HARMONICAL
, or Harmonic, something relating to Harmony. Thus,
Harmonical Arithmetic, is so much of the theory and doctrine of numbers, as relates to making the comparisons and reductions of musical intervals, which are expressed by numbers, for finding their mutual relations, compositions, and resolutions.
Harmonical Composition, in its general sense, includes the composition both of harmony and melody; i. e. of music, or song, both in a single part, and in several parts.
Harmonical Interval, the difference between two sounds, in respect of acute and grave: or that imaginary space terminated with two sounds differing in acuteness or gravity.
Harmonical Proportion, or Musical Proportion, 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 2d and 3d; or when the first, the third, and the said two differences, are in geometrical proportion. Or, four terms are in Harmonical proportion, when the 1st is to the 4th, as the difference of the 1st and 2d is to the difference of the 3d and 4th. Thus, 2, 3, 6, are in harmonical proportion, because . And the four terms 9, 12, 16, 24 are in harmonical proportion, because .—If the proportional terms be continued in the former case, they will form an harmonical progression, or series.
1. The reciprocals of an arithmetical progression are in Harmonical progression; and, conversely, the reciprocals of Harmonicals are arithmeticals. Thus, the reciprocals of the Harmonicals 2, 3, 6, are 1/2, 1/3, 1/6, which are arithmeticals; for , and also: and the reciprocals of the arithmeticals 1, 2, 3, 4, &c, are 1/1, 1/2, 1/3, 1/4, &c, which are Harmonicals; for ; and so on. And, in general, the reciprocals of the arithmeticals a, a + d, a + 2d, a + 3d, &c, viz, , &c, are Harmonicals; et e contra.
2. If three or four numbers in Harmonical proportion be either multiplied or divided by some number, the products, or the quotients, will still be in Harmonical proportion. Thus,
| the Harmonicals | 6, | 8, | 12, |
| multiplied by 2 give | 12, | 16, | 24, |
| or divided by 2 give | 3, | 4, | 6, |
3. To find a Harmonical mean proportional between two terms: Divide double their product by their sum.
4. To find a 3d term in Harmonical proportion to two given terms: Divide their product by the difference between double the 1st term and the 2d term.
5. To find a 4th term in Harmonical proportion to three terms given: Divide the product of the 1st and 3d by the difference between double the 1st and the 2d term.|
6. If there be taken an arithmetical mean, and a Harmonical mean, between any two terms, the four terms will be in geometrical proportion. Thus, between 2 and 6,
Also, between a and b,
