T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 499–511, 2009. © Springer-Verlag Berlin Heidelberg 2009

Affinity Spaces and Their Host Set Classes

José Oliveira Martins

Eastman School of Music, University of Rochester jmartins@esm.rochester.edu

Abstract. This paper proposes the organization of pitch-class space according to the notion of affinities discussed in medieval scale theory and shows that the resultant arrangement of intervallic affinities establishes a privileged correspondence with certain symmetrical set classes. The paper is divided in three sections. The first section proposes a pitch-class cycle, the Dasian space, which generalizes the periodic pattern of the dasian scale discussed in the ninth-century Enchiriadis treatises (Palisca 1995). The structure of this cycle is primarily derived from pitch relations that correspond to the medieval concepts of transpositio and transformatio.1 Further examination of the space’s properties shows that the diatonic collection holds a privileged status (host set class) among the embedded segments in the cycle. The second section proposes a generalized construct (affinity spaces) by lifting some of the intervallic constraints to the structure of the Dasian space, while retaining the relations of transpositio and transformatio, and the privileged status of host set classes.2 The final section examines some of the properties of host set classes, and in turn proposes “rules” for constructing affinity spaces from their host sets. The study of affinity spaces will give us insights regarding scalar patterning, inter-scale continuity, the combination of interval cycles, voice leading, and harmonic distance.3

1 Affinities in the Medieval Dasian Scale

The idiosyncratic dasian scale discussed in the late ninth-century Enchiriadis treatises is constituted by a series of tone-semitone-tone tetrachords consistently separated by a tone of disjunction (see Example 1a). This arrangement replicates the four modal qualities (protus, deuterus, tritus, and tetrardus) of the finales tetrachord throughout 1 For a discussion of the interpretative attitudes involving these concepts see Pesce (1986 and

1987). 2 In earlier work, I propose the Dasian and other affinity spaces as suitable analytical

frameworks to address harmonic and melodic aspects that result from the combination of different scales in twentieth-century music. For analytical accounts of Béla Bartók’s polymodality and Witold Lutoslawski’s “12-note” music see Martins (2006a); for the analysis of Igor Stravinsky’s harmonic syntax in a neoclassical work, see Martins (2006b); and for the interpretation of Darius Milhaud’s polymodality see Martins (2007).

3 This study intersects in interesting ways with work developed by several theorists: Philip Lambert and Edward Gollin on the combination of interval cycles and multi-sets, see Lambert (1990) and Gollin (2007); Carey and Clampitt (1996); and Tymoczko (2004) on scalar theory.

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the scale, since the semitone placed in the center of each tetrachord between deuterus and tritus creates equidistant intervallic markers and allows for establishment of intervallic affinities at the perfect fifth throughout the scale. This scale structure allowed the transfer of melodic segments at both the upper and lower fifth to retain their modal identities. Such transfer was referred to as transpositio and preserved the segments’ local intervallic structure, as illustrated in Example 1b, where the arrows point to the preservation of the modal quality deuterus.

Example 1. The dasian scale (Enchiriadis treatises)

The intervallic arrangement of the dasian scale, however, overrides modal affinity at the octave since the same pitch class is assigned to different modal qualities in different tetrachords. In Example 2, the circled pitch class (pc) A appears as deuterus, protus and tetrardus in the scale.4 The use of B-flat in the Graves tetrachord

4 Guido d’Arezzo considered affinity (likeness) to be completed only at the octave, which is the

reason why Guido and subsequent theorists rejected the dasian scale. This is not to say that the Enchiriadis treatises do not acknowledge the privileged status of the octave relation in their musical system, but rather that affinities at the fifth are of a different consonant “quality” (consonae) than octave “hyper-consonant” relations (aequisonae). See Pesce (1986, 337), and Mengozzi (2006).

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inflecting B-natural in the Superiores changes the modal quality of the adjacent pc A from protus in the Superiores to deuterus in the Graves tetrachord; similarly, the use of F# in the Excellentes tetrachord instead of F-natural makes pc A sound as tetrardus. In other words, pc A has different modal qualities in each tetrachord given the different distribution of tones and semitones surrounding each appearance in the scale. The same could be said for other pcs in the scale. In the medieval notation of chant, Guido d’Arezzo warned against the change of modal quality created by note alterations to a pure diatonic framework. He referred to such modal change as transformatio. But even though disapproved by Guido, transformatio became a powerful practice extending the flexibility of the tonal system. Transpositio and transformatio thus capture significant pitch relationships: transpositio refers to the transfer of modal quality (local interval pattern) to various positions in the tonal system; whereas transformatio refers to the change of modal quality (local interval pattern) on a given pitch class effected by a note alteration, as well as to the note alteration itself.

