I venture to conclude that the typical anatomical arrangement of a nervous mechanism is not a cord with two ends-a point of origin and a terminal extremity, but a cord without an end—a continuous circuit. The peculiar structure of the caudate nerve-cells, which I have described, renders it, I think, very improbable that these cells are sources of nervous power, while, on the other hand, the structure, mode of growth, and indeed the whole life-history of the rounded ganglion-cells render it very probable that they perform such an office. These two distinct classes of nerve-cells, in connexion with the nervous system, which are very closely related, and probably, through nerve-fibres, structurally continuous, seem to perform very different functions,—the one originating currents, while the other is concerned more particularly with the distribution of these, and of secondary currents induced by them, in very many different directions. A current originating in a ganglion-cell would probably give rise to many induced currents as it traversed a caudate nerve-cell. It seems probable that nerve-currents emanating from the rounded ganglion-cells may be constantly traversing the innumerable circuits in every part of the nervous system, and that nervous actions are due to a disturbance, perhaps a variation in the intensity of the currents, which must immediately result from the slightest change occurring in any part of the nerve-fibre, as well as from any physical or chemical alteration taking place in the nerve-centres, or in peripheral nervous organs. XXIII. "On the Physical Constitution and Relations of Musical Chords." By ALEXANDER J. ELLIS, F.R.S., F.C.P.S.* Received June 8, 1864. When the motion of the particles of air follows the law of oscillation of a simple pendulum, the resulting sound may be called a simple tone. The pitch of a simple tone is taken to be the number of double vibrations which the particles of air perform in one second. The greatest elongation of a particle from its position of rest may be termed the extent of the tone. The intensity or loudness is assumed to vary as the square of the extent. The tone heard when a tuning-fork is held before a proper resonance-box is simple. The tone of wide covered organ-pipes and of flutes is nearly simple. Professor G. S. Ohm has shown mathematically that all musical tones whatever may be considered as the algebraical sum of a number of simple tones of different intensities, having their pitches in the proportion of the numerical series 1, 2, 3, 4, 5, 6, 7, 8, &c. Professor Helmholtz has established that this mathematical composition corresponds to a fact in nature, that the ear can be taught to hear each one of these simple tones separately, and that the character or quality of the tone depends on the law of the intensity of the constituent simple tones. These constituent simple tones will here be termed indifferently partial * The Tables belonging to this Paper will be found after p. 422. tones or harmonics, and the result of their combination a compound tone. By the pitch of a compound tone will be meant the pitch of the lowest partial tone or primary. When two simple tones which are not of the same pitch are sounded together, they will alternately reinforce and enfeeble each other's effect, producing a libration of sound, termed a beat. The number of these beats in one second will necessarily be the difference of the pitches of the two simple tones, which may be termed the beat number. As for some time the two sets of vibrations concur, and for some time they are nearly opposite, the compound extent will be for some time nearly the sum, and for some time nearly the difference of the two simple extents, and the intensity of the beat may be measured by the ratio of the greater intensity to the less. But the beat will not be audible unless the ratio of the greater to the smaller pitch is less than 6:5, according to Professor Helmholtz. This is a convenient limit to fix, but it is probably not quite exact. To try the experiment, I have had two sliding pipes, each stopped at the end, and having each a continuous range of an octave, connected to one mouthpiece. The tones are nearly simple; and when the ratio approaches to 6:5, or the interval of a minor third, the beats become faint, finally vanish, and do not reappear. But the exact moment of their disappearance is difficult to fix, and indeed seems to vary, probably with the condition of the ear. The ear appears to be most sensitive to the beats when the ratio is about 16:15. After this the beats again diminish in sharpness; and when the ratio is very near to unity, the ear is apt to overlook them altogether. The effect is almost that of a broken line of sound, as the spaces representing the silences. Slow beats are not disagreeable; for example, when they do not exceed 3 or 4 in a second. At 8 or 10 they become harsh; from 15 to 40 they thoroughly destroy the