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Craft of Musical Composition By Paul Hindemith

"Fortunately the situation is not what we would be lead to believe by many harmony textbooks which make of harmony a deep and difficult science, almost a secret art. On the contrary, harmony is a simple craft, based on a few rules of thumb derived from facts of history and acoustics. Rules simple to learn and apply if they are not obscured by a cloud of pseudo-scientific bombast" Paul Hindemith. (1895-1963)

As a performer, Paul Hindemith played every instrument of the orchestra, as a composer he wrote pieces for almost every type of ensemble from opera to guitars, and was one of the most stimulating teachers of his day. As a music theorist, he attempted to construct a thorough, yet concise method by which one can acquire the skills required for musical composition, and produced musical textbooks that have remained essential reading for the student and professional musician since their publication in the 1940's. In The Craft of Musical Composition Book I: Theory, Hindemith purposes a theory that replaces the diatonic scale (the minor mode of which Ramaeu never derives with complete satisfaction) with the chromatic scale as the basis for harmonic and melodic progressions, abandons Ramaeu's tertian system of chord generation, and modifies his theory of chord inversion and the fundamental bass.

Paul Hindemith was born in Germany in 1895. An internationally acclaimed composer, he had a distinguished career as a performer, teacher, writer, and lecturer, and he organized festivals devoted to the performance of modern music. In 1938 as a leading figure in the musical avant-garde movement, his compositions were banned by the Nazis as "decadent art". Hindemith went into involuntary exile in Switzerland and eventually emerged at Yale University in New Haven Connecticut in the United States.

While at Yale, Hindemith produced a series of textbooks devoted to musicians, in which he attempted to construct a thorough, yet concise method by which one can acquire the skills required for musical composition. In Elementary Training for Musicians, Hindemith drew from the great master Bach, and his keyboard exercises for two, three, and four obligato parts. In those exercises, Bach provides a method (exercises) in which one must develop independence of the left and right hand, or in our terminology, independence of the left and right hemispheres of the brain , in order to play the difficult and numerous two part inventions, three part sinfonias and four part fugues. Bach's exercises were intended not just for the player of keyboard instruments but as Bach himself said in the title page of Inventions "at the same time not only to compose good inventions, but to develop them well" and to "acquire a taste for the elements of composition". Hindemith recognized as Bach did, that it is necessary for the composition of counterpoint to develop left and right hemisphere independence in the student, and to this end Hindemith followed Bach in creating a method in which one must develop independence of the left and right hemispheres in order to sing and play the numerous and difficult exercises. Bach's exercises were for, as he put it "lovers of keyboard music, and especially those desiring to learn to play", whereas, Hindemith's exercises are designed for "musicians" in general. Instead of difficult fingering passages for keyboard as in Bach's exercises, Hindemith includes exercises in rhythm, pitch, and musical notation as well as a section devoted to musical dictation. The exercises in Elementary Training for Musicians are difficult, and the sheer number requires a considerable effort. In this work Hindemith's exercises "provide all a musician needs as a preparation for higher theoretical and practical studies, offered without detours and evasion".

To continue in that spirit, Hindemith provides textbooks for the student who "is prepared for higher theoretical and practical studies" in The Craft of Musical Composition Book I: Theory, Book II: Exercises in two Part Writing, and Book III: Exercises in Three Part Writing. In books II an III, Hindemith provides exercises for the composition of two and three obligato parts. Unlike Bach however, Hindemith provides a theoretical description of his exercises in the first book of the series; Book I Theory. Further, in it he takes up the torch that was handed down from Pythagoras, through Zarlino, Descartes, and Rameau, in the discovery of the nature of harmony, from which he could build a comprehensive system of instruction for the development of the craft involved in musical composition.

The preface of Rameau's Traite de l' Harmonie begins "Whatever progress the art of music may have made amongst us, it would appear that the more the ear becomes sensible to its marvelous effects the less is the desire manifested to understand it's true principles, so that one may say that reason has lost its rights, while experience alone has acquired any authority, the writings which remain from the time before Zarlino sufficiently prove that reason alone has procured for them the means of discovering the greater part of the properties of music; nevertheless, although experience makes us still approve of the majority of the rules which they have given us, we today neglect all the advantages that we might derive from reason in favor of empirical methods which relate solely to practice".

