Introduction to “Contributions toward a Quantitative Theory of Harmony”

James Tenney presents the introduction to his 1979 essay “Contributions toward a Quantitative Theory of Harmony.” In this introduction, Tenney discusses the history of consonance/dissonance, paying attention to the semantic problem, relations between pitches, qualities of simultaneous aggregates, and contextual as well as operational and functional senses of consonance/dissonance. He also explores the structure of harmonic series aggregates, focusing on harmonic intersection and disjunction, harmonic density, and harmonic distance and pitch mapping. Finally, he considers problems of tonality by analyzing harmonic-melodic roots and the “tonic effect,” along with harmonic (chordal) roots, the “fundamental bass,” and a model of pitch perception in the auditory system. In an epilogue, Tenney describes new harmonic resources as well as prospects and limitations of his contributions.

CALCULATION OF HO2 DENSITY OF STATES ON THREE POTENTIAL ENERGY SURFACES

Density of states (DOS) in both bound and unimolecular dissociation regime for HO2 system have been calculated quantum mechanically by Lanczos homogeneous filter diagonalization (LHFD) method. Three potential energy surfaces are explored and the results are contrasted for the total angular momentum J = 0 density of states. While two ab initio potential energy surfaces (PESs) (TU PES, J Chem Phys, 115:3621 and XXZLG PES, J Chem Phys122:244) produce the DOSs which are in fairly good agreement, the semi-empirical double many-body expansion (DMBE) IV PES (J Phys Chem94:8073) generates the much higher DOSs in higher energy range. The quantum mechanical DOSs are also compared with Troe et al.'s results from harmonic density, semiclassical density and their early density of states on the same TU ab initio surface.

Sets of zero discrete harmonic density

Let G be a compact, connected, abelian group with dual group Γ. The set E ⊂ has zero discrete harmonic density (z.d.h.d.) if for every open U ⊂ G and μ ∈ Md(G) there exists ν ∈ Md(U) with = on E. I0 sets in the duals of these groups have z.d.h.d. We give properties of such sets, exhibit non-Sidon sets having z.d.h.d., and prove union theorems. In particular, we prove that unions of I0 sets have z.d.h.d. and provide a new approach to two long-standing problems involving Sidon sets.