Tonal Music E-Book

Franz Sauter

• Herausgeber: Books on Demand
• Kategorie: Lebensstil
• Sprache: Englisch

Seitenzahl: 267

Veröffentlichungsjahr: 2020

Beschreibung

This book is about the aesthetics of the tonal music. It therefore deals with sound forms such as consonance, dissonance, tonality, bar, counterpoint or motif. Thereby, it shows that all harmonic, rhythmic, and melodic sound figures are essentially relations in which similar sound components go well together. The whole secret of the musical aesthetics lies in this abstract determination. In this sense, the musical sound forms are systematically built on each other and form an ensemble of eight aesthetic principles, to each of which a chapter of this book is dedicated. The logical progression of these chapters reveals the inner connection between harmony, rhythm, and melody. Musical phenomena that have so far been interpreted differently and controversially are explained and derived in a comprehensible way in this context. The theoretical results of this book are, at the same time, a critique of previously common dogmas in musicology. For example, the prejudice that the difference between consonance and dissonance cannot be objectively grasped clearly contradicts the results of a rational music theory. Nor will the reader find the usual talk about the supposed anachronism or the transience of the tonal music in this book - for good reasons.

Leseprobe

Contents

Preface

I.

Harmonics

1. Consonance

Harmony of the Basic Consonance

Compound Consonance

Theories About Major and Minor Triads

2. Tonality

Cadence

Dissonance

Theories About Consonance and Dissonance

Key

Primitive Tonality

3. Aesthetics of the Modulation

Key Relationships

Process of the Modulation

Tonal Analysis

II.

Rhythmics

4. Equability of the Harmonic Succession

Transformation of Harmonies into Bars

Fragmentation of the Bar Content

Modulating Bar Content

5. Equability of the Bar Division

Segmentation of the Bar

Formation of the Division Levels

Merging of Bar Segments

Accentuation Ratios

Harmonic Contrast

The Thorough Bass Era

Theories About the Bar

III.

Melodics

6. Aesthetics of Scale Degree Relations

Transformation of Tonal Relations into Intervals

Concept of the Melody

The Myth of the Leading Tone

Alteration

The Construct of the Altered Chord

7. Counterpoint

Polyphony

Parallel Motion of the Voices

Similar and Contrary Motion

Oblique Motion

Digression: Suspension

Asynchronous Pausing

Primitive Counterpoint

8. Aesthetics of the Motiv

Concept of the Motif

Theories About the Motif

Something About Categories of Reflection

The Disregard of Sound Enjoyment

Epilogue by the Translator

List of Used Note Examples

Bibliography

“For some time, efforts have been made to explain our music theoretically. Still we have to admit to ourselves that we do not really have a true system ... The musical Galilei, we are still missing ...” (Heinrich Josef Vincent, Die Einheit in der Tonwelt, Leipzig 1862, preface)

“... a real theory, a scientifically founded music theory that deserves its name ... does not yet exist, however.” (Martin Vogel, Die Lehre von den Tonbeziehungen, Bonn 1975, p. 358)

Preface

Harmony, rhythm, and melody are the three basic features of our present-day music that are usually mentioned in one breath. No one would seriously assert that they only happen to meet in music by chance. Nevertheless, there is to date no scientific representation of the music which demonstrates that these three aesthetic characteristics necessarily cohere and which develops the musical concepts accordingly. Instead, music theory juxtaposes harmony, rhythm, and melody as subfields without revealing any inner connection. This is symptomatic of a state of musicology to which still applies what was already noted in the 19th century about the efforts to establish its theoretical foundation:

“All attempts of this kind have, up to now, not been able to create a really tenable scientific-musical system according to which all phenomena in the musical field are presented to be always necessary consequences of one basic principle … But what is laid down in musical textbooks with scientific justification has so far proved a failure, partly … because it was just as little able to create a self-contained system with undoubted conclusions, partly because, as a fantastic construction, it lacked any scientific basis.”1

The present book on the tonal music and the laws of its beauty emerged from a project initiated in 1980 to overcome the deficiency mentioned by Richter and to present the basic musical forms systematically and coherently. The reader can now be informed about what harmonic, rhythmic, and melodic structures have in common, how they differ, and how they are connected internally. The fact that such an analysis also yields some surprising results lies in the nature of a project that did not want to settle for the previous state of knowledge in musicology. The reader is therefore cordially invited to verify the validity of a whole series of unfamiliar findings and conclusions.

