Central Common Values

Definition: The durational values or pitch-classes that function as pivot points or axes of symmetry in palindromic structures, connecting mirrored sections and serving as the structural center around which symmetry organizes.

Messiaen's Treatment: In nonretrogradable rhythms, the central common value is the durational value around which the palindrome is constructed—the middle element in odd-numbered palindromes, or the point of reflection in even-numbered palindromes. This value is "common" because it belongs equally to both halves of the mirrored structure, serving as the axis of symmetry.

Similarly, in modes of limited transposition, the last note of each symmetrical group is common with the first of the following group—these pitch-classes serve as pivot points connecting the modal segments. These common values frame the symmetrical groups and enable their interlocking structure.

The concept of "central common value" emphasizes that symmetrical structures require anchor points—elements that either sit at the center of the palindrome or connect adjacent symmetrical segments. These values are structural, not merely decorative—they define the axes around which symmetry operates.

Modern Context: Contemporary theory recognizes pivot elements in various contexts:

• Modulation: Common tones or chords connecting two keys
• Set theory: Pitch-classes shared between different sets
• Transformation theory: Fixed points under transformations (elements that map to themselves)
• Symmetry analysis: Axes of reflection or rotation in symmetrical structures

The central value in a palindrome represents a fixed point under retrograde transformation—it occupies the temporal center and maps to itself when the structure is reversed. This is analogous to the axis of a visual palindrome or the center point of a geometrical mirror reflection.

Examples: Example 32 explicitly identifies the quarter tied to a sixteenth-note as the central common value connecting the two retrograded groups.