Wednesday, October 06, 2004

Troubling Terms

Today is the birthday of “downtown” composer and hero Glenn Branca. Check out his symphonies if you haven’t heard them yet.

Here’s a list of some harmonic terminology that strikes me as unfortunate. Much of my quarrel with these terms stems from three basic reasons: (1) there’s a bias toward diatonic tonality that obscures less traditional concepts, (2) these terms are so common that they take on a vivid, concrete meaning early in the study of music and (3) the numerical values within the names are at odds with the physics of the sound they describe.


This is perhaps the most maddening term as it is such an important and common interval in music globally. The oct- prefix that implies that the number 8 is an important descriptor of the interval. There are 8 diatonic steps between the root and the “octave.” Which renders the term quaint in non-diatonic contexts. It is the origin of the “8va” notation that indicates something is to be played an octave higher than written. However, when something two octaves higher the notation is “15va” as opposed to “16va” as two octaves are actually 8 + 7 diatonic steps (the “root” of the second octave isn’t counted whereas the “root” of the first is).

Harry Partch avoided using the term “octave” by simply referring to its frequency ratio: 2/1. This is descriptively more accurate. One wishes that this interval had a name that emphasized the number 2 such as “bi-ative” or “di-ative.” But for better or worse the term “octave” has stuck and even I can’t escape using it and having it mean 2/1 in a vivid manner within my own conception of harmony.

The harmonic sound of the 2/1 is strongly "consonant." Only the unison sounds more "consonant." The harmonic relationship between pitches an octave apart is so strong that they share the same letter name, or pitch-class. The harmonic implication is that they share the same harmonic function. This is true in about 99.9% of all music in the world. Scales replicate their interval patterns at the octave and set-theory is built upon the concept that octaves are "equivalencies" suitable as an orchestration device that adds no new harmonic content.

It is possible to theorize and practice in harmonic environments that do not extend this property to the 2/1 (1200.00 cent octave). My Piano Concerto No. 1 is an example of this. The piano is tuned so that there is no pair of notes 1200.00 cents (2/1) apart. Some of the open strings in the string section are tuned 1200.00 or 2400.00 cents apart from notes found on the re-tuned piano. But these “octaves” are not treated as harmonic equivalencies. Instead, the triative (3/1, a.k.a. the 1901.96 cent just perfect twelfth) takes on this harmonic role. Scales replicate their sequence of intervals at the triative and the piano is tuned so that every perfect twelfth (the octave + perfect fifth) is exactly 1901.96 cents apart. I’ll admit that this is a contrived harmonic system (which is part of the thrill of composing with it). It is a great exercise to compose something that completely renders the traditional meaning/concept of the term “octave” meaningless. It forces a different facility with the “letter names” of pitches.


Why is the 200.00-cent interval (the major second) regarded as “whole?” How arbitrary is that? Again, there’s a diatonic prejudice at work here. The first diatonic interval away from the root of a “major” scale happens to be this interval and it happens to be the most common sequential interval in a scale made up of five “whole-tones” and two “semi-tones.” Unfortunately, this term has taken on a vivid meaning in my ears and I can’t escape calling the scale that divides the octave into six equal parts as the whole-tone scale.

Semi-tone or Half-step

Why is the 100.00-cent interval (the minor second) regarded as “half” or “semi?” It’s an extension of the fallacy of the “whole-tone.” And again it has a vivid meaning to my ears and conception of harmony even if an octave divided into twelve equal parts is the “chromatic” scale as opposed to the more consistent name: semi-tone scale.


This is an intriguing fallacy. Again it’s the extension of the “whole-tone” issue being applied to an interval made up of three such “whole tones.” But there’s an even deeper fog at work with this interval. At 600.00 cents wide it’s an enharmonic stand-in for both the augmented fourth and the diminished fifth. In an equal tempered system these two intervals are sonically identical even if they serve radically different harmonic functions. And this ambiguity is put to good use in tonal syntax when the augmented or diminished chord is used to modulate to different key centers.

In just intonation the “tritone” takes on an interesting property. Typically, the augmented fourth and the diminished fifth have an equivalent harmonic relationship to the tonic (that is, in harmonic space they are the same distance away from the 1/1). However, they are relatively unrelated to each other (the harmonic distance between the augmented fourth and diminished fifth typically being twice the distance between the augmented fourth and the 1/1 or the diminished fifth and the 1/1). This is true of any pair of intervals that are inversions of one another (as is the case with the augmented fourth and the diminished fifth). But it is the only pair of intervals related by inversion that also share an enharmonic equivalence in equal temperment.

Intervals as Seconds, Thirds, Fourths, etc.

All the interval names that attach diatonic numerical values to them share the same problem of diatonic bias. And anyone who spends time learning music develops vivid conceptions of the sound associated with seconds, thirds, fourths, etc. I’m not sure what solution for this terminology would be pragmatic. There are an infinite number of intervals out there and it’s easier to think in terms of a finite set. The “solution” isn’t as simple as the substitution 2/1 for octave. A major second could be 200.00 cents, 9/8, 8/7, 11/10, etc. And this is true for all interval classes. It's the diatonic bias that seems unfortunate here.


Quarter-tone is a further extension of the “whole-tone” problem. Perhaps it should just be called the 50.00 cent interval.


Actually, micro-tone is not a bad term. I’m just sad that there’s no vocabulary for discussing intonation other than as a “deviation” from dominant common practice. It seems inherently more valuable to describe things additively by what they are rather than subtractively by what they are not.

Perhaps I’ll become “radicalized” enough to adopt a language that is mathematically “pure.” But I’ve basically absorbed the good and the bad terms born from a predominantly diatonic tradition. “Octave,” “semi-tone,” “tritone,” etc. all evoke something conceptually concrete as I’ve developed a working familiarity with how they sound. It becomes a choice of abandoning a flawed language in favor of something “accurate” at the expense of being the only person fluent (and thus defeating the communicative properties of language) or striving to communicate ideas within a vocabulary inherently hostile to my own sensibilities. In the end it seems best to strive toward sound while maintaining a delicate truce with verbal language. Words always seem to come up short compared to the directness of experience. There will always be a vast body of ideas that cannot be conveyed by words alone.

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