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A Response to Julia Werntz by David B. Doty is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License.

A Response to Julia Werntz

This essay originally appeared in 1/1 11:2, Winter 2003.

In her article "Adding Pitches, Some New Thoughts..." (Perspectives of New Music, 39:2), Julia Werntz devotes nearly ten pages to explaining why Just Intonation is a bad idea and how Just Intonation composers and theorists have deluded themselves, before undertaking to describe her own preferred methods for adding pitches (in divisible-by-twelve microtonal temperaments). In her arguments against Just Intonation, Werntz quotes extensively from the introduction of my book, The Just Intonation Primer.1 I stand behind those statements of mine that she has quoted 100%, and I must therefore thank her for republishing my words for an audience that might not otherwise read them, and for spelling my name and the Just Intonation Network URL correctly.

Having carefully read Werntz's arguments against Just Intonation, I conclude that one of two things must be true: either her knowledge of Just Intonation theory is superficial and her knowledge of the practice of contemporary composers working in Just Intonation is almost nonexistent, or she has deliberately chosen to misrepresent Just Intonation theory and practice because a distorted view of the subject better serves her arguments—in other words, to set up a straw-man version of Just Intonation. The former case would be regrettable, the latter, deplorable.

In the section of her article headed "The Theoretical Premise of Just Intonation and the General Issue of Dissonance" (pp. 163–167), Wertz sets out to demonstrate either that the small integer-ratio intervals that form the basis of Just Intonation lack the consonance that is claimed for them or that, to the extent that such a property exists, it is not musically significant. Curiously, she chooses to frame the fundamental premise of Just Intonation as a negative:

... the intervals available to us through the great compromise of standard Western equal temperament are merely phony representation of the "natural," "pure" intervals, and are incorrect, flawed and dissonant, whereas the pure intervals are the correct ones, are truly consonant, and must be—indeed are—better to listen to.

Although I don't seriously disagree with this statement (one can dispute the "better to listen to" part—better for what?), I prefer a positive statement: the small integer-ratios that form the basis of Just Intonation are, particular when sounded with timbres consisting of harmonic or nearly harmonic partials, maximally consonant. This is true without reference to 12-tone equal temperament or any other tuning system.

As James Tenney has pointed out, the terms "consonance" and "dissonance" have been used in Western culture over the past 1500 years to express a variety of distinct meanings.2 Therefore, when one uses such terms to underpin an argument about tuning theory, it is essential to understand clearly what one means by them and to make one's usage clear to the reader. To do otherwise is to court confusion. When I use the term "consonance" in reference to tuning theory, I mean what I term "psychoacoustic consonance" (roughly equivalent to Tenney's "consonance-dissonance concept [CDC] number five") This is a perceptual property of intervals or chords that is independent of any particular musical style and of the cultural background of the listener. Of course, as with any property of sensory perception, some people may perceive more acutely than others, and perception may be sharpened by experience and/or training. Nevertheless, the consonance to which I refer is essentially an objective phenomenon that can potentially be detected by anyone with normal hearing. It results from the absence (or, at least, the minimization) of disturbance from beating partials and the reinforcement of interval identity by supportive difference tones and periodicity pitch. The resulting quality has been variously described as clarity, purity, stability, or smoothness.

Psychoacoustic consonance is distinct from any matter of liking or disliking a particular interval in a particular musical context. One may very well prefer a dissonance over a consonance in a particular location in a composition, or even, I suppose, prefer dissonances over consonances in general, but this does not alter the essential nature of the psychoacoustic phenomena, any more than liking a red shirt better than a blue one would cause the red shirt to be perceived as blue. Werntz, in her arguments against Just Intonation, seems to conflate this definition of consonance and dissonance with others, perhaps unwittingly.

In the opening of her attack on the consonance of just intervals, Werntz focuses on one phenomenon, interference beats, while ignoring two others that make significant contributions to interval affect: difference tones and periodicity pitch. When she argues against the significance of beats or their absence as essential determinants of interval quality, she is clearly speaking of psychoacoustic consonance. I do not intend to expend a lot of verbiage defending the psychoacoustic significance of beats as a determinant of interval quality. This is a phenomenon that anyone with a tunable instrument that produces harmonic partials and the means to measure frequency can test and judge for him or herself. The physicist Arthur H. Benade describes such an experiment that he frequently performed with his students:3 the subject is asked to tune a variable oscillator relative to a second, fixed-pitch oscillator so as to produce a "special relationship," which is defined as "a beat-free setting, narrowly confined between two restricted regions in which a wide variety of beats takes place." Benade's students consistently identified the following frequency ratios as "special relationships": 2:1, 1:1, 3:2, 4:3, 5:3, 5:4, 6:5, 7:4, 7:5, 8:5, and 7:6; that is, they selected exactly and only those frequency ratios equal to or narrower than the octave that are typically identified as consonances in seven-limit Just Intonation. One may debate the significance for music of these phenomena, but it seems futile to argue against their existence. Given her interest in expanded atonality based on 72-tone equal temperament, it should not, perhaps, be surprising if Werntz is uninterested in psychoacoustic consonance, which naturally and almost inevitably tends to promote tonality.

