Equivalent Union of Subsets as Progression: "Theme Song and Variations for Six Saxophones", an Example (1)


Thomas Putsché

Despite the pedantic title, the basic format of this presentation is to play a tape for composer-friends. But, after all, we are all pedants. In fact, I hope to bring up before the National Council a suggestion for change in our title. I propose : Society for the Pedantic Composers of America. We could then replace our current acronym, with its somewhat unsavory connotations, with SPCA which might gain us more sympathy.

This will, though, be a bit more formal: there are the examples, there will be a few general comments about the piece and then one aspect of the piece, the title, which may be of general interest, will be considered.

First of all, with the number of saxes, you will not be surprised that this is a hexachordal piece. The hexachords chosen are those which by formation omit the tritone, the ones designated on the handout, Example 1, as Class A. Here the intervallic content is shown in descending size by half step, 6 5 4 3 2 1. A 1 has no 6's, one 5, two 4's, etc., in retrograde of how some do it, since in this classification the number of tritones is the most significant property.


                                                            Example 1: Hexachordal Classes

These four hexachords are displayed in the Prime form in their more familiar conjunct arrangements in Example 2.


                                                                                Example 2: Basic Hexachords

While this talk is not supposed to be about hexachordal theory, perhaps some words may be said about the derivation of these sets. In Example 3 you will see twelve notes designated Universal set, in three tetrachords as the three odd intervals 3-9, 1-11 and 5-7 a tritone apart. There are reasons for this arrangement, of course, but will not be discussed here.

                        Example 3: Universal Set

Class A hexachords are formed by taking an odd interval from each of the three tetrachords thus omitting the 6; Class B, by taking a 6 from one tetrachord and odd intervals from the remaining two; Class C, by taking a 6 from two tetrachords and an odd interval from the remaining one; and Class D, by taking a 6 from each tetrachord. You will note that D3 and D4, while formed analogously to A3 and A4, have the same intervallic content. These nineteen represent all hexachords which have inversional properties, that is, which have an inversion giving the other six elements.

Class A has other unique properties, the most obvious that a transposition at the tritone yields the other six elements, actually not too important here. Another property is that being symmetric, each may be conceived of as four sets of two trichords, Prime and Inversion. These may be arrived at by taking an element from each of the three odd intervals derived from the tetrachords. The four permutational possibilities are shown below, Example 4, upon A2, the Guidonian.

Example 4: Trichordal Permutations of Class A2, P Hexachord

The same permutations may also be made upon the other Class A hexachords with four trichordal arrangements of each. The method of identifying trichords here, you may see, is based upon the intervallic content, Example 5.

Example 5: Trichordal Permutations of Class A1-4, P Hexachord

These sixteen trichordal arrangements are the basis for the formal scheme of the piece. The order of the trichords as employed in the piece are given in Example 6 after the "poetic titles." You will see that the overall form is "Ma Fin et Mon Commencement," the music generally backwards from the middle, but, of course, with different hexachords and different but analogous trichords. This plan, incidentally, was not preconceived. I did not think of going backwards until halfway through.

Theme Song :

Slow, Sentimental

A2, 253

Variation 1 :

Bouncy, Fast

A2, 275

Variation 2 :

Driving, Smooth

A2, 374

Variation 3 :

Soft, Moving

A2, 242

Variation 4 :

Bright, Sharp

A1, 143

Variation 5 :

Singing, Flowing

A1, 121

Variation 6 :

Pushing, Plaintive

A1, 132

Variation 7 :

Noble, Strict

A1, 242

Variation 8 :

Strict, Noble

A3, 242

Variation 9 :

Plaintive, Pushing

A3, 253

Variation 10 :

Flowing, Singing

A3, 154

Variation 11 :

Sharp, Bright

A3, 132

Variation 12 :

Moving, Soft

A4, 484

Variation 13 :

Smooth, Driving

A4, 143

Variation 14 :

Fast, Bouncy

A4, 154

Theme Song :

Sentimental, Slow

A4, 374

Example 6: Form in "Theme Song and Variations for Six Saxophones"

The above is obviously highly abstract and many pieces might be made of these relationships. While still relatively abstract, we shall now suggest a procedure, the main subject of this presentation, Equivalent Union of Subsets as Progression.

Subsets, that is, since equivalent unions may be done with the entire universal set. What this means is: given an ordered set, designated P, at a given transposition, there may be found an inverted set, designated I, at a given transposition, in which at this union of P and I defined elements of P and I will coincide. Further, each element of these corresponding pairs may be considered first elements of the entire ordered sets of P and I. There will thus be twelve transpositions of P and I where the interval between first elements of P and I will necessarily be different, but where corresponding elements between P and I will be identical regardless of the order in which they occur. These twelve transpositions of P and I are designated as equivalent unions.

