126 



DISCOVERY 



in the jazz band : but the vast bulk of music which 

 has stood the test of time contains an admixture of 

 all three. The question we want to propound here is, 

 have we, in order to explain the rather abrupt change 

 or development in modern music, as opposed to 

 classical, to add another factor to these three, or are 

 we still within their bounds ? Has a fourth dimension 

 invaded the realms of music ? 



The Problem of Tuning 



Before answering this we must consider some of the 

 other facts of the case. The composer, in building his 

 structure out of melody, harmony, and rhythm, is not 

 limited by the innumerable combinations he can make 

 out of these individually permutable elements — he can 

 further diffuse his subjects among all the vast numbers 

 of instruments of transmission, the voices, strings, 

 winds, reeds, and brasses. Now these instruments all 

 have very fundamental differences which arise largely 

 from their different methods of tuning, and from the 

 different overtones which they unavoidably produce. 

 Let us consider for a moment the piano and the violin. 



The tuning of the piano is a compromise by necessity 

 from its rigid construction, for any one note has to do 

 duty in all possible scales. Middle C must be the 

 tonic in the scale of C, the major third in the scale of 

 A[>, the dominant in the scale of F, and so on. It can 

 only be nearly right for all these at once. Simple 

 mathematics will show up this weakness of the piano 

 in a very striking manner. The relationship between 

 notes which combine to form a musical consonance is a 

 simple arithmetic one ; thus octaves bear the ratio 

 2 : I, fifths the ratio 3 : 2. Now, if we start from the 

 bottom A of a piano and impose fifth on fifth, we 

 shall after twelve such intervals of a fifth arrive at 

 top A, a super-octave of the note we started from, 

 thus : 



A E B F:: CS Gi E7 B[7 F C G D A. 



This process gives us an interval ^-.^ which should 



represent a multiple of an octave, i.e. a power of 2. 



But —^ can never equal 2' where x and y are integers, 



so that a piano, by virtue of the problem it has to face, 

 can never be exactly in tune. 



The violin presents quite a different problem, for 

 here the player can alter his C slightly at will, to suit 

 the scale he is playing, and is only prevented by an 

 imperfect ear or faulty technique from being always 

 perfectly and mathematically in tune. His overtones, 

 too, are variable by his use of the bow, while in a piano 



• This interval works out actually to 1297 instead of 128, 

 which would accurately represent seven octaves. 



the string is struck at a fixed point which determines 

 a " node " - for that string once and for all. Thus there 

 arise two different systems of tuning — " equal " (or 

 compromised) tuning and true tuning — which give 

 another variable factor to music ; may we call it the 

 fourth musical dimension ? 



The Trend of Modern Music 



To revert to our modern music. The music of to-day 

 does not probably make much conscious use of this 

 fourth dimension. From a practical point of view 

 music still exists fundamentally on melody, harmon}-, 

 and rhythm, and its chief distinction from classical 

 music arises from progress rather than revolution. 

 Harmonic forms are less restricted, the use of harmony 

 and rhythm is more fully understood — consider Ravel's 

 perfect mastery of rhythm — and modern ears are 

 trained to appreciate more obscure melodies. The 

 modernism of Schonberg and Bax are presaged in 

 Schumann and Bach, but the former can write for a 

 more highly trained audience, can make elisions and 

 " musical assumptions " which would have been un- 

 lawful at an earlier stage in the development of the 

 art. Modern orchestras are more varied. But the 

 finer shades of dissonance are asserting themselves, 

 and composers already appreciate the fact that com- 

 binations which sound right on the organ or piano, 

 sound wrong on the orchestra, and vice versa. Quarter- 

 tones are being talked of, and there can be little doubt 

 that, as ears become more finely developed, the question 

 of tuning \vill be an all-important factor. 



This point may be illustrated roughly in another 

 way. The extraordinarily thrilling effect of a number 

 of violins in unison must have been felt by many — an 

 effect which cannot be produced by any mechanical 

 magnification of one violin. This, then, must be due 

 to the fact that all the violins are slightly out of tune 

 with each other, and the ear appreciates only sub- 

 consciously this fine shading. The curious jar of a 

 piano or an organ in a concerto is another example 

 of the same effect. 



This clash of sounds of different shades rather than 

 of different pitch undoubtedly plays a most important 

 part in the emotional appeal of music, but this brings 

 us to an aspect of music which is quite outside the 

 scope of this article, an aspect which avowedly 

 remained a mystery to Beethoven and Wagner them- 

 selves. For above and beyond the phases which 

 we have attempted to analyse lies the great power 

 which music has in common with all art, of transcend- 

 ing the substance whence it is created, and reaching 

 to the secret places of the heart. 



' A string vibrates in segments, the points of maximum 

 oscillation being divided by points of rest or minimum oscilla- 

 tion called " nodes." 