The preface of Hindemith's Craft of Musical Composition begins with the a quote from the forward of Johann Joseph Fux's eighteenth century textbook on counterpoint called Gradus ad Parnassum. Fux begins "Perhaps some will wonder at my undertaking to write about music, when there are at hand the opinions of so many men who have written learnedly and sufficiently about it, and particularly at my doing so at a time when music has become an almost arbitrary matter, and composers will no longer be bound by laws and rules, but avoid the names of School and Law as they would death itself...".

Hindemith continues his remarks with the question "what is the simplest, most practical raw materials for composition". Hindemith concludes that this musician would consider a scale, undoubtedly the major and minor scales, with its seemingly inexhaustible supply of melodic and harmonic combinations, as the supplier of all that is known to him, and can be classified by him. However, as Hindemith points out, the ancients made use of other scales and other cultures today continue to use scales that bear little resemblance to our major and minor scales. Rameau attempted to discover the natural origins of the major and minor scales, with the diatonic function of chords, and Dominant-Tonic resolutions Hindemith does not, instead he embarks on a search for a new scale that will serve not exclusively melodic purposes as with the ancients, but also to accommodate our modern harmonic needs. A scale that cannot accommodate harmonic relationships, such as the Medieval modes, are useless for our needs. Polyphonic music demands that notes must be able to change their "tonal significance". A tone that has already served as a third must also be able to become a root, fifth or seventh in a succession of chords. Because it is impossible for one tone perform all those functions without varying it's pitch, purity of intonation must be sacrificed.

Hindemith continues in his treatise, as did Rameau, with the division of the string of a monochord. The intervals, Hindemith tells us, are embedded in the tonal raw materials that nature provides for us. The first 6 notes of the overtone series or the "scenario" clearly outline the extended major triad. Hindemith proposes a new theory in which he uses Rameau's methods to demonstrate the natural origins of the chromatic scale. He arrives at a formula that is the key to all the calculations necessary to build his scale: "To arrive at each tone of the scale, divide the vibration number of each overtone successively by the order number of the preceding tones in the series." In other words the original tone, in this case C, must be able to be treated as a root, or a third, or a fifth, so that C is the third of Ab and the fifth of F. By applying this formula to the note C, that is, treat it as a third or fifth as well as a root, Hindemith generates the new notes Ab and F. Hindemith then applies the formula to the note G, being the closest note to C, and treating G as a root, third, or fifth of a chord he generates a new note an Eb. E is the next closest note to C, (it is the fifth overtone) and Hindemith applies his formula to generate the new note A. The overtones above the seventh partial cannot be used and Hindemith devotes several pages of explanation as to the reasons, but suffice it to say, he rules out their use for the derivation of the remaining scale tones. Hindemith then treats each of the new notes in turn, in the same way he treated the original notes. Thus he completes a scale of adjacent half steps through to the octave.

For practical purposes the ear hears all the half steps as equal in size, although they are in proportions that occur in the third and fourth octave of the overtone series, and are not all even throughout. Guided by their subconscious feelings for the intervals, singers and players of non fixed pitch instruments differentiate quite largely between large and small whole steps and half steps. In equal temperment, "there is no such adaptation to the fundamental tone, and thus to the sensitive ear, music performed on keyboard instruments lacks the fine luster of the light that falls at ever changing angles as it is cast by different generators" There follows a detailed discussion of the "comma" as it relates to the proper intonation of intervals. With this, Hindemith has established the chromatic scale with its temper tuned equivalent; as a natural basis for the development of a system of harmonic generation.

Hindemith had revised the "scenario" and derived what he called series 1, in which each note of the chromatic scale is arranged in descending order according to the significance of each tone in relation to the progenitor tone. The strength, or "perfectness" of each note is represented by its relative position in the series. The first interval in the series is the octave which is the strongest followed by the fifth which is not as strong, followed by the fourth, which is less strong again and so on. This is a pattern that would be familiar to Zarlino, Descartes, or Rameau.

