harmonic 
taining or relating to harmony of sounds ; of or 
pertaining to music ; in general, concordant ; 
consonant; in music, specifically, pertaining to 
harmony, as distinguished from melody and 
rhythm. 
With heavenly touch of instrumental sounds, 
In full harmonic number join d, their SOURS 
Divide the night, and lift our thoughts to heaven. 
Milton, P. L., iv. 687. 
Forever seeking out and rescuing from dim dispersion 
the rarities of melodic and harmonic form. 
Nineteenth Century, XIII. 441. 
2. In acoustics, noting the secondary tones 
which accompany the primary tone in a com- 
plex musical tone. See II., 1. 
The sounds of the Eolian harp are produced by the di- 
vision of suitably stretched strings into a greater or less 
number of harmonic parts by a current of air passing over 
them. Tyndall, Sound, iii. 
3. In math., involving or of the nature of the 
harmonic mean; similar to or constructed upon 
the principle of the harmonic curve. The first ap- 
plication of the adjective harmonic (in Greek) to mathe- 
matics was in the phrase harmonic proportion, said to 
have been used by Archytas, a contemporary of Plato. 
Three numbers are said to be in harmonic proportion 
when the first divided by the third is equal to the quo- 
tient of the excess of the first over the second divided by 
the excess of the second over the third ; or, otherwise 
stated, when the reciprocal of the second is the arith- 
metical mean of the reciprocals of the first and third, 
the second number is said to be the harmonic mean of 
the first and third. Pythagoras first discovered that a 
vibrating string stopped at half its length gave the oc- 
tave of the original note, and stopped at two thirds of 
its length gave the fifth. Now, as 1, JJ, and 4 are in har- 
monic proportion, and as this phrase arose among the 
Pythagoreans, the word harmonic has always been held to 
have reference here to this fact (although Nicomachus ex- 
plains it otherwise, from the properties of the cube, as 
ap^oi'La, or norm). The harmonic proportion or ratio, as 
thus denned, plays a considerable part in modern geom- 
etry as an important case of the anharmonic ratio, and 
has given rise to the phrases harmonic axis, center, pencil, 
etc. (See below.) A harmonic curve is the figure of a vi- 
brating string. It can assume many forms, but all may 
be regarded as derived, by summation of displacements, 
from simple harmonic curves, or curves of sines. The 
development of this idea has given rise to the theory of 
harmonics, which is one of the great engines of mathe- 
matical analysis. This gives the phrases harmonic analy- 
sis, Junction, motion, etc. 
4. In anat., forming or formed by a harmonia: 
as, a harmonic articulation or suture. 
Also harmonical. 
Center of the harmonic mean of a number of points 
A, B, C, etc., in a line with reference to another point, O, 
in that line a point M, such that 
MA MB MC 
Harmonic analysis, (a) In math., the calculus of har- 
monic functions ; especially, the calculation of the con- 
stants involved in the expression of a phenomenon as a 
sum of harmonic functions. ('<) In music, the analysis of 
the harmonic structure of a piece. Harmonic arith- 
metic, the arithmetic of musical intervals. Harmonic 
articulation. See def. 4, above. Harmonic axis, a ray 
the intersection of which with any curve is the harmonic 
center of the Intersections with the same curve of all the 
rays of a plane pencil. This term was introduced by 
Maclanrin. Harmonic center of the nth order, of a 
number of points lying in one line, a point such that, if 
the reciprocal of its distance from a fixed pole be subtract- 
ed from the distances of the points of which it is the har- 
monic center, and if all products of n of these differences 
be added, the sum is zero. Harmonic conies, two con- 
ies, (a, b, c, f, g, hVu, v, w)3 and (A, B, C, F. 3, HYx, y, 
z)2, such that aA + bB + cC + f F + gG + hH = 0. Har- 
monic conjugates. See conjugate. Harmonic curve. 
