unison there happens a coincidence; whence the interval of the fifth is a perfect concord.

To assist the imagination, I have made. this plainer in fig. I. plate IV. wherein AB represents the unison, and CD its fifth. They are supposed to start together at E and F, and the successive asterics in each string mark the coincidences; by means of which their motions may be compared without any difficulty. In the case of the diatessaron, or fourth, where the lengths are as 3 and 4, the vibrations respectively are as 4 and 3; and the coincidences happen at the second recourse of the unison, and the third course of its diatessaron, as also at the fourth course of the unison, and fifth course of its diatessaron, in which they return to their first po

sition.

My design being rather to shew the principle of consonance, than to pursue it through all its varieties, I shall refer the reader for farther satisfaction, in this branch of the subject, to Mr. Holder's Treatise on the Natural Grounds and Principles of Harmony, a piece which is clearly and learnedly written and when its doctrines are understood, the seven ancient

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ancient Greek writers upon music may be examined, of whose works there is an excellent edition printed by Elzevir, with many useful notes and commentaries by Meibomius of these authors, the most concise and clearest is Euclid. But here the student must expect to find many things which are now obsolete, others which are very obscure, and some which will never find a sufficient interpreter. And I must also warn him not. to expect more than he will find. The principle of coincidence in vibrations, as explained by Galilæo, from the doctrine of pendulums, was wholly unknown to the ancients. They contented themselves with the simplicity of ratios, as if there had been some magic in proportion, and looked no farther. They knew, that the more simple the ratio, the more perfect the consonance; and that, as they became less simple, they degenerated into dissonance. The Pythagoreans laid all their stress upon the ratios themselves: "po"tiores rationes potioribus consonantiis as

signabant," says Dr. Wallis, in his Appendix to Ptolemy, which contains a very good and clear account of almost all that relates to the music of the ancients. Ptolemy him

self

unison there happens a coincidence; whence the interval of the fifth is a perfect concord.

To assist the imagination, I have made. this plainer in fig. I. plate IV. wherein AB represents the unison, and CD its fifth. They are supposed to start together at E and F, and the successive asterics in each string mark the coincidences; by means of which their motions may be compared without any difficulty. In the case of the diatessaron, or fourth, where the lengths are as 3 and 4, the vibrations respectively are as 4 and 3; and the coincidences happen at the second recourse of the unison, and the third course of its diatessaron, as also at the fourth course of the unison, and fifth course of its diatessaron, in which they return to their first position.

My design being rather to shew the principle of consonance, than to pursue it through all its varieties, I shall refer the reader for farther satisfaction, in this branch of the subject, to Mr. Holder's Treatise on the Natural Grounds and Principles of Harmony, a piece which is clearly and learnedly written and when its doctrines are understood, the seven ancient

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