In laying before the Public the result of eight years' labour, I must first pay a debt of gratitude. The following investigations could not have been accomplished without the construction of new instruments, which did not enter into the inventory of a Physiological Institute, and which far exceeded in cost the usual resources of a German philosopher. The means for obtaining them have come to me from unusual sources. The apparatus for the artificial construction of vowels, described on pp. 121 to 126, I owe to the munificence of his Majesty King Maximilian of Bavaria, to whom German science is indebted, on so many of its fields, for ever­ready sympathy and assistance. For the construction of my Harmonium in perfectly natural intonation, described on p. 316, I was able to use the Soemmering prize which had been awarded me by the Senckenberg Physical Society (die Senckenbergische naturforschende Gesellschaft) at Frankfurt-on-the-Main. While publicly repeating the expression of my gratitude for this assistance in my investigations, I hope that the investigations themselves as set forth in this book will prove far better than mere words how earnestly I have endeavoured to make a worthy use of the means thus placed at my command.

The present Third Edition has been much more altered in some parts than the second. Thus in the sixth chapter I have been able to make use of the new physiological and anatomical researches on the ear. This has led to a modification of my view of the action of Corti's arches. Again, it appears that the peculiar articulation between the auditory ossicles called 'hammer' and 'anvil' might easily cause within the ear itself the formation of harmonic upper partial tones for simple tones which are sounded loudly. By this means that peculiar series of upper partial tones, on the existence of which the present theory of music is essentially founded, receives a new subjective value, entirely independent of external alterations in the quality of tone. To illustrate the anatomical descriptions, I have been able to add a series of new woodcuts, principally from Henle's Manual of Anatomy, with the author's permission, for which I here take the opportunity of publicly thanking him.

I have made many changes in re-editing the section on the History of Music, and hope that I have improved its connection. I must, however, request the reader to regard this section as a mere compilation from secondary sources; I have neither time nor preliminary knowledge sufficient for original studies in this extremely difficult field. The older history of music to the commencement of Discant, is scarcely more than a confused heap of secondary subjects, while we can only make hypotheses concerning the principal matters in question. Of course, however, every theory of music must endeavour to bring some order into this chaos, and it cannot be denied that it contains many important facts.

For the representation of pitch in just or natural intonation, I have abandoned the method originally proposed by Hauptmann, which was not sufficiently clear in involved cases, and have adopted the system of Herr A. von Oettingen [p. 276], as had already been done in M. G. Guéroult's French translation of this book.

[A comparison of the Third with the Second editions, shewing the changes and additions individually, is here omitted.]

If I may be allowed in conclusion to add a few words on the reception experienced by the Theory of Music here propounded, I should say that published objections almost exclusively relate to my Theory of Consonance, as if this were the pith of the matter. Those who prefer mechanical explanations express their regret at my having left any room in this field for the action of artistic invention and esthetic inclination, and they have endeavoured to complete my system by new numerical speculations. Other critics with more metaphysical proclivities have rejected my Theory of Consonance, and with it, as they imagine, my whole Theory of Music, as too coarsely mechanical.

I hope my critics will excuse me if I conclude from the opposite nature of their objections, that I have struck out nearly the right path. As to my Theory of Consonance, I must claim it to be a mere systematisation of observed facts (with the exception of the functions of the cochlea of the ear, which is moreover an hypothesis that may be entirely dispensed with). But I consider it a mistake to make the Theory of Consonance the essential foundation of the Theory of Music, and I had thought that this opinion was clearly enough expressed in my book. The essential basis of Music is Melody. Harmony has become to Western Europeans during the last three centuries an essential, and, to our present taste, indispensable means of strengthening melodic relations, but finely developed music existed for thousands of years and still exists in ultra-European nations, without any harmony at all. And to my metaphysico-esthetical opponents I must reply, that I cannot think I have undervalued the artistic emotions of the human mind in the Theory of Melodic Construction, by endeavouring to establish the physiological facts on which esthetic feeling is based. But to those who think I have not gone far enough in my physical explanations, I answer, that in the first place a natural philosopher is never bound to construct systems about everything he knows and does not know; and secondly, that I should consider a theory which claimed to have shewn that all the laws of modern Thorough Bass were natural necessities, to stand condemned as having proved too much.

Musicians have found most fault with the manner in which I have characterised the Minor Mode. I must refer in reply to those very accessible documents, the musical compositions of A. D. 1500 to A. D. 1750, during which the modern Minor was developed. These will shew how slow and fluctuating was its development, and that the last traces of its incomplete state are still visible in the works of Sebastian Bach and Handel.

HEIDELBERG: May, 1870.

In the essential conceptions of musical relations I have found nothing to alter in this new edition. In this respect I can but maintain what I have stated in the chapters containing them and in my preface to the third [German] edition. In details, however, much has been remodelled, and in some parts enlarged. As a guide for readers of former editions, I take the liberty to enumerate the following places containing additions and alterations.^[1]

P. 16d, note footnote 13. — On the French system of counting vibrations.

P. 18a. — Appunn and Preyer, limits of the highest audible tones.

Pp. 59b to 65b. — On the circumstances under which we distinguish compound sensations.

P. 76a, b, c. — Comparison of the upper partial tones of the strings on a new and an old grand pianoforte.

P. 83, note footnote 24. — Herr Clement Neumann's observations on the vibrational form of violin strings.

Pp. 89a to 93d. — The action of blowing organ-pipes.

P. 110b. — Distinction of Ou from U.

Pp. 111d to 116a. — The various modifications in the sounds of vowels.

P. 145a. — The ampullæ and semicircular canals no longer considered as parts of the organ of hearing.

P. 147d. — Waldeyer's and Preyer's measurements adopted.

Pp. 150d to 151d. — On the parts of the ear which perceive noise.

P. 159b. — Koenig's observations on combinational tones with tuning-forks.

P. 176d, note. — Preyer's observations on deepest tones.

P. 179c. — Preyer's observation on the sameness of the quality of tones at the highest pitches.

Pp. 203c to 204a. — Beats between upper partials of the same compound tone condition the preference of musical tones with harmonic upper partials.

Pp. 328c to 329d. — Division of the Octave into 53 degrees. Bosanquet's harmonium.

Pp. 338c to 339d. — Modulations through chords composed of two major Thirds.

P. 365, note footnote 5. — Oettingen and Riemann's theory of the minor mode.

P. 372. — Improved electro-magnetic driver of the siren.

P. 373a. — Theoretical formulae for the pitch of resonators.

P. 374c. — Use of a soap-bubble for seeing vibrations.

Pp. 389d. to 396d. — Later use of striking reeds. Theory of the blowing of pipes.

Pp. 403c. to 405d. — Theoretical treatment of sympathetic resonance for noises.

P. 417d. — A. Mayer's experiments on the audibility of vibrations.

P. 428c, d. — Against the defenders of tempered intonation.

P. 429. —Plan of Bosanquet ’s harmonium.