continuity of tone, and are discordant. After 40 they become less annoying. Professor Helmholtz thinks 33 the beat number of maximum disagreeableness. As the beats become very rapid, from 60 to 80 or 100 in a second, they become almost insensible. Professor Helmholtz considers 132 as the limiting number of beats which can be heard. They are certainly still to be distinguished even at that rate, but become more and more like a scream. Though ƒ and g should give 198 beats in a second if c=264, and the interval is that for which the ear is most sensitive, I can detect no beats when these tones are played on two flageolet-fifes. Hence beats from 10 to 70 may be considered as discordant, and as the source of all discord in music. Beyond these limits they produce a certain amount of harshness, but are not properly discordant. When the extent of the tones is not infinitesimal, Professor Helmholtz has proved that on two simple tones being sounded together, many other tones will be generated. The pitch of the principal and only one of these combinational tones necessary to be considered, is the difference of the pitch of its generating tones. It will therefore be termed the differential tone. Its intensity is generally very small, but it becomes distinctly audible in beats. The differential tone is frequently acuter than the lower generator, and hence the ordinary name "grave harmonic" is inapplicable. As its pitch is the beat number of the combination, Dr. T. Young attributed its generation to the beats having become too rapid to be distinguished. This theory is disproved, first, by the existence of differential tones for intervals which do not beat, and secondly, by the simultaneous presence of distinct beats and differential tones, as I have frequently heard on sounding f1, f, or even f2, f together on the concertina, when the beats form a distinct rattle, and the differential tone is a peculiar penetrating but very deep hum. The object of this paper is to apply these laws, partly physical and partly physiological, to explain the constitution and relations of musical chords. It is a continuation of my former paper on a Perfect Musical Scale*, and the Tables are numbered accordingly. Two simple tones which make a greater interval than 6:5, and therefore never beat, will be termed disjunct. Simple tones making a smaller interval, and therefore generally beating, will be termed pulsative. The unreduced ratio of the pitch of the lower pulsative tone for which the beat number is 70 to that for which it is only 10, will be termed the range of the beat. The fraction by which the pitch of the lower pulsative tone must be multiplied to produce the beat number, will be termed the beat factor. The ratio of the pitches of the pulsative tones, on which the sharpness of the dissonance depends, will be termed the beat interval. A compound tone will be represented by the absolute pitch of its primary and the relative pitches of its partial tones, as C (1, 2, 3, 4, ....). .). As generally only the relative pitch of two compound tones has to be considered, the pitches will be all reduced accordingly. Thus, if the two primaries are as 2: 3, the two compound tones will be represented by 2, 4, 6, 8, 10, ...., and 3, 6, 9, 12, 15 .... The intensity of the various partial tones differs so much in different cases, that any assumption which can be made respecting them is only approximative. In a well-bowed violin we may assume the extent of the harmonics to vary inversely as the number of their order. Hence, putting the extent and intensity of the primary each equal to 100, we shall have, with sufficient accuracy Harmonics... 1, 2, 3, 4, 6, 7, 8, 9, 10. 3, It will be assumed that this law holds for all combining compound Proceedings of the Royal Society, vol. xiii. p. 93. The following misprints require correction:-P. 97, line 7 from bottom, for c2 read b. Table I., p. 105, diminished 5th, example, read f: B; minor 6th, logarithm, read ·20412; Pythagorean Major 6th, read 27: 16, 33: 21; Table V., col. VI., last line, read te tg th tones, the intensity of the primary in each case being the same. The results will be sufficient to explain the nature of chords on a quartett of - bowed instruments, but may be much modified by varying the relative intensities of the combining tones. On examining a single compound tone, we may separate its partial tones into two groups: the first disjunct, which will never beat with each other; the second pulsative, which will beat with the neighbouring disjunct tones. Thus When any compound tone therefore developes any of the harmonics above the 6th, there may, and probably will, be beats, producing various degrees of harshness or shrillness, jarring or tinkling. These, however, are all natural qualities of tone, that is, they are produced at once by the natural mode of vibration of the substances employed. But if we were to take a series of simple tones having their pitches