♦ KNOAVLEDGE ♦ 



[Aug. 28, 1885. 



at page 275, distinctly states that it is an organ for the secretion 

 of an oily fluid which the bird spreads over its feathers by the 

 operation of pruning, and speaks of it aa generally understood and 

 without a doubt. Besides which a duck and a dissecting knife are 

 quite sufficient to prove the correctness of the above quotatioca, or 

 even without the knife. G. A. 



ILLUSORY THAXSFOiniATIOX OF INTAGLIO SCULPTURE 

 INTO APPARENT BAS-RELIEF. 



[1897]— A very pretty object for experimenting in tliis illusion 

 is found in the reverse side of the embossed " Queen's head " or 

 postage-stamp impressed on stamped envelopes, and it will be inte- 

 resting to mauy of your readers to examine such an object both by 

 daylight and by artificial light. 



Unless my experience is exceptional, it will be easy for observers 

 to cause the image of the features to appear to assume the more 

 familiar aspect of a convex or rounded surface instead of their 

 true shape of a concave or hollow, and, aided by this illusion, to 

 fancy that the hollowness of the engraved oval lines by which it is 

 surrounded is transformed to an untrue solidity. In all these 

 effects there is a false supposition of the illumination being in the 





nial 



angle, but on the opposite side of a perpendicular to the gem 

 surface of the paper. The human face being such a con- 

 stant object of observation not only during our own lives, 

 but throughout the long experience of the race, while 

 hollow-moulds of it or of other familiar solids are rarely seen, 

 it is much easier to fancy that the hollow cast is solid than 

 the reverse. In fact, though in close proximity to the intaglio, 

 and while it appeared to stand up in beautiful relief, yet it was im- 

 possible to depress mentally the features on a coin laid upon the 

 paper. It will also be observed that contradictory results may be 

 observed by introducing a few indentations in the paper or throw- 

 ing sharp shadows upon it, when, though the over-mastering fami- 

 liarity of the features enables us to hold them to the fancied relievo 

 appearance, yet these mouldings, wrinkles, depressions, &c., retain 

 their true aspects, and wholly or for a time refuse to be transformed, 

 and the surface appears to be lighted from opposite quarters simulta- 

 neously. It would also appear that monocular vision and slightly 

 unfocussed vision are favourable to the inversion of the perception, 

 and that slight crossing of the axes of the eyes, or causing them to 

 be directed to vision "at infinity," helps to form the illusory con- 

 ception. J. Gbeevz Fishek. 



MUSICAL TEMPERAMENT. 

 [1898]— In my last letter I left it to be inferred that to take 

 any notice of such a difference among the ratios of vibrations as 

 that between 8 : 9 and 9 : 10 was something as irrelevant to music 

 considered as an appeal to the understanding, as it would be in 

 criticising an architectural work, to make a microscopic examina- 

 tion of the bricks to see whether they actually corresponded to 

 the beau idial of a brick. But it may now be well to collect and 

 compare a few conflicting anthorities. 



The late General Perronet Thompson contrived a moat ingenious 

 and complicated organ for playing in a variety of keys, with scales 

 justly intoned ; and he wrote a book upon it full of curious infor- 

 mation, which went through several editions. To his feelings, just 

 intonation and equal temperament seem to have been opposed, as 

 truth to falsehood, right to wrong. He says : " If a sculptor, 

 instead of making fingers of the various lengths nature has made 

 them, should determine to make one medium length serve for all, 



this would be a temperament." " It is as if sailors and 



astronomers were invited to employ telescopes fixed to one length 

 for all eyes and distances, like those sold for children in the toy- 

 shops" (General P. Thompson's "Just Intonation," 7th edition, 



p. 45). " The misery of temperament must be got rid of " 



(p. 50). " Singers sing to the pianoforte because they have bad 

 ears, and they have bad ears because they sing to the pianoforte " 

 (p. 94). " The tenth of an inch may in some senses be called 

 small ; but it is a mountain in the edge of a razor or the nose of a 

 profile." . ..." As it is, the singers and violists live in a constant 

 struggle with the temperament, and get rid of it as far as they 

 are able, though a portion sticks by them in the shape of accus- 

 toming their ears to what is out of tune." .... " Instead of being 

 a ' slight imperfection,' it is ruinous to all approaches to perfection ; 

 a man might as well try to shave himself with a handsaw, and 

 persuade himself he liked it " (p. 95). 



General Thompson had his sympathisers. He quotes from De 

 Morgan, " The system of equal temperament is, to my ear, the 

 worst I know of." One enthusiastic young lady said, " But for this 

 organ" (General Thompson's) " I should never have known music. 

