Does Music Represent Feelings?

Chapter II of The Beautiful in Music 

  1. The theory that the ultimate aim of music is to excite feelings is, in an amended form, partly responsible for the proposition that feelings are the subject which music represents. 
  2. A philosophical examination of an art demands a clear definition of its subject matter. The various arts display a diversity of subject matter. The fundamental differences in their mode of treatment are a natural result of the dissimilarity of the senses to which they severally appeal. Every art has a range of ideas which it expresses after its own fashion in sound, language, color, stone, etc. So each work of art embodies a definite idea of beauty in material form. This idea, its embodiment, and their unity are the conditions of beauty involved in a critical examination of each art.
  3. The subject of a poem, a painting, or a statue can be expressed in words that express concepts. We say, for instance, this picture represents a flower girl, this statue represents a gladiator, this poem presents one of Roland's exploits. The more or less perfect embodiment of the particular content in the artist's production underlies our judgment of the beauty of the work of art.
  4. There has been almost unanimous agreement that the whole gamut of human feelings is the subject of music, because the emotions were thought to be opposed to the definiteness of intellectual conceptions. This opposition was supposed to be the feature by which music is distinct from the ideal of the other fine arts and poetry. According to this view, sound and its elaborate combination is only the material, the medium of expression, by which the composer represents love, courage, piety, and delight. These various emotions constitute the idea which, on being embodied in physical sound, takes on the earthly form of a musical composition. The lovely melody and the skillful harmony do not charm us for their own artistry, but only for what they refer to: the whispering of love, or the violence of combatants. 
  5. In order to make progress, we must first get rid of all these old clichés. The whispering may be in the music, but not the whispering of love; violence may be reproduced, undoubtedly, but not the battle of two combatants. Music may reproduce whispering, storming, and roaring. But the feelings of love or anger are only present in the mind of the listener.
  6. The presentation of specific feelings and emotions is not among the powers of music.
  7. Our emotions are not sufficiently isolated in the mind in order to be presentable by an art which is incapable of representing the other relevant mental activities. They are, on the contrary, dependent on physiological and pathological conditions, and dependent on ideas and judgments--indeed, on all the cognitive processes which so many people regard as opposed to the emotions. 
  8. What, then, makes an indefinite feeling into a definite one, for instance, into the feeling of longing, hope, or love? Is it the mere degree of intensity, or the fluctuattions of our inner activity? Certainly not. These inner states are similar in very different feelings, and the same feeling will vary from time to time and from person to person. Only by virtue of a number of ideas and judgments (unconscious ones when our feelings are strong) can our state of mind embody one or another definite feeling. The feeling of hope is inseparable from the conception of a future state which is happier than our present state. The feeling of sadness requires a comparison of a past state of happiness with the present. These are definite representations, involving specific concepts. In the absence of  this cognitive apparatus of thought, we cannot label a feeling as "hope" or "sadness." Cognition alone gives a feeling its definite character. If we take away the cognitive element, nothing remains but a vague stirring, which at best counts as a general sense of satisfaction or distress. Love cannot be conceived apart from the thought of a beloved person or being, or apart from the desire and the longing for the happy possession of the particular object of our affections. It is not simply a case of agitation, but its cognitive basis, its subject, constitutes it as a feeling of love. Love's dynamic may be gentle or impetuous, buoyant or depressed, and yet it remains love. This fact alone is sufficient to show that music can only express the qualifying [dynamic] adjectives, and not the substantive thing, love itself. A specific feeling (a passion, say, or an affect) never exists without a definite content, which can only be communicated in conceptual terms. Now, it music is an "indefinite form of speech," then there is no compromise rendering it capable of expressing definite feelings. For the definite character of an emotion lies in its conceptual basis.
  9. How music can excite feelings (can, not must!) such as sadness and joy we shall discuss later, when we examine the subjective effects of music.  At this stage of our inquiry our goal is to determine whether or not music is ever capable of representing a definite emotion. This question must receive a negative answer. The definite specification of an emotion is inseparable from actual representations and concepts, and it is altogether beyond the power of music to present these concepts. However, a certain sort of ideas can be adequately expressed by the resources of music proper. 
