**copyright ©** **Larry J. Solomon,
1973, revised 2002, 2005**

**Download
Windows program for generating Quadrate transformations**

Note: This chapter appeared in *Perspectives
of New Music *in Spring/Summer 1973 in a slightly different form. (Vol.11/2,
pp. 257-263)

"Our inventive resourcefulness discovered the following forms: cancrizans, inversion, inversion of the cancrizans. Four forms altogether. There aren't any others. However much the theorists try."{Webern, 54}

The forms to which Webern referred can be represented graphically, and, as such, are known as the symmetry group of a rectangle. Consider the ordered pitch set of a well known theme.

Example 92. An ordered pitch set of a famous theme

This theme uses only two pitches with four ordered events which, graphically, can be shown as follows:

Example 93. A graphic representation of example 92

The number of events here is unequal to the number of pitches, resulting in its graphic representation as a rectangle, which has sides of unequal numbers of units. This rectangle can undergo only three reflection transformations to yield the- traditional four forms to which Webern refers.

Example 94. The symmetry group of a rectangle performed upon the theme in example 92-93

How are these symmetric according to our definition? The theme is not, by itself, symmetric. It is so only if it is considered in relation to or combination with the newly generated form, i.e., inversionally related sets, such as P (prime) and I (inversion).

Example 95. The reflection relation of P and I.

The same is true of temporally reflected sets, such as P and R (retrograde).

Example 96. The reflection relation of P and R.

Any rectangular thematic pitch set or pitch class set has only four possible reflectively related forms: P, I, R, RI. These are commonly generated by our two reflection operations, but, as we have seen, RI may be generated by 180° rotation.

The square is a special case of the rectangle and occurs as a graphic representation of a pitch ordering when the number of events is equal to the number of different pitches, analogous to the equivalence of the units in the sides of a square.

Example 97. The ordered pitch set of a theme in which the number of events equals the number of different pitches and its graphic representation in a square.

This theme is shown in a two-dimensional matrix as: [0,0] [1,2] [2,3] [3,1], where the first number in each pair indicates the temporal order t (horizontal axis), and the second number represents the pitch p (vertical axis); thus, the Prime form may be represented P(t,p).

The symmetry group of the square is of a higher order than the rectangle and has, in fact, double the number of related forms. The first four of these correspond with the symmetry group of the rectangle and are P, I, R, and RI. The last four are new and are identified by the prefix Quadrate. The Quadrate Prime, or QP, is derived from the Prime by reflection about the ascending diagonal. The other Quadrate forms, QI, QR, and QRI, may then be derived from QP by the same operations performed on P to get I, R, and RI.

All eight tranformations, including the Quadrate forms may be derived directly from P by the following operations:

Example 98. The possible reflection operations performable on a square resulting in symmetry.

These may be defined mathematically as transformations of P as follows, where n = z-1 and z = the total number of pcs:

P(t,p) = P(t,p) | QP(t,p) = P(p,t) |

I(t,p) = P(t,n-p) | QI(t,p) = P(p,n-t) |

R(t,p) = P(n-t,p) | QR(t,p) = P(n-p,t) |

RI(t,p) = P(n-t,n-p) | QRI(t,p) = P(n-p, n-t) |

Our theme in example 97 results, then, in the following forms by these operations (n=3):

P=0231, I=3102, R=1320, RI = 2013, QP=0312, QI=3021, QR=2130, QRI=1203

Example 99. The theme of example 97 generates the following forms by symmetry operations on a square.

The symmetry of two of these forms may be easily observed by their combination.

Example 100. Reflection in the combination of P and QP.

With any four pitches the number of different ways to order them is 4! (4 factorial), or 24. One out of three is reflectively related since there are eight possible reflective forms as in our example. For a five note line there are 8 out of 120 forms which are reflective. There are a maximum of eight different lines which are reflective for a twelve note row with twelve factorial possible orderings. Therefore, the percentage of possible symmetric forms decreases with increasing numbers of pitches.

A twelve tone row with a small number of internal relationships normally has eight different forms exclusive of transposition.

Example 101. A twelve tone row with eight different reflective forms.

However, not all sets will have eight different forms. Sets with a large number of similar elements will generally have less than eight. Some themes and rows are known to have even less than four different forms, excluding transpositions, e. g., those with P= R and those with P= RI.

Example 102. A row whose four common forms are reduced to two.

It is commonly held that P cannot equal I, but under special conditions even this is possible. If,for instance, a row is considered cyclic (the end linked with the beginning), the following results.

Example 103. A row in which P= R= I= RI cyclically

In this row P=RI and R=I, but P is also equal to I and R=RI if the row is cyclical; i.e., starting from the first note and from the seventh note in a circle. This cyclic row, then, seems to have only one form. However, it has at least one other cyclically and two others non-cyclically. These are the Quadrate-prime and the Quadrate-retrograde, where QP=QRI and QR=QI.

In the following row the Quadrate forms are hidden inside the regular forms. Here, P = QRI, I = QR, R = QI, and RI = QP.

Example 104. The four traditional forms of a row containing hidden Quadrate forms

In another example, P=QP, I=QI, R=QR, and RI=QRI.

Example 105. Another row with hidden Quadrate forms

The symmetries of the rows of examples 104 and 105 would be not be noticeable without recognition of the Quadrate forms.

The symmetry of the Quadrate forms is sensitive to the transposition choice for the prime. Although P=QP, R=QR, etc., in example 105, if the prime were set to begin with D and assigned the pitch number 2, this symmetry is destroyed. This is equivalent to using an asymmetrical axis.

Example 106. The above row transposed to an asymmetrical axis

Although there has been no thorough search for occurrences of Quadrate forms in existing music, it is probable that they would occur with some affluence in some thematic manipulations or in tone row variants even though the latter is more rationally restricted. Quadrates will, at least, occur as hidden forms identical to other variants in highly organized rows. The following row by Webern is an example.

Example 107. A row by Anton Webern for Op. 30, where P=RI=QP=QRI and I=R=QI=QR

What kind of changes are being made in a set's structure by the quadrate operations, and what are the perceptual implications? The operation itself is logically consistent with those of the traditional transformations. The difference lies in the exchange of parameters in the Quadrate transformations; that is, the operations of I, R and RI do not exchange pitch and order axes, whereas the Quadrates do. The psychological question of audibility arises which is still discussed about the traditional forms. Experienced musicians rarely question the audibility of I, R and RI as related to P. However, identification of the specific operation does depend upon conscious knowledge of the properties of the operation, e. g., it helps to know that P and RI are directly related by being mutual intervallic retrogrades.

For the composer who seeks new ways to communicate, it normally does not matter that the listener consciously identify these operations. What does matter is the ability of a compositional procedure to create unity. The psychological ramifications of symmetrical variation has not been tested scientifically, but the following chapter speculates on these.

As Webern stated in "The Path to Twelve-Note Composition:

"An example: Beethoven's "Six easy variations on a Swiss song." Theme: C-F-G-A-F-C-G-F, then backwards! You won't notice this when the piece is played, and perhaps it isn't at all important, but it is unity." {Webern, 52}