Ramsay
which seems to show that not only has one part of a vibrating string sympathy with another part of it so as to go into harmonic partials, as we have just seen, but as if the very air itself had sympathy with harmoniously vibrating strings; for Tartini observed that two harmonious sounds being produced and sustained as they can be, for example, by a strong bow on the violin, a third sound will be heard. Tartini's name for it was simply "a third sound." This is not an overtone, as Helmholtz has called the harmonic partials of one sounding string, but an undertone, because it is a "grave harmonic," away below the sounds of the two strings which awaken it. The subject of these undertones has been carefully studied since Tartini's day, and more insight has been obtained since we are now able to count and register the vibration of any musical sound. Helmholtz has called these third sounds of Tartini's "difference sounds," because when awakened by two strings, for example, the vibration-number of the third tone is the difference of the vibrations-numbers of the two tones which awaken it. The note C with vibration-number 512, and another C whose vibration-number is 256, the octave, awakened no third sound, because there is no difference between the two numbers - the one is just the doubled or halved; but if we take C256 and G381, its fifth, the difference number is 128; this being a low octave of C256, it has the effect of strengthening the upper one. Helmholtz found this to be the law of the third sound as to its producing, and the effect of it when produced. This third sound, mysteriously arising in the air through the sympathy it has with all concordant things, is another among many more suggestions that the whole Creation is measured and numbered to be in sympathy one part with another. The Creation is a universe. [Scientific Basis and Build of Music, page 60]

Nine-tenths of a string, if stopped and acted on, gives a tone the ratio of 9:10, but if touched and acted on as a harmonic it gives a note which is three octaves and a major third above the whole string. If the remaining tenth of the string be acted on either as a stopped note or a harmonic it will give the same note which is three octaves and a major third above the whole string the ratio of 1:10, so that the stopped note of one-tenth and the harmonic of nine-tenths are the same. Indeed the bow acting on stopped note of one-tenth, on harmonic of nine-tenths, or on harmonic of one-tenth, produces the same note, as the note is the production of one-tenth in each case; for in the harmonic, whether you bow on the nine-tenths or the one-tenth, while it is true that the whole string is brought into play, yet by the law of sympathy which permeates the entire string, it vibrates in ten sections of one-tenth each, all vibrating in unison. This is what gives the harmonic note its peculiar brilliancy. [Scientific Basis and Build of Music, page 92]

"The three notes of the dominant chord resolve by each note going to the next note upward - G soars to A, B to C, D to E. The three notes of the subdominant resolve by each note going to the next note downward - C sinks to B, A to G, F to E. The two upper notes of the dominant resolve into the tonic chord according to the Laws of Proximity and Specific Levity; and the two lower notes of the subdominant resolve into the tonic chord according to the Laws of Proximity and Specific Gravity. And in this way Nature, in chord-resolution, has two strings to her bow." [Scientific Basis and Build of Music, page 96]