Nov. 8, 1877] 



NATURE 



33 



which we hear every day, and to show how these are 

 explained by the fundamental laws of the science. 



Although music has appeared to many persons a diffi- 

 cult subject, it is really one of the most easily intelligible 

 and one of the most firmly grounded of sciences. It is 

 purely a science of numbers. 



The consonances which charm the ear, such as the 

 octave, twelfth, fifth, fourth, and the major and minor 

 thirds, have two concurrent sets of vibrations ; the one 

 set produced by the lower string or pipe, and the other 

 by the upper. Although they vibrate at different rates, 

 yet there are periodical coincidences of vibration between 

 them, and these coincidences sound with much more 

 power upon the ear than the vibrations which are non- 

 coincident, or sound apart. It has been calculated that 

 two hammers striking simultaneously upon an anvil have, 

 through the greater displacement of air, fourfold loudness, 

 instead of merely double. The same law applies to 

 musical sounds. Coincidence of vibration is more briefly 

 expressed by its synonym, "consonance;" and all non- 

 coincident vibrations, are included in "dissonances," 

 meaning only that they sound apart. In a musical sense, 

 dissonance is the medium between concord and discord, 

 running from one into the other ; for, in the most pleasing 

 intervals, there are some non-coincident vibrations, and 

 when these become very numerous, they overpower all 

 concord. This will be shown in the sequel. 



Suppose we take one long pianoforte string or an organ- 

 pipe. The lowest sound it can' produce will be that of 

 its whole length, and this may be made the foundation of 

 an entire scale of consonant notes, for every aliquot part 

 of the length, being such as will measure without any re- 

 mainder, will be also a multiple of the vibrations of No. 

 I. Thus No. 2, the octave, is half the length and vibrates 

 twice as fast as the whole string. No. 3, the so-called 

 twelfth, or octave and fifth, is a third of the length of 

 No. I, and it vibrates thrice as fast. Then, if we sound 

 No. 3 with No. 2 instead of No. i, we throw off the lower 

 octave and have the fifth only, or 3 to 2. It is essential 

 for consonance that the intervals should be aliquot parts 

 of No. I, for if otherwise, we should only create discord. 

 The musical law is expressed very simply, that the 

 number of vibrations is in inverse ratio to the length of a 

 string. 



The scale of all consonances is called the harmonic 

 scale, copies of which are before you. It is exemplified 

 by string or pipe. Let us consider, first, the ^olian 

 harp, on which the winds alone produce the consecutive 

 sounds. The strings are tuned in unison, except the two 

 outmost, one on each side, and those are covered with 

 wire, and tuned an octave lower. When the wind blows 

 quickly enough to sound the bass strings, which we will 

 suppose to have tuned to C on the bass clef, with 128 

 vibrations in a second of time, it is the whole string 

 which sounds first, and the rapidity of the wind must be 

 doubled before the harp will sound any change of note, 

 and that note will be the octave^^above the first. It has 

 already been said that the octave is produced by half the 

 length of a string, and that it vibrates twice as fast as the 

 whole — but mark the coincidence between the music and 

 consecutive numbers ; i and 2 have no note between 

 them, although the sound jumps from the whole length 

 to that of the half ! When the bass strings sound the half 

 length they have divided themselves into equal halves 

 by a node, and that node creates tension in opposite 

 directions, the one ventral segment pulling, as it were, 

 against the other. These self-forming nodes may be 

 easily seen by daylight, and at night by throwing a light 

 upon the string. They were shown at our first conver- 

 sazione in these rooms by Mr. Spiller, and at the Edin- 

 burgh meeting of the British Association by Mr. Ladd. 

 The gust of wind which sounds the octave, or half length 

 of the bass strings of the ^olian harp, sounds at the same 

 time the whole length of the gut strings, because they are 



tuned to that pitch. Then, as the wind rises, subdivision 

 goes on in both with every multiple of 128 vibrations for 

 the bass, and of 256 vibrations for the tenor strings. 



The reason for tuning the yEolian harp to a low pitch 

 is, that the strings may be more easily acted upon by the 

 wind. We read, poetically, of hanging one in a tree, but 

 it requires a much stronger draught than it will get there, 

 except during a hurricane, when no one will care to go 

 to listen. Our late lamented Vice-President, Sir Charles 

 Wheatstone, F.R.S., fixed a single violin string under a 

 very draughty door, as an yEolian harp, and he calculated 

 the increase of draught caused by lighting a fire in the 

 room, and by the opening of an outer door, by the rising 

 pitch of the note. The varieties produced by this string 

 have been described as " simultaneous sounds," but they 

 were purely consecutive. Anyone may satisfy himself 

 that it could only be so, by repeating the experiment with 

 a good violin string. The change of note is simultaneous 

 with the change of nodes in the string. Mere undula- 

 tions, or irregularities of vibration, will not change the 

 note, but injure the quality of the tone. All the curves 

 that a string may describe in vibration have been cal- 

 culated by mathematicians, but only when nodes are 

 formed are they of any importance in music. 



Often have I experimented upon harmonics or natural 

 sounds, in former years, and have watched the changes of 

 node, and have heard the simultaneous change of note. 

 The experiments may be tried by any one who has access 

 to a harpsichord, or a very old grand pianoforte. The 

 tension is too great in modern instruments to allow free 

 play to the string. Raise the damper and strike one of 

 the longest uncovered strings with a hard pianoforte 

 hammer near the bridge. The changes follow in nu- 

 merical order, i, 2, 3, 4, 5, as in the paper before you, and 

 the sounds ascend by octave, fifth, fourth, major and 

 minor third, harmonic seventh, to the third octave, and 

 then to the major and minor tones. It is difficult to 

 attain the highest of these numbers, but the harmonic 

 seventh. No. 7, is readily distinguished by its unusual 

 sound. 



In the yEolian harp the rising pitch of the sounds is 

 caused by the increasing rapidity of the wind ; but it is 

 not so on a pianoforte. It is there due to gradual 

 contractions of the string till it ceases to vibrate, and 

 sinks to rest. The vibrations of a long string are widely 

 discursive, but they become gradually more and more 

 contracted as the nodes of the string diminish in length. 

 The point to be remarked is that the sounds jump over 

 intermediate discords — all are consonances — all aliquot 

 parts : all the sounds are multiples of No. i. It matters 

 not whether it be wind, string, or pipe ; in each of them 

 nature teaches us the scale which is to resolve all musi- 

 cal doubts, all disputed chords. She indicates all the 

 basses for musical intervals, the more remote ones adapted 

 only for melody, and the nearest for consonant harmony. 



To prove the case further we may take an illustration 

 from a pipe. It must not be from those which have 

 lateral openings, or keys, because they shorten the 

 column of air artificially, but from such instruments as 

 the coach horn, or hunting horn, the so-called French 

 horn, or the trumpet without valves. 



The fundamental tone. No. i, or lowest sound it can 

 produce, is derived from the whole column of air within 

 the tube. To produce No. 2 the rapidity of the breath- 

 ing must be doubled, and then the column of air within 

 the horn divides itself into two equal halves, and the 

 sound is an octave above ; so that, if the first note be 

 tenor C with 256 vibrations in a second of time, this 

 treble C requires to be blown at the rate of 256 vibrations 

 to produce it. Here, again, we arrive at the identification 

 of sounds with numbers ; for, just as there is no inter- 

 mediate number between i and 2, so is there no inter- 

 mediate sound between i and 2, its double in vibrations, 

 produced by half its length, upon the horn. The 