 I did not like it, and could not understand it." " It's quite a 



treat! " exclaimed a professional tuner, after tuning the just instru- 

 ment : " When I have tuned my chords iu a tempering organ I have 

 to make them say It'ou-, wow, v:ou-, for the temperament." (" Just 

 Intonation," pp. 96, 47 and preface.) 



To turn away from General Thompsou, I quote llelmholst, in hia 

 chapter xvi., "The System of Keys," of his great work, the " Sen- 

 sations of Tone," &c., as translated by Mr. Ellis:—" There can be 

 no question that the simplicity of tempered intonation is extremely 

 advantageous for instrumental music .... But it must not be 

 imagined that the difference between tempered and just intonation 

 is a mere mathematical subtlety without any practical value. That 



... the early musicians, who were still accustomed to the perfect 

 intervals of vocal music .... felt the same, is immediately seen 

 by a glance at the musical writings of the latter half of the seven- 

 teenth and the earlier part of the eighteenth centuries, at which 

 time there was much diacuesion about the introduction of different 

 kinds of temperament" .... "Whoever has heard the difference 

 between justly-intoned and tempered chords can feel no doubt 

 that it would be the greatest possible gain for a large organ 

 to omit half its stops, which are mostly mere toys, and 

 double the number of tones in the octave, so as to be 

 able, by means of suitable stops, to play correctly in all keys." 

 (N.B. — The translator demurs to the absolute accuracy of this.) 

 " Down to the seventeenth centui-y, singers were practised 

 .... with a degree of care of which we have at present no 

 conception." . . . . " But it is impossible not to acknowledge that 

 at the present day few even of our opera singers are able to execute 

 a little piece for several voices .... unaccompanied .... in a 

 manner suited to give the hearer a full enjoyment of its perfect 

 harmony." Professor Helmholst here instances the trio for the three 

 masks in " Don Giovanni.") . . . . " When the intonation of con- 

 sonant chords ceased to be perfect .... it was necessary .... 

 to have recourse to a frequent employment of harsh dissonances 

 .... to replace the characteristic expression .... These are 



unpleasant symptoms Our last greatest composers, Mozart 



and Beethoven, were yet at the commencement of the reign of 

 equal temperament." (Ellis's " Helmholst," 1875 ; pp. 498, 504, 

 508, 510, 511.) 



#*# An opposed view will require a future letter. 



A. 0. D. 



THE CORRELATION OF THE TWO COMMON CHORDS OF 

 MUSIC, AND THE BINARY CHORDAL STRUCTURE 

 OF THE MUSICAL SCALE, AS ILLUSTRATING EVOLU- 

 TION. 

 [1899]— In letter 1850, reference is made by " Gamma" to the 

 serviceable use which is frequently made of music for the purpose of 

 illustrating the character and mode of nature's processes ; and as, in 

 connection with that subject, I think I am able to add something 

 to the existing store of knowledge, I am induced to offer my humble 



If not well-known to all, it is at least easily found that the first 

 six numbers, viz., 1, 2, 3, 4, 5, 6, represent the ratios of the major 

 chord in a complete form. Take, for instance, the major chord of 

 C ; Ci, C, G, c, e, g. Observe that there are present in it, in 

 ascending order, the harmonic intervals the octave, the fifth, the 

 fourth, the major third, and the minor third. When it is stated 

 that the six numbers above given represent the ratios of the 

 major chord, it is of course understood to be signified that with one 

 vibration of the fundamental note Ci, two vibrations of C coincide; 

 three vibrations of G, four vibrations of c, five vibrations of e, and 



So far we have come upon nothing new. To what follows, how- 

 ever, I think I must ask for a little heedful attention, for though 

 it is not a matter presenting much difficulty, it is one, I have no 

 doubt, now presented to readers fur the first time. 



As the numbers 1, 2, 3, 4, 5, 6, are the ratios of the major chord, 

 so are their reciprocals 1, h, k, i, I, \, the ratios of the minor chord. 

 To instance the minor chord as represented by these ratios, let us 

 take that called the minor chord of A, which appears as e", e. A, E, 

 C, Aj. In this chord the series of the harmonic intervals again 

 appears, but in contrary, that is in descending order, viz., the 

 octave, the fifth, the fourth, the major third and the minor third. 

 The interpretation of these reciprocal ratios in relation to the 

 minor chord is this : the uppermost note e' is the note of origin, 

 or, of the unit, and there coincide with its single vibration one 

 half of a vibration of e, one third of a vibration of A, one fourth of 

 a vibration of E, one fifth of a vibration of C, and one sixth of a 

 vibration of Aj. 



It is thus shown that the relations of both the major and minor 

 chords arise equally from the simplest origin, that is, the unit, the 

 single vibration; and that these relations with this unit are of 

 equal simplicity. And we here observe that of these two chords 



