**OVERVIEW** Serial composition, a development of the twentieth century,
evolved from the use of ideas about the mathematics of sets in musical
construction.

**Diatonic Sets** A **set** is a collection of like objects, a
class of elements. The set can be made up of pitches as easily as any
other kind of object. A scale can be seen as a collection of pitch
objects that form a pitch set. A key signature (or consistent use of
certain accidentals) defines what notes are members of a collection of
pitches. This collection can be divided into smaller segments
*(subsets)*.

Any segment of a larger collection can be treated as an independent collection of pitches or as a fragment of a larger set. Notes within one of these subsets can be presented in a particular order, given a rhythm, then transposed to another level within the parent set. In other words, a motive can be created from a small group of pitches then transposed to other levels within the key.

A **trichord **consists of three different pitches. Like other
collections of three objects, the notes of a trichord can be presented
in six different orders; 123, 231, 312, 321, 213, and 132. The
notes E-F-G form a diatonic set because this pattern can be found in a
diatonic scale. Because of this, it can also be regarded as a subset of
a diatonic scale. In the next example, each permutation was created by
shifting the notes to the left. The fourth pattern was created by
switching the order of E and F.

Example 1: Permutations of the subset EFG

The interval content of EFG is m2, M2, and m3 does not change no matter how the notes in the pattern are arranged. The same pattern of intervals is present in the notes BCD, the one transposition of EFG possible in the diatonic set of no accidentals.

Example 2: intervals in EFG and BCD

These two patterns and all of their permutations share common traits that distinguish them from other patterns. Thus, both are like objects that belong to the same collection, a set of sets. This set of sets includes all trichords that contain the intervals m2, M2, and m3. Membership in this set can be restricted to the natural pitches (white-keys) or be expanded to include all eleven transpositions of the set.

The pattern EFG also resides in the key of one flat, BCD in the key of one sharp. Because they can appear in more than one pitch set, these two patterns form an intersection between two diatonic pitch sets.

Example 3: Set intersection

The pitches of a trichord need not be adjacent notes of a scale. Major, minor and diminished triads, for example, are diatonic subsets that contain __no__ steps. The next example contains a diatonic subset that includes both a step and a skip. The first measure can be found in keys from zero to three flats. CDF forms unions (intersects) with four different diatonic scales. This pattern occurs at three other locations in the set of white keys, DEG, GAC, and ABD. Since all of these sets share the same interval content, they and all their permutations are members of a collection of like objects, a set of sets.

Example 4: set membership by interval content

**Combinations of Notes in Ordered Subsets ** The number of
possible combinations of notes depends on the number of notes in the
pattern. For example, the notes of a particular trichord can be
combined 6 ways, expressed mathematically as 3! (three factorial). The
arithmetic of three factorial is 1 X 2 X 3. A tetrachord can be
presented 24 unique ways (expressed 4!). Four factorial (4!) is 1 X 2 X
3 X 4. Members of a pentachord can be combined 120 ways (expressed 5!).
Five factorial (5!) is 1 X 2 X 3 X 4 X 5. The interval content of a
particular pattern remains unchanged regardless of how its notes are
ordered.

**The Diatonic Scale as a Set** Key signatures define
transpositions of diatonic collections. The traditional modes under
one key signature are permutations of a single set of pitches.

**The Diatonic Set as a Collection of Intervals** Intervals in a diatonic set can be inventoried as illustrated in example 5. The inventory accounts for every interval possible between every pitch in the set. Each interval column includes an interval and its inversion. The octave and unison were not tallied because the notes in these intervals are pitches of the same name.

Example 5: Interval Inventory (interval vector) in a Diatonic Set

A |
B |
C |
D |
E |
F |
G |
m2 /M7 | M2 /m7 | m3 /M6 | M3 /m6 | P4 /P5 | tt |

M2 | m3 | P4 | P5 | m6 | m7 | 2 | 1 | 1 | 2 | |||

B |
C |
D |
E |
F |
G |
|||||||

m2 | m3 | P4 | o5 | m6 | 1 | 1 | 1 | 1 | 1 | |||

C |
D |
E |
F |
G |
||||||||

M2 | M3 | P4 | P5 | 1 | 1 | 2 | ||||||

D |
E |
F |
G |
|||||||||

M2 | m3 | P4 | 1 | 1 | 1 | |||||||

E |
F |
G |
||||||||||

m2 | m3 | 1 | 1 | |||||||||

F |
G |
|||||||||||

M2 | 1 | |||||||||||

total | 2 | 5 | 4 | 3 | 6 | 1 |

*The inventory of all the intervals possible among the white
keys remains unchanged regardless of the order of the notes.* The
same pattern of intervals can be found in any collection of pitches
defined by any diatonic key signature. Thus, *all diatonic modes*
in *all keys* belong to *a collection of like objects*, a set
of diatonic sets.

Go to Terms and Problems for Part 1

Go to Set Theory, Part 2

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Copyright © 2004, Kenneth R. Rumery, all rights reserved.