- Shopping Bag ( 0 items )
On Music and Sound
Music is the science of sounds; therefore sound is the principal subject of music.
Music is generally divided into harmony and melody, but we shall show in the following that the latter is merely a part of the former and that a knowledge of harmony is sufficient for a complete understanding of all the properties of music.
We shall leave the task of defining sound to physics. In harmony we characterize sound only as grave and acute, without considering either its loudness or its duration. All knowledge of harmony should be founded on the relation of acute sounds to grave ones.
Grave sounds are the lowest, such as are produced by male voices; acute sounds are the highest, such as are produced by female voices.
The distance from a low to a high sound is called an interval, and from the different distances that may be found between one sound and another, different intervals are formed; their degrees are named after the numbers of arithmetic. Thus, the first degree can only be named after the unit, so that two sounds of the same degree are said to be in unison. Likewise, the second degree is called a second, the 3rd a third, the 4th a fourth, the 5th a fifth, the 6th a sixth, the 7th a seventh, the 8th an octave, etc. The first degree is always assumed to be the lowest, and the others are formed by raising the voice successively according to its natural degrees.CHAPTER 2
On the different ways in which the relationship between Sounds can be known to us
In order to understand the relationship between sounds, investigators took a string, stretched so that it could produce a sound, and divided it with movable bridges into several parts. They discovered that all the sounds or intervals that harmonize were contained in the first five divisions of the string, the lengths resulting from these divisions being compared with the original length.
Some have sought an explanation of this relationship in that relationship existing between the numbers indicating the [number of] divisions. Others, having taken the lengths of string resulting from these divisions, have sought an explanation in the relationship between the numbers measuring these different lengths. Still others, having further observed that communication of sound to the ear cannot occur without the participation of the atmosphere, have sought an explanation in the relationship between the numbers indicating the vibrations of these various lengths. We shall not go into the several other ways in which this relationship may be known, such as with strings of different thicknesses, with weights which produce different tensions in the strings, with wind instruments, etc. It was found, in short, that all the consonances were contained in the first six numbers, except for the methods using thicknesses and weights, where the squares of these fundamental numbers had to be used. This has led some to attribute all the power of harmony to that of numbers; it is then only a matter of applying properly the operation on which one chooses to base one's system.
We must remark here that the numbers indicating the divisions of the string or its vibrations follow their natural progression, and that everything is thus based on the rules of arithmetic. The numbers measuring the lengths of string, however, follow a progression which is the inversion of the first progression, thus destroying some of the rules of arithmetic, or at least obliging us to invert them, as we shall see later. Since the choice between these operations does not affect the harmony, we shall use only those in which the numbers follow their natural progression, for everything then becomes much clearer.CHAPTER 3
On the origin of Consonances and on their relationships
Sound is to sound as string is to string. Each string contains in itself all other strings shorter than it, but not those which are longer. Therefore, all high sounds are contained in low ones, but low ones, conversely, are not contained in high ones. It is thus evident that we should look for the higher term in the division of the lower; this division should be arithmetic, i.e., into equal parts, etc. [...] Let us take AB [Ex. I.1] as the lower term. If I wish to find a higher term so as to form the first of all the consonances, I divide AB in two (this number being the first of all the numbers), as has been done at point C. Thus, AC and AB differ by the first of the consonances, called the octave or diapason. If I wish to find the other consonances immediately following the first, I divide AB into three equal parts. From this, not only one but two higher terms result, i.e., AD and AE; from these, two consonances of the same type are generated, i.e., a twelfth and a fifth. I can further divide the line AB into 4, 5, or 6 parts, but no more, since the capacity of the ear extends no further.
To make this proposition clearer, we shall take seven strings whose [number of] divisions are indicated by numbers. Without inquiring whether they are equal in any other respect we assume that the strings are all tuned at the unison. We then put the numbers in their natural order beside each string, as in the following demonstration [Ex. I.2.]. Each number indicates the equal parts into which the string corresponding to it is divided. Notice that number 7, which cannot give a pleasant interval (as is evident to connoisseurs), has been replaced by number 8; the latter directly follows 7, is twice one of the numbers contained in the senario, and forms a triple octave with I. This does not increase the quantity of numbers put forth, since 6 and 8 give the same interval as 3 and 4, every number always representing the number that is its half.
