HARMONICS, GRAVE. These are low sounds, of small intensity or loudness, which, if attentively observed by au experienced ear, will be heard to accompany every accurate or perfect consonance of two sounds, whose ra tio is expressible in small numbers, whether the same may be composed of the Diatonic 'mimes 1, 2, 3, and 5, or of somewhat larger prime numbers.
A grave harmonic of a different species from the above is also heard, whenever the BEATS of an imperfect or tem pered concord (see that article) exceed 12 or 13 in a se cond of time.
About the beginning of the last century, M. Sauveur considered the reinforcement of sound which must peri odically take place, while we hear a consonance of two mu sical sounds, whose ratios are in small numbers, occasioned by the coincidence at short intervals of the pulses of the two sounds. These reinforcements he called beats . and, as Dr Robert Smith has observed, improperly confounded these, which are grave harmonic sounds, (because rarely occurring so seldom as twelve times in a second,) with the beats (of imperfect concords) properly so called, which can be separately heard, and distinctly counted, in every instance where the degree of temperament or imperfection is sufficiently small.
About the year 1754, M. Rameau and M. Tartini first made observations on the coincident pulses of consonances with small ratios ; but each of the ten grave harmonics which the latter has mentioned, have been found by later observers to be an octave too high ; that is, the new sounds really heard to result from the rapid coincidence of the pulses of the single sounds, are an octave lower with re gard to these sounds than Tartini supposed ; errors which fcw will wonder at, who have experienced the difficulty of avoiding errors of an octave, or sometimes more than one, in estimating sounds that are either very high or very low.
In 1807, Mr John Holder published a work on music, in which he attempted to build a good deal of the theory of composition and harmonica) effect on these grave harmo nic sounds, accompanying consonances as their " depend ants ;" but most of his speculations have failed, and many of them led to very absurd conclusions, owing to his hav ing set up and used a defective or false rule, for assigning the grave harmonic of a given consonance.
It is the more necessary, therefore, to give here a true rule for finding the grave harmonic of any given con sonance, viz. Find the vibrations made by each of the sounds of the given consonance in one second, divide these Successively by the reversed terms of the given ratio of the consonance, and the quotient (in each case) will give the vibrations for 1" of the grave harmonic ; the ratio of which vibrations, to the vibrations of either of the given sounds respectively, will give the interval of the harmonic below such given sound.
If, for example, the major second CD were given in the middle of the scale, where the ratio is and the sound of the lowest note (on the tenor cliff line) makes 240 vibra tions per 1", then lx240=270 is the vibrations nf D, and or =30, the vibrations of the harmonic note ; and its interval below the lower note C is or 3 VIII ; and below the upper note D is 3 VIII+ II. The principal intervals of the scale, and some others, calculated as above, have their grave harmonics shewn in the following Table, viz.
The first column of the above Table shews the ratios of the given consonances ; the second, tho intervals ex pressed in numerals ; the third, the vibrations per second, supposing the lower part to be the note on the tenor-cliff line of the stave; the fourth column contains the calcu lated vibrations of the harmonic note ; column five shews the ratio, and column sir the interval of this harmonic, be low the lowest of the given notes or C ; and columns seven and eight shew the same things with regard to the highest of the given notes.
For the sake of more ready comparison with Mr Hol der's defective rule for calculating grave harmonics, the errors of which it seemed of some importance to place in as clear a view as possible, we have given above a far less simple rule for obtaining the ratios of the new sounds, with relation to either of their generators, than the one which we are now about to add, viz. The ratio of any given consonance above a bass or fundamental note being a being the least term of the ratio ; then is the ratio of the grave harmonic below the bass note, and — the ratio of the same harmonic below the upper note of the conso nance ; which is too evident, from an inspection of the above Table, to need a particular example.
With regard to the other kind of grave harmonics, the results of tempered concords, which beat too fast to be se parately counted or perceived : If, for instance, we were to consider the grave minor third 34 in the above Table to be a tempered concord, we should have, by our fourth method in the article BEATS, 240x 6-2844x5=1440-14224— 174, the beats per second, or vibrations of this grave har monic, being just double, or an octave higher, than those in the Table above, and so of others ; but our limits will not admit of our enlarging further on this subject.