1 Ernst Friedrich Richter, Lehrbuch der Harmonie, Leipzig 1886, preface to the first edition (1853).

I. Harmonics

1. Consonance

a) Harmony of the Basic Consonance

Tonal music is music on a harmonic basis. The key to its aesthetics lies in the analysis of the major and minor triads, the basic sound forms of the tonal music. In the progress of the analysis, it will become apparent how any harmony, rhythm, and melody of the tonal music is built upon the harmony of these two sound figures.

Major and minor triads

Major and minor triads are triads whose tones harmonise with each other. What characterises these triads are the relations between the three tones called the root, third, and fifth. Only because of and within these relations, musical tones are defined as a root, third, and fifth. It is therefore quite appropriate to name the tones third and fifth after their relation to the root, which in turn would not be a root tone without these relations.

The relation between the tones third and fifth is again a third. Each major or minor triad contains in its basic form a major and a minor third, which complete each other to form a fifth. The major triad has the major third in the bottom part, the minor triad has it on top. Both sound forms have something in common: a specific harmony, which is called consonance. The essence of this harmony shall now be explained.

The harmony of the major and minor triads is obviously related to the frequency ratios that are characteristic of these sound forms:

Tone frequencies of two triads

The frequency ratios of these triads are:

Fifth:

660 Hz : 440 Hz

3:2

Major third:

550 Hz : 440 Hz

5:4

Minor third:

660 Hz : 550 Hz

6:5

The mathematical ratios between the frequencies of harmonising tones are of interest in instrument making, where these proportions have to be considered with regard to the dimensions of resonating cavities or strings. An organ pipe, for example, has to be the longer the lower its tone and the smaller therefore the related tone frequency. The frets on a guitar neck must be positioned in such a way that, for example, the tones of a major triad can be played one after the other on a single string, which presupposes that the vibrating string parts have the frequency ratio 4:5:6. Since the lengths of vibrating string parts are inversely proportional to the number of vibrations per second, the numerical ratios that are decisive for the harmonising tones have a clear appearance. It is therefore not surprising that the proportion 2:3 as a length ratio of string parts was known long before the knowledge of tone frequencies, namely, already in ancient times.

However, the quantitative relations observable in connection with major and minor triads (the proportions 3:2, 5:4, and 6:5) merely indicate the external relations between the tone frequencies. The question of the harmonic character of these sound relations is thereby in no way answered. Harmony, as a characterisation of what is perceived during the sounding of major and minor triads, means, namely, an inner relation of the tones respectively sounding together: a relation in which the musical tones go well together.

The basis for the fact that contents of perception go well together – and, by the way, that is what any aesthetics is all about – lies always in the properties of these things. In the case of major and minor triads, it is obviously the tones themselves that have something about themselves that makes them go well together. This property of the musical tones has therefore to be examined more closely.

A tone with a frequency of 500 Hz oscillates by definition 500 times per second, that is, once every two milliseconds. On an oscillograph, these oscillations are displayed, for example, as follows:

Four oscillations of a tone of 500 Hz

As the physicist and mathematician Jean Baptiste Fourier has generally demonstrated, regular oscillations are composed of simple sine oscillations. In music, these are referred to as partial tones of a musical tone.2 A single vibration of the above form consists of four partial tones:

Superposition of partial tones

While the first partial tone executes one oscillation, the second does two, the third three, and the fourth four vibrations. Accordingly, the partial tones vibrate at the basic frequency, double frequency, triple and quadruple frequency. The linear superposition of these oscillations results in a total vibration of the displayed form.

The illustration shows only a simple example. Musical tones normally consist of many more partial tones so that even the fifth, sixth, seventh etc. partial tones are contained in the sound. The sound spectrum of a tone contains a whole series of partial tones that are integer multiples of the basic vibration. Individual partial tones can vibrate more or less powerfully so that different waveforms are the result, which are perceived as specific timbres and acoustic colours. With the use of the Fourier analysis, it can be determined how strongly each single partial tone oscillates in the sound of a musical tone. This gives the characteristic sound spectrum, in which every partial oscillation is displayed with its frequency and volume.

