Tone rows and tropes

Abstract

Applying various methods based on group actions we provide a complete classification of tone rows in the twelve-tone scale. The main objects of the present paper are the orbits of tone rows under the action of the direct product of two dihedral groups. This means that tone rows are considered to be equivalent if and only if they can be constructed by transposition, inversion, retrograde, and/or time shift (rotation) from a single row. We determine the orbit, the normal form, the stabilizer class of a tone row, its trope structure, diameter distance, and chord diagram. A database provides complete information on all $\mathrm{836,017}$ pairwise non-equivalent tone rows. It can be accessed via http://www.uni-graz.at/∼fripert/db/. Bigger orbits of tone rows are studied when we allow further operations on tone rows such as the quart-circle (multiplication), the five-step (multiplication in the time domain), or the interchange of parameters.

Keywords:

• tone rows
• trope
• database
• group actions
• normal form
• stabilizer type
• complete systems of representatives of non-equivalent tone rows
• 12-tone series

Mathematics Subject Classification 2010:

• 05E18
• 00A65

Acknowledgements

The authors thank the Guest Editors Jay Hook and Robert Peck and the anonymous referees for carefully checking all details, for suggesting necessary corrections, and for finding the elegant proof of the first assertion of Theorem 37. The authors are also grateful to Thomas Fiore and Marek Žabka for the suggestion to publish the full-length paper in a Special Issue.

ORCID

H. Fripertinger http://orcid.org/0000-0001-7449-8532

Notes

1 Already in 1924 (cf.  Hauer [1924]) J. M. Hauer introduced the circular representation of tone rows. Instead of the chromatic order of the pitch classes he used the order according to the quint-circle.

2 Already in 1924 (cf.  Hauer [1924]) J. M. Hauer introduced this matrix representation for tone rows.