Remember that in every instance the numbers mark both a division of the unit and of the undivided string, which corresponds to I.
The order of origin and perfection of these consonances is determined by the order of the numbers. Thus, the octave between 1 and 2, which is generated first, is more perfect than the fifth between 2 and 3. Less perfect again is the fourth between 3 and 4, etc., always following the natural progression of the numbers and admitting the sixths only last.
The names of the notes should make it apparent that the string 1, its octave 2, and its double and triple octaves 4 and 8 yield, so to speak, only a single sound. Furthermore, this arrangement of notes, conforming to the order of the numbers and the divisions of the string, gives the most perfect harmony imaginable, as everyone may judge for himself. As for the properties peculiar to each sound or consonance, we shall discuss each of these in a separate article, in order to provide a clearer notion of them.
On the source of Harmony or the fundamental Sound
We should first assume that the undivided string corresponding to 1 produces a given sound; the properties of this sound must be examined by relating them to those of the single string and even to those of the unit, which is the source of all numbers.
(1) The different divisions are marked on all the strings equal to the first and are determined by the magnitude of the number alongside the strings. These divisions clearly prove that each part of the divided strings arises from the first string, since these parts are contained in that first, unique string. Thus, the sounds which these divided strings produce are generated by the first sound, which is consequently their source and their fundamental.
(2) From the different distances found between this fundamental sound and those it generates by its division, different intervals are formed. The fundamental sound is consequently the source of these intervals.
(3) Finally, from the union of these different intervals, different consonances are formed. The harmony of these consonances can be perfect only if the first sound is found below them, serving as their base and fundamental, as was seen in the demonstration [Ex. I.2.]. Thus, the first sound remains the source of these consonances and of the harmony they form.
In the following articles, we shall see which sounds are most closely related to this source and what use the source makes of them.
On the Unison
Properly speaking, the unison is only a single sound which may be produced by several voices or by several instruments. We observed this in the preceding demonstration, before the seven strings were divided. Thus, the unison is not called a consonance as it does not fulfill the necessary condition for one, i.e., a difference in the sounds with regard to low and high. It has the same relationship to consonances, however, as the unit has to numbers.
On the Octave
The proportion of the whole to its half or of the half to the whole is so natural that it is the first to be understood. This should predispose us in favor of the octave, whose ratio is 1:2. The unit is the source of numbers, and 2 is the first number; there is a close resemblance between these two epithets, source and first [Fr. principe and premier], which is quite appropriate. Likewise, in practice, the octave is characterized by the name "replicate," all replicates being intimately connected to their source, as was apparent from the names of the notes in the preceding demonstration. This replicate should be regarded less as a chord than as a supplement to chords; therefore it is sometimes compared to zero. Male and female voices naturally intone the octave, believing themselves to be singing a unison or the same sound. In flutes, this octave depends only on the force of the breath. Furthermore, if we take a viol whose strings are long enough for their oscillations to be distinguished, we shall notice when making a string resonate with some force that the strings an octave higher or lower will vibrate by themselves, while only the upper sound of the fifth [i.e., the string a fifth above] will vibrate, not the lower [i.e., the string a fifth below]. This proves that the source of the octave is intimately connected to both the sounds which form it, while the source of the fifth and consequently of all other intervals resides solely in the lower and fundamental sound. Descartes was led astray here by the false proof with regard to the octave that he based on the lute.
Excerpted from Treatise on Harmony by Jean-Philippe Rameau, Philip Gossett. Copyright © 1971 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Posted September 11, 2008
if this is the title I have been looking for since studying it as a youth, I am astounded.This has to be the single most important and practical book for the study and understanding of harmony, and western music in general. Do not miss it!!Was this review helpful? Yes NoThank you for your feedback. Report this reviewThank you, this review has been flagged.