SECTION III

After choosing two set-classes X and Y and generating their chord classes, the first task in launching the chord pairing process is to select a referential chord from chord class X. This chord can be picked randomly or chosen because it has some desired properties. After this, suitable chords will be searched for and processed in chord class Y as follows:

(2) Transpose each selected Y-chord so that its outer pitches match those of the referential chord.

(3) Find out the largest subset classes Z1, Z2, ... Zn being in common between set-classes X and Y.

(4) From the group of transposed Y-chords, select the ones having a maximum of common pitches with the referential chord, that is, chords whose common pitches constitute one of the Z-classes.

Comparability criteria (1) and (2) together guarantee that the highest and lowest pitches between each referential chord/Y-chord pair are always the same. The two criteria were adopted because in my subjective judgment movement between the highest and/or the lowest pitches is a more discernible notion than movement between the inner voices. Note that criterion (1) filters out all Y-chords if X and Y do not have any 2-element or larger subset classes in common. In these cases the referential chord/Y-chord pair cannot share even the outer pitches. For SC pairs like these, similarity measures assign the value indicating maximal dissimilarity.

Criteria (3) and (4) guarantee that every selected Y-chord has as many pitches in common with the referential chord as the current SCs allow. If, for example, X and Y are 5-element classes and have one 4-element class Z in common, it is possible to find Y-chords with 4 pitches in common with the referential X-chord. (This does not mean, however, that every Y-chord would have 4 pitches in common with the referential chord). The obvious reason to maximize the number of common pitches is to minimize the amount of movement between the non-common ones. If SCs X and Y are of different cardinalities and not in the inclusion relation, or if they are of the same cardinality, there are always one or more non-common pitches between each chord pair. If X and Y are in the inclusion relation, criterion (4) guarantees that only the larger chord will have non-common pitches.

Examples 1-4 illustrate how the selection process worksin practice. Let us start with SC 5-21A, generate its chord class of 120 pentachords and select the one in Ex. 1 to be the referential chord.[10]

The SC to provide pairs for the referential chord is the inversionally related SC, 5-21B. The RECREL value between the two is 4, the very lowest value among pairs of 5-pc classes.

After generating the 120 chords in the chord class 5-21B, criterion (1) requires us to select those being of the same width as the referential chord. In this case, the width is 20 semitones. There are 16 5-21B chords meeting this requirement. They are shown in Ex. 2.

As the chords are derived from permutations of the 5-21B prime form {0,3,4,7,8}, each consists of PCs C, E flat, E, G and A flat in various order. Criterion (2) requires the chords to be transposed to the same level as the referential chord. The transposed chords are in Ex. 3.

Criterion (3) requires the largest common subset classes of 5-21A and 5-21B to be ascertained. The three 4-pc classes are 4-20, 4-17 and 4-7. Out of the 16 5-21B chords in Ex. 3, criterion (4) accepts only those having four pitches in common with the referential chord. Five chords qualify. They are shown in Ex. 4 a-e, each paired with the referential chord. In Ex 4 a and b the four common pitches are {60,67,71,80}. They constitute SC 4-7. In examples 4 c and d the common pitches are {60,63,71,80}, the SC being 4-17. In the last pair, 4 e, the common pitches are {60,63,67,80}, constituting SC 4-20.

EXAMPLE 4: The five 5-21B chords sharing four pitches with the referential 5-21A chord.

[10] In this study and in the demonstrational program associated with it, the following conventions are used: Pitch-class C = 0; Pitches are numbered according to the MIDI standard, middle C being 60; SCs are labeled as in (Forte 1973:178-181), with one exception: Forte’s T_n/I-classes will be divided into two inversionally related transpositional set-classes that receive extra labels A and B. The minor and major triad classes, for example, will be 3-11A {0,3,7} and 3-11B {0,4,7}, respectively. If neither A nor B appears in the name of a SC, it is inversionally symmetric.

If a chord contains a minor second written with both natural and sharp or flat versions of the same note, the note with a sharp or flat sign is given first and the untransformed one after this, without a natural sign. Each sharp or flat sign affects only the note immediately to its right. This is to save space. (See, for example, the third chord in Ex. 2, containing both E and E flat).

To the right of each chord is its INT, giving distances between successive pitches (in semitones. T = ten, E = eleven). Below each chord is an 11-entry vector giving the chord’s total interval contents. Intervals wider than 11 semitones are represented by their mod-12 equivalents. The rightmost entry is for instances of 6s. Top row, l to r: instances of intervals 1-5. Bottom row, r to l: instances of intervals 7-11.