Hindemith then turned his attention to an examination of Rameau's theory of the fundamental bass and the inversion of chords. Hindemith reasoned that music arises out the combined effect of at least two notes, performed simultaneously or consecutively. The interval, he continues, is the basic unit of musical construction. He then introduces series 2, which corresponds to the relative strength values of individual intervals, as they relate to each other. Series 1 differs from series 2 in that series 1 determines the relative strength of each note as it pertains to the original progenitor note, or the relative distance to a tonic note. Series 2 evaluates the relative distances of the two notes to each other.

When any instrument or instruments, including voices, play any simultaneous group of tones, additional tones are involuntarily produced which Hindemith identifies as combination tones. The presence of combination tones have been known since before the time of Rameau and were similar to the fundamental bass notes he perceived present in all intervals and chords. Hindemith further discovered that the combination tone of any interval can be determined with the following formula: "the frequency of the combination tone is always the difference between the frequencies of the directly produced tones of the interval" Combination tones are real tones that "as component parts of the sounding intervals, they produce further combination tones, which are, of course, less intense than the first ones".

Hindemith, using combination tones, demonstrated that the intervals pair up according to their combination tones. For example, the octave and the unison have the combination tone at zero point, the fifth and the fourth have the same combination tone but in the interval of the fourth, the combination tone is above the lower sounding note, whereas in the fifth, it is below the lowest sounding note, providing acoustical proof of the invertability of intervals. For example, the interval C - G produces the combination tone C, an octave below the sounding C, whereas the interval G - C produces the combination tone C above the G. The lower position of the combination tone provides the fifth with its stability, whereas, in the interval of the fourth, the combination tone occurs above the lower note. While the root of the fourth is the same as the root of its "inverted" counterpart the fifth, in the interval of the fourth, the strength of the root is weakened because of its placement in a higher octave. All of the other intervals are similarly paired with their inversions in a manner that has been taught to every first year student of music theory since the days of Rameau.

The combination tone of the tritone produces a fifth which combines with the tritone to make a Dominant Seventh chord, in which the two notes of the tritone make up the third and seventh of the chord. The tritone always has a dominant effect and is which note of the tritone is the leading tone without first having established its family relation. On the one hand the tritone has an ambiguous, and opaque character, and on the other it has a most unambiguous tendency toward resolution. The presence of combination tones, which were known of at the time of Rameau, has provided Hindemith with empirical evidence that the principle of the inversion of intervals is of a natural origin. Hindemith established the principle that all intervals have a root, determined by the total of the frequencies of the two notes and the combination tones in the formula given above. Again, Hindemith has furnished mathematical evidence of the physical presence of a fundamental root of all intervals and chords.

Hindemith isolates several points in which he feels that the conventional theories of harmony appear too narrow a system for the construction and analysis of chords. First, construction in thirds must no longer be the basis of chord construction. He replaces the tertian system with a system derived from series 2, where chords can be placed into groups and subgroups according to the values of the intervals that comprise those chords. In this way any combination of intervals can be measured for effect according to its intervalic makeup. The first category divides all chords into those with tritones and those without tritones. The chords without tritones do not have the same tendencies for as, chords with seconds and or sevenths, chords with two or more tritones in them, etc...

Another important aspect of conventional harmony is the inversion of chords. In traditional theory, the inversion of a chord is never as strong as the chord in its original position. The fact that one chord had to be related to another chord as the inversion of the original "perfect" chord has "prevented the comprehensive use of the principle of root transposition". The ranking of all chords into two main categories where they are further ranked according to their component intervals, and, the position of the root within the chord, allows for any combination of tones and eliminates all ambiguity in their classification, description and analysis. No longer are chords considered with varying interpretations depending on its supposed root as in Rameau's theory of chords by supposition. Hindemith continues with detailed analysis into the values of chords, and includes a table with all categories clearly laid out, in which any combination of tones can be evaluated on the relative values of the constituent intervals that comprise that tonal combination.