See curve. Harmonic division of a line, the division 
of a line by four points forming two pairs of harmonic 
conjugates. Harmonic engine. See engine. Har- 
monic figuration, in music: (a) A melodic figuration 
produced oy using in succession the tones that constitute 
the harmonies or chords of a piece : as, 
(b) The amplification of a harmonic passage by the intro- 
duction of passing-notes, etc. Harmonic flute. See 
harmonic stop. Harmonic function, a series composed 
of terms each the product of a function into the sine 
of a variable angle, these angles being in arithmetical 
progression; the general formula being % cos (ni( c). 
^0 
Also, an analogous function of two or three indepen- 
dent variables. Harmonic mark, in musical notation 
for the harp and instruments of the viol family, a small 
circle () placed over a note that is to be played so as to 
produce a harmonic tone. Harmonic mean, the recip- 
rocal of the arithmetical mean of the reciprocals of the 
quantities concerned. Harmonic modulation. See 
modulation. Harmonic note. See harmonic tone. 
Harmonic pencil, four rays lying in a plane and meeting 
in a point so as to divide harmonically every fourth line 
lying in the same plane. Harmonic progression, in 
math., a series of numbers the reciprocals of which are in 
arithmetical progression : so called because they are pro- 
portional to the lengths of a string vibrating to the har- 
monics of one musical tone. Also called musical progres- 
Hon. Harmonic proportion, the proportion existing 
between three numbers which form successive terms of a 
harmonic progression. Harmonic reed. See harmonic 
2723 
stop. Harmonic row, four points forming two pairs of 
harmonic conjugates. Harmonic scale, in music, the 
scale or tone-series formed by the harmonics of a tone. 
See II., and the illustration there given. Harmonic 
stop, in organ-building, a stop having pipes of twice 
the usual length, with a small hole at the mid-point, 
so that the halves of the air-column vibrate synchro- 
nously. The tone is sonorous and brilliant, and is not 
readily disturbed by overblowing, so that such stops may 
safely be subjected to an extra pressure of wind, and 
thus be utilized for striking solo effects. A harmonic flute 
is a flute thus constructed, and a harmonic reed a reed stop 
thus constructed, as, for example, a tuba mirabilis. Har- 
monic suture. See def. 4, above. Harmonic tele- 
graph. See telegraph. Harmonic tone, in playing the 
harp or instruments of the viol family, a tone produced by 
lightly touching one of the nodes of a vibrating string, or by 
changing the place of the contact of the bow, so as to sup- 
press the fundamental tone, leaving certain sets of its har- 
monics unaffected. The result is a tone much higher than 
the fundamental, and very clear and pure in quality. To 
produce the first harmonic, the string must be touched at 
its half-way point ; to produce the second harmonic, at a 
point one third of its length ; etc. Harmonic tones made 
on an open string are called 
natural (see fig. IX those on a 
stopped string artificial (see 
fig. 2) ; only those of the for- 
mer variety are possible on the 
harp. The white notes rep- 
resent the tone of the string, 
open or stopped ; and the 
black notes, the harmonic 
Artificial (4th string). 
tones actually produced. 
Also called flageolet - tones. 
Harmonic tones are not 
strictly harmonics of the 
fundamental tones with which they are related, because 
they are themselves complex. Harmonic triad, in mu- 
sic, a major triad. See triad. Harmonic triads, in math., 
two triads of points, a b c, A B C, taken on the same line, 
such that aA. bB. cC + aB. bC. cA + aC. bA. cB + aC. bB. 
cA + aB. bA. cC + aA. be. cB = 0. Simple harmonic 
function, a harmonic function consisting of a single term. 
Simple harmonic motion, a motion expressible as a 
simple harmonic function of the time. Also called a har- 
monic motion or harmonic vibration. 
II. n. 1. In acoustics: (a) A secondary or 
collateral tone involved in a primary or fun- 
damental tone, and produced by the partial vi- 
bration of the body of which the complete vi- 
bration gives the primary tone. Nearly every tone 
contains several distinct harmonics, which are always 
taken from a typical series of tones the vibration-numbers 
of which, beginning with that of the fundamental tone, are 
proportional to the series 1, 2, 3, 4, 5, 6, 7, etc. The inter- 
val from the fundamental tone to the first harmonic is, 
therefore, an octave ; to the second, an octave and a fifth; 
to the third, two octaves ; to the fourth, two octaves and a 
major third ; to the fifth, two octaves and a fifth ; to the 
sixth, two octaves and somewhat less than a minor sev- 
enth ; to the seventh, three oc- 7th 
taves; etc. (See illustration.) 