Later in her argument (pp. 166–167), Werntz shifts definitions and uses the terms "consonance" and "dissonance" with quite a different meaning:

For example, even in tonal music with equal temperament on an instrument with fixed tuning, such as the piano, G-sharp to B has a consonant sound when the pitches are members of either a major or a minor triad, and a different, tense (even dissonant) sound as the augmented second A-flat to B in a C harmonic minor scale. Even open perfect fifths or octaves may seem "jarring" in some contexts.

Neither of these examples has much to do with consonance vs. dissonance as I use the terms (or with tuning). In the first case, she is talking about interval usage in common practice music, and the roles that listeners have learned to expect intervals to play in that usage: G-sharp to B is a "minor third" whereas A-flat to B is an "augmented second"; both spellings represent the same 300-cent interval in 12-tone equal temperament, with necessarily the same psychoacoustic properties, but composers of the common practice period treat them differently and listeners familiar with this style have come to expect such usage and react accordingly. As to the matter of open fifths and octaves being "jarring," this may be so, especially when they are contrasted with a richer and more complex harmonic texture, but that will not make such intervals dissonant, at least not if they are pure 3:2s and 2:1s.

When Werntz comes to describe compositional practice in Just Intonation ("Practical Limitations," pp. 167–170) she describes an imaginary world that is wholly unfamiliar to me (and one that would be difficult for anyone familiar with a cross section of recent compositions in Just Intonation to recognize). She seems to have drawn the erroneous conclusion that, because the tuning of certain consonant intervals is central to the theory of Just Intonation, there exists a commandment for Just Intonation users along the lines of "Thou shalt use consonances only; dissonances thou shalt not use." Of course no such rule, either explicit or implicit, exists. The frequency with which one uses consonant vs dissonant intervals or chords, whether in Just Intonation or any other tuning system, is a matter of style and taste. Of course, if one wishes to use dissonant intervals exclusively, it is not obvious what advantage Just Intonation offers over microtonal temperaments, especially those that fail to closely approximate just intervals; nevertheless, I have know some composers working in Just Intonation to write pan-dissonant music. In my personal compositional style, my rule regarding the usage of consonance can be expressed as "if the music requires a consonance, use a real one"; that is, if you want a perfect fifth, use a 3:2, not a 32:21 or a 40:27 or some other, more complex interval; if you want a major third, use a 5:4; if you want a major triad, tune it 4:5:6, and so on. In other words, it is my practice to make a clear distinction between consonant and dissonant intervals, a practice that would be impossible in12-tone equal temperament, where all of the supposed consonances other than the octave are, to varying degrees, dissonant. Much of what Werntz says about what composition in Just Intonation must be like seems to be pure invention:

The most significant aspect of this theory with regard to its effect on composition is the belief, shared by Rameau and modern just intonation composers, that music is generated by harmony. Just Intonation is specifically a vertical view of music, in which the composer's choice of pitches is governed by strict, outside (even scientific) criteria concerning all pitches' relationship to a fundamental. The effect on the compositional process is that the notes of the music will be determined by these pre-existing harmonic structures, and melodies and voice leading will not be the primary forces producing harmonic relationships. ...

Beyond this fundamental creative restriction, it almost goes without saying that just intonation, if truly adhered to in order to maintain the simple consonant harmonic language that is its credo ... also limits the composer stylistically to a sparse, simple triadic idiom...."

