Anomalies I: voice leading
An anomaly here means something unusual, a musical passage that does not conform to contemporary rules of counterpoint or otherwise deviates from the norm. It also entails any incongruities or inconsistencies in melody, harmony, counterpoint, or texture. The concept thus entails interpretation and assumes some knowledge of historical musical practice. In a piece with the rigorous formal construction of the one under consideration, we might expect the voice leading, the treatment of harmony, the handling of texture, etc. to be just as rigorous. Yet there are a number of ambiguities that are noteworthy and curious.
The hexachord theme wanders from voice to voice. This is unusual for the simple reason that the theme is treated as a cantus firmus; cantus firmi, as a rule, do not wander. It is especially peculiar that the wandering takes place in the middle of thematic statements. Only once does it happen at the beginning of the theme (at unit 79) and that is remarkable for another reason. The hexachord theme first begins as the soprano voice. It wanders to the alto voice at unit 10 (see Example 14).
The theme only remains in the alto voice for the duration of nine units before returning to the soprano (see Example 15). Only four units later it wanders again to the alto (see Example 15). The switch in voices takes place on paper at unit 22 where the upper two voices use the same pitch, however it is not audible until unit 23.
The cantus firmus theme returns to the soprano beginning at unit 31, where the ambiguity of which notes represent which of the two upper voices is extended until unit 33 (see Example 16).
Another switch to the alto voice occurs during the space of units 37-38 (see Example 17).
During the first four statements the hexachord theme wanders between the soprano and the alto six times; one hears the theme moving through the fabric of the composition. The return to the soprano, the last until the final section of the piece, is effected at unit 43 (see Example 18).
The shift, at unit 79, of the hexachord theme from the soprano voice to the bass voice, the leap from f1 to A-flat, is one of the more dramatic shifts of the piece, moving very audibly from one outer voice to the other. As we have noted above, the leap occurs as a result of the transposition scheme. After that, the theme continues to wander somewhat. It moves to the tenor at unit 133 (see Example 19). It also moves to the alto voice at unit 163, but that voice crossing is as good as inaudible and is accentuated by Bull in another manner discussed below.
Anomalies II: texture/harmony
Example 20 shows how the hexachord theme, for the space of two and one half whole notes, audibly becomes the highest voice without actually leaving the alto voice on the page. Bull draws attention to this fact by leaving the leading tone, b-natural0 of the cadence on C, hanging without resolution; the soprano voice does not end, but is suddenly abandoned in mid-cadence. This is an abruptio, the sudden cessation of a musical thought. Right at that point, unit 13 of the piece, the effect of being left hanging at the cadence is further emphasized by the open sonority of the fifth, C-G, an anachronism, sounding for an instant in two-voiced texture due to the voice leading of the bass and tenor voices. The effect is one of a hole in the music, and it is one that allows, for a few moments, the theme to sound as if it were the soprano voice.
Example 21 shows another curious passage. One can see that the half-note e-flat1 in the alto at unit 100 is followed by a rest and that the alto proper doesn't reenter until the second half of unit 102. There is, however, right at unit 101, the entrance of an extra voice in the left hand, f0, which sounds for the duration of a whole note only and then disappears. It can only be the alto voice which suddenly plunges down into the tenor area, crossing below the actual tenor at unit 101. The open space between the tenor c1 (which, at unit 101, sounds like the alto voice) and soprano f2 disobeys the rules of counterpoint which stipulate that there be no more than an octave between upper neighboring voices. Not only does the space call attention to itself, but the apparent extra voice, f0, does so as well. The musical-rhetorical figure signified by the space, called longinqua distancia, continues into unit 102 and is found between tenor and soprano (b-flat0 to e-flat2) and then between alto and soprano (c1 to e-flat2).
Example 22 shows, at unit 144, the 12th entrance of the hexachord theme, beginning on f-sharp0. The sonority is unusual, to say the least, and eminently avoidable. First, the natural movement of the soprano voice would be to a1 at unit 144, completing the cadential figure begun around unit 141. The bass voice is expected to drop to A at unit 144 for the same reason, but it does the unexpected also. The alto voice withdraws itself from the affair with a rest, leaving the other voices to form the unusual harmony: octave f-sharps over an e0 in the bass. I have no doubt that Bull knew that the 12th entrance of the theme, using the 12th pitch of the scale, was taking place at the 144th (12 x 12) unit of the piece.