Harmonics result from the 
elasticity of the tone-produ- 
cing body, which leads it to vi- 
brate, not only entire, but in its 
aliquot parts ; thus, a violin- 
string tends to vibrate through- 
out its whole length, and also 
at the same time in each of its 
halves, thirds, quarters, etc. 
The vibration of the whole, be- 
ing much the greater, gives the 
Fundamental. 
primary or fundamental tone ; while the several partial vi- 
brations, which diminish rapidly in force as they rise in 
pitch, give the harmonics. In a given tone the harmon- 
ics may usually be roughly detected by the unaided ear ; 
but for precise and minute analysis specially construct- 
ed resonators are necessary. Tuning forks and large 
stopped organ-pipes give only insignificant harmonics; 
certain reed-instruments, like the clarinet* give only se- 
lected sets of harmonics, ad the second, fourth, sixth, etc. ; 
while the human voice is capable of the greatest richness 
of harmonics. What is technically known as quality or 
timbre in a tone is due to the number and the relative 
strength of the harmonics contained in it. Different in- 
struments and voices are thus distinguished from each 
other, and different uses of the same instrument or voice. 
In the voice, in particular, the essential difference between 
different vowel-sounds is a matter of harmonics. In any 
tone the lower harmonics are strictly consonant both with 
the primary tone and with each other : hence the use in the 
organ of mutation- and mixture-stops, whereby the conso- 
nant harmonics of a given tone are much emphasized. 
Many of the higher harmonics, on the other hand, are 
strongly dissonant both with the primary tone and with 
each other: hence the discordant quality of such instru- 
ments as cymbals, and the peculiar construction of the 
pianoforte, whereby dissonant harmonics are suppressed. 
In instruments of the viol and harp classes very beauti- 
ful effects are produced by suppressing the primary tone, 
leaving one set of its harmonics to sound alone. Such 
tones are called harmonic tones, or simply harmonics 
(though they are themselves compounded of a primary 
tone and its harmonics). In instruments of the trumpet 
class, like the horn, all the tones ordinarily used are 
really harmonics of the natural tone of the tube, and are 
produced by varying the pressure of the breath and the 
method of blowing. The same is true to a less degree of 
instruments of the wood-wind group. Harmonics are also 
called oaertones. All the tones, primary and secondary, en- 
tering into the constitution of an actual tone are often call- 
ed partial tones, or partials, the fundamental tone being 
the /rat partial, and the harmonics the upper partials. 
(b) A harmonic tone. 2. In math., a function 
expressing the Newtonian potential of a point 
in terms of its coordinates Artificial harmonic, 
natural harmonic. See harmonic tone, under I. 
Grave harmonic, the low tone generated by the simul- 
taneous sounding of two concordant tones. See combina- 
harmonics 
tional tone, under tone. Sectorial harmonic, a spherical 
surface-harmonic the axes of which lie equidistant in the 
plane of the equator. Solid harmonic, any homogeneous 
function of x, y, and z which satisfies Laplace's equation. 
A solid harmonic usually expresses the potential due to 
pairs of equally ami infinitely attracting and repelling 
points placed infinitely near to one another. Spherical 
harmonic. See Laplace's function, under function. 
Spherical surface-harmonic, or Laplace's coefficient, an 
expression of the variation of the potential over the sur- 
face of a sphere. Every such harmonic supposes the ex- 
istence of a certain number of fixed axes through the 
sphere. It is obtained by taking the product of the cosines 
of the angular distances of the variable point from some 
of these axes, together with the cosines of the angular dis- 
tances of pairs of the other axes, until each axis has been 
used once, and once only, in forming the product, and then 
summing all possible products of this sort. Zonal har- 
monic, a spherical surface-harmonic which has all its 
axes coincident. 
harmonica (har-mon'i-ka), n. [NL., fern, of L. 
liartnonicus, musical: see harmonic.] 1. Same 
as musical glasses (which see, under glass). 2. 
