In 1998, Richard S. Parks wrote the article Pitch-Class Set Genera: My Theory, Forte’s Theory which, as implied by the title, discusses the pitch-class set genera theory of Allen Forte. This theory contains the idea that the 220 pitch-class set-classes, all including three to nine elements, could be classified in genera, based on the structural likenesses between sets. Forte states that the system of genera offers us an objective skeleton to refer harmonical material to; which is independent with regard to music and only based on the Pythagorean tradition. The genera serve models of longing inclusion, that, according to Forte’s theory, can be compared to the musical material until the ‘right fit’ is found. The difference with Parks’ theory, as he suggests, is finding a good fit, on the one side, and limiting the amount of possible genera of which just a few perfectly fit to a graspable number, on the other side. In 2003, Richard C. Pye went into these articles by testing Parks’ theory with reference to the harmonical duality in William Schumans Sixth Symphony in the essay The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony. I will start my report by laying down Parks’ theory based on Forte’s theory, after which I’ll discuss the analyses and findings of Pye. I’ll criticize if necessary to establish the functionality of these analytical methods.[1]

Parks starts his article by giving several definitions to support the concept. A ‘cynosural set-class’ is the set-class to which a collection pitch-class set-classes is related by the means of inclusion, in which a ‘simple genus’ functions as the term of this collection pitch-class set-classes. A simple genus, then, contains ‘primary members of a simple genus’, the pitch -class set-classes that are bound to the ‘cynosural set-class’ as subset or superset, and ‘ancillary members of a simple genus’ that are Z-related[2] to members of the genus, but aren’t a subset or superset of the cynosural set-class. Besides, there is a ‘complex genus’ which contains a collection of pitch-class set-classes related to two or more cynosural set-classes by inclusion, as subset or superset of these cynosures, which puts them together into an ‘union’. Of this, there are, again ‘primary members of a complex genus’, as subset or superset of all cynosural set-classes, which creates an ‘intersection’ of simple genera; the ‘secondary members of a complex genus’ all being a subset or superset of at least one, but not all, cynosural set-classes and the ancillary members that are only Z-related. Subsequently he mentions the ‘characteristic members of a genus’ that contain three to nine elements, meeting the following criteria: i) they are all subsets or supersets of each other, ii) within their interval vectors they contain uniform patterns in interval class or iii) within the consecutive interval series they’ll show similar interval patterns.[3]

Besides, Parks states that the main problem with this theory, is the fertility, as the number of genera to be created would be uncountable following into its uselessness. According to Parks this number should be reduced, based on the practical idealistic lines of simpleness, manageability and symmetry. For this, he formulated four preference rules that relate to the musical object of research. And so, the preference goes out to the genera with the most pitch-class set-classes that can be found in the composition, of which the genus, with the biggest amount of set-classes in the primary or characteristic member form, smallest number of (primary) members and the genus most comparable with recognizable tonal structures, has to be picked.[4]

The usability of this theory, is seen when a pitch-class set-class genera fitting the musical object, is found. The structural quality of the genus then also includes the musical object; or so, what can be said about the genus, can be said about the composition. In Pye’s essay he outs Parks’ theory to the test based on an analyses of Schuman’s Sixth Symphony, that I’ll discuss here. Pye states, before starting his analyses, that;

the real difficulties behind the further progress of genus theory are to be found […] in their application to real musical objects. Anyone seeking to deploy the genus theories of either Forte or Parks to model anything other than small-scale structures soon discovers that such models often form far-from-perfect interfaces with the music in question. However […] it is precisely in the perceived gap between model and object that meaningful analytical discoveries lie.[5]

He focusses his analyses on highlighting the difficulties in the practical application of the theory, but also on offering a tool to produce a meaningful interpretation from the results.[6]

The symphony would contain a harmonic duality on two levels; one including chromatic elements, and the other with diatonic elements. This contrast could be played out in the form of two triads plus one note. The first is the major-minor chord set-class 4-17 [0347] or C-Dis-E-G and the second one is 4-14 [0237] or C-D-Dis-G. He shows, through his analyses, how these two set-classes could function as a basis of the whole composition, in which 4-17 functions as a kind of tonic. In example 1, we see the first measures of the work, in which 4-17 occurs vertically in the first two chords and overall four times. Besides, it is a melodic segment of the theme (C#-Bb-F-D), as [1,10,5,2] is [12510], but in ascending form. This contains the same distances of intervals as [0347], namely 3-1-3-5. 4-14 is, however, only heard in the first verticality of measure 3 and the first bass tones of the work.[7]