Example 2. Different modal qualities for the same pitch-class in the dasian scale

2 The Dasian Space

Example 3 proposes a pitch-class generalization of the dasian scale by extending the species of fifth beyond the medieval scale’s boundaries (the disjunct tetrachords of the scale are marked by brackets and Roman Numerals). This generalization closes the space in two ways: by exhausting the 12 transpositions of the tetrachordal interval pattern, and by creating a pitch-class cycle.5 This framework renumbers the modal quality of each note in the dasian space, such that tritus is 0, tetrardus is 1, protus is 2, and deuterus is 3. In this arrangement, semitones always fall between adjacent positions 3 and 0, and each of the 12 semitones occurs only once in the space.

5 Edward Gollin (2007) refers to pitch-class cycles that exhaust the aggregate multiple times as

multi-aggregate cycles. Although Gollin does not address specifically the structure of the Dasian space, the space exhausts the aggregate four times.

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The affinity of the space is expressed by the interval of transpositio 7, which encompasses as a modular unit of three recurrent whole tones and a semitone, i.e., 7 = 3 • 2 + 1 (mod 12).

Example 3. The Dasian space and the transpositio relation.

Example 4 shows that the transformatio relation has two outcomes in the Dasian space: it either produces a note substitution, by ascending a pitch class by a semitone; or it retains the same pitch class. In both cases, transformatio corresponds to moving seven steps in clockwise direction.

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Example 4. The transformatio relation in the Dasian space.

3 Four Properties of the Dasian Space

The examination of continuous segments embedded in the Dasian space allows us to establish the following four properties:

(1) Set-class consistency: every continuous 7-note segment is a diatonic collection. Segments of different cardinalities yield different set-classes, but 7-note segments yield a single set class, which I call the host set class of the space.

(2) Set-class completion: The space contains all 12 transpositions of this scale, which are related by perfect fifth, the interval of transpositio (t).

(3) Set locality: Each diatonic scale appears in a single region of the space, extending through a major-10th segment. Regions are defined by Carey and Clampitt (1996, esp. 124-125) and constitute the largest space in which both parallel octaves and fifths can be maintained.

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(4) Every single “step” or “shift” of a 7-note segment in clock- or counterclockwise direction either substitutes a pitch class for its replication or for a pitch class a semitone away. Both of these relations are structured by transformatio.

In Example 5a, successive clock- or counterclockwise steps of 7-note segments yield the same collection four times before a new pitch is introduced. Example 5b shows that the transformatio relation that produces note change corresponds to what David Lewin refers to as a Cohn flip, i.e., when inverted around a given axis, a diatonic set produces a near symmetrical mapping, where one of the notes is substituted by another adjacent note in the space (Lewin 1996, 181-216). In the example, the white-note diatonic collection can Cohn-flip in two familiar ways, each retaining six of the pitches and exchanging B for Bb, and F for F#. The transformatio relation thus has an important function in defining the host set of the Dasian space as the number of steps spanned by the relation coincide with the cardinality of the host set class.

Example 5. “Step” motion and transformatio in Dasian space

The role of the diatonic collection as a host set class, however, is not unique to the Dasian space. Example 6 shows three additional cycles: the Guidonian space, the Hauptmann’s line-of-thirds, and the circle-of-fourths.6 In all of these spaces, diatonic collections are maintained as 7-note segments and the transformatio relation spans seven contiguous steps. As with the dasian space, transformatio either retains the same pitch-class seven steps away, or as shown by the dotted arrows, connects pitch-classes related by semitone belonging to adjacent host sets in the space. In addition, the four properties that characterize the pitch structure of the Dasian space also apply to these spaces. 6 The Guidonian space derives its name from maximally overlapping the Guidonian hexachordal

arrangement.

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Example 6. The diatonic collection as the host set-class of several cycles

4 Affinity Spaces

Example 7 proposes a formula (t = n • r + s (mod 12)) that models the interval relations in the modular unit of a generalized construct I call affinity spaces. Let us call affinity spaces to any concrete realization of such construct. The formula lifts the constraints on interval size and interval recurrence specific to the Dasian space, but retains the roles performed by the relations of transpositio and transformatio, and continues to grant privileged status of host set classes.7 In this formula, t corresponds to the interval of transpositio, measuring the affinity of the space; r is the interval of recurrence, and generalizes the “whole tone” of the dasian space; n counts the recurrence of r; and s generalizes the “semitone,” the unique interval of the modular unit.