Sound spectrum of a violin tone of 440 Hz

The sound of a tone results from its inner composition. The fact that a tone always has a sound character is, on the one hand, quite natural because mechanical oscillations have a more or less pronounced sound spectrum, simply for physical reasons. On the other hand, the fact that musical tones are rich in partial tones is not a mere natural phenomenon, because it is the result of a sophisticated instrument making. The sonority of the tones is achieved by various techniques: by the positioning of the violin bow when stroking the strings, by resonance bodies, by octave couplers of an organ, etc. The musical sound is cultivated for its own sake; it is designed as an object of enjoyment.

The quality of the tones by which they are predestined to harmonise can thus be summarised as follows: It is a matter of sonorous tones insofar as they are composed of a whole series of partial tones. Obviously, the harmony between the tones does not depend on their particular timbre and sound colour. The sonorous tone can therefore be represented schematically, by abstracting from the sonic intensity of the individual partial tones:

Schematic representation of the sound spectrum of a sonorous tone

In this illustration, the frequencies are scaled logarithmically. This representation has the advantage that equal frequency ratios appear as equal distances, as comes closest to the perspective of musicians who are familiar with this view due to the musical notation, the piano keyboard, etc.

By comparing the sound spectra of two tones that have the frequency ratio 3:2, one can see that every second partial tone of the higher tone oscillates at the same frequency as every third partial tone of the lower tone.

Consonance schema of the fifth

The two sonorous tones have thus coinciding partial tones. In the sounding-together of the tones, the oscillations of their common partial tones overlap to form a single partial oscillation in each case, in which even the Fourier analysis cannot indicate to which tone belongs which share in the sound intensity of a partial tone. Only the schematic separation of the tones can reveal the different origin of the partial tone in question.

Now it becomes apparent in what way both of the tones of a fifth harmonise: They match as sonorous tones, based on an accordance of their sound parts. This effect shall be called basic consonance.3

As said before, the harmony in the sounding-together of the sonorous tones is in any case independent of their specific timbre and acoustic colour. In the sound spectrum of tones, individual partial oscillations can therefore be completely absent without this diminishing the effect of the consonance. This applies, for example, to the clarinet, whose tones do not have even-numbered partials in its sound spectrum. The coincidence of partial tones also includes the case where a partial oscillation of a tone lies on a frequency that is a multiple of the basic frequency of another tone. That this other tone lacks a partial tone at the concerning position, is no reason for the ear to ignore the principle assignment of partial vibrations to a basic frequency. The detectable vibration has therefore partial tone character for both tones, even if it has its origin in only one of the tones. Basic consonance is therefore, strictly speaking, the harmony of a sound combination in which vibrations occur that have partial tone character for both of the sounding together tones alike.

Experiments have shown that, after an adaption phase, test persons can consider relations between sinus tones – even if they are offered separately for each ear through headphones – to be consonances. Testers and test persons have the freedom to call acoustic phenomena consonances if they remind them of consonances, although individual sinus tones cannot harmonise at all. Such experiments do not disprove the concept of the consonance, but are at most evidence for the susceptibility of the perception for illusions, which can come about under the participation of other mental capacities like memory, imagination, mind, and interest.4

When schemata of different consonances are compared, one can see that the extent of the coincidence of partial tones can be expressed by the ratio of the total vibrations of the tones: In the case of the fifth, the frequency ratio 3:2 equals a harmony in which every second partial tone of the one tone coincides with every third one of the other tone. The frequency ratio of the major third (5:4) corresponds to a harmony in which every fifth partial tone of the one tone coincides with every fourth one of the other tone.

Consonance schema of a major third

The conspicuous numerical proportions, which are characteristic for harmonic sound combinations and which can be read off accordingly from the length ratios of vibrating strings or air columns, have therefore their reason in the sound architecture of the tones. The proportions that can be observed in the external tone relations are the necessary manifestation of the inner relations that the sonorous tones have to each other because of their internal structure. The essence of the harmony lies definitely not in the numerical ratios, but in the going well together of the tones that is based on their sound character.

The difference in the degree of the harmony, which can be seen at the consonance schemata, is intuitively felt when comparing tone ratios acoustically. Since the weaker or stronger harmony of the sounding-together tones is related to the proportion of their frequencies, which is easily determinable, the illusion arises as if the frequency ratios themselves have a harmonic character. The theoretical deepening of this confusion consequently ends up in number mysticism, which subscribes to the believe that the harmony that is inherent to the sounding-together of the tones is a property of the associated external numerical proportions.

