Hence we have altogether for the inferior term, log, mod. = 2-1838; amp. = + 193o 45'.5 Hence reducing each imaginary result from the form p (cos + √ 1 sin ) to the form a+-1b, we have for the final result, obtained from the descending series: From superior term - 14:98520+ 43-81046-1;-45 43.360 – 8 92767 √ – 1 Had the asserted discontinuity in the value of the arbitrary constant not existed, either the inferior term would have been present for amp. 90°, or it would have been absent for amp. x = 150o, and we see that one or other of the two results would have been wrong in the second place of decimals. In considering the relative difficulty of the calculation by the ascending and descending series, it must be remembered that the blanks only occur in consequence of the special values of the amplitude of r chosen for calculation: for general values they would have been all filled up by figures. Hence even for so low a value of the modulus of x as 2 the descending series have a decided advantage over the ascending. VII. On the Beats of Imperfect Consonances. By AUGUSTUS DE MORGAN, F.R.A.S. of Trinity College, Professor of Mathematics in University College, London. [Read Nov. 9, 1857.] THE subject of this paper was treated in full, for the first and only time, by Dr Robert Smith, in the two editions of his Harmonics (Cambridge, 1749, 8vo.; London*, 1759, 8vo.). The results are the same in both editions, but the improvements of the second edition add considerably to the learned obscurity in which the subject is involved. Dr Smith presents, so tuners. far as I know, the strongest union of the scholar, mathematician, physical philosopher, and practical musician, who ever treated of mathematical harmonics: and his book is not only the most obscure and repulsive in its own subject, but it would be difficult to match it in any subject. The consequence has been that the point in which Robert Smith made an important addition to acoustics has been little more than a result in the hands of some of the organDr Young certainly did not understand Smith's theory. He was also a remarkable union of the scholar, mathematician (a character in which he deserves to stand much higher than he is usually placed), and physical philosopher: and was a successful student in music; but he wanted a musical ear (Peacock's Life, pp. 59, 79, 81). I have my doubts whether Robison had read more of Smith's theory than its results. For myself, I made out what ought to have been the theory from the formula, and then was successful in mastering Smith's explanations. Before proceeding to the subject, I make some remarks upon the method of dividing the octave. Should this paper fall into the hands of any mathematician unused to musical measurement, he must be informed that proximity and longinquity are measured by ratio, not by difference. Thus notes of p and q vibrations per second are at the same interval as notes of kp and kq vibrations per second, be k what it may. Consequently, an interval remains constant, not with p q, but with log p-log q. The octave of any note, which has with that note a sort of identity of effect which no words can describe, makes two vibrations while the note makes one vibration. Any note makes p vibrations while its upper octave makes 2p vibrations: hence log 2p log p, or log 2, is the measure of every interval of an octave. It is worthy of note that at this period the book bears the name of the place where it is printed, not of the place where the publisher sells it. Both these editions are printed for Cambridge publishers (the Merrills). + So long as unequal temperament was in use, and even VOL. X. PART I. now when it is adopted, the beats were and sometimes are used in tuning: but when equal temperament is required (and this system has gained ground rapidly) the tuners have nothing to do with beats, except to get perfect octaves by destroying them. I speak of the organ, and of this country. 