Imaginary transformations

Abstract

We investigate classes of discrete musical operations that function in ways that correspond to the imaginary units i, j, and k of the quaternions (i.e. orthogonal square roots of−1). In particular, we focus on musical transformations of order 4 that are mutual square roots of a central involution. We define imaginary transformations as members of groups of operations acting on the set of pitch classes in an octatonic collection and as members of subgroups of various triadic-transformational systems. These structures include quaternion, Pauli, (almost) extraspecial, and dicyclic groups, among others.

Keywords:

• transformation theory
• neo-Riemannian theory
• Lewin
• octatonicism
• uniform triadic transformations
• complex numbers
• quaternions
• group theory
• quaternion group
• Pauli group
• extraspecial group
• dicyclic group

AMS Classification :

• 00A65 – Mathematics and music

Acknowledgements

I would like to thank Guerino Mazzola, Julian Hook, Jack Douthett, Stephen Smith, Thomas Noll, and the anonymous readers of this article for their many valuable suggestions.

Notes

A Schritt in neo-Riemannian theory transposes major triads in one direction while transposing minor triads by an equal distance in the opposite direction. We call the operation s ₃=(0, 3, 6, 9)(1, 10, 7, 4) Schritt-like, as it sends the pitch-class members of one fully-diminished seventh chord in O by +3 mod 12, while it sends the members of the other by −3 mod 12.

Whereas certain other operations on O exist that share the same square as t ₃, they do not generate together with t ₃ groups that are isomorphic to . Hence, we do not include them in this class.

The Frattini subgroup of a group G contains the intersection of G and all its proper maximal subgroups (where a maximal subgroup H of G is a proper subgroup, such that no other proper subgroup K contains H strictly). Then, an elementary abelian group is a finite abelian group, where every non-trivial element has order p, such that p is a prime.

Some authors, such as 13, give a stronger requirement for central products, i.e. that .

In neo-Riemannian theory, a Wechsel exchanges a major triad with root x with the minor triad with root x+n mod 12, for all triads in K. The operations w ₁ and w ₂ (and w ₁ t ₆ and w ₂ t ₆) on O are Wechsel-like, in that they exchange each pitch-class member x of one fully-diminished seventh chord in O with the one x+n in the other fully-diminished seventh chord.

Our notation for a UTT differs from Hook's 9 10. He gives our as . We incorporate the present notation to be consistent with the notation for members of the ‘Mother Group’ used in 11, of which the UTTs form a subgroup.

For example, under the UTT , a major triad with a root x transposes to a major triad with root x+2 mod 12, and a minor triad with root x transposes to one with root x+10 mod 12, for all triads in K. Because in this case σ=+, no further action is required. On the other hand, if we had σ=−, the major triad with root x would first transpose to the one with root x+2 mod 12, and would then exchange with the minor triad with root x+2 mod 12. A minor triad with root x would similarly transpose by +10 mod 12 and exchange with the major triad with that root, again for all triads in K.

A Wechsel is characterized in our notation for UTTs by . A skew-Wechsel, then, has the form .

Again, our notation differs from Hook's 9 10, to conform to that of 11.

The action of a QTT such as on a major triad with root x is similar to that of a UTT. The triad transposes first to the major triad with root , but then we multiply x+2 by m ⁺ mod 12 and take the major triad with the resulting root. The action on minor triads uses instead transposition by +10 and multiplication by m ⁻ mod 12. As before, σ=+ preserves modes, and σ=− exchanges them.