The consonance of the fifth and thirds is summarized in the harmony of the major or minor triad:

Consonance schemata of a major and minor triad

The mere summing-up of the consonances to a triad intensifies the effect of the basic consonance. The reason for this is that the combination of two consonances automatically includes a third consonance, whose harmony thus comes in addition ‘for free’. This is equally true for major and minor triads. However, the schematic representation of their consonance also shows a specific difference in the harmonic structure of the major and minor triads.

In case of the major triad, the coincidences of partial tones extend over more frequencies than in case of the minor triad. This is, for one thing, due to the fact that these coincidences are more widely dispersed in the major triad, whereas, in the minor triad, they are more concentrated on frequencies where all three tones have common partial oscillations. For another, there is more accordance in total in the major triad. Within the bandwidth illustrated above, the major triad has twelve times common partial oscillations, whereas the minor triad shows only ten such coincidences. This harmonic difference is the reason why musicians are able to intuitively distinguish major and minor triads on the basis of their sound. Also, the greater consonance of the major triad is noticeable when compared acoustically.

The consonant triad perfects the principle of the harmony that is contained in its components, the consonant dyads (fifth and thirds). It follows from this that the actual harmonic potency of the fifth and the thirds comes into effect in their determination as subordinate moments in the harmony of the consonant triads. Their harmonic role in tonal music is entirely in accordance with this: The fifth is a sound that is always defined by the sound relations of major and minor triads; it is divided in one way or another by thirds. Likewise, major and minor thirds are not autonomous harmonies, but are in each case components of either a major or a minor triad.

b) Compound Consonance

The musician distinguishes the basic form of a major and minor triad from the other forms in which individual tones are shifted to sound one or more octaves higher or lower or in which tones are supplemented by accordingly shifted tones. This distinction is determining for the practical handling of the harmonies in tonal music.

Basic form, inversions and extended forms of a C major triad

All of these sound shapes are named C major, even though they are quite different: Tones of varying numbers and positions sound together in it. On the other hand, it is not a matter of a random name equivalence, as is the case with persons, for example, who are all called Paul. Rather, it is a matter of an appropriate subsumption of sounds under a collective term. The use of the term ‘C major triad’ in the above example shows that musicians consider an equal treatment of these sounds as reasonable. They therefore abstract from the differences of these sounds; and they do rightly so: They take into account that an actual commonality of these sounds do exist. But what is it that is being abstracted from and what is the commonality of all these sound shapes?

First of all, all of these chords sound very similar. In this respect, the commonality of the forms of a major or minor triad is their belonging to a family of sounds that differ only slightly when listening to. However, this similarity, which the musician intuitively grasps, lies on the level of the immediate appearance and does not justify a conceptual abstraction which can last against errors and misperceptions. In biology, for example, the external similarity of animal species is a starting point for a classification, but it does not necessarily reveal the animal’s true nature. Accordingly, whales are, despite all appearances, not fish but mammals.

When musicians identify a major or minor triad independently of its particular form, they abstract from a difference that in fact only represents a harmonic nuance. It is a matter of that difference that makes itself felt when a tone within a sound combination is substituted or supplemented by another tone that lies one or more octaves higher or lower. Obviously, the sound relation of the octave does not create by itself any significant harmonic differences.

Consonance schema of the octave

Tones sounding together in an octave ratio appear as a dyad with an unsurpassable consonance because the complete sound of the upper tone coincides with half the sound of the lower tone. What presents itself as a similarity in the harmonic comparison of the different shapes of a triad is the same thing that appears as an extra strong harmony in the perception of the octave: the accordance of the sounds in regard to the frequencies of their partial tones. In realising major and minor triads and, accordingly, in the entire world of harmony that is based on major and minor triads, one thus abstracts from the harmony of the octave.

However, the neglect of the octave when determining major and minor triads is, in substance, not a disregard of the octave, but its consideration. The musical treatment of the octave fully corresponds to its harmonic property. That the octave causes only a negligible harmonic difference is only the flip side of a superlative of the consonance. The insignificance of the octave in a sphere, where it is about the harmonic identity of the sounds, is the form in which its absolutely dominant position in the harmonics of the tonal music comes into its own. The octave is therefore a substantial part of the consonant dyads which in their combination to major or minor triads perfect the principle of the consonance.

