Volume 13, Number 3, September 2007
Copyright © 2007 Society for Music Theory
Without a Safety (k)NetPhilip StoeckerREFERENCE: http://www.mtosmt.org/issues/mto.07.13.2/mto.07.13.2.buchler.html

[1] When analyzing a passage of music with Knets, the transformational operation from one Knet to the next is just as important as the construction of each individual Knet. To ensure that different Knets are related, various authors have proposed that either an interval (Lewin 1990 and Klumpenhouwer 1991) or an inversional relationship (Stoecker 2002) from one Knet to the next must remain invariant. And with all the isographic tools that are available—strong, positive, negative, and axial isography—virtually every trichordal Knet can be isographically related to any other trichordal Knet. This high degree of isographic relatedness is a concern for Michael Buchler. In his critique of Knets and Knet analyses, Buchler points out that
[2] Though Buchler warns us about the promiscuous nature of trichordal isography, there are times when adjacent, trichordal collections do not share an interval or an inversional relationship. As a result, the Knets that model these sonorities cannot be isographically related, creating a gap in our transformational pathway. These isographic holes are the focus of my article. Trichordal Knets that cannot be conventionally and axially isographic are extremely rare in the literature. It is unusual to find two collections that do not share an interval class or an inversional relationship.^{(2)} To better understand how these analytical gaps occur, I will focus on the voice leading between Knets that cannot be isographically related. [3] In his first published study of Knets, David Lewin analyzes a few brief passages from “Eine blasse Wäscherin,” the fourth song from Arnold Schoenberg’s Pierrot Lunaire, op. 21.^{(3)} The Pierrot passages that Lewin analyzes are good ones for illustrating the strengths of a Knet analysis. Lewin effectively demonstrates how to construct coherent, transformational pathways and introduces Knet recursion. But if we analyze the entire song with a Knet lens, we will find that the isographic tools that Lewin introduced cannot relate a few Knets. Example 1. Schoenberg, “Eine blasse Wäscherin,” Pierrot Lunaire, op. 21, mm. 1–4 (click to enlarge) Example 2. Schoenberg, “Angst und Hoffen,” Das Buch der hängenden Gärten, op. 15, mm. 1–2 (click to enlarge) Example 3. Voiceleading scenarios from Schoenberg’s “Angst” trichord (click to enlarge) [4] Example 1 includes the score for the opening instrumental phrase of “Eine blasse Wäscherin.” Most of the trichords in this passage contain a dyad of intervalclass 3, and the Knets that model these collections could be configured in such a way that T_{3} and T_{9} arrows are drawn. From an isographic perspective, virtually every trichordal Knet in this passage is strong, positively, or negatively isographic to its adjacent Knet. In m. 3, however, the Knet that interprets the [036] diminished trichord cannot be isographically related to the Knet that interprets the [048] augmented trichord. Since these two collections do not share an interval class, the Knets cannot be related by strong, positive, or negative isography. And since these particular diminished and augmented trichords do not share an inversional relationship, they cannot be related by axial isography. As a result, the Knets that interpret these two collections cannot, in any way, be isographically related. [5] An even more remarkable passage to explore nonisographic Knets occurs in Schoenberg’s song “Angst und Hoffen” from his Buch der hängenden Gärten, op. 15, no. 7, given in Example 2.^{(4)} The first three collections of the piano accompaniment are members of three different set classes: [048], [016], and [03] respectively. The augmented triad accompanies the word “Angst” while setclass [016] accompanies the word “Hoffen.” The A4/C5 dyad on the downbeat of the second measure accompanies the word “wechselnd.” Throughout the song, the “Angst” [048] and “Hoffen” [016] sonorities alternate with one another, reflecting the anxious longing for love by the narrator. What makes the opening accompanimental passage so remarkable from a Knet perspective is that these three adjacent collections do not share an interval or an inversional relationship, two requirements to ensure Knet relatedness.^{(5)} [6] Example 3 presents models that review voiceleading properties for three different types of isographies—the example does not include negative isography—and the last progression is Schoenberg’s “Angst” trichord to the “Hoffen” trichord. When trichords are strongly isographic (Ex. 3a) the registral, voiceleading lines feature three moves, all by the same interval class. When trichords are positively (Ex. 3b) or axially isographic (Ex. 3c) the registral, voiceleading lines will always feature two different intervalclass moves. When trichordal Knets cannot be isographically related, the registral lines will always feature three different voiceleading moves.^{(6)} In Example 3d, the progression from the “Angst” trichord to the “Hoffen” trichord features three different registral voiceleading moves. [7] An effective way to measure the voice leading of nonisographic Knets is to use Joseph Straus’s “total displacement,” which is the sum in absolute value of the three voiceleading moves from one trichord to the next.