Then, Pye draws on a formal design on how the composition structurally can be described, based on harmonical and thematical material, as to be seen in example 2. He argues that the given genera are picked on the basis of harmonic common occurrence, playing a part in contradictions and coherence, while thematical material remains crucial for the definition of form. The broader formative distinctions in the work would dynamically run parallel to the thematic material in measure 50, 169, 421, 495 and 688, as a result of which a traditional scheme of four measures arises including an introduction and coda. However, when I look at the material of the indicated borders, the drastic thematic differences aren’t to be found. The introduction and coda contain, with an exception of the beginning of the Moderato con moto-part and the transition between Leggeramente and Adagio, as only sections material B. After which it stands out that ‘A’ is to be found within the whole piece and ‘C’ and ‘D’ are alternated with a big block of ‘D’ in the middle, almost continuously until the end, with an exception of measures 515-517. This distribution, however, doesn’t seem to related to the duality between 4-14 and 4-17. Pye, unfortunately, only offers an analyses with the introduction of the second movement, causing example 2 to be the only basis for assessing the boundaries of the complete composition.[8]

To clarify the harmonic duality, he supports himself with a matric of pitch-class set-classes of the two principal tetrachords, see example 3. The harmonical duality is, according to hem, clearly seen in the opening, where the chromatic 4-17 harmonization by the means of theme A are dualized by theme B. This causes the introduction of 4-14, harmonized with theme A in measure 9, to be highlighted. By supporting subsets of 4-14, as 3-2 and 3-4, the priority of contrasting 4-14 is secured from the beginning. By then inserting non contextual dissonant set-classes that dissolve chromatic formation to diatonic set-classes, this contextual duality and the priority of the diatonic, specifically the 4-14, is confirmed. See example 4.[9]

By letting the horns and trombones play the 4-17 material, followed by strings in contrasting 4-14 material and the return of brass in 4-17, Schuman focusses, again, on the duality after introducing the thematical material of ‘C’ in measures 35-36. Such sections in the form of separated block could, according to Pye, be seen as a signature of Schuman; instrumental groups playing crucial parts for defining structures and oppositions. ‘From this perspective, the idea that the principal tetrachords may be fulfilling a cynosural role in the context of a clearly defined generic duality is a promising one.’[10] It is of importance that not all complex members of 4-14 are diatonic, and the other way around, that not all 4-17 complex members are chromatic, besides the use of diatonic and chromatic material which isn’t a part of 4-14 or 4-17. Pye calls this an obstacle; as it makes analyzing based on genera more complex. I see it, however, as a logical consequence of working out the musical material while composing, and thus, as slightly naïve of Pye.[11]

The movement Moderato con moto would consist of three parts; namely the 4-14, 4-17 and 3-11 movement. The beginning, of measure 50 to 80, actually keeps trotting on the tension between 4-14 and 4-17 supported by theme A. However, with the return of A and B in measure 80, 4-14 is reconfirmed, namely by mean of the vertical application. In measure 94, 4-14 is seen as melodically applied as a transition between A and B; see example 5. Measure 94 is the only moment in which 4-14 specifically occurs in a melodic manner besides the vertical use. The rest of the thematical material consists of 4-14 complex set-classes; see example 6.[12]

A harmonical turn is noted with the recurrence of theme C in measure 123 with the help of an accelerated tempo, the shift of strings to brass and the syncopated rhythms of theme C. Instead of using 4-14 or 4-17 material, Schuman uses the shared material of 3-11, as seen in example 7. 3-11 has the function of a transition in this movement, as seen in example 2. This transition is closed on measure 132 with a statement of a 4-17 chord in brass and strings. The measure 132 to 149 then show a reduction of notes, so melodical repetition can be avoided. Chords are alternated with fast melodic material, in which theme C is further developed. The tetrachords are, again, put at the front with major triad F-A-C in timpani. The 4-17, in addition, isn’t only found in the bass chords, but also in the melody of the upper strings and brass, causing the harmonic neutral setting to be resumed after the transition; see example 8.[13]

