AUTHOR'S PREFACE

FOURTH GERMAN EDITION.

In the essential concepts of musical relations, I have found nothing to alter in this new edition. In this respect, I can only 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 remodeled, and in some parts enlarged. As a guide for readers of former editions, I take the liberty to list the following places containing additions and alterations.*

P. 16d, note.—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 †.—Herr Clement Neumann's observations on the vibrational form of violin strings.
Pp. 89a to 93b.—The action of blowing organ pipes.
P. 110b.—Distinction of Ou from U Referring to vowel sounds, specifically the difference between the "oo" sound as in "soup" versus the "u" sound as in German "über"..
Pp. 111b to 116a.—The various modifications in the sounds of vowels.
P. 145a.—The ampullæ The sensory organs in the inner ear associated with balance. and semicircular canals no longer considered as parts of the organ of hearing.
P. 147b.—Waldeyer's and Preyer's measurements adopted.
Pp. 150b 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 the 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 for musical tones with harmonic upper partials.
Pp. 328c to 329b.—Division of the octave into 53 degrees. Bosanquet's harmonium A keyboard instrument designed to play in just intonation..
Pp. 338c to 339b.—Modulations through chords composed of two major thirds.
P. 365, note †.—Oettingen and Riemann's theory of the minor mode.
P. 372.—Improved electro-magnetic driver of the siren An instrument used to determine the frequency of sound vibrations..
P. 373a.—Theoretical formulas for the pitch of resonators.
P. 374c.—Use of a soap bubble for seeing vibrations.
Pp. 389d to 396b.—Later use of striking reeds. Theory of the blowing of pipes.
Pp. 403c to 405b.—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 A tuning system that slightly adjusts intervals to allow playing in all keys..
P. 429.—Plan of Bosanquet's harmonium.

H. HELMHOLTZ.

BERLIN: April, 1877.

• [The pages of this edition are substituted for the German throughout these prefaces, and omissions or alterations regarding the first edition of this translation are mostly pointed out in footnotes as they arise.—Translator.]