There is little of truth to be found in the above paragraphs. The claim that Just Intonation composers "believe that music is generated by harmony" sounds more like the nineteenth-century common-practice (equally tempered) view of the relation of melody to harmony, that is, that melody is "the surface of harmony" or "harmony expressed horizontally," than anything I have heard expressed by a contemporary Just Intonation composer. As for being a "specifically vertical view of music," I am aware of only one significant contemporary Just Intonation composer who has embraced this view: La Monte Young. Young has carried this view to its logical extreme, both in his live-performance instrumental and vocal drone pieces and his long-duration electronic sound environments. Lou Harrison, in contrast, has shown very little interest in chordal harmony in his Just Intonation compositions. Melody and rhythm take precedence, and chords, if present, play a decorative or coloristic role. ("I'm a 'melode' in the old sense." "I'm a song and dance man in the abstract."4) As for others, my experience is that composers working with Just Intonation are just as likely to use melody or voice leading or rhythm as they are to use harmonic structures (pre-existing or original) as the "primary force" in organizing a composition. Certainly there is nothing inherent in Just Intonation to prevent them from doing so. I have used all of these approaches myself at various times.

Of course, the primary intervals of Just Intonation are, in some sense, harmonically determined, in that interval perception is more acute in the case of simultaneously sounded pitches than with successive ones, and it is primarily the psychoacoustic properties of simple-ratio intervals sounded simultaneously that leads to their selection as the basis of Just Intonation. When these intervals are concatenated in various ways, they give rise to a great variety of intervals of all sizes, which can be used to construct a variety of scales, of however many tones one may desire or can be used in what Harrison calls "freestyle," combined without reference to any predetermined scale.

The notion that an "indisputably simple consonant harmonic language" is the "credo" of Just Intonation is simply another of Werntz's fictions. I wish she would tell me where this "credo" is published, though I doubt that she can do so. As far as justly tuned music being limited to "a sparse, simple triadic idiom," this, as a generalization, is without foundation. Can anyone tell me where to find significant pieces in this idiom in the works of Partch (one might find a few minutes of this style here or there, when he is parodying a hymn or a popular song) or of Harrison? Can anyone name any Just Intonation composer who consistently obeys this supposed credo? I confess that I sometimes write in a simple triadic idiom because I like the emotional affect of this style (simple five-limit harmony in Just Intonation is capable of unequaled serenity and sweetness, quite unlike anything that can be produced by the equivalent harmonic style in tempered tuning), not because I subscribe to some imaginary credo, nor because I am compelled to do so by the limitations of the material (since I do not hold a faculty position in a university music department, or aspire to one, I am free to make my music just as consonant and tonal as I wish). I am just as likely at other times to use harmonies generated by products of the primes (2, 3, 7) or (or 2, 3, 7, 11), which produce no conventional major or minor triads at all.

When she mentions a few actual compositions in Just Intonation, Werntz, unsurprisingly, finds something quite different from the imaginary simple pan-consonant style she has described. For example, she singles out the use of the intervals 7:6, 16:15, and 10:9 in Harrison's Simfony in Freestyle and Cinna as though the use of such intervals was somehow remarkable, when in fact such intervals are commonplace in the works of most contemporary Just Intonation composers.

There is one passage in Werntz's article to which I must take particular exception, both personally and on behalf of all Just Intonation composers (p. 168):

Finally, I also find just intonation problematical on a more personal level simply because simply because its origins lie in extra-musical ideologies rather than musical experience." (my italics)

The implication of the above passage is that those of us working in Just Intonation learned about it from some musty tome (whether by Ptolemy, Rameau, Helmholtz, or Partch) and, dazzled by the numbers and the pseudo-scientific terminology, concluded that it must be true, and have since been in a trance wherein we failed to hear the music we were making or make any aesthetic judgements about it. This is insulting nonsense. I will wager that my musical experience is as good as, and at least as extensive as Werntz's. I became aware of Just Intonation from Partch's work, but it was the experience of performing music on Lou Harrison and Bill Colvig's first American Gamelan in 1975 that convinced me that Just Intonation was the right road for me, a decision that I have never regretted.

Ironically, near the end of her attack on Just Intonation, Werntz (p. 170) opines, "Certainly, the need to engage in the intensely negative act of denouncing is a dubious foundation for a new music." I cannot but wonder, then, why she found it necessary to devote ten pages of her article denouncing composers and theorists working in Just Intonation as a prelude to a description of her own theories and practice with regard to 72-tone equal temperament.


1. San Francisco: The Just Intonation Network. 1993, '94. It is interesting that Werntz quotes only from the first chapter available free on the JIN website. Presumably, she did not bother to acquire and read the entire book. Had she done so, she would have found little evidence to support the view of Just Intonation put forward in her article.

2. Tenney, James. A History of 'Consonance' and 'Dissonance'. New York: Excelsior Music Publishing Co. 1988.

3. Benade, Arthur H. Fundamentals of Musical Acoustics. New York: Dover Publications. 1990. pp. 274–275.

4. Doty, David B. "The Lou Harrison Interview." 1/1 3:2 (Spring 1987).