34 



NATURE 



\Nov. 8, 1877 



numbers run both ways. They are fractions as to length 

 of tube, and multiples as to vibrations. Again, just as 

 there ts an intermediate number between 2 and 4 (the 

 second octave), so is there one intermediate sound, and 

 one only ; it is No. 3, which is produced by a third of 

 the length of the tube, and is the fifth above No. 2. 

 The fifth and fourth divide the vibrations of the octave 

 equally between them, so that the fifth is three times 

 No. I, and the fourth immediately above it is four 

 times ; — this, notwithstanding the diminution of the 

 musical interval. The names which we have 

 adopted for musical intervals are usually calculated 

 from the keynote, as from C to E a third, from C to F a 

 fourth, and from C to G a fifth, but these names are not 

 real quantities, and are rather confusing than an assist- 

 ance. The octave is not an eighth, but half, and the 

 double octave is not a fifteenth, but a quarter of the 

 length of No. i, and vibrates four times as fast. Octaves 

 are powers of 2, thus 2, 4, 8, 16, and 32 are successive 

 octaves. But the octave 4 to 8 has only four sounds, 

 and these are our major and minor third, and two 

 others, divided by the harmonic seventh, which we do 

 not use. From 8 to 16 are eight sounds, of which we use 

 three, the major and minor tones, and the so-called 

 diatonic semitone, as from B to C. It is really the 

 smallest of the eight tones, and not a semitone. The 

 next octave is from 16 to 32, and that is all of semitones, 

 while 32 to 64 is all of quarter-tones. After that, the 

 octave is divided into eighths, sixteenths, and thirty- 

 second parts of tones, among which it is only useful to 

 note (and that only among musicians and mathemati- 

 cians, that the so-called " comma," having the ratio of 

 80 to 81, is the eighth of a tone above the third of any 

 key — as it is above E in the key of C. We have lately 

 had mathematicians among us who are not fiovacKol, and 

 who have, therefore, proposed to divide an octave into 

 " twelve egna/ semitones." This is pure geometry, and 

 not music. In music there cannot be even two equal 

 semitones within an octave. If our friends will only 

 change their theme from twelve equal semitones into 

 twelve eqically tempered semitones, and give us their 

 experience of the proposed sounds when heard with the 

 bass (which seems not to have yet been taken into ac- 

 count), we shall gladly avail ourselves of their research, 

 on the grounds of modern expediency. In the meantime 

 we must be content to leave the tempering of a scale in 

 the hands of experienced practical men, who, judging 

 only by their ears, as they always will, have hitherto 

 satisfied onr immediate requirements. 



The interval of a fifth is 2 to 3 in ascending and 3 to 

 2 in descending, but, as the figures are usually placed 

 over the upper note in scales, the 3 is written above the 

 2 as in the scale in your hands (the third of them), where 

 it appears over G, referring to C as 2. 



And now for the practical use of these figures, for 

 although the harmonic scale may be referred to, they 

 are most easily remembered. All young pupils are taught 

 the difference between an octave, a fifth, a fourth, and a 

 third, upon the pianoforte, and it is only to associate the 

 numbers with those intervals, to find out the best bass, 

 and every adtnissible bass. All octaves are in the ratio of 

 2 to I, whether it be 4 to 2, 8 to 4, or 16 to 8. All fifths 

 are in the ratio of 3 to 2, all fourths in that of 4 to 3, all 

 major thirds 5 to 4, and minor thirds 6 to 5. 



For instance, in the key of C, C to the F above it is a 

 fourth, and F is No. 4, therefore, the F, two octaves 

 below, is the consonant bass ; whereas, if we strike G 

 with the C above, C becomes the natural bass to that 

 interval. The most consonant basses are always found 

 in the lowest numbers, because the proportion of con- 

 sonant vibrations is there greatest. Thus, from D to G 

 is also a fourth, in the key of C, but the numbers are 9 

 to 12, with a remote bass in C, and there will be 21 vibra- 

 tions, of which only two will coincide in every cycle — i 



of the 8, with i of the 9. Then, the proportion of non- 

 coincidence will be so great as to make the sound un- 

 pleasing to the ear. But as 9 to 12 is in the ratio of 3 to 

 4, we have the best bass in these lowest numbers, and 

 take G. By the various basses to intervals we modulate 

 into other keys. 