Example 23 illustrates a passage set audibly apart at unit 151 by the manipulation of texture. The four-voiced texture found at the first unit of the example, the alto having first a half-note rest, is thinned out as the soprano drops out at unit 150, while the bass drops out right at unit 151 leaving just a two-voiced texture. This is a unique moment for two reasons. The two-voiced texture is the thinnest used by Bull in this work, and rarely used at that. But that fact alone is not enough. What makes this striking is the open sound of the fifth occurring right at the point where the texture is thinnest. The listener cannot miss the anachronistic sonority; Bull literally diverts our attention away from everything else directly to it as he also did at unit 13 (see Example 20).
Example 24 shows another voice crossing, but one that does not belong to the more audible events of the work. However, beginning at unit 163, we have an occurrence which acts as an accent and thus draws the ear to it. The alto voice here carries the hexachord theme. One can observe how Bull accentuates this particular event. First, the thematic note, e1, is only a half-note long; this is the only passage in the work which has a thematic note which is not a whole note. Second, the chord played at this point has an added voice in the right hand making it five-voiced; at no other point, excepting the final chord of the work, do more than four voices sound simultaneously. Third, the three inner voices have the same length, a half note, and all are followed by a half-note rest; i.e. the thickest texture is immediately followed by the thinnest texture used by Bull in this piece. Fourth, the five-voiced texture is further emphasized rhythmically by the quarter-note A found in both of the outer voices; the outer voices are the most audible and the A found here presages the cadence on A found across the next bar line. Fifth, the five-voiced texture is restored for the brief period of a half note one unit later. It is an e1 in the right hand, the only tone found both in the A (tonic) and E (dominant) triads which are here forming a cadence--the d1 of the hexachord theme is the seventh of the dominant-seventh chord--though E is conspicuously missing from the a-minor sonority of unit 165.
Order
Hexachord
John Bull uses the hexachord as a cantus firmus. It is not a theme that is developed as the theme of a fugue might be and does not itself undergo transformation. It is a building block with which Bull constructs the framework of his piece. It goes through a number of statements, but each statement is clearly identifiable as a hexachord. The hexachord system reckons with hexachords on G, C, and F although it is clearly a system based on flexible pitch, i.e. it is the relation between each of the hexachord members which remains inviolate; a particular ut may have any G, C, or F pitch. Bull, however, seemingly drawing the consequences of a movable ut, places the hexachord on all 12 chromatic tones, establishing a comprehensive system of relative pitch. Mutation from one hexachord to the next, by way of a pivot tone, a tone which has a function in two successive hexachords and facilitates the transition of one hexachord to the next, does not occur here. Rather, each hexachord stands on its own and demands its unique right to existence independent of the previous hexachord. By setting up his piece in this manner, Bull does away with the entire hexachord theory. The hexachord is used as a tool to dismantle the theory based upon it.
Modulation
Having done away with mutation, Bull employs transposition. Each of the first 12 entrances of the hexachord theme thus demands a harmonic response, a modulation. Bull forces himself to write a music which touches on 12 keys. If we regard major and minor as modes, the entire gamut of keys in the tonal system of the common practice period is utilized--all in one piece of music. And his use of the whole-tone scale in transposing the hexachord theme is truly astounding and sets this piece apart from anything else in the keyboard literature of the time. So, too, did Debussy use the whole-tone scale when, at the turn of the 20th century, traditional tonality was increasingly becoming problematic as a system and composers were experimenting with new systems of harmonic organization.
13
The hexachord theme, used as a cantus firmus in this composition, i.e. unchanged, though transposed, is organized into a 13-unit phrase. The transposition scheme of the piece requires 13 statements of the theme in order to include all of the 12 tones and return to the starting point. The correspondence of the length of the theme and the number of repetitions it undergoes in the transposition scheme represents order on the highest level. This produces a first section of 169 whole-note units (13 x 13).
