Treatise on Harmony

by Jean-Philippe Rameau

Overview

The Traité de l'harmonie of Jen-Philippe Rameau is one of the most important books in the history of Western music. Written while Rameau was still a relatively obscure organist and music master at Clermont-Ferrand, the book received but one printing during Rameau's life, in 1722, shortly before he settled in Paris. The Traité was immediately recognized as a profound advance in musical theory, however, and it established Rameau's reputation as a theorist. His book was the first to codify those principles of tonality that were to dominate the music of the West for almost two centuries. Even today the theories of Rameau remain the basis for the study of harmony.
Rameau's Traité de l'harmonie is divided into four books, the first of which presents the mathematical basis from which Rameau sought to derive his theories. Book Two may be considered the most important section of the Traité; in it Rameau generates his entire harmonic system from fundamental principles, explaining intervals, chords, and modes — everything, in fact, essential to musical composition in tonal style. Working from the principles developed in Books One and Two, Book Three treats the practical rules of composition, including such topics as harmonic modulation and chord progressions. Book Four concerns the practical art of accompaniment on harpsichord or organ, including the realization of a figured bass. Corrections added by Rameau in a supplement are included in the text, and all the musical examples have been reset in modern musical notation. In addition, two pages from a unique copy of the first issue of the first edition are given in facsimile. The translator's introduction discusses the history of the work, Rameau's mathematics, and his place in the history of music theory.

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ISBN-13:	9780486171371
Publisher:	Dover Publications
Publication date:	04/03/2012
Series:	Dover Books On Music: Analysis
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Format:	eBook
Pages:	512
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Treatise on Harmony

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By Jean-Philippe Rameau, Philip Gossett

Dover Publications, Inc.

Copyright © 1971 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-17137-1

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CHAPTER 1

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.

ARTICLE I

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.

ARTICLE II

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.

ARTICLE III

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.

(Continues...)

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Excerpted from Treatise on Harmony by Jean-Philippe Rameau, Philip Gossett. Copyright © 1971 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents

Contents

Title Page,
Copyright Page,
Dedication,
TRANSLATOR'S INTRODUCTION,
PREFACE,
TABLE OF TERMS,
BOOK ONE - On the Relationship between Harmonic Ratios and Proportions,
CHAPTER ONE - On Music and Sound,
CHAPTER TWO - On the different ways in which the relationship between Sounds can be known to us,
CHAPTER THREE - On the origin of Consonances and on their relationships,
CHAPTER FOUR - Remarks on the properties of the Harmonic and Arithmetic Proportions,
CHAPTER FIVE - On the origin of Dissonances and on their relationships,
CHAPTER SIX - On doubled Intervals, and especially on the Ninth and the Eleventh,
CHAPTER SEVEN - On Harmonic Division, or on the origin of Chords,
CHAPTER EIGHT - On the inversion of Chords,
CHAPTER NINE - Remarks on all the preceding Chords,
CHAPTER TEN - Remarks on the different Ratios which can be given to a single Chord,
CHAPTER ELEVEN - How to relate the Ratios given by the Divisions to the Vibrations and to the Multiplication of Lengths,
BOOK TWO - On the Nature and Properties of Chords and on Everything which may be used to make Music perfect,
CHAPTER ONE - On the fundamental Sound of Harmony and on its progression,
CHAPTER TWO - On the Chords appropriate to fundamental Sounds and on their progression,
CHAPTER THREE - On the nature and properties of the Octave,
CHAPTER FOUR - On the nature and properties of the Fifth and the Fourth,
CHAPTER FIVE - On the Perfect Cadence, in which the nature and properties of all the Intervals are found,
CHAPTER SIX - On the Deceptive Cadence,
CHAPTER SEVEN - On the Irregular Cadence,
CHAPTER EIGHT - On the imitation of Cadences by inversion,
CHAPTER NINE - On how to avoid Cadences while imitating them,
CHAPTER TEN - On Chords by supposition with which we may also avoid Cadences while imitating them,
CHAPTER ELEVEN - On the Fourth and the Eleventh,
CHAPTER TWELVE - On Chords by borrowing with which we may avoid Perfect Cadences while imitating them,
CHAPTER THIRTEEN - Rule for the progression of Dissonances, derived from the progression of fundamental Chords,
CHAPTER FOURTEEN - Remarks on the progression of Thirds and Sixths,
CHAPTER FIFTEEN - On occasions when the Seventh should be suppressed from the Ninth Chord,
CHAPTER SIXTEEN - On dissonant Consonances, in which the Fourth is discussed together with the false idea of it that exists because of superfluous Rules,
CHAPTER SEVENTEEN - On License,
CHAPTER EIGHTEEN - Observations on establishing Rules, in which the method of composing a Fundamental Bass is taught,
CHAPTER NINETEEN - Continuation of the preceding Chapter, in which it appears that Melody arises from Harmony,
CHAPTER TWENTY - On the properties of Chords,
CHAPTER TWENTY-ONE - On the Modes,
CHAPTER TWENTY-TWO - On the origin of our liberty to pass from one Mode or from one Key to another,
CHAPTER TWENTY-THREE - On the properties of Modes and Keys,
CHAPTER TWENTY-FOUR - On Meter,
CHAPTER TWENTY-FIVE - On the usefulness of this new way of indicating different Meters,
CHAPTER TWENTY-SIX - On the number of Measures each Air should contain, and on their characteristic Movements,
CHAPTER TWENTY-SEVEN - How to proceed when setting Words to Music,
CHAPTER TWENTY-EIGHT - On Design, Imitation, Fugue, and on their properties,
CHAPTER TWENTY-NINE - On those Intervals which should be classified as major and minor; as just or perfect; as augmented and diminished,
BOOK THREE - Principles of Composition,
CHAPTER ONE - Introduction to practical Music,
CHAPTER TWO - On the Fundamental Bass,
CHAPTER THREE - On the Perfect Chord, with which Composition in four Parts begins,
CHAPTER FOUR - On the succession of Chords,
CHAPTER FIVE - On several Rules which must be observed,
CHAPTER SIX - On the Seventh Chord,
CHAPTER SEVEN - Remarks on Dissonance,
CHAPTER EIGHT - On Key and Mode,
CHAPTER NINE - On how to Modulate harmonically when the Bass is given a diatonic progression,
CHAPTER TEN - On the Basso Continuo,
CHAPTER ELEVEN - On the progression of the Bass, which simultaneously determines the progression of the Chords; how we may relate a derived Chord to its Fundamental,
CHAPTER TWELVE - Continuation of the Rules drawn from the preceding Example,
CHAPTER THIRTEEN - On the Perfect Cadence,
CHAPTER FOURTEEN - On the Leading Tone, and on how all Dissonances are resolved,
CHAPTER FIFTEEN - On the Eleventh, called the Fourth,
CHAPTER SIXTEEN - On the irregular Cadence,
CHAPTER SEVENTEEN - On the different progressions of a Bass which are related to one another in such a way that the Harmony in the upper Parts does not change at all,
CHAPTER EIGHTEEN - On how to prepare Dissonances,
CHAPTER NINETEEN - On occasions when Dissonances cannot be prepared,
CHAPTER TWENTY - A precise enumeration of the different progressions of the Bass, according to the different. Dissonances used there,
CHAPTER TWENTY-ONE - On the Chord of the Second,
CHAPTER TWENTY-TWO - On Keys and Modes in general,
CHAPTER TWENTY-THREE - On how to pass from one Key to another; i.e., on how to Modulate,
CHAPTER TWENTY-FOUR - Continuation of the Rules contained in the preceding Chapter,
CHAPTER TWENTY-FIVE - How to know which Chords must be given to the Bass Notes in any progression,
CHAPTER TWENTY-SIX - How to use the Seventh on every Note of a Key in a diatonic progression,
CHAPTER TWENTY-SEVEN - How the same Dissonance may occur in several consecutive Chords on different Notes; how it may be resolved by Notes which appear to be foreign,
CHAPTER TWENTY-EIGHT - On all Licenses, beginning with the Deceptive Cadence,
CHAPTER TWENTY-NINE - On the Chord of the augmented Fifth,
CHAPTER THIRTY - On the Ninth Chord,
CHAPTER THIRTY-ONE - On the Eleventh Chord, called the Fourth,
CHAPTER THIRTY-TWO - On the Chord of the augmented Seventh,
CHAPTER THIRTY-THREE - On the Chord of the augmented Second and on its derivatives,
CHAPTER THIRTY-FOUR - On Chromaticism,
CHAPTER THIRTY-FIVE - On how to make use of everything we have discussed hitherto,
CHAPTER THIRTY-SIX - On Composition in two Parts,
CHAPTER THIRTY-SEVEN - On False Relations,
CHAPTER THIRTY-EIGHT - On how to write a Melody above a Bass,
CHAPTER THIRTY-NINE - On ornamented Melody or Supposition,
CHAPTER FORTY - On how to compose a Fundamental Bass below a Treble,
CHAPTER FORTY-ONE - How to compose a Basso Continuo below a Treble,
CHAPTER FORTY-TWO - Useful Remarks concerning the preceding Chapter,
CHAPTER FORTY-THREE - Rules to be observed in a Composition in two, three, and four Parts,
CHAPTER FORTY-FOUR - On Design, Imitation, and Fugue,
BOOK FOUR - Principles of Accompaniment,
CHAPTER ONE - How to recognize the Intervals from the arrangement of the Keyboard,
CHAPTER TWO - On the difference between major and minor Intervals; and between those which are perfect and those which are augmented or diminished,
CHAPTER THREE - On the Position of the Hand and on the Arrangement of the Fingers,
CHAPTER FOUR - On how to find Chords on the Keyboard,
CHAPTER FIVE - Useful Remarks concerning all the Chords,
CHAPTER SIX - On Keys and Modes,
CHAPTER SEVEN - On the order which must be followed for the succession of Chords found within the Octave of each Key,
CHAPTER EIGHT - General Rules,
CHAPTER NINE - On the different Chords which should follow the Seventh Chord when the Bass Note remains on the same degree,
CHAPTER TEN - On the Chord of the Second,
CHAPTER ELEVEN - On Chords of the Sixth,
CHAPTER TWELVE - On the Chord of the augmented Second and on its derivatives,
CHAPTER THIRTEEN - On Chords by Supposition,
CHAPTER FOURTEEN - Observations on the relations between all the preceding Chords,
CHAPTER FIFTEEN - On how to prepare and resolve all Dissonances, from which we shall come to know the Key in use and the Chords which each Note of this Key should bear,
CHAPTER SIXTEEN - On Chromaticism,
CHAPTER SEVENTEEN - Recapitulation of the various successions of Chords,
CHAPTER EIGHTEEN - Rules which are necessary in order to accompany properly,
CHAPTER NINETEEN - On how to figure a Basso Continuo, and on how to know which Chords each figure denotes,
CHAPTER TWENTY - How to tell which Bass Notes should bear a Chord,