What applies to the major and minor triads as a whole, also applies to their components – the thirds and fifths. These, too, are to be understood abstractly and grasped in accordance with their harmonic essence against their specific manifestations:

Inversion of the fifth and of the major third

The fourth (4:3) is harmonically the same as the fifth (3:2), and the minor sixth (8:5) is the same as the major third (5:4), as Jean-Philippe Rameau already stated.5 The reason is obviously the harmony of the octave, which determines the sound image of the fourth and sixth. The harmony of the in such a way modified shapes of the fifth and thirds shall be called compound consonance.

The fourth as a compound consonance

The dashed lines in the consonance schema of the fourth indicate an imaginary tone that forms an octave along with the upper tone of the fourth, and a fifth along with the lower tone. These lines illustrate the intermediate step in the harmonic constitution of a fourth. The harmony of the fourth is not reduced to the immediate coincidence of the partials in this sound combination, but unites the harmony of the octave and fifth:

The practical equation of the various forms of major and minor sounds thus allows a conclusion to be drawn on the specific capacity of the musical perception to intuitively break down sound relations into their simple components.

If one wanted to adequately express what the fourth is and what it represents for the perception, then one could write the frequency ratio in the following form: 22:3. But this is merely a mathematical illustration of the compound consonance at the level of its external frequency ratios. At this level, the consonance cannot be grasped anyway, and the formula appears as a mathematical nonsense. Accordingly, the difference between the basic and compound consonance appears to be erased in the form of the frequency ratios. Theorists who confuse the outward appearance of the harmonies with their essence locate the fourth harmonically between the fifth and the third because they are fascinated and dazzled by the mathematical series

2:1, 3:2, 4:3, 5:4, 6:5.

If the fifth and fourth are presented in such a series, then the difference in their kind of consonance is ignored as well as the commonality in their abstract harmonic essence. It is then not surprising that such a theory construction gets into trouble when, at first, it misinterprets the consonance as a relation of small numbers, to then speculate and wonder whether the relation 16:5 (a variant of the major third) can be a consonance or not.6

The abstract identity of the consonant sounds is finally reflected by the tones of which they consist:

The tones that sound together in a ratio of 2:1 are identical because and insofar it is abstracted from their insignificant contribution to the harmonic differentiation in all sound combinations. Thus, in addition to the concrete tone appears the tone in its abstract definition so that because of this double meaning of the term tone it can be said that the octave is a harmonic relation between identical tones. It is therefore quite consistent and appropriate if the respective tones are given the same name.7

The musician usually imagines the matter the other way round: The identity of two harmonies seems logical to him insofar as the harmonies consist of the same tones. And that these tones are equal is not familiar to him as a harmonic abstraction, but by the names under which the tones were introduced to him. That works very well in practice. Why racking one’s brain over the absurdity that language rules are supposed to have an effect on the identity of harmonies?

The abstract definition of the tones is also reflected in the designation of the tones of major and minor triads: Root tone, third, and fifth are, regardless of their respective octave position, the only components of these sounds, which exactly therefore are consonant triads in the abstract sense, and not only in the basic form analysed at the beginning, but in all other forms as well. The lowest tone of a chord is therefore not the same as its root tone. This also applies to the harmonic components of the triads: The fourth has its root tone on top, that is, it is the fifth turned upside-down.

Root tone, third and fifth in C major triads

c) Theories About Major and Minor Triads

Gioseffo Zarlino was the first one to express the opinion that major and minor triads are the basic shapes of the harmonics. He opposed the then usual treatment of the thirds as “imperfect” sound combinations. This treatment was based on the Pythagorean tuning, which deduces all tone relations from fifths so that the frequency ratio of the major third does not have the numeric value of

but

In order to prove the harmonic character of major and minor triads, however, he lacked the necessary fundamentals, namely the knowledge about the inner structure of the tones. Actually, he should have realised that he was not capable of explaining the harmonic principle of the consonance. Nevertheless, he realised that consonant tone relations can be demonstrated by dividing a vibrating string by 2, 3, 4, 5, and 6. As a matter of fact, this results in the oscillation ratios of the octave, fifth, fourth, and thirds. This induced Zarlino to assume that the reason for the correlation between the consonance and the observed numerical proportions lies in a specialty of the number six. To that end he applied two traditional thought patterns of the mystical numerology:

First, he claims to know a mathematical quality of the number six, which he calls ‘perfect’. He, thereby, creatively complements the presentation of the known types of numbers like prime numbers etc.: “Perfect numbers are those which can be added up from their parts ...”8 In order to make the assumed perfection clear, he says that the number six has “a certain measure whereby nothing is too much and nothing too little.”9

Second, he calls attention to the role of the number six in many important contexts. This starts in the Holy Scriptures right off with the story of creation and is then soon broken off for good reasons: “It would take a long time to recount all the things in detail that come to a completion with the number six.”10

The effort to prove specific tone relations to be necessary manifestations of the consonance never gets to the essence of the consonance in this way because it does not thematise its property, but instead the property of a number. As an explanation of major and minor triads, it is offered that they be attested by laws that are supposed to prevail regardless of any music. This is an interpretation within the framework of a worldview.