There are two outcomes for transformatio in any affinity space (Example 7b): it either retains the pitch class, in which case the order position of that pitch class descends one order position within the modular unit. When transformatio is applied to a pitch class in the zero order position within a modular unit, it substitutes it for another one in a position n, such that the interval between the two pitch classes is given by r – s. When the interval of transpositio t is co-prime with 12 (i.e., when it is 1, 5, 7, or 11), we can then calculate the number y of steps a transformatio spans (Example 7c). y is also the cardinality of the host set class. In the case of the Dasian space as addressed above, a transformatio corresponds to moving seven steps.

7 Although conceived independently, this generalization intersects in a number of ways with the

work of Gollin (2007).

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Example 7. Affinity spaces

Let us now examine the embedding of host set classes in some affinity spaces. Example 8 shows twelve different affinity spaces, which share the same interval of

transpositio (t = 5), the recurrence n = 2, and consequently also the cardinality of space (mod 36). A circle-of-fourths in each space is highlighted in order to point out the affinity at-the-fourth in all spaces.

In each space, arrows indicate the number of STEPs spanned by transformatio.8 Solid arrows indicate pitch-class retention and dotted arrows indicate note substitution. In each space, the cardinality of the host set-class equals the number of STEPs involved in a transformatio. Notice that six of the spaces have positive STEP values, and the other six have negative values. I have represented the spaces this way in order to keep the shortest number of STEPs spanned by the spaces’ transformatio operation.

8 Let us define STEP as the operation that moves clockwise in a cycle, such that STEP1 moves

clockwise one station, between adjacent elements; STEP2 moves clockwise 2 stations, etc. STEP-1 moves counterclockwise one station, etc.

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Example 8. Twelve affinity spaces structured by 5 = 2 • r + s

The table of Example 9 organizes the twelve affinity spaces presented in Example 8. Going down the list of spaces in the left column shows that the interval r (the recurrence within the modular unit) increases by five semitones (or a perfect fourth) and the interval s (the unique single interval) increases by two semitones. The second column shows that are only three values for the interval of transformatio that produces note substitution, i.e., 1, 4, 7, and 10 (mod 12). The third column shows that the increasing number of STEPs involved in a transformatio increases by three for positive values and then decreases by three for negative values. The fourth column is subdivided, by the dotted line, pairing complementary host set classes for each space.

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Example 9. The table for the twelve affinity spaces under 5 = 2 • r + s

Example 10 examines this complementary relation in two spaces: 5 = 2 • 2 + 1 (mod 12) (the Guidonian space), and 5 = 2 • 3 + 11 (mod 12). Example 10a shows that starting with pc C in position 0, the transformatio relation spans STEP -7 from C to C#. This relation signals the boundaries of two adjacent host sets, which in the case of the Guidonian space is the diatonic collection. Another way of inflecting C to C# in the same positions of the space is to go clockwise by STEP 29. This path produces another set class that is also consistent throughout the space. The host set corresponding to STEP 29 is the pentatonic collection plus two-times-the-aggregate. In other words, the space in fact has two host set classes, which are complements of each other plus twice the exhaustion of the aggregate. Example 10b presents another space, whose transformatio spans STEP 8, and the host set class has cardinality 8. The alternative counterclockwise path of STEP -28 produces a complementary set class (plus twice the aggregate) to the host set produced by STEP 8. In short, the transformatio relations produced by clock- and counterclockwise paths signal complementary host sets (plus a number of aggregates) for a given affinity space.9

9 It is also interesting to notice that those spaces that share the same host set class (plus or

minus the aggregate) also share the interval mod 12 of the transformatio.

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Example 10. Complementary host set classes for 5 = 2 • 2 + 1 and 5 = 2 • 3 + 11

5 Three Properties of Host Set Classes

We are now in a position to ask the question: What are the properties of host set classes? Can any set class be a host in an affinity space? Generalizing from the host sets addressed so far, (1) all are inversionally symmetrical (i.e., they map into themselves under some axis); and (2) all produce a near self-mapping “flip,” that is, they map into themselves except for one note. The interval between unmatched notes is the interval between substituted and substituting notes in a transformatio. However, unlike what we observed for Cohn-flips in the diatonic collection, the flip that corresponds to a transformatio does not have to substitute adjacent notes in other host sets (i.e., it produces a kind of unrestricted Cohn-flip). Finally, (3) host sets are either interval-cycle segments or a combination of i-cycle segments.10

6 Generating Affinity Spaces

Let us now address the reverse case: Given a set class with the three host-set properties, what are the “rules” for generating an affinity space?11 We first arrange the host set into an ascending i-cycle, and partition it into ascending i-cycle segments of cardinality c, and one segment of cardinality c - 1. Then, we arrange all segments to reflect a constant transpositional relation between their first elements (there might be a few ways of doing this). These transpositional relations correspond to possible

10 Host set classes can be arranged into i-cycle segments when t = 1, 5, 7, or 11; or arranged

into a combination of i-cycle segments when t = 2, 3, 4, 6, 8, or 10. 11 I’ll address here only the cases for t = 1, 5, 7, or 11.

