Ratio Of Small Integers Research Articles

Responses of Single Fibers in the Auditory Nerve of the Squirrel Monkey to Paired Phase-Locked Low-Frequency Tones

Data were analyzed on-line and from analog tape using a LINC computer. Stimuli were 10-sec complex sounds generated by summing two phase-locked low-frequency sinusoids. Both frequencies were within the response area of the unit and were related in ratios of small integers. Periodograms, i.e., distributions of unit responses over one period of the complex waveform summed for all repetitions of the waveform stimulus, were studied. The findings suggest that units discharge preferentially at times when the displacements in one direction of the cochlear partition are at or near maximal values. When intensity or phase of either stimulus component is varied with a resulting change in the complex waveform, there is a corresponding change in the periodogram. Concurrently, interspike intervals are grouped around values which are integral multiples of the time between the peaks of the periodogram, and the frequency of occurrence of such intervals is a function of the amplitude of these peaks (NB-06225).

On pointed minima in the interfacial energy of bicrystal systems

Abstract A parabolic Frenkel-Kontorova model was used to investigate the misfit dependence of the interfacial energy of a bicrystal system. An exact solution was obtained substantiating the conclusions drawn by other authors from approximate solutions of more complicated models. The main conclusion is that pointed minima of various orders of strength exist in the graph where the lattice constants are in the ratio of small integers and where previous less accurate calculations predicted a smooth variation. A correlation between the complexity of the misfit and the order of strength of the minima is also obtained.

Crystal Interfaces

An expression for the energy of an interface of general form between two crystals of arbitrary structures and relative orientations is derived in a form suitable for a variational calculation. This variational approach is applied to a one-dimensional interface between two two-dimensional crystals of differing lattice constants. In general there is a minimum in the interfacial energy when the lattice distances in the two crystals are in the ratio of small integers. Particular cases in which the surface potential is either sinusoidal or parabolic are discussed and detailed curves of interfacial energy as a function of lattice misfit are calculated.

Some Characteristics of the Tuning of Valved Wind Instruments

Experimental evidence on the actual tuning of “brass” instruments demonstrates the magnitude of the discrepancies in tuning resulting from the dual use of valves singly and in combination. The effectiveness of the valve tubing in lowering the frequency varies somewhat with different modes of vibration. Moreover, the frequencies of the normal modes are shown to be not necessarily in the ratio of small integers. This refutes the explanation often given in treatises on the acoustics of wind instruments that the series of sounds elicited, for example from the cornet without the use of valves, forms the harmonic series. Actually the fifth, sixth and eighth modes are likely to be found sharper than the “harmonic” values. For a steady musical tone each mode of vibration has associated with it harmonic overtones whose frequencies are not in general those of the various normal modes. In order to obtain a clear understanding of the action of wind instruments, terms such as harmonic, partial, and overtone used in the description of tone quality should be distinguished carefully from the normal modes of vibration of an arbitrarily shaped air column whose frequencies may or may nor be in the ratios of small integers.