in the above ratios, and to vary their intensities at pleasure, we should produce a variety of artificial qualities of tone, some of which might be coincident with natural qualities, but most of which would be new. This method of producing artificial qualities of tone is difficult to apply, but has been used with success by Professor Helmholtz to imitate vowel-sounds, &c. If, however, instead of using so many simple tones, we combine a few compound tones, the pitches of which are such that their primaries might be harmonics of some other compound tone, then the two sets of partial tones will necessarily combine into a single set, which may, or rather must be considered by the ear as the partial tones of some new compound tone, having very different intensities from those possessed by the partial tones of either of the combining compound tones. That is, an artificial quality of tone will have been created by the production of these joint harmonics. Such an artificial quality of tone constitutes what is called a musical chord. The two or more compound tones from which it is built up are its constituents. The primary joint harmonic is the real root or fundamental bass of the chord, which often differs materially from the supposititious root assigned by musicians. If the primaries of the constituents are disjunct, and all their partial tones are disjunct, then the joint harmonics will be also disjunct, unless some pulsative differential tones have been introduced. If, however, the constituents have pulsative partial tones, the chord will also have them. Such chords, which are generally without beats, and are only exceptionally accompanied by beats, are termed concords, and they are unisonant or dissonant according as the beats are absent or present. Their character therefore consists in having the pitches of their constituents as 1, 3, 5, or as these numbers multiplied by various powers of 2, that is, as 1, 3, 5, or their octaves. If any of the constituents is pulsative the chord will generally have beats, but may be exceptionally without beats. Such chords are termed discords. Their character consists in having two or more of the pitches of their constituents as 1, 3, 5, or their octaves, and at least one of them as 7, 9, or some other pulsative tones, or their octaves. What pulsative tones should be selected depends on the sharpness of the dissonance which it is intended to produce, and therefore on the interval of the beat which is created, Thus, since 7: 6=1·16667 and 8: 7=1·14286 are both near the limit 6:5=1.2, the discord arising from 7 would be slight. Some writers have even considered the chord 1, 3, 5, 7 to be concordant. Again, 9:8-1.125 is rather rough, but 10:9-111111 is much rougher. Hence, if 9 is introduced, 10 should be avoided, that is, the octave of 5 should be omitted, which generally necessitates the omission of 5 itself, as in the chord 1, 3, 9. But 11:10=1'1 and 12:11=1·09091 are both so sharply dissonant, that if 11 is used neither 10 nor 12 should be employed. Now 10 is the octave of 5, and 12 is both the 3rd harmonic of 4 and the 4th harmonic of 3, and would therefore be produced from 3 and 4. Hence the use of 11 would forbid the use of 3, 4, and 5, that is, of the best disjunct tones. Hence 11 cannot be employed at all. Similarly, 13: 12=1.08333 and 14:13=1.07692 are both extremely harsh. The latter is of no consequence, because 7 can be easily omitted. But even 15: 13=1•15384 is more dissonant than 7: 6. Hence 13 would also beat with the harmonics of 3, 4, and 5. Consequently 13 must be also excluded. All combinations in which the differential tones 11 and 13 are developed will also be extremely harsh. As we therefore suppose that 14: 13=1.07692 never occurs, and as 14:12=7: 6, the mildest of the dissonances, 14 may be used if 15 is absent, and thus 15:14=1·07143 avoided. When 14 and 15 are developed as harmonics of 7 and 5, and not as the primaries of constituent tones, their intensity will be so much diminished that the discord will not generally be too harsh. When 15 is used as a constituent, 14 and 16 should be avoided; that is, 7, and 1, 2 and 4, of which 14 and 16 are upper harmonics, should be omitted to avoid 15: 14=1·07143 and 16:15=1·06667, which may be esteemed the maximum dissonance. By omitting 16 and 18, and thus avoiding 17: 16=1·0625 and 18:17= 105882 (that is, by not using 4, 8, or 9 as constituent tones), 17 becomes useful; for 17:15=1·13333 is milder than 9:8=1·125, which is by no means too rough for occasional use. The other pulsative harmonics, which are represented by prime numbers, are not sufficiently harmonious for use; but those produced from 2, 3, 5 (such as 25, 27, 45) may be sometimes useful, provided that the tones with which they form sharp dissonances are omitted. The result of the above investigation is that the only pulsative tones suitable for constituents are 7, 9, 15, 17, 25, 27, 45, and their octaves. |