**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, |

Variation 1 : |
Bouncy, Fast |
A2, |

Variation 2 : |
Driving, Smooth |
A2, |

Variation 3 : |
Soft, Moving |
A2, |

Variation 4 : |
Bright, Sharp |
A1, |

Variation 5 : |
Singing, Flowing |
A1, |

Variation 6 : |
Pushing, Plaintive |
A1, |

Variation 7 : |
Noble, Strict |
A1, |

Variation 8 : |
Strict, Noble |
A3, |

Variation 9 : |
Plaintive, Pushing |
A3, |

Variation 10 : |
Flowing, Singing |
A3, |

Variation 11 : |
Sharp, Bright |
A3, |

Variation 12 : |
Moving, Soft |
A4, |

Variation 13 : |
Smooth, Driving |
A4, |

Variation 14 : |
Fast, Bouncy |
A4, |

Theme Song : |
Sentimental, Slow |
A4, |

**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, P_{t8} would be
designated P_{t0} and I_{t10} would be designated I_{t2} 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, P_{t} I_{t}

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
P_{t0} I_{t0}, 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**: P_{t0} I_{t0},
successive

**Example 10**: P_{t0} I_{t0}, 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, P_{t} I_{t}, 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 P_{t0} I_{t0}, element 2 pairs with element 10
in the top voice at the difference of 8 and in P_{t3} I_{t9} 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 P_{t0} I_{t0}, 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 I_{t0}, in P_{t0} I_{t0}, a. successive and b. verticalized

**
Example 15 a **and** b**: T's of P_{t0}, in P_{t0} I_{t0}, 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 P_{t0} I_{t0} = I_{t0} P_{t0}, P_{t5} I_{t7} = I_{t7}
P_{t5} and P_{t8} I_{t4} = I_{t4} P_{t5}, 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 P_{t0} I_{t3} but also P_{t3} I_{t0}, 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 P_{t0}
I_{t3} equal the second three 's beginning with P_{t3} I_{t0} by inversion, and the second three 's
beginning with P_{t0 }I_{t3} equal the first three 's beginning with P_{t3} I_{t0} by inversion.
Individual voices will, of course, also have these relationships. For example, the top voice
of the first three 's at P_{t0} I_{t3} equal the bottom voice of the second three 's at P_{t3} I_{t0},
exact elements exchanged.

**
Ex****ample 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 P_{t0} I_{t0}, Example 18a, at P_{t0} I_{t3},
Example 18b, and at P_{t3}
I_{t0}, 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, P_{t0}
I_{t0} and P_{t3} I_{t3}, forming the first part, and the next two systems are the odd additions, P_{t0}
I_{t3} and P_{t3} I_{t0}, 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.