Page images
PDF
EPUB

ed, ƒ ft 9 g, a g‡, bat. Hence it is easy to compare the different effects of the two systems as applied to the same quality of tone, for harmonies which are common to both. Having two concertinas so tuned, and a third tuned to just intervals, I have been able to make this comparison, and my own feeling is that the Mesotonic is but slightly, though unmistakeably, inferior to the Just, and greatly superior to the Hemitonic.

There are two other points in which the complete Mesotonic system possesses advantages over the Hemitonic. The Mesotonic VIIth is rather flat, but by using the flat VIIIth in its place, when the harmony will allow, the effect of an extremely sharp VIIth is produced, which is sometimes desirable in melodies. Thus log Mesotonic VIII)=28195, which is sharper even than log Pythagorean VII=27840. The ordinary and flatter VIIth can be used when necessary for the harmony. Again, by using the German sharp VIth in place of the dominant 7th, that is, by using the chords GB Dɔ e, Dɔ FAɔ b, Aɔ C Eb ft, Eɔ G Bɔ c‡, Bɔ D F g➡, FAC, CEG a, GB D e‡, D F- A b‡, A CEƒ ×, E G‡ B c ×, B D F 9 ×, in place of G B D ƒɔ, Dɔ F Aɔcɔ, Aɔ C Eɔ gh, Eɔ G Bɔ ɖɔ, B D F, FAC, CEG, GB Dƒ, DFA c, A C Eg, E G B d, B D F a, when the progression of parts will allow, an almost perfect natural seventh, better than that obtained by using the corresponding just tones, will result, producing beautiful harmony; for log Mesotonic VI=24228, log=24304, and log Just VI 24497. The ordinary sharper 7th can be used when necessary. Neither of these effective substitutions is possible on the Hemitonic system.

=

Considering that singers and violinists naturally intone justly (Delezenne, loc. cit.), and that the interval errors of the Mesotonic system seldom exceed the natural errors of intonation which may be expected from the inability of the ear to appreciate minute distinctions of pitch, it appears desirable to tune harmoniums at least, and perhaps organs, mesotonically. Except as an instrument for practising singers, however (for which purpose it would be superseded by a Mesotonic harmonium), it would be unnecessary to alter the Hemitonic tuning and arrangement of the piano. But it would be best to teach the Mesotonic intonation on the violin in preference to the Hemitonic, as proposed by Spohr *. As, however, it would be useless to tune mesotonically with only 12 tones to the octave, it is necessary to have some practical arrangement for 27, 24, or 21 tones at least. I propose the following plan for 24 tones, and as these are exactly twice as many as on pianos, &c. of the usual construction, I call my arrangement the

"Unter reiner Intonation wird natürlich die der gleichschwebenden [Hemitonic] Temperatur verstanden, da es für moderne Musik keine andere giebt. Der angehende Geiger braucht auch nur diese eine zu kennen; es ist deshalb in dieser Schule von einer ungleichschwebenden [defective equal, or unequal] Temperatur eben so wenig die Rede, wie von kleinen und grossen halben Tönen [c c‡ & c d = Bc, that is, ], weil durch beides die Lehre von der völlig gleichen Grösse aller 12 halben Töne nur in Verwirrung gebracht wird."-Violinschule, p. 3.

=

[subsumed][merged small][graphic][subsumed][subsumed][subsumed][subsumed][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Let the black and white manuals remain as at present, and let a yellow manual, of the same form as the black, be introduced between B and C, and E and F. Cut out about the middle third of each black and yellow manual, up to half its width, on the right side only, and introduce a thin red manual rising as high above the black or yellow as these do above the white. Over G, A, and D, each of which lies between two black manuals, introduce three yellow metal manuals (lacquered or aluminium-bronze) shaped like flute keys, and standing at the height of a red manual above the white one, which can therefore, when necessary, be reached below it. The 7 white manuals are the naturals; the 5 black manuals are the 5 usual sharps, c d f ga; the 2 long yellow manuals are the unusual sharps et 6, and the 3 metal yellow manuals are the double sharps fx gx cx; and the 7 thin red manuals are the 7 flats, c, d, e, f, g, ab, b. The shapes of the red and metal manuals were suggested by those of General T. Perronet Thompson's quarrils and flutals. The 24 levers opening the valves on the organ or harmonium would lie side by side, being made half the width of those now in use, and metallic, if required for strength. The organ pipes or harmonium reeds would be arranged in two ranks of 12 for each octave, the first rank containing the 7 naturals and 5 usual sharps, and the back rank containing the 7 flats, 2 unusual and 3 double sharps. The use of this finger-board is accurately pointed out by the ordinary musical notation which distinguishes the sharps from the flats, and is therefore in no respect adapted to the Hemitonic fusion of sharps and flats into mean semitones.

Side Elevation.

On the Calculus of Symbols.-Fourth Memoir.

With Ap

plications to the Theory of Non-Linear Differential Equations." By W. H. L. RUSSELL, Esq., A.B. Received July 31, 1863*.