As seen, 4-17 isn’t the only material that is used. Diatonic formations are applied to form a kind of cadential contradiction. These are then resolved in the context of the harmonic norm of 4-17. And so, the triad of timpani’s (F#-A-C#) of measures 134, 136 and 137 is a mediation between such cadences. In measure 143 it forms a part of the 4-17, while, in measure 136, it is combined with a 4-14 chord, namely the Fb-Ab-A-Cb, which results into the hexachord 6-23 [024579]. The solution is, then, achieved through the diatonic bass chord 3-5 A-Bb-Eb, combined with the same triad of 4-17, 3-11, to create the superset 5-31 [01469]. Such transitions are noted more often, as in measures 143 to 146, by the shift of E (4-17) to D (4-14) in the trombones. Such shifts are often strengthened by dynamics and instrumentation.[14]

According to Pye, the three-part structure, of measure 80-122, 123-132 and 132-168 in the 4-14, 3-11 and 417 distribution, contains the harmonical play found back through the entirety of the symphony. The 3-11 functions as connecting element between the two opposites. The connection of theme C with the neutral triad is, however, more significant, as the moment brings ‘something else’. Namely a moment of distance from duality. ‘The projection of this clearly defined and manageably compact three-part structure makes the movement ideal subject upon which to base a generic model of pitch structure.’[15] This model should concisely present, the relationship between the two tetrachords and their bound character, in which, naturally, the language of set-class genera should be central and their characteristics clearly drawn, so that the set-class structures of the music can be characterised. Based on this model, then, a broader observation of the music can be made.[16]

Reducing the number of possible cynosures onto a treatable amount is made by Pye according to Parks rules of preference, as described in Alinea three. He decides to, instead of firstly looking at the genera containing as much as possible members within the composition, look at the genus containing the most likeness in tonal structures. He insures, so to say, that the model, because of that, has the most compact possible relationship possible with the set-classes characterizing the composition. With this, he ignores Park, who mentions always to start at rule one, after which the following three could randomly be applied. By starting with rule four, Pye takes a risk to fail in result. It seems, however, to ensure that rule one, in a later application, takes on less work, as a lot of genera are already ruled out, due to a different tonal structure. Following to this, he writes that an important consideration would be, that he isn’t looking to one genus best representing the matrix of example 3, but two genera best representing the most extreme differences of tetrachords 4-14 and 4-17. Because of this, you’ll notify the biggest differences, but also the clear areas of agreements. And so, a clear overlap of set-classes between 4-14 and 4-17 exists, but also the shared characteristic of inaccurate agreement of 4-14 with a broader diatonic classification of material and the reversed less extensive, deviation of the 4-17 complex regarding chromatic material.[17]

Disadvantageously, set-classes playing a more prominent role in the harmonic duality by putting rule four at the front, are isolated, while the model should make the distinction between the two chromatic and diatonic extremes clear. By the means of these critical criteria, set-classes that don’t characterize the broader and closer orbits of duality are locked out. For example, the chromatic complex members of 4-14, like 6-z48 [012597], or the chromatic set-classes that fall outside the 4-17 complex, like 5-22. Off course, it is desirable that these classes are brought along by the generic model, but I believe that the minor relevance in the bigger picture will eventually eliminate these sets as well, as a result of which it leaves Pye with thirteen potential cynosures.[18]

Example 9, then, display three measurements. The first column is the percentage of agreement with the matrix per genus. The second relates to the size of the cynosural sent complex per genus, based on a statistical calculation of Forte from 1988. This amount is of an importance, as small set-class complexes in the context of bigger matrixes don’t necessarily give out a lot of information. The calculation for Squo is; Squo (Ga) = ((X/Y)/Z) x 10. X represents the number of representative genera in the matrix, Y the total amount of set-classes within the matrix and Z de biggest of the regarding genus. The final column comes forth from the thesis The Harmonic Species of Frank Bridge: An Experimental Assessment of the Applicability of Pitch-Class Generic Theory to Analysis of a Corpus of Works by a Transitional Composer by Chris Kennet in 1995 in which the limit of smaller matrixes in the case of a bigger set-class complex is shown regarding the possibility to fulfil the number of Squo. The percentage, thus, shows us to what extent the Squo can be achieved against the maximum possible Squo percentage of 100%.[19]

