Harmony and nonharmonic partials

We have explored some musical effects of tones with nonharmonic partials. The spacing of partials can be stretched so that each partial frequency f_ij present in tones sounded singly or together is given by f_ij=A^i/12+log2j. Here i is the scale step in semitones, j is the partial number, and A is the frequency ratio of a pseudo‐octave (A=2 for a true octave). We find that subjects can match the keys of stretched (A=2.4) as well as unstretched passages. Stretched cadences (A=2.4) do not seem final. But, a stretched ’’cadence’’ with equally spaced partials that goes from closely spaced (tonally dissonant) to widely spaced (tonally consonant) partials does seem final. Our experiments do not decide finally among three views of harmony: that harmony depends on a fundamental bass or periodicity pitch (Rameau), that harmony depends on the spacing of partials (Helmholtz and Plomp) or that harmony is a matter of brainwashing.