D12 × D12-normal form of a tone row

The normal form h of the tone row f is the lexicographic smallest element in the orbit of f. Thus it is the tone row which occurs in first position in the listing of all elements of the D12 × D12-orbit. Knowing h we can search the database for all information on the orbit of f. For more details see Section 3.2 and Remark 3.2 of [1].

In order to compute the normal form of f it must be input either in numeric form or by choosing note names. Not only the normal form h is computed, but also an element (φ,π) of the acting group D12 × D12 so that h=(φ,π) f=φ o f o π-1 or, equivalently, f=(φ-1-1) h=φ-1 o h o π. The position of the input tone row f in the orbit of h with respect to the lexicographic order is indicated. Moreover we present the orbit of h in the chromatic circular representation and indicate the particular position of f in this polygon or, if necessary, in its mirror which is the polygon of I o h. Since f is an individual tone row we indicate its first note (or pitch class) by a circle around the vertex corresponding to this note (or pitch class), and the connection between the last note and the first note of f is not drawn in the polygon. Moreover we indicate how the pitch classes are labeled. Finally, we draw the trope structure of f according to this labeling of the pitch classes.

There are further buttons which allow to search the database for this tone row, to obtain all information about this tone row, or to show the orbit of this tone row.


Database on tone rows and tropes
harald.fripertinger "at" uni-graz.at
January 2, 2019