17 Many writers, from Sauveur downwards, have seen the convenience of using the figures of *3010300, the common logarithm of 2. Thus Sauveur, for one method, divides the octave into 301 parts, so that if the higher of two notes make m vibrations while the lower makes n, the integer in 1000 (log mlog n) is the number of subdivisions contained in the interval, quam proximè. The tuner of the pianoforte is required to estimate half a subdivision: for the fifth of equal temperament is 175.60 subdivisions, and the perfect fifth is 176'09 subdivisions. Even in practice, then, a smaller subdivision is required: and theory will hardly be content without the representation of the 50th part of the smallest interval in common practical use. I should propose to divide the octave into 30103 equal parts, 2508-6 to a mean semitone. Each part may be called an atom; and we have the following easy rules, which suppose the use of a table of five-figure logarithms. To find the number of atoms in the interval from m to n vibrations per second, neglect the decimal point in log mlogn, or in log n log m, whichever is positive. To find the ratio of the numbers of vibrations in an interval of k atoms, divide by 100,000, and find the primitive to the result as a logarithm. To find the number of mean semitones in a number of atoms, divide the number of atoms by log 2 × 100000, which may be done thus. Multiply by four; deduct the 300th part of this 1 12 - product and its 10,000th part, adding one-ninth of this 10,000th part; make four decimal places, and rely on three. Thus a perfect fifth has 100,000 (log 3 – log 2) atoms, or 17609, which multiplied by 4 is 70436. The 300th part of this is 235, and the 10,000th part is 7, of which one-ninth may be called 1. And 70436 241 is 70195, whence 7:0195, say 7:020, is the number of mean semitones in a perfect fifth. To find the atoms in a number of mean semitones, multiply by 10,000; add to the result its 300th part and its 10,000th part, and divide by 4. Thus 12 mean semitones gives 120,000 increased by 400 + 12, or 120412, which divided by 4 gives 30103. This rule is as accurate as the value of log 2; the one which precedes is a near approximation. Both are consequences of the equation Dr Smith found that the D of his organ, the first space below the lines of the treble, gave 254, 262, 268, double vibrations† in the common temperatures of November, September, and August. Euler, and after him Lambert, suggested the use of acoustical logarithms; and proposed systems, of which the bases are 2 and 12/2. Prony gave both tables in his Instructions Élémentaires sur les moyens de calculer les intervalles musicaux, Paris, 1832, 4to. The second table shows at once, in log m-logn, the number of mean semitones in the interval whose ratio of vibrations is m: n. Prony has also calculated, but I cannot give the reference, a table of logarithms to the base which gives the number of commas in m :n, by log m-logn. The atom which I have proposed, which is the 540th part of a comma, gives the commas by division by 60 and 9. I have my doubts whether any tables will be so convenient as those of common logarithms, used in the way I propose. Special tables, 81 80' for purposes which do not often occur, are of value only when they save complicated operations. Such tables are not in the way when wanted; and when they are found, their structure and rationale have to be remembered. It is a sufficient proof of the state of knowledge of the theory of beats that a work which goes so deeply into the formulæ connected with musical vibrations as Prony's makes no allusion to beats. Previously to the use of logarithms, the arithmetical calculations of the scale were very laborious. Mersenne makes 58 commas in the octave, the true number being 553. Nicolas Mercator corrected this in a manuscript seen by Dr Holder, and then proposed an artificial comma of 53 to the octave, which gave all the intervals very nearly integer. + Writers are very obscure in their use of the word vibra Here we have intervals of 54 and 39, altogether '93, of a mean semitone. Mr Woolhouse's experiment gives 254 double vibrations to the C immediately below; and other experiments give nearly the same, for our day. The common tradition is that concert-pitch has risen about a note in the last century. The change can be traced in its progress. Robison, at the end of the last century, found the ordinary tuning-forks gave 240 vibrations for C, that is, 270 vibrations for D, a little higher than Dr Smith's organ at its warmest. Possibly some of this effect may have arisen as follows. The organs being tuned in the cold to the usual pitch of the day, the orchestras, on tuning with them after the air had been warmed by a crowd, would find it necessary to raise their pitch. This would have a tendency to cause a permanent rise, which the organ-tuners would of course follow, and then the same effect would be repeated. The convenience of representing the Cs by powers of 2 has led many writers to choose 256 as the number of double vibrations in the first C below the lines of the treble: I trust this power of 2 will be enough to prevent the pitch from making any further ascent. The subject to which I now come has been perplexed from the beginning by a confusion