In the preceding memoirs on the Calculus of Symbols, systems have peen constructed for the multiplication and division of non-commutative ymbols subject to certain laws of combination; and these systems suffice for linear differential equations. But when we enter upon the consideration of non-linear equations, we see at once that these methods do not apply. It becomes necessary to invent some fresh mode of calculation, and a new notation, in order to bring non-linear functions into a condition which admits of treatment by symbolical algebra. This is the object of the following memoir. Professor Boole has given, in his Treatise on Differential Equations,' a method due to M. Sarrus, by which we ascertain whether a given non-linear function is a complete differential. This method, as will be seen by anyone who will refer to Professor Boole's treatise, is equivalent to finding the conditions that a non-linear function may be externally divisible by the symbol of differentiation. In the following paper I have given a notation by which I obtain the actual expressions for these conditions, and for the symbolical remainders arising in the course of the division, and have extended my investigations to ascertaining the results of the symbolical division of non-linear functions by linear functions of the symbol of differentiation.

Let F(x, y, Y1, Y29 Yз

[ocr errors]

y) be any non-linear function, in which Y1, y2 y3... y, denote respectively the first, second, third, .... nth differential of y with respect to (x).

Let U, denote fdy,, i.e. the integral of a function involving x, y, y1, Y2.... with reference to y, alone.

d

Let V, in like manner denote when the differentiation is supposed

dy,

effected with reference to y, alone, so that V, U,F=F.

The next definition is the most important, as it is that on which all our subsequent calculations will depend. We may suppose F differentiated (m) times with reference to yn Yn-1, or yn-2, &c., and Yn, Yn-1, or Yn-29 &c., as the case may be, afterward equated to zero. We shall denote this entire process by Zm), Z), Z", &c.

The following definition is also of importance: we shall denote the ex>pression

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Having thus explained the notation I propose to make use of, I proceed d to determine the conditions that F may be externally divisible by

[ocr errors]

or,

in other words, that F may be a perfect differential with respect to (x). It will be seen that the above notation will enable us to obtain expressions for the conditions indicated by the process of M. Sarrus.

It is obvious that if we expand F in terms of y,, in order that the sym

d

bolical division with reference may be possible, the terms involving

yn, yn, &c. must vanish.

dx

[blocks in formation]

Z-1ZF-Z-Y-1U-1ZF+Yn-1 (Z1-„ZF-Z-1Y-1U-1Z',F).

Now

d

=

dxUn-2(Z'n-1Z, F—Z'n-1Y2-1U„-1Z'„F) =

Y-2U-2Z-1ZF-Y-2U-Z-1Y-1U-1ZF
+Yn-1(Z-1ZF-Z-Y-1U-1Z',F);

and if R be the second remainder, we find

R2=Z-1ZF-Z-1Yμ-1Un-1Z'„F

-Y-2Un-2Zn-1ZF+Y-2U2-2Z'n-Y-1U-Z'„F:

the next condition is V-2R2=0, and therefore

R2=Z-2Z9-1Z%F

-Z-Z-1Y-1U-12',F-Z-Y-2Un-2Z'-1ZF

+Z-2Yn-Un-2Z'n-1Yn-1Un-1Z,F

+Yn-2(Zn-2Z-1ZF-Z-Z-1Y-1U-1Z'„F

-Z'n-2Y-Un-2Zn-1ZF+Z-2Y-2Un-2Z'n-1Y-1U-1Z',F).

But

d

dx

{ Un-3(Z'n-2Z-1ZF-Z'n-2Z-1Y2-1Un-1Z'„F

— Z' „-2Yn-2Un-:Z'n-1Z«F + Z'n-2Yn-2Un-2Z'n-1 Yn-1Un-1 Z',F) }

=Yn-3Un-3 (Z'n-2Zn-1ZF-Z'n-2Z-1Y-1U2-1Z'„F

-Z'n-Y-U-Z'n-1ZF+Z'n-2Yn-Un-2Zn-1Y-1U-1Z',F)

+Yn-2(Zn-2Z-1ZF-Z-Z-1Y-1U-1Z'„F

-Z'n-2Y2-2Un-2Z'n-1ZF+Z'n-2Yn-2Un-2Z'n-1Yn-1Un-1Z'„F);

whence we find

R3-Z-Z-1ZF-Z-Z-1Y-1U-1ZF

-Z-2Y-2Un-2Zn-1ZF+Z-2Yn-2Un-2Z'n-1Y2-1U-1Z'F -Y-3U-3Z-Z-1ZF+Y-3Un-3Z'n-2Z-1Yn-1U-1Z',F +Yn-3Un-3Z'n-2Yn-2Un-2Z'n-1Z F

-Y-3Un-3Z'n-2Yn-2Un-2Z'n-1Yn-1U-1Z'F.

Hence we infer the following rule for the formation of R..
Construct the term

[ocr errors][merged small][merged small][merged small]

In any symbol Z'm the accent may be changed into a zero, i. e. we may at pleasure substitute Z, anywhere for Z; but in such case the previous symbolical factor Ym-1Um-1 must be omitted. This term is positive or negative according as the symbol Z' occurs an even or an odd number of times in it; the aggregate of all the terms thus formed constitute the remainder R,, and the conditions that F may be externally divisible by

[blocks in formation]

V1F=0, V-R1 =0, V,-2R2=0, V1-3R3=0, &c.

d

dx

We shall now investigate the conditions that +P may externally divide F where P is a function of (x) and (y).

As before, V,F=0, and in consequence

[ocr errors][merged small][subsumed][merged small][merged small]

Hence we shall have

R1=ZF-Y-1U-1Z'„F-PU-1ZF.

We have V-R1=0 in order that this remainder may contain only the

[blocks in formation]

Yn-19 and

R-Z-ZF-Z-Y-U-Z',F-Z-PU-ZF
+Yn-1(Zn-1ZF-Z-1Y-1Un-1Z',F-Z-1PU-1Z',F),

« PreviousContinue »