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values for t (transpositio). Finally, we determine the values for f (transformatio) by producing the two near self-mapping flips.12 For instance, consider the i-cycle segment {1, 2, 3, 4, 5, 6, 7}, which has the three host set properties. Example 11a arranges the set as an i-cycle segment and partitions or de-cycles it in eight different ways. In Example 11b, the two near self-mapping flips give us the values of the two possible transformatio relations. Example 11c shows three spaces that result from the eight possible arrangements of i-cycle segments.13 Several of the arrangements give rise to what amounts to equivalent affinity spaces: the if r or s = 0 then t = s or t = r in (1) and (2); retrograde cyclic orderings produce the same space when “read” clock or counterclockwise—for instance (3) and (4), or (7) and (8); and the permutation of r and s in a 2-element modular unit results in the same ordering of elements in the space—for instance (3) and (5), or (4) and (6).

Example 11. Generating affinity spaces from the set {1, 2, 3, 4, 5, 6, 7}

12 As we’ve seen for the case of the diatonic collection, there’s no one-to-one correspondence

between a host set class and an affinity space. 13 The 7-note set is not the only contiguous segment that can serve as host set in the complete i-

cycle 1 or “chromatic scale” of Example 11c; rather, any contiguous segment can function as a host set for the i-cycle. In general, complete i-cycles are trivial realizations of affinity spaces and do not exclusively embed a single host set class. A similar condition applies to the circle-of-fourths in Example 6.

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7 Conclusion

The transformatio relation enables a gradual harmonic “modulation” across members of the host set class by retaining all but one note in the near self-mapping flip between adjacent host sets of an affinity space. The note change involved in the harmonic modulation can take the form of parsimonious voice leading when the value of (the note change in) transformatio is 1 (or 11), as is the case for diatonic or pentatonic host sets.14 Example 11 demonstrates, however, that harmonic modulation can rely on less parsimonious transformatio values, which nevertheless entail a harmonic gradation across set-class members. The several possibilities for de-cycling a given host set create a relation (one-to-several: between a given host set class and the corresponding affinity spaces) that is theoretically rich and analytically useful.

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Carey, N., Clampitt, D.: Aspects of Well-Formed Scales. Music Theory Spectrum 11(2), 187–206 (1989)

Carey, N., Clampitt, D.: Regions: A Theory of Tonal Spaces in Early Medieval Treatises. Journal of Music Theory 40(1), 113–147 (1996)

Gollin, E.: Multi-Aggregate Cycles and Multi-Aggregate Serial Techniques in the Music of Béla Bartók. Music Theory Spectrum 29(2), 143–176 (2007)

Lambert, J.P.: Interval Cycles as Compositional Resources in the Music of Charles Ives. Music Theory Spectrum 12(1), 43–82 (1990)

Lewin, D.: Cohn Functions. Journal of Music Theory 40(2), 181–216 (1996) Martins, J.O.: The Dasian, Guidonian, and Affinity Spaces in Twentieth-century Music. Ph. D

diss., University of Chicago (2006a) Martins, J.O.: Stravinsky’s Discontinuities, Harmonic Practice and the Guidonian space in the

‘Hymne’ for the Serenade in A. Theory and Practice 31, 39–64 (2006b) Martins, J.O.: Diatonic reorientation in dual-organization spaces: interpreting polymodality in

works of Milhaud. In: Presented at the annual meeting of the Music Theory Society of New York State (2007)

Mengozzi, S.: Virtual Segments: The Hexachordal System in the Late Middle Ages. Journal of Musicology 23(3), 426–467 (2006)

Palisca, C.V. (ed.): Musica enchiriadis and Scolica enchiriadis. Trans. Raymond Erickson. Music Theory Translation Series. Yale University Press, New Haven (1995)

Pesce, D.: B-Flat: Transposition or Transformation? The Journal of Musicology 4(3), 330–349 (1986)

Pesce, D.: The Affinities and Medieval Transposition. Indiana University Press, Bloomington (1987)

Tymoczko, D.: Scale Networks and Debussy. Journal of Music Theory 48(2), 219–294 (2004)

14 The value transformatio f = 1 can take the form of an augmented unison (or chromatic

inflection) as in the case of the diatonic collection or of a diatonic semitone as in the case of a pentatonic collection. In both of these cases the transformatio relation corresponds to what Carey and Clampitt (1989, 192–93) refer to as “primary intervals” of well-formed scales.