200 years later, when it was known that the musical tones consist of partial tones, nothing more would have stood in the way of exploring the consonance. In fact, the major and minor triads were associated with partial tones, but merely as a continuation of the endeavour to substantiate the aesthetic quality of these triads with outside existing phenomena. The inner structure of the tones serves from now on in speculative music theory as a model, as a pattern of nature, to which harmony is imitated. The appeal of this model lies in its character as an undoubted and experimentally verifiable natural phenomenon. This is supposed to guarantee a harmonic quality separately from the examined harmonies.

Very common is the interpretation of the major triad as a replication of the fourth, fifth, and sixth partial tones. In the case of the minor triad, one discovers either a direct model that is given in the relation of the 10th, 12th, and 15th partial tones; or one believes that only dyads are “naturally” prefigured in the overtone series, namely the thirds, from which major and minor triads are then “artificially” put together; or one imagines a reverse “undertone series”, which, however, is not detectable in nature, but instead fits well into the imagination of a “truth of nature of the dual harmony principle ...”.11 The speculative musicology can draw on an inexhaustible pluralism of such theories, which classify themselves within a factional dispute between monists and dualists and which are supported in this classification by an academic reporting system that is rather striving for a rubrication of theories than for a scientific explanation.12

The contrasts of the theories on major and minor triads are of a purely ideological nature. The opponents agree in the ideological starting point of the debate: What they do not notice about harmony is that it is the work of aesthetically interested subjects and their materialism eager for sound enjoyment; but instead they assume the effective force of a higher principle given in nature, according to which the whole world is constructed in the truest sense of the word. The most popular arguments are therefore confessions to the respectively own world view, as can be seen from the following polemic against dualism:

“Everything earthly is looking for a support point ... However, if the nature only builds upwards and lets grow, but never also ‘symmetrically-contrarily’ downward, then the ‘will of nature’, presented for proof, is non-existent, and the proof is untenable.”13

2. Tonality

a) Cadence

Major and minor triads, the completed shapes of the consonance, enter for their part into harmonic relations: As triads that stand in fifth ratios to each other, they form the elements of tonality. This kind of harmony initially manifests itself in a succession of consonant triads that is called cadence.14

Cadences in A major and A minor

Jean-Philippe Rameau has coined the terms tonic, dominant, and subdominant for the sounds that stand in such a constellation. In the above example, the tonic (T) has the root tone A, the dominant (D) has – a fifth higher – the root tone E, and the subdominant (S) has – a fifth below the tonic root tone – the root tone D. The essence of the cadence, however, does not lie in the relation of their root tones, but in the harmony between the complete triads.

Taken by itself, these triads are merely the – basic or compound – consonances analysed in the first chapter: They have their foundation in the immediate harmonising of the tones, but exactly therefore, they are determined abstractly, that is, independently of the form of the triads and the octave position of the tones. The consonance form as such is exactly not what characterises the tonic, dominant, and subdominant, since these are defined by their interrelation, in a similar way as the tones of a triad are defined by their interrelation.15

If one considers only the relation of the root tones of the dominant and subdominant to the root tone of the tonic, one again has to do with a consonance here. It is the same consonance that also in the major and minor triads themselves provides the strongest contribution to the harmony of the triads. The fact that these root tones do not sound simultaneously, but one after another, does not change anything in terms of their harmony, whose essence, as already stated, does not lie in the mere vibration ratio of 3:2, but in the accordance of the sound parts of different tones. But, as said before, it is not about the mere relation between the root tones, but about a harmony between complete major and minor triads.

Partial tones of a tonic and dominant in major

In the above example, in which the sound spectra of two successive major triads are compared, 10 out of 15 partial oscillations of the dominant coincide with those of the tonic. In this respect, one is tempted – and the author himself succumbed to this temptation in the beginning – to characterise tonality as a consonant relation of consonances.16 But one must actually rather speak of a quasiconsonant