As Rameau said, a real science of harmony must explain harmonic progressions, not just chords as single entities. Hindemith begins his discussion of chord progressions with a discussion about counterpoint and the two-voice framework. Of the two-voice framework, he calls it the "skeleton which gives the chords their necessary contour", and as he says, "this framework is constructed by the bass voice and the most important of the upper voices". He further analyses the root progressions, determines the groups and sub groups of each of the chords, and discusses the fluctuation of the increases and decreases of tension according to their ranks in the chord table, of each of the chords in a progression. In addition there is a discussion on progressions involving tritone chords.

Hindemith concludes this part of his discussion with an investigation harmonic family relationships. He extracts the roots of the chords and works with them instead of the chords themselves. The succession of roots are now analyzed. A succession of roots forming broken chords are susceptible to the influences of the overall effects of the combination tones that they generate. This relationship can be effected by rhythmic considerations, and the various chords erected on the roots in question. A chord of group I will always try to act as the tonal center to the chords of lower values. The root progression of successive chords in group III and IV are more susceptible to the effects of rhythm and of harmonic fluctuation, and chords of group V or VI are almost always effected by so as to dispel their uncertainty.

A discussion of the family relationship and the construction of tonal spheres ensues. A chord progressions that contains a succession of chords from group A (triads in root position) must consist of at least three chords to represent a tonal entity. The final chord of progression enjoys a prominent position of a group. When using chords exclusively from group II (chords with tritones and major seconds or minor sevenths) the tonal center is not very stable because the dominant effect does not completely satisfy as a tonal center.

Hindemith devotes several pages to a discussion of cadences, which he describes as "chord progressions of which the effect is strongly final", followed by an analysis of larger harmonic relations, in which the roots "are the supports of a wide tonal structure of which the thrusts and tensions act in accordance in with the family relationships of series 1". He continues "The roots which support the burdens of larger harmonic groupings may be called degrees, and their succession in accordance with the demands of series 1 the degree progression".

After a discussion about modulation, Hindemith makes comments concerning music that is usually referred to as atonal. Phrases like "we have seen that tonal relations are founded in nature and "tonality is a natural force, like gravity", and "we may assert that there are but two kinds of music: good music, in which the tonal relations are handled intelligently and skillfully, and bad music, which disregards them and consequentially mixes them in aimless fashion", expresses his feeling on the popular use of the term atonality. Hindemith adds "There are today a considerable number of composers who issue works that they call atonal. To what extent the atonality of these compositions rests upon the lack of a convincing degree progression and to what extent it is a more or less developed tonality concealed by an uninterrupted succession of sharp sonorities, the reader himself can determine by extracting the degree progression of such pieces".

The remainder of the work is devoted to the practical application of his principles, and corresponds to Book II: Exercises in two part Writing, in which the exercises for two voiced settings are used as a two-voice framework (constructed by the bass voice and the most important of the upper voices) in Book III Exercises in Three Part Writing. "Fortunately the situation is not what we would be lead to believe by many harmony textbooks which make of harmony a deep and difficult science, almost a secret art. On the contrary, harmony is a simple craft, based on a few rules of thumb derived from facts of history and acoustics. Rules simple to learn simple to learn and apply if they are not obscured by a cloud of pseudo-scientific bombast". To this end he published two volumes of exercises devoted to Traditional Harmony with Emphasis on Exercise and a Minimum of Rules vol. I and II . The title of the work provides an ample description of its contents.

A German composer, theorist, teacher, violist, and conductor, Paul Hindemith purposed a theory that replaces the diatonic scale with the chromatic scale as the basis for harmonic and melodic progressions. He further abandoned Ramaeu's system of chord generation by means of added thirds, and modified his theory of chord inversion and the fundamental bass. He attempted to construct a thorough, yet concise method of exercise by which one can acquire the skills required for musical composition, and produced musical textbooks for students and professional musicians that are considered one of the outstanding works of our time.