When the conditions of the room are such that a system of stationary waves are so formed that a region of great intensity coincides with the source, the phenomenon of resonance oc curs. That is to say the emission of that par ticular note will be increased in comparison with other notes of such pitch that their great est nodal intensity, in their own interference sys tems does not coincide with the source. This phenomenon is called resonance. Both interfer ence and resonance result in the destruction of chordal balance.
The ear is ordinarily divided, in anatomical work, into three parts, the outer, the middle and the inner ear. The drum of the ear separates the outer from the middle ear, the middle ear being an air cavity connected by the eustachian tube with the nasal cavity. The sound is conducted across the middle ear by a system of three bones which connect the drum with another membrane separating the middle ear from the inner ear. The inner ear is a somewhat complicated cavity in the solid bone of the skull. It consists essentially of three semi-circular canals, and a much longer and larger snailshell-like cavity called the cochlea. This inner ear is separated from the middle ear not merely by the membrane already referred to pressed against and vibrated by the bones, but also by another small membrane. Starting from between these two membranes a diaphragm runs the length of the cochlea. This diaphragm, somewhat intricate in its complete structure, has as an essential part a system of numerous stretched fibres, varying in length and probably also in tension. When the sound is conducted from the outer ear to the inner ear by the three bones in the middle ear, the vibration is communicated to a liquid which fills the inner ear. This liquid in vibrating causes the small fibres of the diaphragm, called the fibres of Corti, to vibrate. As the fibres of Corti are of different lengths and of a different tension, dif ferent regions of the diaphragm respond to dif ferent notes. On this diaphragm terminate the auditory nerves which are stimulated by the vibration of the fibres, and communicate the corresponding sensation to the brain. The vari ation in pitch sensation is due to variation in the stimulated region of the diaphragm. When the sound is not a pure tone the various partial tones excite the corresponding parts of the diaphragm. When two notes are sounded, each with its system of overtones, there are regions of the diaphragm more or less excited simul taneously by the two systems. When the two partials which excite overlapping regions of the diaphragm are not of exactly the same pitch beats occur between them. These beats when
slow are not wholly disagreeable, and having a tremulo effect in moderate use are not without musical value. When, however, the beats are more rapid, and this occurs when the over lapping partial tones differ more in pitch, the beats lose their distinct character as such and produce the effect known as discord. If the two partial tones differ still more in pitch the regions which they excite overlap less and less and the discord diminishes. Following out this line of argument Helmholtz was able to show that when the fundamentals having harmonic upper partials bear to each other simple ratios in their vibration frequency their discord is a minimum, deriving in this way a complete ex planation of the musical scale as used in har monic composition. The scale thus obtained is the true or natural scale. The intervals be tween the successive notes are not equal, but fall into two groups of so-called whole and half tone intervals. The whole tone intervals are not equal among themselves and are not twice the half tone intervals. Therefore, even after inserting sharps and flats to subdivide the whole tone intervals the resulting chromatic scale is not one of equal interval. While this is the scale which would be employed by instru ments without fixed key-boards, and by the hu man voice accurately trained, it cannot be em ployed in instruments with fixed key-boards if such instruments are to be used in different keys.
For this purpose Bach invented a scale called the equally tempered scale in which all the half tone intervals are made equal. On this scale no key is accurate, but no key is so inaccurate as to result in serious discord. The following table gives the vibration frequency of the eight notes of the middle octave on the natural and on the tempered scale: Natural Tempered Scale Scale C 258.7 258.7 D 291.o 290.3 323.4 325.9 F 344.9 345.3 G 388.0 387.6 A 431.1 435.0 B 488.2 C 517.3 517.3 As the invention of the musical scale long preceded its use in harmonic composition, and during the period preceding the 11th century was used only in melodic composition — that is for notes sounded in sequence — the simul taneity necessary for harmonic effect was ob tained by the prolongation of one note into the other. Consult Franklin, 1". and Macnutt, B. 'Light and Sound' (1909) ; Hamilton, 'Sound and Its Relation to Music' (Boston 1911); Sabine, 'Architectural Acoustics' 1 Boston 1906) ; Saeltzer, 'Treatise on Acoustics in Con nection with Ventilation' (New York 1908).