Pye, then, considers the highest possible ranked pitch-classes based on characteristic agreements with 4-14 and comes to the conclusion that the complex 7-35, despite it only corresponding with the matrix for 36,1%, agrees the most with the composition. 8-22 Seems to be a more suitable candidate, but contains totally different characteristics. With that, the complex 7-35 is representable for the wider track of 4-14 associated set-classes. He does the same with 4-17, in which 8-17 seems to emerge as the winner, but fails in the field of bringing harmonic duality in the collection of set-classes. 3-3, however, contains the set-classes that 7-35 do not contain, adding that the similarities are found in set-classes containing 8 and 9 tones. By comparing the 3-3/4-17 and the 4-14/7-35 genera complexes, the overlapping sets can be found and characteristics of the compositions can be bound to the genera. A somewhat confusing twist Pye then makes, is that he still looks at the set-class 8-22 to solve the problem of the missing centrality between the two complexes. The tonal material of 8-22 would contain familiar sounds for Schumans diatonic work and other compositions of his time. He leaves the set-class, after one Alinea, however, for what is it, which results in an unclear image of added value to me.[20]

The remaining task, with this, is only determining the characteristics of the set-classes genera 4-14/7-35 and 3-3/4-17, after which the characteristics could be reflected onto the set-classes within the composition. For this, Pye takes the characteristic set-classes, or so; the set-classes closest to the cynosure(s). In example 10, you’ll see the characteristic set-classes of 3-3/4-17; with ‘iv’ for interval vector and ‘sia’ for successive-interval arrays.

He then refines this selection by looking at the set-classes with a high level of intersection with the cynosures and equal interval patterns; in this case vector 1, 3 and 4. As a result he points to set-classes 5-21, 6-15, 6-20 and 7-21, as they all emphasize iv 1, 3 and 4 and, except for 6-15, the sia sequence 1-3-1-3. 6-15 contains, however, a mediation-function between 7-21 and 5-21, as seen in example 11.[21]

When he tries to match the set-classes with the music, Pye finds out that most of the characeristicset-classes won’t come back in the musical material, considering 6-20 and 6-15. This problem represents the distance between the practical and theoretical side. The genus 3-3/4-17’s hexachord 6-z19 represents the final statement of theme D, but is theoretically less characteristc. By putting the genera onto the music, Pye formulated a definitve characteristic set genus; see example 12. With the help of this complex genus, the chromatic side of the harmonic duality could eventually be read within quantitive terms, but also the theoretical association with the music and their placement considering the centre of the genus 4-17. The complex genus of 4-14/7-35 is, logically, formulated in the same manner; see example 13. Terminology ‘R2’ means in this ‘maximum similarity with respect to interval class’ and ‘Rp’ for ‘maximum similarity with respect to pitch class’.[22] Logically, a set-class is characteristic for a genus, when the innermost represents elements of set-classes, as well as 4-14 and 7-35 or 3-3 and 4-17. This leaves us with the characteristic set-classes of 4-14/7-35 in example 14.[23]

Most striking is the relationship between the set-classes as equal instead of additional, off course due to the assembly of two cynosures. An expected conclusion is then, that a cynosural tetrachord has the biggest influence over small set-classes, as they are projected within the music. While the bigger classes deserve their spot by the means of diatonic chrystallisation; a characteristic of the bigger cynosures. A demonstration of this duality is, for example, the less focused group of characteristic set-classes in the heart of genus 4-14/7-35 confirming the influence of the chromatic tetrachord 4-17 as ‘type tonica’ and lay the connection between the genus 3-3/4-17. The analist is, with this, capable to better understand the internal dynamics and relations between pitch-class set-classes, but does have to take into consideration the degree of suitability of the musical piece and the chorsen theory. Forte’s theory could bring more insight considering the highlighting of differences and movements within works and between repertoires, while Parks’ theory suits itself more for compositions with a big accumulation of structural meaning. What I miss, after all, in Pye’s conclusion is a tool to process the results into a meaningfull formulation, besides its structural function.[24]