17
After the transposition scheme has run its course, there are four more statements of the hexachord theme in the soprano voice, bringing the total number of thematic statements to 17. However, the total number of whole-note units which comprise this second section of the work is not 52 (4 x 13). The 13th unit of the last thematic statement is omitted. That leaves us with one unit fewer than anticipated. But 51 = 3 x 17.
11
The two main sections of the work, comprising all of the statements of the hexachord theme, make up the body of the piece. It contains 220 whole-note units of music, instead of the 221 (17 x 13) units it would have had, had the last statement of the hexachord theme contained its 13th unit. 220 (20 x 11 or 2 x 2 x 5 x 11 or 5 x 44) contains no factor of 13 or 17. However, we know that at this period, as part of a very long history, gematria, the theory of numbers and their meanings, was a branch of knowledge in which artists not only dabbled, but used with impunity. One of the common uses of numbers was the representation of names: A = 1, B = 2, C = 3, etc. ("I" and "J" were considered one letter, as were "U" and "V"). Both "John" and "Bull" are represented by the number 44 (4 x 11). By leaving out the final rest of the final statement of the hexachord theme, Bull changes 52 to 51, thus relating the second section of the work to the number 17, but also changing 221 to 220, bringing the body of the piece into relationship with his own name through the common factor of 11.
Almost as an aside I would also like to mention that the number of anomalies, both of voice-leading and of texture-harmony, is 11. In other words, the entire work is regularly constructed employing the accepted rules of counterpoint/composition except those passages which Bull has given his personal stamp by deviating from the norm. His personal stamp also extends to the number of those stamps.
7
The peroration (or coda) comes last. It is extra, the icing on the cake, the statement that ends the oration. It is fitting that it has its own identity. The number seven has great significance in many cultures of the world. It is traditionally related to the length of the week and thus also to creation and the concept of completeness. It has even been maintained that the number seven is responsible for bringing everything into existence, a thought not necessarily misplaced in this context since Bull is calling a new system of musical order into existence. The seven units of peroration bring the total length of the composition to 227 whole-note units.
Primes
The structural numbers of the composition, 7, 11, 13, 17, 227, are all prime numbers. In addition, most of the anomalies take place at or are centered on whole-note units which are prime (10, 13, 23, 31, 37, 43, 101, 133, 144, 151, 163). The only exceptions are units 10 (example 14) and 144 (example 22). Unit 144 has been related to the fact that the 12th statement of the hexachord theme is taking place using the 12th member of the chromatic scale. The passage at unit 10 is the first one in the series of those that appear unusual. It is in fact the case that the numbers 10, 100 and 1000 have an intimate relationship to the number 1. As beginnings of new orders of numbers (the tens, the hundreds, the thousands) they have the same function as the number 1 itself, the beginning of all numbers. Music theorists/philosophers from all over Europe, from Italy to north-Germany, from Zarlino to Kircher to Descartes, recognize in 1 not a number, but rather the concept of unity from which all numbers, indeed all existence has its origin. Even as late as 1722, Rameau makes the statement that the number 2 is the first number, not the second. Observed from this point of view it is not inappropriate that the first of the anomalies should occur at unit 10. Given the nature of music, both the fact that it occurs in time and that it is context-bound--i.e. a context needs to be established before events can be perceived--unit 10 is the first point at which a unique event could occur "at the beginning" without simply being perceived as the beginning of the music. Though the numbers 10 and 144 are not prime, I don't think that one can deny Bull's interest in prime numbers (see Table 1.)
The whole-tone scale
One additional aspect of the piece remains to be mentioned, one that is for me a particularly savory morsel. The whole-tone scale is made up of just that: whole tones. How does the use of this scale tie in with the order found in the rest of the composition? The whole tone is mathematically represented by the proportion 9:8. We can now answer the question why there are four concluding statements of the hexachordum durum, instead of three or five, bringing the total number of statements of the hexachord theme to 17 (13 + 4). The connection between the overall form of the work, which consists of 17 statements of the hexachord theme, and the transposition scheme is given by the whole-tone scale itself: 9 + 8 = 17.
This composition is not to be disposed of as a mere curiosity. It is a clear statement by a serious composer. Ut, re, mi, fa, sol, la: a treatise on a system of music which it methodically, through the organization of its discourse, declares obsolete and actively replaces, utilizing admirable intellectual rigor, with a new order. n