^{(7)} In Example 3d, the registral voice leading from the “Angst” to the “Hoffen” trichord features three different interval classes (2, 0, and 1), and these three voiceleading intervals sum to a total displacement of 3. A total displacement of three is significant since this is the smoothest possible voice leading for trichordal Knets that cannot be conventionally and axially isographic. Since nonisographic Knets will always feature three different voiceleading moves, a total displacement of 0 (0+0+0), 1 (0+0+1), and 2 ((0+1+1) or (0+0+2)) is not possible.^{(8)} Figure 1. Cohn’s Directed VoiceLeading Sum (DVLS) applied to “Angst” and “Hoffen” (click to enlarge) Figure 2. The [016] presets that cannot be isographically related—strong, positive, negative, and axial—to {G,B,D}, the “Angst” sonority (click to enlarge) Figure 3. Setclass pairs that are not isographic organized into DLVS categories (click to enlarge) [8] Another way to measure all the possible voiceleading lines between nonisographic Knets is to use Richard Cohn’s
directed voiceleading sum [9] If the original “Hoffen” trichord is transposed, different DVLS numbers emerge. Figure 2 shows that the “Angst” trichord cannot be conventionally or axially isographic with twelve of the twentyfour members from the [016] “Hoffen” family.^{(12)} The figure also illustrates that four different DVLS numbers are possible when the “Angst” trichord is followed by one of the nonisographic, [016] pitchclass sets: 7 and its complement 5, 1 and its complement 11.^{(13)} [10] One of the advantages of using Cohn’s DVLS is that it helps catalog the different types of voice leadings for nonisographic Knets, which I have included in Figure 3.^{(14)} Each DVLS number is paired with its complement and listed along the top. Beneath each DVLS category I have included all setclass pairs that cannot be conventionally or axially isographic. For example, DVLS 0 contains only two setclass pairs that cannot be isographic: [015][036] and [024][036]. What the figure does not tell us is which [015] is not isographic with which particular [036]. Note that each nonisographic progression listed in the figure has at least one symmetrical trichord; these sonorities contain a limited number of different interval classes, which increases the chances that it will not be isographic with its adjacent sonorities. In addition to the symmetrical set classes, another manifestation of symmetry occurs in Figure 3: the trichordal progressions that are listed in DVLS 0 also appear in DVLS 6; the setclass pairs listed in 1/11 also appear in 5/7, and so on.^{(15)} [11] The setclass pairs highlighted in bold (Fig. 3) represent examples I found in the literature. Of the twentyfour possible pairs of nonisographic Knets listed in the figure, only six are represented here. And in every case, an augmented triad plays a role in establishing the nonisographic relationship. In fact, the chance that an augmented triad plays the spoiler is quite high. Recall that in Figure 2, the [048] “Angst” trichord cannot be isographic with twelve different pitchclass sets from the [016] family.^{(16)} In addition to its intervallic redundancy, all four members of the [048] family feature only even I_{n} labels. This is a significant property since all the other trichordal set classes feature two odd I_{n} labels and only a single even I_{n} label.^{(17)} So when it comes to network isography, [048] functions as an isographic renegade. [12] Though the augmented triad is responsible for most of the transformational gaps, trichordal Knets that are not conventionally or axially isographic are quite rare. If we were to analyze the rest of the trichords in Schoenberg’s “Eine blasse Wäscherin,” network isography will be easily achieved. It should come as no surprise that in all the analyses that use trichordal Knets, the analyst has presented coherent, transformational pathways since Knet relatedness is so common. Still, there is a definite limit to the abundant isographic relations among trichordal Knets. But what do these isographic holes tell us about the music it models? As Figure 3 illustrates, all nonisographic progressions include at least one trichord that is symmetrically organized. In addition, nonisographic Knets will always feature three different voiceleading moves, creating a maximally diverse pitchclass counterpoint. Recall that in Schoenberg’s “Angst und Hoffen,” the opening [048] to [016] progression features a variety of different voiceleading lines, and it is not difficult to imagine Schoenberg choosing two trichords that do not share any similarities; for me, Schoenberg’s diverse voiceleading lines appropriately accompany the words fear and hope of the text. So when it finally happens that Knets cannot be isographically connected, the transformational paths are momentarily thwarted and the voice leading from one trichord to the next is maximally diverse. Rather than search for cues elsewhere to bridge these transformational gaps, I wish to highlight and celebrate those extraordinarily unique moments when our transformational safety (k)net has been taken away.