of different things under one word. By a beat, I mean any acoustical cycle derived from composition of ordinary vibrations; whether the returns can be distinguished by the ear as separate occurrences, or whether they are rapid enough to cause a sound. The first kind of beats were used by Sauveur: but as there is a confused discussion about them in which his name occurs, it will be more convenient to call them Tartini's beats, because, when they become rapid enough to give a note, that note is the grave harmonic detected by Tartini in or tion; they make it difficult to know whether they mean the single wave, be it of condensation or of rarefaction, or the double wave made up of one condensation and one rarefaction. Much confusion might have been saved in many subjects if terms of contempt, or of slang, had been seriously adopted: for such terms are very often more expressive than the solemn words which they are directed at. The previous examination" is very feeble compared with the "little-go." For the present case, when the pendulum was brought into use, it was called in derision a swing-swang. If this word had been adopted by writers on acoustics, all the confusion I speak of would have been prevented; for no writer would have left it in doubt whether he reckoned in swings, or in swing-swangs, as I shall do. There is the same difficulty in medical descriptions, occasionally: some have counted inspiration and respiration as one, most as two. * The organ tuners must in all time have known the beats which disappear when the concord becomes perfect. The first writer who is cited as having mentioned them is Mersenne (Harmonie Universelle, Paris, 1636, folio, book on instruments, p. 362). But Mersenne does not attempt any explanation. He observes that two pipes which are nearly unisons tremble, and make the hand which holds them tremble. But the trembling goes off when the unison is made perfect; which, says Mersenne, is the exact opposite of what takes place in strings. That is, he imagined the beats were to be compared with the sympathetic vibrations. Dr Smith, with that habit of indistinctive citation which is one of the manias of much learning, cites Mersenne and Sauveur together as his predecessors in the subject. There is another writer who is better qualified to be classed as the immediate predecessor of Sauveur, because he distinctly opposes the sympathy of consonant vibrations, and its effects, to the clashing of dissonant vibrations. I mean Dr Wm. Holder, F.R.S., who died in January 1696-7, and was the opponent of Wallis on a question of priority in the method of teaching the deaf and dumb. In his Natural Grounds and Principles of Harmony, published in 1694, he describes beats in a manner which is worth quoting, were it only as an instance of the poetry of explanation which science has driven out (pp. 34, 35, ed. of 1731): "It hath been a common Practice to imitate a Tabour and Pipe upon an Organ. Sound together two discording Keys (the base Keys will shew it best, because their Vibrations are slower), let them, for Example, be Gamut with Gamut sharp, or F Faut sharp, or all three together. Though these of themselves should be exceeding smooth and well voyced Pipes, yet, when struck together, there will be such a Battel in the Air between their disproportioned Motions, such a Clatter and Thumping, that it will be like the beating of a Drum, while a Jigg is played to it with the other hand. If you cease this, and sound a full Close of Concords, it will appear surprizingly smooth and sweet.... .... Being in an Arched sounding Room near a shrill Bell of a House Clock, when the Alarm struck, I whistled to it, which I did with ease in the same Tune with the Bell, but, endeavouring to whistle a Note higher or lower, the Sound of the Bell and its cross Motions were so predominant, that my Breath and Lips were check'd, that I could not whistle at all, nor make any sound of it in that discording Tune. After, I sounded a shrill whistling Pipe, which was out of Tune to the Bell, and their Motions so clashed, that they seemed to sound like switching one another in the Air." about 1714. And even when they give a sound, it will still be convenient to call them Tartini's beats. These beats are in their perfect theoretical existence when a consonance is quite true, and they owe their usual existence to its approximate truth. Tartini* used to tell his pupils that their thirds could not be in tune when they played or sang together, unless they heard the low note: assuming, doubtless, that their perceptions were as acute as his own. The second kind