Chapter 3. Some Additional Relationships
l When a pitch-class set is transposed or inverted, its content will change entirely, partially, or not at all.
l Common Tones under Transposition (Tn) à For every occurrence of a given interval n, there will be one common tone under Tn. Look at the interval class vector.
l The Tritone (interval class 6) is an exception. Because the triton maps onto itself under the transposition at T6, each occurrence of interval class 6 in a set will create two common tones when the set is transposed at T6.
l *** The interval-class vector for the major scale, (0123568T) is 254361. It has a different number of occurrences of each interval class. As a result, it will have a different number of common tones at each transpositional level (except at the semitone and the triton, where it will have 2 common tones). This property is called “unique multiplicity of interval class.” Due to this property, the major scale (and the natural minor scale, a member of the same set class) can create the hierarchy of closely and distantly related keys that is so essential to tonal music. When the scale is transposed up a P5 (T7), six are held in common. That’s why V is considered closely related, and VII is a relatively remote tonality.
l Whole-tone scale has the interval class vector of 060603. For each transpositional level, there will either be six common tones (that is, a duplication of the original scale) or none. No gradual hierarchy exists here, just a stark either/or.
l Sets that are capable of mapping entirely onto themselves under transposition are called transpositionally symmetrical. If the interval-class vector contains an entry equal to the number of notes in the set (or half that number in the case of the triton), then the set has this property. There are 12 set classes.
3 (048) +
4 (0268) Fr+6
4 (0369) –
6 (013679) Petrushka
6 (014589) hexatonic scale
6 (02468T) whole-tone scale
8 (0134679T) Octatonic scale
l They have a degree of transpositional symmetry greater than 1.
l Common tones under Inversion (TnI) is similar to transposition, only now concerned not with intervals (differences) but with index numbers (sums). We need to consider the index numbers (sums) formed by each pair of elements in a set.
l (1,3,6,9) 1+3=4, 1+6=7, 1+9=10, 3+6=9, 3+9=0, 6+9=3. Each of these sums represents two common tones. T7I of (1,3,6,9) is (10,1,4,6) and 1 maps onto the 6 and 6 maps onto the 1. This double mapping will occur nder TnI for any value of n that is a sum of elements in the set.
l ***There is an additional factor. Unlike Tn, TnI can map a pitch class onto itself. Any pitch class will map onto itself under TnI where n is the sum of that pitch class with itself. (1,3,6,9) 1+1=2, 3+3=6, 6+6=0, 9+9=6. Each of these sums represents one common tone.
l We can compile all of the sums and make index vector, remembering that the sums of different elements will yield two common tones, while the sums of elements with themselves will yield one common tone. The number of common tones under TnI for every value of n can be read from the vector. (더하는 방식, strauss.p.84)
l Common tones under Tn and TnI can be an important source of musical continuity.
l Some set classes that can map entirely onto themselves under inversion are said to be inversionally symmetrical, and of the 220 set classes, 79 have this property. The index vector for a set with this property will have an entry equal to the number of notes in the set.
l Sets that are inversionally symmetrical can be written so that the intervals reading from left to right are the same as the intervals reading from right to left. Usually, but not always, this intervallic palindrome will be apparent when the set is written in normal form.
l In inversionally symmetrical sets, all the notes in the set are mapped either onto other notes in the set or onto themselves under some TnI.
l Pitch symmetrical à they are symmetrical in register an their pitch intervals form a palindrome. Any symmetrical pitch-class set can be arranged symmetrically in pitch space à pitch-class symmetrical.
l Schoenberg Orchestra Piece Op.16 No.3 begins with the five note chord. (B,C,E,G#,A) His chord is pitch class symmetrical, but not pitch symmetrical. The inversional symmetry is still there—every note in the chord has its inversional partner also in the chord—but it is felt only in the more abstract pitch-class space, not in the more concrete, immediate pitch space.
l Boulez, Le Marteau sans Mattre, “L’Artisanat furieux,” mm.6-9, vocal part à Inversionally symmetrical sets. Some of these are arranged pitch-symmetrically, but most are not. The two forms of (0134) at the beginning and at the end. The first is asymmetrical in pitch space, while the second is pitch symmetrical—its pitch intervals are the same reading from bottom to top as from top to bottom. (p.88)
l Varese, Hyperprism, m.1-12 à while inversional symmetry is an important compositional resource generally, it can play a particularly decisive role when the symmetry is realized in pitch space. For example, (012) is abstractly symmetrical no matter how it is deployed, but its symmetry can be dramatically reinforced by the arrangement of the notes in register. C# enters in a middle register in the tenor trombone, at the end of m.4, D comes in 23 semitones below, then in m.12, the low D is balanced symmetrically by the high C, 23 semitones above the middle C# à a vast pitch space is articulated symmetrically, and the symmetry is reinforced by the grace notes at m.4, and the quick embellishment in m.11. à C# is literally the central tone; the role of inversional symmetry establishes a sense of pitch centricity.
l Many sets cannot map onto themselves under inversion, and thus have a degree of inversional symmetry that is 0. Some sets can map onto themselves at one or more than one inversion levels. The set (024) has a degree of symmetry of (1,1). It maps onto itself at one transpositional level, T0, and one inversional level at T4I. The most symmetrical set of all is the whole tone scale; it maps onto itself at 6 transpositional and 6 inversional levels.