Eventually, this analytical theory rqeuests discretion and attention, as wrongfull noted set-classes could be extended to misplaced conclusions considering the whole composition. This makes that, for a detailed and complete understanding of the analyess, without temporal interim assumptions, each observation should be explained arithmetically. The theory doesn’t provide any meaning or interpretation of the music, but leaves information about the often applied (tonal) structure of the musical object, which could say something about the compositional technique. Through pythagorean methods, the sounds of the composition appear to be far out of sight, which results in a theoretical conclusion regarding musical characteristics. The question, thus, wuold be how this information can be applied on the musical meaning of a specific compoistion, especially with regard to a meaning for the listeners. The analytical method, with that, needs many steps come to the endresult, while the meaning of the results aren’t neccesarily found. This makes the method cumbersome; as a result of which we could better put more ‘traditional’ analytical methods based on thematic and motivic material first. Or does this method bring more detailed insights, specifically in the area of interval structures? My answer to this is; yes. The analytical method takes more care of detailed insights in the area of interval structures and underlying relations, which causes the compositiontechniques to be explained better. For an interpretative meaning, however, which focusses more on the sounds and functions of the music, other, more traditional methods of analyses, should be combined.

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[1] Richard S. Parks, “Pitch-Class Set Genera: My Theory, Forte’s Theory,” in Music Analysis 17, no. 2 (juli 1998), 206; Allen Forte, “Pitch-Class Set Genera and the Origin of Modern Harmonic Species,” in Journal of Music Theory 32, no. 2 (herfst 1988), 187, 188.

[2] Definitie Z-gerelateerd; verwijst naar een relatie van sets die dezelfde interval-class content bevatten, maar niet gerelateerd zijn door transpositie of inversie. John Roeder, 3. Synopsis., in “Set (ii),” in Grove Music Online, Red. Deane Root (Oxford: Oxford Music Online), http://www.oxfordmusiconline.com/grove

music/view/10.1093/gmo/9781561592630.001.0001/omo-9781561592630-e-0000025512.

[3] Parks, Pitch-Class Set Genera: My Theory, Forte’s Theory, 207, 209.

[4] Parks, Pitch-Class Set Genera: My Theory, Forte’s Theory, 209-211.

[5] Richard C. Pye, “The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony,” in Music Theory Spectrum 25, No. 2 (herfst 2003), 243, 244.

[6] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 243, 244.

[7] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 246.

[8] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 247.

[9] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 249.

[10] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 252.

[11] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 249-251.

[12] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 249, 253-254.

[13] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 254, 255.

[14] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 255, 258.

[15] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 258.

[16] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 258, 259.

[17] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 259-261.

[18] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 261.

[19] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 262.

[20] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 263, 264, 266.

[21] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 266.

[22] Allen Forte, The Structure of Atonal Music (New Haven, Yale University Press, 1973), 46-60.

[23] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 269, 271.

[24] Pye, The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony, 272, 273.

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Used Literature

Forte, Allen. “Pitch-Class Set Genera and the Origin of Modern Harmonic Species.” Music Analysis 17, no. 2 (juli 1998): 187-270. The Structure of Atonal Music. New Haven: Yale University Press, 1973.
Parks, Richard S.. “Pitch-Class Set Genera: My Theory, Forte’s Theory.” Music Analysis 17, no. 2 (juli 1998): 206-226.
Pye, Richard C.. “The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony.” Music Theory Spectrum 25, No. 2 (herfst 2003): 243-274.
Roeder, John. “Set (ii).” in Grove Music Online, Red. Deane Root (Oxford: Oxford Music Online). http://www.oxfordmusiconline.com/grovemusic/view/10.1093/gmo/9781561592630.001.0001/omo-9781561592630-e-0000025512.

Examples 1 to 14 are copied from:

Pye, Richard C.. “The Construction and Interpretation of Bespoke Pitch-Class Set Genera as Models of Harmonic Duality in William Schuman’s Sixth Symphony.” Music Theory Spectrum 25, No. 2 (herfst 2003): 243-274.

Pitch-Class Set Theory Genera considering William Schumans Sixth Symphony

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