of beats I shall call Smith's beats, because Dr Smith first made use of them, and gave their theory. They are entirely the consequence of the imperfection of a consonance, and become more rapid and more disagreeable as the imperfection increases, vanishing entirely when the consonance is perfectly true. I cannot find the means of affirming that Smith was acquainted with Tartini's grave harmonic. In the place in which one would have expected him to mention it, namely, when he mentions the flutterings, as he calls them, which I name Tartini's beats, he does not make the slightest reference to those flutterings becoming rapid enough to yield a note, though he complains that he could hardly count them. Smith accuses Sauveur of confounding the beats of an imperfect consonance with the flutterings of a perfect one. It is true that Sauveur makes the same use of Tartini's beat • Tartini published his treatise on harmony at Padua in 1754. D'Alembert's account of this work is so precisely what he might have written of Smith, that I quote it. "Son livre est écrit d'une manière si obscure, qu'il nous est impossible d'en porter aucun jugement: et nous apprenons que des Savans illustres en ont pensé de même. Il seroit à souhaiter que l'Auteur engageât quelque homme de lettres versé dans la Musique et dans l'art d'écrire, à développer des idées qu'il n'a pas rendues assez nettement, et dont l'art tireroit peut-être un grand fruit, si elles étoient mises dans le jour convenable." M. Romieu, of Montpellier, published a memoir in 1751, in which he described Tartini's grave harmonic: and hence some have made him the first discoverer. But Tartini had been teaching the violin, on which instrument he was the head of a celebrated school, a great many years: that he should not have published the grave harmonic to every pupil whom he taught to tune by fifths, is incredible. He himself affirms in his work that he always did so from 1728, when he established his school and further, that he made the discovery on his violin, at Ancona, in 1714; this was the year after he dreamed the Devil's Sonata. As it is stated that he told how the devil played to him in his sleep, many years after, to Lalande, who could make astronomical gossip of any thing, I should not be at all surprised if a certain four-volume work contained evidence of the date of the grave harmonic. : Rameau, not Romieu, is the natural counterpart of Tartini. In 1750 he published his celebrated treatise on harmony, the completion of a system which he had sketched in previous works and he and Tartini are thus related. Tartini makes his grave note the natural and necessary bass to the consonance which produces it: Rameau makes the harmonics of any given note the natural and necessary treble of the given note as a bass. These contemporary counter-systems are now exploded: they have an uncertain connexion with the truth, no doubt; but the are demands and obtains a great number of combinations which neither system will allow. It is due, however, to Rameau to observe that his discovery, which appears independent of Tartini's, is that of a physical Chladni (Acoustique, p. 253) says that the first mention of the grave harmonic which he knew of is by G. A. Sorge (Anweisung zur Stimmung der Orgelwerke, Hamburg, 1744), who asks why fifths always give a third sound, the lower octave of the lower note, and concludes that nature will put 1 before 2, 3, that the order may be perfect. If Tartini's evidence in his own favour be disallowed, then Sorge becomes the first observer. But to me the uncontradicted assertion of a teacher whose pupils were scattered through Europe, and included men so well known on the violin as Nardini, Pugnani, Lahoussaye, &c. &c., that he had pointed out the third sound to all his school from 1728 to 1750, is real evidence. Chladni's mention of Tartini is as uncandid as possible:- Tartini, auquel on a voulu attribuer cette découverte, en fait mention dans son Trattato... Mentions it! No one knew better than Chladni himself (as he proceeds to show, the moment the paragraph about priority is finished) that Tartini's whole book is a system founded upon it. D'Alembert, La Borde, Rousseau, &c. do not dispute Tartini's claim; and the common voice of Europe gives no other name to the discovery. On this subject in general see the Article Fondamental in the Encyclopædia, by D'Alembert; Rousseau's Musical Dictionary, Harmonie and Systême; Matthew Young's Enquiry, &c. + Smith does not, so far as I can find, attempt to explain these Autterings; though I think it may be collected that he knew their cause. |