Abstract
All-interval structures are subsets of musical spaces that incorporate one and only one interval from every interval class within the space. This study examines the construction and properties of all-interval structures, using mathematical tools and concepts from geometrical and transformational music theories. Further, we investigate conditions under which certain all-interval structures are Z (or GISZ) related to one another. Finally, we make connections between the orbits of all-interval structures under certain interval-groups and the sets of lines and points in finite projective planes. In particular, we conjecture a correspondence that relates to the co-existence of such structures.
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- 1.
N.B.: The condition of triangularity is necessary, but not sufficient. For instance, 21 is a triangular number, but there are no all-interval heptachords in \(\mathbb {Z}_{43}\), as 6 is not a power of a prime (see below).
- 2.
Among the 46 groups of order 32 that are not generalized dihedral, only 2 have half or more of their elements that are not involutions, but the number of interval classes for both these groups is 25, which is not a triangular number. Similarly, of the 257 groups of order 64 that are not generalized dihedral, half or more than half the elements of three of these groups are involutions, but none of them have a triangular number of interval classes: 49 for one such group, and 51 for the other two.
- 3.
This situation does not imply that all Z relations are the result of affine transformations for sets with flat interval distributions other than 1. For instance, \(\mathbb {Z}_{31}\) contains several different (31,15,7) difference sets; hence, they are Z-related. However, they are not all related to one another by multiplication (from personal correspondence with Jonathan Wild). See [8, p. 420] for more information in this connection.
- 4.
“In an abstract GIS, the operation of GIS-transposition by interval i is well-defined by the formula int(s, Ti(s)) = i. That is, for any object s, the Ti-transform of s is that unique object Ti(s) which lies the interval i from s.” [11, p. 42]. “In an abstract GIS, given any objects y and v (where v may be the same object as y), the operation I of y/v GIS-inversion is well-defined by the formula int(v, I(s)) = int(s, y). The formula expresses a pertinent intuition: given any object s, its inverted transform I(s) lies intervallically in relation to v, exactly as y lies intervallically in relation to s.” [11, p. 43].
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Peck, R.W. (2015). All-Interval Structures. In: Collins, T., Meredith, D., Volk, A. (eds) Mathematics and Computation in Music. MCM 2015. Lecture Notes in Computer Science(), vol 9110. Springer, Cham. https://doi.org/10.1007/978-3-319-20603-5_29
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DOI: https://doi.org/10.1007/978-3-319-20603-5_29
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