ΤΟ SIR WILLIAM STERNDALE BENNETT (PRINCIPAL), AND TO THE PROFESSORS AND STUDENTS, OF THE ROYAL ACADEMY OF MUSIC, THIS VOLUME IS, WITH SINCERE RESPECT, DEDICATED. PREFACE. THE following treatise, portions of which have been delivered in lectures at the South Kensington Museum, the Royal Academy of Music, and elsewhere, aims at placing before persons unacquainted with Mathematics an intelligible and succinct account of that part of the Theory of Sound which constitutes the physical basis of the Art of Music. No preliminary knowledge, save of Arithmetic and of the musical notation in common use, is assumed to be possessed by the reader. The importance of combining theoretical and experimental modes of treatment has been kept steadily in view through out. The author has incorporated the chief Acoustical discoveries of Professor Helmholtz, but adopted his own course in explaining them and developing their connection with the previously established portions of the subject. The present volume, therefore, even where its obligations to the great German philosopher are the deepest, is not a mere epitome of his work, but the result of independent study. TRINITY COLLEGE, CAMBRIDGE, 1 Die Lehre von den Tonempfindungen. Dritte Ausgabe. Braunschweig. 1870. Of this profound and exhaustive treatise it is not too much to say that it does for Acoustics what the Principia of Newton did for Astronomy. CONTENTS. ON SOUND IN GENERAL, AND THE MODE OF ITS TRANSMISSION. Sensation of Sound, and its cause, § 1-Connection of Sound with motion, § 2-Velocity of Sound, § 3-Stationary media of Sound, § 4-Motion of sea-waves, § 5-Description of a wave, § 6-Length, amplitude, and form of wave, § 7-Length of wave and time of vibration, § 8-Periodic vibrations, $$ 9, 10-Form of wave and mode of vibration, §§ 11, 12-Extension of term 'wave' to solid bodies, § 13-Wave on the surface of a field of standing corn, § 14-Longitudinal vibrations, § 15, 16-Conden- sation and rarefaction, § 17-Associated wave, § 18-Law of pendulum-vibration, § 19-Pressure and density of air; Mariotte's law, § 20-Transmission of sonorous waves along a tube of uniform bore, § 21-Unconstrained motion of Sound-waves, § 22- Three elements of a musical sound, loudness, pitch, and quality, § 24-Loudness and extent of vibration, § 25-Pitch and rapidity of vibration; the Syren; continuity of pitch, §§ 26, 27-Measure of pitch; vibration-numbers, § 28-Limits to the pitch of musical sounds, § 29-Relative pitch; intervals, § 30-Tonic intervals of the Major scale; concords and discords, § 31-Additional notes required for the Minor scale, 32-Measure of intervals, § 33— Vibration-fractions, § 34-Table of vibration-fractions for the tonic intervals of the Major and Minor scales, § 35-Calculation of the vibration-numbers of all the notes in a scale, from the |