HAR 
HAR 
sien ; and, conversely, the reciprocals of 
liarmonicals are arithmetical®. Thus, the 
reciprocals of the liarmonicals 2, 3, 6, are 
111 1 
-, -, which are arithmetical ; for -> 
— and 
- = ~ also : 
6 6 
and the 
reciprocals of the arithmeticals 1, 2, 3, 4, 
&c. are 
till 
&c. which are harmo- 
nicals; for 
and so 
3’ 4’ 
1 _ 1 _■ 1 . 1 _ 
1 ' 3 " 1 2 : I 3 ’ 
on. And, in general, the reciprocals of the 
arithmeticals a, a -}- d, a -j- 2 d, a -}- 3 d, &c. 
11 1 1 
liiz. ~, 
&c. are 
a' a-|— iT a ■*!“ 2 <i — [- 3 u 
harmonicals; et e contra. 2. If three or 
four numbers in harmonical proportion be 
either multiplied or divided by some num- 
ber, 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, 
which are also harmonicals. 3. To find a 
harmonical mean proportional between two 
terms : divide double their product by their 
sum. 4. To find a third term in harmonical 
proportion to two given terms ; divide their 
product by the difference between double 
the first term and the second term. b. To 
find a fourth term in harmonical proportion 
to three terms given: 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 — r4 ; the third harmonical pro- 
u-j -b 
portion is ■ ■ ■ " also to a, b, c, the fourth 
harmonical is - 
, . 6. If there be taken 
2 a — b 
an arithmetical mean and a harmonical 
mean between any two terms, the four terms 
wall be in geometrical proportion. Thus, 
between 2 and 6 the arithmetical mean is 4, 
and the harmonical mean is 3 ; and hence 
2 : 3 :: 4 : 6. Also, 'between a and b the 
a-\-b 
arithmetical mean is 
2 
and the harmo- 
nical mean 
aJ r b . i. 
2 ah 
a + b ’ 
but a : 
2 a b 
a-j-b 
, Harmonical series, a series of many 
numbers in continual harmonical proportion. 
Thus, if there are four or more numbers, of 
which every three immediate terms are har- 
monical, the whole will make an harmonical 
VOL. III. 
series : such is 30 : 20 : 15 : 12 : 10. Or, 
if every four terms immediately next each 
other are harmonical, it is also a continual 
harmonical series, but of another species, as 
3, 4, 6, 9, 18,36, &c. 
Harmonical sounds, an appellation given 
to such sounds as always make a determi- 
nate number of vibrations in the time that 
one of the fundamentals, to which they are 
referred, makes one vibration. 
Harmonical sounds are produced by the 
parts of chords, &c. which vibrate a certain 
number of times, while the whole chord vi- 
brates 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 ex- 
amined the relations of sounds arising there- 
from. The result is, that the first interval, 
1 : 2, is an octave ; the second, 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 fire assistance of art. 
HARMONICS, that part of music which 
considered the differences and proportions 
of sounds, with respect to acute and grave; 
in contradistinction to rhyme and metre. 
HARMONY, in music, the agreeable 
result, or union, of several musical sounds, 
heard at one and the same time ; or the 
mixture of divers sounds, which together 
have an effect agreeable to the 'ear. As a 
continued succession of musical sounds pro- 
duces melody, so does a continued combi- 
nation of these produce harmony. See 
Music. 
Harmony of the spheres, or Celestial 
Harmony, a sort of music much talked of 
by many of the ancient philosophers and 
fathers, supposed to be produced by the 
sweetly-tuned motions of the stars and pla- 
nets. This harmony they attributed to the 
various proportionate impressions of the 
heavenly globes upon one another, acting 
at proper intervals. It is impossible, ac- 
cording to them, that such prodigious large 
bodies, moving with so much rapidity, 
should be silent ; on the contrary, the at- 
mosphere, continually impelled by them, 
must yield a set of sounds proportionate to 
the impression it receives ; consequently 
as they do not all run the same circuit, nor 
Ff 