A musical toy consisting of a set of small me- 
tallic reeds so mounted in a case that they may 
be played by the breath, certain tones being 
produced by expiration, others by inhalation. 
Also called harmonicon Somzee's harmonica, a 
device for preventing accidents from fire-damp in a mine. 
The draft upon a flame burning in a glass chimney is 
so regulated that while the air remains pure the flame is 
silent, but when its density is altered by the mixture of a 
certain proportion of fire-damp it gives a musical tone, as 
in the chemical harmonicon. 
harmonical (har-mon'i-kal), a. [< harmonic + 
-?.] Same as harmon ic. " 
It were but a phantasticall deuise and to no purpose at 
all more then to make them harmonicall to the rude eares 
of those barbarous ages. 
Puttenham, Arte of Eng. Poesie, p. 11. 
After every three whole notes, nature requireth, for all 
harmonical use, one half note to be interposed. Bacon. 
harmonically (hiir-mon'i-kal-i), adv. 1. In a 
harmonic manner; harmoniously; specifically, 
in music, in a manner suitable to the rules of 
harmony, as distinguished from melodically or 
rhythmically. 
Plato therefore intending to declare harmonically the 
harmony of the four elements of the soul, ... in each 
intervall hath put down two medieties of the soul, and 
that according to musicall proportion. 
Holland, tr. of Plutarch, p. 1022. 
2. In acoustics, by or in harmonics. See har- 
monic, n., 1. 
They may heat absorbent gases, such as ammonia, and 
cause them to do mechanical work, or to produce sound, 
if the incident beam be intermittent or harmonically vari- 
able. A. Darnell, Physics, p. 612. 
3. In math., in a harmonic relation. Thus, two 
segments, AB, MN, of the same straight line are said to 
be harmonically situated when AM. BN + AN. BM = 0. 
The three diagonals of a four-side cut each other har- 
monically. Encyc. Brit., X. 392. 
4. In anat., so as to make a harmonia. 
harmonichord (har-mon'i-kord), n. [< Gr. ap- 
fiavia, harmony, + xpfy, a chord. ] A musical 
instrument having a keyboard and strings like 
a pianoforte, in which the tone is produced by 
the pressure against the strings of small revolv- 
ing wooden wheels covered with resined leather. 
The tone resembles that of a violin. The principle of the 
tone-production is the same as that of the hurdy-gurdy. 
Also called piano-violin, violin-piano, tetrachordon, xan- 
orphica, etc. 
harmonic! (har-mon'i-si), n. pi. In one. music, 
theorists who reached harmonic rules by in- 
duction from subjective aural effects, as op- 
posed to canonici, or those who deduced rules 
from a mathematical theory of intervals. Also 
called harmonists, and, from their leader (Aris- 
toxenus, a Greek peripatetic philosopher, a 
disciple of Aristotle), Aristoxenians. 
harmonicism (har-mon'i-sizm), n. The state 
of being in harmonic proportion. 
harmonicon (har-mon'i-kon), ;.; pi. harmonica 
(-ka). [NL., < Gr. dpfiovwov, neut. of dpftovtuof, 
musical: see harmonic."] 1. See harmonica, 2. 
2. An orchestrion. 3. An acoustical ap- 
paratus consisting of a flame of hydrogen burn- 
ing in a glass tube so as to produce a mu- 
sical tone. See Singing-flame. The principle has 
been used in a musical instrument, sometimes called chem- 
ical harmonicon, but better pifrophone (which see). 
harmonics (har-mon'iks), re. [PI. of harmonic, 
after Gr. ap^ovin&, the theory of sounds, music, 
neut. pi. of apfiovticof: see harmonic.'] 1. The 
science of musical sounds: a department of 
acoustics. [Bare.] 
During the era in which mathematics and astronomy 
were . . . advancing, rational mechanics made its second 
step; and something was done towards giving a quanti- 
tative form to hydrostatics, optics, and harmonics. 
H. Spencer, Universal Progress, p. 175. 
2. The mathematical theory of harmonics (see 
harmonic, n., 2), or the development of expres- 
sions for the Newtonian potentials. 