As an example from the literature, Variations for Piano, Op. 27, Second Movement by Webern, chosen because it is in the "Top Forty", that is, in the Burkhart Anthology, we may see this principle in operation. The sets of the entire short piece are given in Example 7.


   Example 7: Sets in Webern's Variations for Piano, Op. 27, II

In this display, the numbers above each note of I and below those of P represent the fixed elements: C as 0, C as 1, D as 2, etc. You will notice that the transposition numbers of the entire sets of P and I are also represented by fixed elements rather than "movable doh" According to most practice, Pt8 would be designated Pt0 and It10 would be designated It2 to show the interval between, the difference, 2. The same result, though, may be achieved by simple subtraction, 10-8.

The reason for designating in this manner is to show another way to conceive of inversion, that is, that corresponding elements of P and I will add to the same number. In the Webern example the pairs add to 6, Modulo 12 of course: the first pair; 8 + 10 = 18 (6); the next pair, 9 = 9 = 19 (6) etc. The point is that any pair of elements adding to this number may be taken as first elements of the entire sets of P and I. Here the last pair of one union becomes the first pair of the next union. Obviously, if this is the case, corresponding elements of a pair adding to the same number because it is the same pair, then all other pairs of the next union must also add to the same number. Whether Webern thought of these as equivalent unions and then placed them as above, or whether he simply made the last pair of one union the first pair of the next, and then noticed the relationships that come about, is immaterial. He did make use of the relationships, treating the elements that occur together, regardless of the order in the sets, in the same manner. For example, you will recall that where the two elements are both 9, forming the interval 0-12, the two A's are always a one, a unison, as well as staccato and piano, and where the elements are 10 and 8, 10 is B two and 8 is small G, always in the same register as well as slurred and forte. Also elements 10 and 8 are obviously placed as the first and last "idea" of the piece.

In unions of the U-set, any given element must exist in any position in any of the twelve positions, and therefore, with a given addition, any element of P will be paired with the same element of I in all equivalent unions. With subsets, however, equivalent unions may exist only from the number of positions in the set, the number of elements. We may, however, also have sets of transpositions of subsets with the same general properties of the U-set, for reasons perhaps too involved to discuss here, in which a given element may be in successive positions, 1 through 6. In Example 8, we have taken Class A2, the Guidonian, where element 0 in both P and I is in each successive position. Any element may be in each position in both P and I, but here both at 0 for exemplification, the addition of elements then to 0-12. You will note that the differences between P and I are to one each of the even intervals, instead of two each with the U-set. If at an odd difference, an odd addition, then to one each of the odd intervals, another special property of Class A hexachords.

Example 8: Equivalent Unions of Class A2, Pt  It

Usually the sets of P and I are presented as separate melodic statements; in the Webern example the corresponding elements immediately follow one another. Another way to consider union is as progression. We may take a given union, say Pt0  It0, Example 9, corresponding elements numbered, and then dispose the two sets vertically where now the corresponding elements will progress in each of the six voices, numbered from lowest to highest in Example 10.

Example 9: Pt0 It0, successive


Example 10: Pt0 It0, verticalized

This will also be true for all equivalent unions as displayed in Example 11, which you may see are the same unions as in Example 8, now as progression.

Example 11: Equivalent Unions of Class A2, Pt  It, verticalized

Thus in each of the six progressions, each voice moves from a given element to its paired equivalent in the same way at a defined difference. With the exception of element 0, occurring in each T of P and I as common tone in Example 11, other elements of P and I will not occur in each progression but, if not, will occur t6 at the same difference. In Example 11, in Pt0 It0, element 2 pairs with element 10 in the top voice at the difference of 8 and in Pt3 It9 this difference is represented by the elements 8 and 4, the top voice of the bass clef. Still, however, with Class A hexachords each voice will progress at one each of the even differences. The top voice is given in Example 12. The lowest voice in this form also indicates the differences between the T's of the entire hexachords of P and I, also to each of the even numbers.

Example 12: Top voice of veticalized P I Equivalent Subset Union

Likewise, if at an odd addition, then each voice will progress at one each of the odd differences, the lowest voice again representing T's of P and I of the entire hexachord as set forth in Example 13.


                                        Example 13: Top voice of verticalized, P I Equivalent Subset Union, at odd difference

The sets of P and I of these unions may also be thought of as being expressed vertically by the first union. The first chord, P, forms elements of t's of the P form by inversion; and the second chord, I, forms elements of the I form by inversion. From Example 11, where Pt0 It0, the t's of I, Example 14a, are elements of the first chord P, Example 14b; and the T's of P, Example 15a, are elements of the second chord, I, Example 15b.

