Mat 25, 1883.] 



♦ KNOV/LEDGE ♦ 



3K 



follows from the law that the aquares are proportional to the odd 

 numbers. 



Let it be required to extract the square root of any number — say, 

 "I'lU. Square a number that is somewhere near it — say, 90. This 

 ves 8100. Then square the next higher number, 91. We obtain 

 ^.'Hl ; difference by subtraction, 181. 



N'ow subtract in turn all the odd numbers from 181 upwards 

 im our orifpnal square 9004, till we obtain a remainder of 8100, and 

 ■' I + number of subtractions) will be the root required. If the 

 uare be not a perfect one, we, of course, only obtain the nearest 

 Jole number by subtracting till the remainder is nearest to that 

 • require. 



WoRKixG OF Problem. 

 90^ = 8100, 91= = 8281, 8281-8100 = 181 



9601- (181 + 183 + 185 + 187 + 189 + 191 + 193 + 195) 

 = 8100 

 -■ . sqroot of 9001 = 90 + 8 = 98. 



T. U. B., Lieut. R.A., F.U.A.S. 



MUSICAL QUESTION, 808. 



[832] — Mr. Walter Jones and his musical friends fall into the 

 common error of confounding musical instruments with music. 

 Tliere is no difference in the same chord if struck in two different 

 keys, C major or C sharp major — naturally. There are but two 

 classes of instruments in this world that give out true music, and 

 they are the vocal organs of men, women, and children, and the 

 fiddle family. The over-lauded organ and the modern pianoforte 

 have much to answer for in destroying our n.itural ear for music. 

 In my boyhood I used to run after a man who played the triangles 

 with his feet, the cymbals with his knees, the big drum with his 

 «lbowg, the accordion with his hands, the pandean pipes with his 

 mouth, and a bunch of bells with his head — sis instruments at once, 

 no doubt all out of tunc. Not so bad. But pretty much the same is 

 the pianoforte, which endeavours, by the touch of one performer, 

 to represent the effect of an orchestra, and, by an arrangetnent of 

 black and white keys, to produce all the chords and modulations of 

 nature. 



JIusic consists simply of eight sounds, the first and last being 

 alike, separated by seven intervals — three great tones, two small 

 tones, and two so-called semitones. 



Expressed in figures, the difference between any sound and its 

 octave may be represented by 474. 



From 1 to 2 80 deg; 



2 to 3 72 



3 to 4 45 



4 to 5 80 



5 to 6 72 



G to 7 80 



7 to 8 45 



474 



With the voice or violin, where there are no key-boards and fixed 

 sounds to bother yon, it is quite possible, when the brain and 

 stomach are in fair working order, to sing or play any passage in 

 tune. But, with the pianoforte tuned (or, one might say, mistuned) 

 to equal temperament, it is impossible for Mr. Walter Jones to play 

 cnt; chord in tune, no matter whether he commences on C or 

 C sharp. 



In equal temperament, the scale is divided into twelve equal 

 eeraitones ; that is to say, five whole tones, all alike, and two semi- 

 tones, thus : — 



Eqiuil N.tnn.1. 



Temperament. 



From 1 to 2 79 instead of 80 



2 to 3 79 „ 72 



3 to 4 39i „ 45 



4 to 5 79 „ 80 



5 to 6 79 „ 72 



6 to 7 79 „ 80 



7 to 8 39i „ 45 



474 474 



To place the matter before you at a glance, I send you enclosed 

 a long piece of paper showing two octaves of natural sounds side 

 by side with two octaves of equal tempered pianoforte sounds, and 

 a; short piece of paper with one octave of nattire's sounds drawn 

 to the same scale. 



If you use the small piece as a sliding sqale by laying it on the 

 larger, you will see at a glance where the sounds don't fit. This 

 will perhaps explain why the verse parts of a cathedral anthem 

 sound like heavenly music when unaccompanied by the organ, 

 why " Cast thy bsrden upon the Lord," in " Elijah," when sung 



by such artists as Titiens, Dolby, Reeves, and Santley, causes a 

 feeling of satisfaction vastly different to that of, say a German 

 band, or an average congregation when each one fills up the in- 

 tervals in the scale according to his temperament. 



Why Mr. Jones finds the common chord of C more grateful to 

 his ear than that of C sharp, is because the tuner lias shown hia 

 judgment in throwing the necessary errors into those keys where 

 they are less likely to be discovered. J. H. Chattertox. 



[833]— Your correspondent Walter Jones (808) may be more 

 Burpriscd to learn that no two keys on the piano have the same 

 sound. Each has its own distinct individuality, and arises from the 

 fact of the great difference between the enharmonic notes — i.e. 

 G sharp is not A flat, F sharp is not G flat, &c. ; but on the piano 

 each black key has to serve the double purpose, therefore is tem- 

 pered, or tuned in between, which makes the intervals, thirds and 

 fifths of each key all different in distance. Tliis is so equally dis- 

 persed by good tuners as to be almost imperceptible, but is quite 

 suflicient to account for the fact. 



While on the subject, may I point out another curious fact. The 

 notes of the chords of the sharp sixth and flat seventh on the piano 

 are identical, and yet have a different sound. The following is an 

 illustration ; — 





(m^^^mm^^^ 



The chords above the cross are the same on the piano, and yet 

 produce a totally different effect. The reasons are they both occur 

 in different progressions and have both different resolutions. 



William Godhex, Prof cssor of Harmony. 



[834] — This question has been considered a good many times. 

 even amongst musicians, and it has been asked, consequently, Why 

 do composers often write their pieces in such a lot of flats or 

 sharps, when the more simple keys would answer the same purpose, 

 and make a piece of music so much less diflicult for the general 

 public ■'" Now, I am of opinion that, speaking of the pianoforte, 

 the chord of C shar]', or a piece written in that key, would produce 

 exactly the same effect upon an audience as if played in C natural, 

 because in our modem system of keys the relations of one key are 

 the same as those of another. 



Supposing you had two pianofortes, the second tuned a semitone 

 higher than the first, then C natural on the one would sound 

 C sharp on the other, and a piece in C played on the iirst, C sharp 

 on the second, and rice versd. 



But as soon as the orchestra is concerned, matters alter alto- 

 gether, where there are combinations of a very different 

 character to those on the piano. On a stringed instrument, 

 for instance, the chord (1) would, on account of the empty 

 strings and their harmonies, give a very different effect — or 

 Klang-tint, as Helmholt'. calls it— to chord 2, where every string 



is covered by the fingers and the vibrations interrupted. Here the 

 tone not only alters as to pitch, but also as to intensity and quality. 

 Hector Berlioz has, in hi.^ celebrated " Instrumentations-lelire," 

 assigned special and individual characters to each major and minor 

 key ; but, in a great measure, this will be more or less a matter of 

 subjective feeling — on the piano, absolutely imaginary. 



A nocturne of Chopin, written in D flat and played in D natural, 

 loses certainly all poetical expression for the performer ; but for 

 an audience 'unaware of the transposal, I dare say none— or very 

 few, perhaps — wonic! be cognisant of the fact. This would simply 

 prove what you wish to know, viz., that there is in reality no 

 difference between the chord of C and C sharp on the pianoforte, 

 except, of course, that of pitch. Gcstave Wolff. 



THE DIVIDED SKIET. 



[835]— As the divided skirt was brought out by the Rational 

 Dress Society, will yon allow me to say that it is meant to be worn 

 as the dress itself, and not merely as an under petticoat ? Wearing 



