Hostname: page-component-788cddb947-jbjwg Total loading time: 0 Render date: 2024-10-15T07:13:15.410Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Sensitiveness of the Ear to Pitch and Change of Pitch in Music

Published online by Cambridge University Press:  01 January 2020

Alexander J. Ellis
Get access

Extract

The principal aim of the present paper is to bring before the Musical Association some results obtained last autumn from numerous and very careful experiments by Dr. W. Preyer, Professor of Physiology in the University of Jena, because those results are not accessible in an English form, and seem to have a very practical bearing upon tuning and singing. My subordinate purpose is to exhibit a method of representing pitch and intervals, which shall be more consonant to the habits of musicians than that adopted by Dr. Preyer for the statement of his results. For the idea of this method I am indebted to Mr. Bosanquet, though I have worked it out somewhat differently from him.

Type
Research Article
Information
Proceedings of the Musical Association , Volume 3 , 1876 , pp. 1 - 32
Copyright
Copyright © Royal Musical Association, 1876

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

∗ Ueber die Grenzen der Tonrvahrnehmung (on the Limits of the Perception of Musical Tone), Jena, 1876; preface dated ‘Autumn 1875,’ 8vo, pp. viii. 72.Google Scholar

∗ In verifying Appunn's Tonometer in the Loan Exhibition of Scientific Instruments, I counted the beats during 20 seconds. An error of one beat could therefore have only produced an error of one-twentieth of a vibration in a second. Any decent tuning-fork will give audible beats with a neighbouring tone for 5 or 10 seconds, and hence allow of measurements to one-fifth or one-tenth of a vibration.Google Scholar

∗ This selection of 1 as the initial pitch is made on purely arithmetical grounds. The mode of treating other cases will be explained immediately. There is here no intention of deciding on the best standard pitch for practical purposes.Google Scholar

† See also the bottom of Table I. In higher arithmetic the number of the Octave is the index of the power of 2; that gives the number of vibrations with which the Octave begins. Thus 256 = 28 begins the 8th Octave.Google Scholar

∗ Of course this has been obtained by higher arithmetic, and is a close approximation to the 12th root of 2. Observe that gives the well-known approximations (too large) and (too small). It will be seen by Table II., col. B, below, that the vibrations of A are to those of A♯ as 1,681,793 to 1,781,797, which gives the above ratio. A whole Tone is very nearly indeed 10000÷8909, which is obviously less than the Major Tone 9÷8, and greater than the Minor Tone 10÷9.Google Scholar

† When the paper was read, a diagram, having the dimensions here indicated, was exhibited. It is by no means large or unwieldy, but for the purposes of printing it, the length of the imaginary key-board has been reduced to 48 inches, divided into 12 parts of 4 inches each, that is, it appears as one-tenth of its actual size. The statement of the actual dimensions is, however, retained in the text.Google Scholar

∗ Tithe, of course, means a tenth, but the word tenth itself had to be omitted because it is the name of well-known intervals, the major Tenth, or , and the minor Tenth, or .Google Scholar

† To reduce these relations to one common standard, suppose that the starting note makes 1,000,000 vibrations in some known interval of time. Then a note which is a sem higher than the starting note makes 59,462 vibrations more in the same time; a note which is a tithe higher makes only 5,793 vibrations more; one a cent higher only 678 more; one a mil higher only 58; and one a dime higher only 6 vibrations more than the starting note, in the same time.Google Scholar

∗ In order to be visible, the pointers were really about -inch wide at the bottom, but the measurements were determined by the vertical side.Google Scholar

† See examples of this mode of expressing pitch in Tables IV. and V.Google Scholar

‡ In Table III. a means is given for finding 0C from any value of 8A or 9 C that is usually employed.Google Scholar

∗ See Dr. Stone's observations on the difficulty of distinguishing a difference of pitch in the Oboe and Clarinet, and Mr. Hipkins's remark on pianos tuned to the same pitch but having different qualities of tone. Dr. Stone, in Concordia, ‘On Tuning an Orchestra.’Google Scholar

† This instrument was purchased by the South Kensington Museum at the close of the exhibition.Google Scholar

‡ Dr Preyer gives the following as the final result of experiments repeated hundreds of times when the listener did not know what number of vibrations he had to hear, but was somewhat acccustomed to observe, and listened with his ear close to the wood of the instrument as the wind was allowed to die out:—

8, 9. No musical tone; an intermittent fractional noise is heard, and the intermittences can be counted.

10, 11, 12, 13, 14. No musical tone; the tremor is felt, and the vibrations are visible; the rattle is feebler.

16. No musical tone; some perceive an obscure sensation of sound.

16, 17, 18. The sensation of musical tone commences; in addition to the tremor of the air, which is sensible to the touch, many hear an obscure sound.

19, 20. With many the sensation of tone is distinct; the tone itself buzzes gently.

21, 22. Many hear a humming tone.