At the International Exhibition, held at South Ken- 

 sington in 1862, Mr. Saxe, the eminent inventor of Saxe 

 horns, exhibited an immense horn with an exceedingly 

 long coil of tube, and perhaps standing six feet in height. 

 When asked by the jury the object of this excesssive 

 size and length, he answered, " Cest pour jouer dans le 

 cinquieme ^tage " — " It is for playing in the fifth octave," 

 and he produced with facility any of the sixteen tones and 

 semitones of that octave from it. Half the length of any 

 open conical tube is expended upon its second note, the 

 octave. No human power could have blown the low 

 notes of that horn. Supposing it to have been tuned to 

 the lowest C upon the pianoforte, with thirty-three vibra- 

 tions in a second, as the usual French pitch, it would 

 have had 66, 132, 264, and 528 for its first, second, 

 third, and fourth octaves, while its fifth octave would 

 commence on treble C, with 528, and extend to C above 

 the lines with 1056 vibrations in a second of time. It 

 would thus be within the power of the lungs. He 

 utilized only from the i6th to the 32nd part of his 

 enormous tube, but it gave him the command of the 

 semitones. 



This great incumbrance of length is not necessary in 

 a cylindrical stopped tube. It will take up its own 

 octave according to the ratio of its length to its 

 diameter. We have here an example in a resonating 

 tube invented by Charles Wheatstone just fifty years 

 ago. The lecture for which he invented it was after- 

 wards reported in the twenty-fifth volume of the 

 Quarterly Jour7tal of Science, Literature, and Art, 

 January to March, 1828. Both he and I knew Eulen- 

 stein, an accomplished musician, whose admirable skill 

 in playing upon the Jew's harp was the inducing cause of 

 that particular lecture. Eulenstein had a peculiar facility 

 for contracting and expanding the cavity of his mouth, 

 through the pliability of his very thin cheeks and by the 

 management of his tongue, so that he could fit them for 

 any harmonic note within a certain compass. Wheat- 

 stone then gave the law, that a perfect harmonic scale 

 might be drawn from a single tuning-fork, or from the 

 vibrating tongue of a Jew's harp, by resonators adapted, 

 or adapting themselves, to multiples of the original 

 number of vibrations. " I took," said Sir Charles, " a 

 tube, closed at one end by a movable piston, and placed 

 before its end the branch [or prong] of a vibrating tuning- 

 fork of the ordinary pitch — C. The length of the column 

 of air [within the tube] was six inches. On diminishing 

 the length of the column of air to three inches [by moving 

 up the piston], the sound of the tuning-fork was no longer 

 reciprocated [in unison], but its octave was produced." 

 " It is therefore evident from experiments," says he 

 " that a column of air may vibrate by reciprocation, not 

 only with another body whose vibrations are isochronous 

 [or in unison] with its own, but also when the number of 

 its own vibrations is any multiple of the sounding body." 

 Again, he says : " No other sounds can be produced by 

 reciprocation from a column of air, but those which are 

 perfectly identical with the multiplications of the original 

 vibrations of the tuning-fork or the tongue of the Jew's 

 harp." I produced the original tube in this room about 

 two years ago, to check a recent theory — that reso- 

 nators strengthened the ear, and answered only in 

 unison, and Sir Charles ordered this one for me, made 

 by Mr. Groves, under his own superintendence. The 

 improvement in this is, that the piston now works in a 

 groove and is not liable to stick. Two octaves are pro- 

 duced from the tongue of one Jew's harp as rapidly as 

 the piston can be moved up and down. There is 