Philip Stoecker Works CitedBuchler, Michael. 2007. “Reconsidering Klumpenhouwer Networks.” Music Theory Online 13.2: 1–69. Buchler, Michael. 2007. “Reconsidering Klumpenhouwer Networks.” Music Theory Online 13.2: 1–69. Cohn, Richard. 1998. “Square Dances with Cubes.” Journal of Music Theory 42.2: 283–296. Cohn, Richard. 1998. “Square Dances with Cubes.” Journal of Music Theory 42.2: 283–296. Klumpenhouwer, Henry. 1991. “A Generalized Model of VoiceLeading for Atonal Music.” Ph.D. dissertation, Harvard University. —————. 1991. “A Generalized Model of VoiceLeading for Atonal Music.” Ph.D. dissertation, Harvard University. Lambert, Philip. 2002. “Isographies and Some Klumpenhouwer Networks They Involve.” Music Theory Spectrum 24.2: 165–195. Lambert, Philip. 2002. “Isographies and Some Klumpenhouwer Networks They Involve.” Music Theory Spectrum 24.2: 165–195. Lewin, David. 1981. “A Way Into Schoenberg’s Opus 15, Number 7.” In Theory Only 6.1: 3–24. Lewin, David. 1981. “A Way Into Schoenberg’s Opus 15, Number 7.” In Theory Only 6.1: 3–24. Lewin, David. 1990. “Klumpenhouwer Networks and Some Isographies That Involve Them.” Music Theory Spectrum 12: 83–120. —————. 1990. “Klumpenhouwer Networks and Some Isographies That Involve Them.” Music Theory Spectrum 12: 83–120. Stoecker, Philip. 2002. “Klumpenhouwer Networks, Trichords, and Axial Isography.” Music Theory Spectrum 24.2: 231–245. Stoecker, Philip. 2002. “Klumpenhouwer Networks, Trichords, and Axial Isography.” Music Theory Spectrum 24.2: 231–245. Straus, Joseph N. 2003. “Uniformity, Balance, and Smoothness in Atonal Voice Leading.” Music Theory Spectrum 25.2: 305–352. Straus, Joseph N. 2003. “Uniformity, Balance, and Smoothness in Atonal Voice Leading.” Music Theory Spectrum 25.2: 305–352. Footnotes1. Michael Buchler (2007, [37]). 2. Though some Knets can only be related by axial isography, the invariant inversional relationship often exists in pitchclass space. For me, axial isography is a much stronger relationship when it unfolds in pitch space. 3. Lewin (1990, 91–95 and 98–99). 4. David Lewin (, 3–24), Philip Lambert (2002, 165–195), and Philip Stoecker (2002, 231–245) have analyzed passages of this song that resonate with my current analysis in meaningful ways. 5. The Knet that models the A4/C5 dyad could be configured in such a way that it shares an I_{0} arrow with the first trichordal Knet. 6. The reverse, however, is not true. Two trichords that feature three different voiceleading moves can, at times, be isographically related. 8. On the other end of the spectrum, 15 will be the highest total displacement for Knets that cannot be isographic. A trichordal progression that features a total displacement of 16 (5+5+6 or 6+6+4), 17 (6+6+5), and 18 (6+6+6) will have two (or three) voiceleading moves by the same interval class. Thus, the Knets that model these trichordal progressions can be isographic. 10. Unlike Cohn’s DVLS, if we choose different voiceleading lines Straus’s totaldisplacement number changes. 11. As Cohn points out, the DVLS number can also be measured by calculating the difference between the SUM of the pitchclass integers for set (X) with the SUM of the pitchclass integers for set (Y). That is, if C = 0, then SUM(Y) – SUM(X) = DVLS. 12. Tritone transpositions for each [016] set class are listed next to each other since a T_{6} operation on any set class preserves all the I_{n} labels. 13. Unlike Straus’s totaldisplacement numbers, DVLS numbers for nonisographic Knets do not have a minimum value of 3. 14. [026] is the only trichordal set class not included in this figure since it can be isographic with all trichordal Knets. 15. The symmetrical organization of Figure 3 can be explained as follows: If pcset X is not isographic with pcset Y, then pcset T_{6}X will also be nonisographic with pcset Y—a tritone transposition of a pcset preserves both the interval content