23, 21. Everyone whose sensation of hearing is not impaired now hears a very deep, mild, beautiful tone.

26, 26, 27, 28, 29, 30. As the pitch of the tone increases it becomes less easily heard, because the length of the time for which it lasts, as the wind dies off, becomes shorter, but it is distinct.

31, 32. The tone is still distinct, but snort.

34, 36, 38. The tone is very short and difficult to hear.

40. Impossible to hear anything, because the vibrations of the reed, as the wind dies out, have become too feeble.

∗ The case is different for beats. If a major Third be sounded, the beats arising from the 4th partial of the lower and the 3rd of the upper note, relatively 16 and 16, are well heard separately, and their number is exactly ¼ of the number of vibrations made by the lower note in one second. But at the same time the differential tone arising from sounding 4 and 6 together is also heard, and it makes exactly the same number of vibrations in a second as the beats which are heard as separate sounds. I have heard this distinctly for the just major Third 266: 320, beating 64 times in a second by its partials, and producing the differential tone 64, on Appunn's Tonometer. Dr. Preyer thinks that different parts of the ear hear the discontinuous beats and the continuous tone. But in the case of an original tone itself the separate beats are not heard. I believe the two phenomena to be totally distinct, and that beats and differential tones have no physical connection. They seem to me to have a different origin and to follow different laws.Google Scholar

∗ These forks were also purchased by the South Kensington Museum at the close of the exhibition.Google Scholar

† Herr Appunn informed me that a hundred guineas would not pay him for the mere labour of making these 31 forks mentioned in the text.Google Scholar

∗ The Tonometer with 33 reeds (proper tune and not altered as for Dr. Preyer's purposes), together with another with 65 reeds (C 256 to C 512 proceeding by 4 vibrations at a time), was exhibited at the Loan Collection of Scientific Apparatus, and both were purchased at the close of the exhibition by the South Kensington Museum. The Differential Apparatus was not exhibited.Google Scholar

∗ Mr. Herrman Smith writes to me that ‘very few persons are able to discriminate between a perfect and a slightly imperfect interval of an Octave. It is only,’ adds he, ‘by interposing another interval that yon can be certain of exactitude. I have often proved the best tuners at fault, to their great astonishment.’ A mistuned Fourth above any note is a mistuned Fifth below the Octave of that note, and makes the same number of beats with each. Harmonium tuners, I believe, use this property to tune Octaves by. But this regards notes which are struck at the same time, not in succession, and merely shows the difficulty of observing the instant of the disappearance of beats, not the sensitiveness of the ear in recognising precision of interval when the notes are sounded successively.Google Scholar

∗ C, D, &c. are used for the equally tempered notes of the arithmetical scale, where 0C marks 1 vibration in a second; C', D'C, &c., are used for the equally tempered notes when the initial pitch 0C' is not 1. Similarly the small letters c, d, &c., with the additional conventions shown in Table V., represent just tones, beginning with 0C and d, d', &c, with the same additional conventions, the just tones commencing with OC'.Google Scholar

∗ See his paper in Nature, for 10th Aug 1876, p. 318.Google Scholar

∗ Galin, the introducer of the method, advocated Huyghens' cycle of 31 notes to the Octave, with nearly perfect major Thirds and flat Fifths (Exposition d'une Nouvelle Methods pour l'Enseignement de la Musique, 1816; reprinted 1862, p. 162). Emile Chevé, its chief propagator, uses the cycle of 29 to the Octave, which gives sharper major Thirds than the Greek, and also sharp Fifths, of which he was apparently not aware, for no one could have endured them (Méthode Élémentaire de Musique Vocale; this division results from p. 292).Google Scholar

∗ The limitation ‘partial’ must not be overlooked. As far as my own opinion goes, a very great deal more is required. But here it was necessary to confine myself to the matter brought out in this paper.Google Scholar

∗ In Messrs. Cornu and Mercadier's experiments, cited in my translation of Helmholtz, there are very few specimens of major Sevenths, but the four instances by amateurs are 21, 17, 32, 29 cents too sharp, and the four of the professional violinists are 34, 40 (thrice) cents too sharp as compared with the tone which makes a just major Third with the Fifth of the scale, or b = 10·88 sem, the actual pitches were 11·09, 11·05 11·20, 11·17 (the 61st harmonic duly reduced) for amateurs, and 11·22, 11·28 (thrice) for professionals. This is almost as bad as the sharp F' (11th harmonic, 53 cents too sharp) on the trumpet, and in all chords of the dominant in which it occurred the effect would be horrific. But it is evident that ears attuned to such intervals are likely to be offended at just major Sevenths intended for harmony, and still more at the major Sevenths of the mean tone temperament which are a quarter of a comma or 5½ cents flatter still, as I had occasion to remark when Mr. Bosanquet played a specimen of mean tone temperament (the only one used by Handel) before the Musical Association. (Proceedings for 1874–6, p. 128, note.)Google Scholar