Inversional symmetry is a reasonably common property, but inversional symmetry at more than one level is rare—only 11 sets.
3 (048) + 3
4 (0167) 2
4 (0268) Fr+6 2
4 (0369) – 4
6 (012678) 2
6 (014589) hexatonic scale 3
6 (02468T) whole-tone scale 6
8 (0134679T) Octatonic scale
l Only the petrushka chord (013679) is missing from the list of tranpositionally symmetrical sets.
l The greater the number of operations that map a set onto itself, the smaller the number of distinct sets in the set class. Most set classes have a degree of symmetry of (1,0) and contain 24 distinct sets.
l 24 divided by the number of operations that will produce each member of the set class equals the number of distinct members of the set class.
l There are several pairs of sets that have the same interval-class content, but are not related to each other by either transposition or inversion and thus are not members of the same set class à Z-related sets à (1 pair of tetrachods and octachords, 3 pairs of pentachords and septachords, and 15 pairs of hexachords) * Z stands for “zygotic” meaning twinned.
l (0146) and (0137) are called all-interval tetrachords, because their shared interval vector is 111111, both tetrachords contain one occurrence of each of the 6 interval classes.
l Elliot Carter’s String Quartet No.2. à in this quartet Carter often differentiates instruments in the way by assigning each a distinctive interval. The vertical combination of those intervals produces either (0137) or (0146). In the second passage, a form of (0146) is stated melodically in the second violin while the other three instruments combine to create a form of (0137), etc. (p.91-92)
l Any set with a Z in its name has Z-correspondent, another set with a different prime form but the same interval vector.
l For any set, the pitch classes it excludes constitute its complement. Any set and its complement, taken together, will contain all 12 pitch classes. For any set containing n elements, its complement will contain 12-n elements.
l It turns out that a set and its complement always have a proportional distribution of intervals. For complementary sets, the difference in the number of occurrences of each interval is equal to the difference between the size of the sets (except for the triton, in which case the former will be half the latter). Ex) if a tetrachord has the interval class vector of 021030, its eight-note complement will have the vector 465472.
l Even if the sets are not literally complementary (i.e., one contains the notes excluded by the other), the intervallic relationship still holds so long as the sets are abstractly complementary (i.e., members of complement-related set classes).
l Complement-related sets do not have as much in common as transpositionally or inversionally related sets, but they do have a similar sound because of the similarity of their interval content.
l Complement-related set classes always have the same degree of symmetry and thus the same number of sets in the class. If set X is Z-related to set Y, then the complement of X will be Z-related to the complement of Y.
l The complement relation is particularly important in any music in which the 12 pitch classes are circulating relatively freely and in which the aggregate (a collection containing all 12 pitch classes) is an important structural unit. (ex) Schoenberg string quartet no.3, little piano pieces, op.19
l Some hexachords are self-complementary à they and their complements are members of the same set class. In other words, self-complementary hexachords are those that can map onto their complements under either Tn or TnI.
l If a hexachord is not self-complementary, then it must be Z-related to its complement. As a result, a hexachord always has exactly the same interval content as its complement. Thus, the hexachords are either self-complementary or Z-related.
l If set X is included in set Y, then X is a subset of Y and Y is a superset of X. A set of n elements will contain 2 to the nth power subsets. However, the null set, the one-note sets, and the set itself will usually not be of particular interest as subsets. à (2 to the nth power-(n+2)), and naturally, the bigger the set the more numerous the subsets.
l Inclusion lattice, lists all of the subsets of a given set as well as the subsets of those subsets. But only a small number will be musically significant in any specific musical context.
l The subsets of a set are a kind of abstract musical potential; the composer chooses which to emphasize and which to repress.
l As with the complement relation, the subset/superset relation can be either literal or abstract. Set X is a literal subset of Set Y if all of the notes of X are contained in Y. Set X is the abstract subset of Set Y if any transposed or inverted form of X is contained in Y, that is, if any member the set class that contains X is found among the subsets of Y.
l Smaller collections can frequently be heard combining into larger ones and larger collections dividing into smaller ones.
l Transpositional combination (TC) à the combination of a set with one or more transpositions of itself to create a larger set. The larger set, which can then be divided into two or more subsets related by transposition, is said to have the TC property, and sets with this property have often proven of interest to composers. (Stravinsky symphony of psalms.)
l Contour relationship. Contour segment à CSEG, and this intervallically varied melody is unified. For example, Crawford String Quartet first movement à <2013> 0 is assigned to the lowest note, and so on.
l Approaching contour permits to discuss musical shapes and gestures with clarity, but without having to rely on more difficult discriminations of pitches, pitch-classes, and their intervals. Also, it becomes possible to discuss similarities of shape in the presentation of different set classes and, conversely, the divergent shapes given to members of the same set class.
l Contour can be useful in dynamics. (fff-ff-pp-p)
l Contour can be valuable as a way of talking about music that is indefinite as to pitch, like a lot of experimental music written since 1950.
l To organize the larger musical spans and draw together notes that may be separated in time, composers of post-tonal music sometimes enlarge the motives of the musical surface and project them over significant musical distances. à composing-out à a motive from the musical surface is composed-out at a deeper level of structure à other related terms are: enlargement, concealed repetition, motivic parallelism, nesting, and self-similarity.
l Voice leading
l Voice leading spaceà defined by semitonal offset.