and the I_{n} labels. From a voiceleading perspective, the DVLS numbers for X to Y and T_{6}X to Y will differ by six, e.g., DVLS 1/11 (X to Y) and DVLS 7/5 (T_{6}X to Y). Thus the nonisographic progressions in the figure appear in two different DVLS categories related by a tritone. DVLS 3/9 is a selfmapping category, i.e., DVLS 3/9 and its tritone partner DVLS 9/3 are the same category. 16. To offer a different scenario, {C,C,E}, a member of set class [014], cannot be isographic with only two pcsets from the [027] family: {C,D,G} and 17. In addition to [048], all members of the [024] and [026] families feature only even I_{n} labels. Michael Buchler (2007, [37]). Though some Knets can only be related by axial isography, the invariant inversional relationship often exists in pitchclass space. For me, axial isography is a much stronger relationship when it unfolds in pitch space. Lewin (1990, 91–95 and 98–99). David Lewin (, 3–24), Philip Lambert (2002, 165–195), and Philip Stoecker (2002, 231–245) have analyzed passages of this song that resonate with my current analysis in meaningful ways. The Knet that models the A4/C5 dyad could be configured in such a way that it shares an I_{0} arrow with the first trichordal Knet. The reverse, however, is not true. Two trichords that feature three different voiceleading moves can, at times, be isographically related. On the other end of the spectrum, 15 will be the highest total displacement for Knets that cannot be isographic. A trichordal progression that features a total displacement of 16 (5+5+6 or 6+6+4), 17 (6+6+5), and 18 (6+6+6) will have two (or three) voiceleading moves by the same interval class. Thus, the Knets that model these trichordal progressions can be isographic. Unlike Cohn’s DVLS, if we choose different voiceleading lines Straus’s totaldisplacement number changes. As Cohn points out, the DVLS number can also be measured by calculating the difference between the SUM of the pitchclass integers for set (X) with the SUM of the pitchclass integers for set (Y). That is, if C = 0, then SUM(Y) – SUM(X) = DVLS. Tritone transpositions for each [016] set class are listed next to each other since a T_{6} operation on any set class preserves all the I_{n} labels. Unlike Straus’s totaldisplacement numbers, DVLS numbers for nonisographic Knets do not have a minimum value of 3. [026] is the only trichordal set class not included in this figure since it can be isographic with all trichordal Knets. The symmetrical organization of Figure 3 can be explained as follows: If pcset X is not isographic with pcset Y, then pcset T_{6}X will also be nonisographic with pcset Y—a tritone transposition of a pcset preserves both the interval content
and the I_{n} labels. From a voiceleading perspective, the DVLS numbers for X to Y and T_{6}X to Y will differ by six, e.g., DVLS 1/11 (X to Y) and DVLS 7/5 (T_{6}X to Y). Thus the nonisographic progressions in the figure appear in two different DVLS categories related by a tritone. DVLS 3/9 is a selfmapping category, i.e., DVLS 3/9 and its tritone partner DVLS 9/3 are the same category. To offer a different scenario, {C,C,E}, a member of set class [014], cannot be isographic with only two pcsets from the [027] family: {C,D,G} and In addition to [048], all members of the [024] and [026] families feature only even I_{n} labels.
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