval, kind of like calling a minor third "sad" or "dark". Moreover, the so-called "neutral" intervals are always doubly comma-shifted versions of some interval and so can be called "wide" and "narrow" intervals - terms which directly refer to their size. Where these are versions of major and minor intervals, the NM (Narrow Major) intervals overlap with Wm (Wide minor) intervals, such that a Wm3 and a NM3 are in fact the same interval which can be called by either name. If a qualitative or functional description is needed for such intervals, the prefix "neutral" does not accurately describe them, but the prefix "ambi" does accurately describe them, as they are derived from two qualities of a given interval and can be used to function either as a major or a minor form of that interval in a given context (here, an "ambithird"); however, such a name does not belong in the systematic taxonomy; it is just an additional possibly useful term for theoretical purposes.

In Schulter's paper there are also variant terms using the prefix "inter", such as "interpental" and "interseptimal" which I find meaningfully descriptive and appropriately used. There is however the problem that they bear structural resemblance to the "sub" and "super" etc. constructions used in the paper, so that they appear to be taxonomically ambiguous. This ambiguity would of course not exist if the prefixes "sub", "super" etc. were not used for intervals, but were relegated to individual tones.

The conclusion of the paper presents an outline of intervals. In my opinion, the most interesting things in the list are the names which do not have connections with overall systemic taxonomy but are instead locally descriptive, such as the "inter" intervals. I feel that these kinds of ideas should be explored as fully as possible, and such ways of naming intervals should be available for their descriptive theoretical usefulness, but it would be helpful if they were not mixed in with systematic interval size descriptors like "large" and "small". A "large" or "small" (comma shifted) interval may be in some "inter" category, after all.

I would rather like to see a discussion of such theoretical issues within the context of a systematically codified intervallic continuum, a foolproof intervallic universe, if you will, wherein interval names are taxonomically correct and have an internal consistency which does not interfere with the qualitative or functional taxonomy proposed in the discussion. No previously existing naming system meets this criteria, but I have defined such a system. For the past several years, the disclosure of this system has been the focus of my lectures on microtonality. For lack of a better name, I have called this system the "Hunt System" (H-System for short), and I have outlined it online as Chapter 7 of the Systematic Music Theory pages.

- Aaron Andrew Hunt

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