Bells have many modes of vibration whose modal shapes are characterized by the numbers of nodal meridians and nodal circles. In Western church bells and carillon bells, the bell is shaped so that the lowest mode frequencies are harmonically related. The subjective strike note is determined by three strong partial tones with frequencies nearly in the ratios 2: 3: 4. The pitch of a handbell, on the other hand, is determined by the frequency of the fundamental tone. In ancient Chinese bells, the vibrational modes occur in pairs, and two different tones result from striking at the sui and gu strike points. A new type of carillon bell has been developed in which the traditional minor third partial is replaced by a major third partial, thus changing the timbre of the bell.

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A method for controlling the inharmonicity of a stiff string by adding a lumped mass (or masses) was originally suggested by F. Miller, Jr. In the present of paper, the same idea is investigated in a more precise manner. The effect of the string stiffness on its inharmonicity is discussed for different support conditions (the simple-simple, simplefixed, and fixed-fixed conditions). The simple-simple support condition gives the largest inharmonicity (deviation from a true harmonic series) among them even though the differences are small for most cases. The theoretical and experimental results of the resonance frequencies of a string with the fixed-fixed condition show very good agreement with each other. For a representative middle C string, a mass of 0.2-0.4 grams, when attached 13-9 mm from one end of the string, reduces the inharmonicity significantly.

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In 1860, H. v. Helmholtz concluded, by observing the movement of each particular point of a bowed string with his vibration microscope, that the sharp bend of the bowed string travels along upper and lower parabolic arcs. Since then, his conclusion has been accepted and little attention has been paid to the shape of the envelope of the bowed string. The experimental work of M. Kondo et al., which focused on this subject, stimulated the present work. This work re-examines the general solution of the wave equation by giving some initial conditions and then, in addition to the normal Helmholtz solution, obtaining two new solutions, whose shapes are of an elliptic arc and of an hyperbolic arc, respectively. These three solutions, parabolic, elliptic, and hyperbolic coincide with each other when specific parameter approaches infinity. Two experimental results found in the literature seem to represent an elliptic solution.

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Reed vibrations and pressure variations in the reed have been observed at the same time for the lowest note of a bassoon artificially blown. For comparison, the observations have been performed also for a cone simulating the bassoon and for a bassoon without tone holes. Behaviors of the bassoon and the bassoon-like systems artificially blown are fairly similar and remarkably complicated.

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This thesis deals with the koto, a 13-stringed wooden instrument belonging to the zither family and one of the most representative instruments used in Japanese music. The following points are discussed: 1) explanation of the physical construction and nature of the instrument, 2) nomenclature, explanation and analysis of the characteristic koto tunings (or modes, termed in Japanese choshi), including an analysis of their frequency of use in traditional music and their interrelationships, 3) explanation of the traditional way of setting up or tuning the choshi, using the most common choshi (hira joshi) as an example, 4) analysis of individual performer's pitch deviations (especially concerning the minor seconds) using this traditional tuning method, and 5) a proposal for a more reliable method of tuning (especially in regards to beginners), based on the results of this research. Included also are data in 4) showing that the minor seconds, in particular, are related to the Pythagorian seconds, and that the tuning over the total range of the koto shows scatter deviations similar to those found in a normally tuned piano.

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It is shown by theory and computer simulation that the decay characteristics of piano sound are dependent upon the degree of coupling between the strings. String and soundboard vibration are calculated using an equivalent circuit. Each string is comprised of resonance circuits that correspond to partials, and the behavior of individual partials is described independently of each other by these circuits. When the most simplified version of only one partial is considered, the equivalent circuit is expressed by the two resonant circuits coupled with the soundboard impedance. The degree of coupling between the two strings is dependent on the ratio of two constants: the degree of mistuning and the ratio of soundboard impedance to string impedance. If the former is smaller than the latter, the two strings are closely coupled with each other and a double decay characteristic results. If the relation between the two constants is reversed, the coupling is loose and a beat type decay characteristic results. The decay characteristics are changed by the above two constants. The results show that the degree of tuning and the characteristics of the strings and the soundboard determine the decay characteristics of piano sound.

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Resonance frequencies and tonal system are investigated on a set of ancient Chinese bells named “Piao-shi bianzhong.” According to the measurement of 12 bells preserved in Kyoto, it is found that a simple relation between frequency and size holds well. The relation tells frequency of a bell inversely proportional to square of its linear measure. It is reasonable to conclude that the set was cast for a heptatonic scale which is very similar to F# major in our days. Preceding studies in history and archaeology on Piaoshi bells are also shortly reviewed.

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