Anomalous Pairs in “Paradigms Lost”
This article was originally published in 1/1 5:1 (Winter 1989). It originally referred to a recording of my composition in extended just intonation, “Paradigms Lost,” which was included in the Just Intonation Network compilation tape Rational Music for an Irrational World (1989). The sound examples included in this version of the article are excerpted from my CD Uncommon Practice (1998).
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My composition "Paradigms Lost," combines harmonic and melodic material derived from late-1960s modal rock with cyclic, additive rhythmic structures similar to certain Indian talas, or to the structures of Balkan dance tunes. "Paradigms Lost" represents my fantasy of what might have happened if psychedelic-era rock bands had pursued their interests in the exotic, free from the constraints imposed by the record industry. The piece is based on harmonic progressions consisting almost exclusively of major triads built on minor or modal patterns. IV–VII♭–I cadences predominate, whereas V–I and V7–I cadences and chains of secondary dominants are avoided. The functions of the four instrumental parts are differentiated in a manner typical of a generic rock band: melodic lead, block-chord accompaniment, bass line, and percussion.
No fixed scale was presumed in composing "Paradigms." The chord progressions, which determine the structure of the piece, were composed first, and the tunings of the chords and the relations among them dictated the intonation of the piece. All major triads are tuned as 4:5:6 and all minor triads are tuned as 10:12:15 (harmonic). Because the harmony is primarily triadic, the resulting tunings are primarily five-limit. The leads and bass lines are derived from the harmonic progressions, as is typical of this style of music. The leads incorporate a few seven-limit intervals, which generally serve to add a harmonic seventh (7:4) to the underlying triad, creating a "blue note" effect. No primes higher than seven are involved.
“Paradigms" comprises three principal sections, which I designate A, B, and C. The respective tonal centers of the three sections are C (1/1), F (4/3), and E♭ (6/5). The three sections are repeated in the following order: introduction (truncated A), A1, A1, A2, B1, B2, A1, A1, A2, B1, B2, A1, A1, transition, C. The numerals indicate alternate endings. Each section uses a distinct tuning, and each of the three tunings, coincidentally, includes three anomalous pairs of tones. That is, each involves three pairs of tones separated by small, comma-like intervals, which pairs would each map to a single tone in twelve-tone equal temperament. These anomalous pairs and their musical functions provide the principal subject of this article.
Lattice 1 shows the tuning of section A of "Paradigms." In this and other tuning lattices in this article, each horizontal line segment represents a perfect fifth (3:2), each vertical line segment represents a major third (5:4), and each diagonal line segment represents a harmonic seventh (7:4). The system of accidentals used in the charts and musical examples and in the text is that proposed by Ben Johnston. For a more detailed discussion of tuning matrix diagrams, see my tutorial in 1/1, Volume 2, Number 2. For Ben Johnston's accidentals, see 1/1 Volume 2, Number 4 or Computer Music Journal, Volume 11, Number 1. This section, as it happens, uses twelve different tones, but in a configuration that bears little resemblance to a standard chromatic scale. The tonality of the section is clearly centered on C, with major and minor elements being freely mixed. The anomalous pairs of tones in this section are D 9/8 and D– 10/9, which differ by the syntonic comma, 81:80; B♭ 9/5 and B♭– 16/9 which also differ by the syntonic comma; and E♭ 6/5 and E
7/6, which differ by 36:35, which I shall call the septimal quarter tone.
Example 1 is the score, minus the percussion parts, of section A1 of "Paradigms" (all parts, including percussion are heard in the examples.) All three of the anomalous pairs described above are heard in this section. B♭ 9/5 serves as the fifth of the major triad on E♭ 6/5, and, in measure two, as the seventh of an incomplete minor seventh chord on C 1/1. B♭– 16/9 serves as the root of a B♭– major triad, which participates in the full and half cadences at the ends of measures two and four. B♭– 16/9 is used in preference to B♭ 9/5 in these cadences for several reasons. First, a progression from a 9/5 major triad to a 1/1 major triad would involve a false fourth (27/20) between the fifth of the 9/5 triad and 1/1. Second, the use of a 9/5 triad in the IV–VII♭–I cadence in measure four would result in an additional false fourth between 4/3 and 9/5. Finally, the 9/8 between 16/9 and 1/1 gives a stronger cadential feeling than the 10/9 between 9/5 and 1/1.