Example 14 a and b: T's of It0, in Pt0 It0, a. successive and b. verticalized

Example 15 a and b: T's of Pt0, in Pt0 It0, a. successive and b. verticalized

Thus one may speculate that the first progression, the first two sonorities, imply the t's of the progressions that follow.

If the hexachord is conceived trichordally, then further relationships come about. A2 as 7/3 s and 7/4 s is placed in Example 16 on the two staves, P and I, symmetrically, the outer two voices of each trichord forming a 3-9, the middle voices forming a 5-7, and the inner voices forming a 1-11. The order numbers in this form are shown accordingly --1 and 4, 2 and 5, 3 and 6 corresponding.

Example 16: Verticalized trichords of PI Equivalent Subset Unions

As the T's are arranged in example 18, where Pt0 It0 = It0 Pt0, Pt5  It7 = It7 Pt5 and Pt8 It4 = It4 Pt5, then, not only will all even differences occur in each voice, but also sets of differences in corresponding voices will result as follows: the top voice in the first three progressions equal the bottom voice in the second three progressions by inversion, exact elements of each pair being switched, and the bottom voice of the first three progressions equal the top voice in the second three progressions analogously. The same relationships exist between middle voices of each trichord and between inner voices.

These relationships come about because of the method of formation. If the sets of T's may be placed as the hexachord, a trichord and its inversion, so also may the set of unions be expressed as three unions and these inverted. Sets of unions with this property we shall refer to as reflexive sets of equivalent unions.

Reflexive sets at the odd additions must be considered a bit differently. With the even addition in the above example, the common element of P and I is itself, thus one set of 's for each addition. With the odd additions, however, the common element of P must be with a different common element of I--in Example 13, element 0 of P and element 3 of I. Since two elements are paired in each , then either element may be in P or I. Thus with this odd addition not only may we begin with Pt0 It3 but also Pt3 It0, two sets for each addition, Example 17. It will be seen that this pair of sets of 's is reflexive just as the sets of 's at the even addition above. The first three 's beginning with Pt0 It3 equal the second three 's beginning with Pt3 It0 by inversion, and the second three 's beginning with Pt0 It3 equal the first three 's beginning with Pt3 It0 by inversion. Individual voices will, of course, also have these relationships. For example, the top voice of the first three 's at Pt0 It3 equal the bottom voice of the second three 's at Pt3 It0, exact elements exchanged.

Example 17: Reflexive Sets of P I Equivalent Subset Unions

From the above we may also see relationships between the even and the odd additions. Taking an individual voice, the top voice of the first three 's and the bottom voice of the second three 's at Pt0 It0, Example 18a, at Pt0 It3, Example 18b, and at Pt3 It0, Example 18c--it will be seen that in comparing Example 18a with Example 18b, that elements of P, the first notes in each bar, are identical, and those of I are t3; and in comparing Example 18a with Example 18c, that elements of I are identical, and those of P are t3. Because of these identities, one might consider that the two sets of 's, odd and even, may substitute for one another.

Example 18a: Top and bottom voices of Example 16


Example 18b: Top and bottom voices of Example 17, first system

Example 18c: Top and bottom voices of Example 17, second system

In the "Six Sax" piece this principle is employed to arrive at a binary form for the theme. As displayed in Example 19, the first two systems are the even additions, Pt0 It0 and Pt3 It3, forming the first part, and the next two systems are the odd additions, Pt0 It3 and Pt3 It0, forming the second part. The same sets of P and I are used but united differently to give odd and even differences with the same elements, the common elements shown with ligatures in this example. In succeeding variations the order of the two parts is exchanged. Variation 1 is odd-even, Variation 2 is even-odd, etc.

In the theme, the progressions are used much in the way that you see them in Example 19, "first species." In the variations inversional relationships are introduced in different ways. These specific procedures could be traced but are outside of this discussion. There is also a transpositional scheme going on which will not be dealt with here. Besides which, much of the above theory came after the piece, and I have forgotten most of the specifics.

But aside from the theory, the artistic concern was to write a straightforward sort of nostalgic piece for which explanations would be unnecessary. As to stylistic influences, I am sure that you will all recognize some middle Glenn Miller as well as late Jimmy Dorsey.


                            Example 19: PI Equivanent Subset Unions for "Theme Song and Variations for Six Saxophones"

1. Paper presented at the former American Society of University Composers (Society for Composers Incorporated) at the Hart School of Music in Spring of 1974.