HARMONICAL proportion or musical pro portion, is that in which the first term is to the third, as the difference of the first and second is to the difference of the se cond and third ; or when the first, the third, and the said two differences, are in geometrical proportion. Or, four terms are in harmonica' proportion, when the first is•to the fourth as the difference of the first and second is to the difference of the third and fourth. Thus 2, 3, 6, are in harmonica] proportion, because 2: 6:: 1 : 3. And the four terms, 9, 12, 16, 24, are in harmonica' proportion, because g 24:: 3: 8. If the proportional terms be continued in the former case, they will form an harmonica) progression, or series. 1. The reciprocals of an arithmetical progression are in harmonica) progres sion: and, conversely, the reciprocals of harmonica's are arithmeticals. Thus, the reciprocals of the harmonica's 2, 3, 6, are 1 1 1 2' 3 6 1, which are arithmeticals ; for 2 • 11 1 1 1 and = also : and the reci procals of the arithmeticals 1, 2, 3, 4, 8cc. 1 are which are harmonica's; 1 1 1 1 11 for 1 3 2' - - — and so on. in general, the reciprocals of the arith meticals a, a+d, 1 a-F2 d, d, &c. viz.
1 1 1 a a- Eti' &c. are harmoni cals ; et e contra. 2. If three or four mum bets in harmonical proportion be either multiplied or divided by some number, the products, or the quotients, will still be in harmonica] proportion. Thus, the harmonicals 8, 12, multipled by 2, give 12, 16, 24, or divided by 2, give 3, 4, 6, which are also harmonicals. 3. To find a harmonica] mean proportional between two terms : divide double their product by their sum. 4. To find a third term in harmonica! proportion to two given terms: divide their product by the difference be tween double the first term and the se cond term. 5. To find a fourth term in harmonica] proportion to three terms giv en : divide the product of the first and third by the difference between double the first and the second term. Hence, of the two terms a and b the harmonical mean is the third harmonical pro portion is ; also to a, b, c, the fourth harmonical is 2 • 6. If there
be taken an arithmetical mean and a har monical mean between any two terms, the four terms will be in geometrical pro portion. Thus, between 2 and 6 the arithmetical mean is 4, and the harmoni cal mean is 3 ; and hence 2 :3 :: 4: 6. Also, between a and b the arithmetical mean is as and the harmonical mean is 2 2 a b a-Fb but a: b.
Hanurosacar, series, a series of many numbers in continual harmonical propor tion. Thus, if there are four or more numbers, of which every three immedi ate terms are harmonica], the whole will make an harmonical series : such is 30 : 20 : 15 : 12 - 10. Or, if every four terms immediately next each other are harrno nical, it is also a continual harmonical se ries, but of another species, as 3, 4, 6, 9, 18, 36, &c.
HasmosticaL sounds, an appellation giv en to such sounds as always make a de terminate number of vibrations in the time that one of the fundamentals, to which they are referred, makes one vi bration.
Harmonic al sounds are produced by the parts of chords, &c. which vibrate a certain number of times, while flue whole chord vibrates once.
The relations of sounds had only been considered in the series of numbers, 1 : 2, 2 : 3, 3 : 4, 4: 5, &c. which produced the intervals called 'octave, fifth, fourth, third, &c. M. Sauveur first considered them in the natural series, 1, 2, 3, 4, 5, &c. and examined the relations of sounds arising therefrom. The result is, that the first interval, 1 : 2, is an octave ; the se cond, 1: 3, a twelfth ; the third, 1: 4, a fifteenth or double octave ; the fourth, 1 : 5, a seventeenth ; the fifth, 1 : 6, a nineteenth, &c.
The new consideration of the relations of sounds is more natural than the old one: and is, in effect, all the music that nature makes without the assistance of art.
HARMONICS;that part of music which considered the differences and propor tions of sounds, with respect to acute and grave, in contradistinction to rhyme and metre.