The B♭– major triad requires D– (10/9) as its major third. Elsewhere, the D in the lead melody at the end of measure one, which serves as a ninth or added second for a C major triad, and the D’s in what may be considered a G suspended-fourth or C added-second chord at the beginning of measure four, are all 9/8's. E♭ 6/5 serves as the root of the E♭ major triad, the fifth of the A♭ major triad, and the minor third of the incomplete minor-seventh chord on 1/1. E 7/6 occurs in only one location, in the reed part at the beginning of measure two, adding a harmonic seventh to the F major triad and lending a "bluesy" inflection to the phrase.
Lattice 2 illustrates the tuning of section B of "Paradigms." This section uses fourteen tones, and is centered on F 4/3. A comparison of this tuning and that of the previous section illustrates that modulation to a closely related tonal center doesn't necessarily require a literal transposition of the tuning. The 9/5–16/9 and 6/5–7/6 comma pairs from section A are retained, and one further anomalous pair is added: G♭– 64/45 and G 7/5, which differ by the septimal comma, 64:63. Example 2 is the score, again without percussion parts, of section B1. The respective functions of 9/5 and 16/9 and those of 6/5 and 7/6 are the same as in section A. G 7/5 serves as an added harmonic seventh for the A♭ (8/5) major triads in measures three and five. G♭– 64/45 occurs only in the ending of section B2 (not shown), where it forms part of a cadence which modulates back to C (1/1) and section A1 via the sequence B♭–, G♭–, D♭–, B♭–, F.
A point to note regarding this section is that a chord change between the major or minor triad on B♭– (16/9) and the major triad on E♭ (6/5) occurs in three places (measure 2, measure 4, and measures 5–6). In this chord change, the root moves by the wolf fourth or fifth (27:20 or 40:27), and the syntonic comma occurs between the root of the 16/9 triad and the fifth (9/5) of the 6/5 triad. In two of these three cases (measure 2 and measures 5–6) the syntonic comma is explicitly present in the organ part, as indicated in the score. The use of syntonic comma and wolf fourth/fifth between successive tones would be condemned by many theorists. Indeed, the existence of these phenomena are often held up as conclusive evidence that it is not possible to compose satisfactory triadic music in Just Intonation (often, it should be added, by theorists who have neither composed nor heard such music). My conclusion is, not surprisingly, the opposite. I find this chord change entirely satisfactory. It is a different musical experience from a chord change by a true fourth or fifth (4:3 or 3:2), but is no less of a valid musical expression for that. Note, however that I would not recommend this chord change as part of a final cadence (see discussion of section A, above). I suspect that the habit of describing chord changes primarily in terms of root movement, as is the convention in traditional music theory, is inadequate to fully explain harmonic progressions in Just Intonation. I think that much more attention needs to be paid to the intervals by which other voices progress. As for the syntonic comma, one has to listen very carefully to even notice its existence, and I doubt that one listener in a thousand will be disturbed by it.
The final section, C, is based on a single two-measure ostinato centered on E♭ (6/5). The tuning of this section, which uses eleven tones, is illustrated in Lattice 3. In this tuning, the anomalous pairs are the now-familiar B♭ 9/5 and B♭– 16/9, F 4/3 and F + 27/20, which differ by a syntonic comma; and D♭– 16/15 and D 21/20, which differ by the septimal quarter tone, 36:35. In this section, a distorted combo organ plays an extended solo over the repeating ostinato. An excerpt from this passage is shown in Example 3. B♭ 9/5 serves as the fifth of the E♭ major triad, which is the tonal center of the ostinato, whereas B♭– 16/9 serves as the major third of the G♭– triad in the turn-around of measure two. D♭– 16/15 plays three different roles: it is the fourth of the A♭ suspended-fourth chord at the end of measure one of the ostinato and the fifth of the G♭– major triad and the root of its own triad at the end of measure two. F 4/3 is heard as the major third of the Db 16/15 major triad. In the combo organ part, D 21/20 and F + 27/20, respectively the harmonic seventh and ninth of E♭ 6/5, are played against the E♭ major triad, as, of course, is B♭ 9/5. B♭– 16/9, D♭– 16/15, and F 4/3 are used as appropriate to harmonize with the chords which require them, as indicated above.
The tunings used in the three main sections of "Paradigms Lost" illustrate once again that even this comparatively simple style of triadic harmony cannot be realized in Just Intonation by means of fixed diatonic or chromatic scales. The tunings must follow the harmonic progressions, wherever they may lead. Commas and other anomalies will inevitably result, but there is no reason to fear these intervals. They are simply part of the vast and varied musical landscape of Just Intonation.