  10. This class of ideas is consistent with the organ which receives them. They involve audible changes of strength, motion, and proportion. They include the ideas of intensity, both increasing and diminishing, and of accelerating and decelerating motion, and of both ingeniously complex and simple progression, and so on. The aesthetic expression of a piece of music may be described appropriately as graceful, soft, violent, powerful, elegant, and fresh. All these ideas have corresponding sensuous embodiments in sound relationships. So we can use these adjectives as to describe music without thinking about their usual ethical implications,  which, from the habit of associating ideas, we readily confuse with purely musical properties.
  11. The ideas which a composer expresses are primarily musical. His imagination apprehends a particular beautiful melody. The melody exists simply as a melody, without purpose beyond itself. Yet as every concrete phenomenon suggests the general class to which it belongs, and then some still more general concept in which the latter is included, and by continuing this process we arrive at the idea of the absolute Idea. So also for musical ideas. This lovely adagio [slow movement], for instance, softly dying away, suggests the general idea of gentleness and harmony. Our imagination is always ready to refer any artistic idea to peculiarly human sentiments, so it will construe the softly ebbing music in another, higher sense, e.g., as the mellow resignation of a person; and perhaps then on to a higher level, treating it as a suggestion of everlasting rest in the world beyond.
  12. Likewise, poetry, sculpture, and painting must first produce a concrete image. The picture of a flower girl can indirectly imply the less specific notion of girlish contentment and innocence. The picture of a snow-covered churchyard can indirectly suggest the transitory nature of earthly existence. Music is different. Music only vaguely and arbitrarily directs the listener to the idea of youthful innocence or that of transitoriness. But these abstract ideas are never the content of the musical work any more than they can be the actual subject matter of the aforementioned pictures. It is absurd to talk of the musical representation of feelings of earthly transitoriness or of youthful innocence. 
  13. There are ideas which are fully expressed by music, but these ideas do not occur as feelings. Conversely, feelings can affect our minds that have no corresponding representation in musical ideas.
  14. What aspect of feelings can music represent, then, if music cannot present their conceptual content? Only their dynamic properties. Music can reproduce the motion of a physical process by presenting is accompanying momentum: its speed, slowness, strength, weakness, increasing and decreasing intensity. But motion is only one aspect of feeling, not feeling itself. It is generally supposed that the representational power of music is sufficiently explained by saying that, although music cannot represent the particular object of a feeling, it represents the feeling itself, e.g., not the person loved, but Love. In fact, music can do neither. It cannot depict Love. It can only whatever motion is associated with love. This motion may occur in other feelings and is incidental to love. Love is as abstract as Virtue or Immortality. Theoretical assurances that music cannot express abstract notions are pointless: No art can do this. Obviously, only ideas that have assumed a concrete form can become artistic content. But instrumental music never represents the ideas of love, anger, or fear, because there is no necessary link between these ideas and particular combinations of sound. So which of the elements inherent in these ideas does music turn to to capture them effectively? The only element available to music is motion (taken in the wider sense, of course, which includes the increasing and decreasing force of a single note or chord ). Motion is the element which music has in common with our emotional states and which it shapes to creatively in an endless variety of forms and contrasts. Though the concept of motion has previously been ignored in theories of the nature and effects of music, it appears to be the most important concept.
  15. When anything else in music seems to depict feelings, it  is symbol. Tones, like colors, have intrinsic symbolical meanings, producing their effects independently of and in advance of any artistic intentions. Every color has its own unique character. It is not a mere cipher into which the artist confers some chosen meaning. It has a power which Nature herself has established a sympathetic connection with some of our mental states. Who does not know the simple meanings of colors, established in the popular imagination, which cultured minds elevate into poetic refinement? Green is associated with a feeling of hope, blue with loyalty. Rosenkranz recognizes graceful dignity in orange, hypocritical niceness in violet, etc. (Psychologie, 2nd ed., p. 102.)
  16. Likewise, the basic elements of music, such as the various tonalities, chords, and timbres, have their own characters. We tend to be too industrious in thus interpreting the meanings of musical elements. Schubart's symbolism of the keys in music forms a sort of musical counterpart to Goethe's interpretation of colors. However, when these elements (sounds, colors) are employed and combined for the purposes of art, their artistic application is subject to laws quite distinct from those governing them in their isolated action. In a historical painting we do not construe each bit of red in it as always meaning joy, nor every spick of white as meaning innocence. Likewise in a symphony. The key of A flat major does not always indicate romantic feelings, the key of B minor always misanthropic ones, every triad a feeling of satisfaction and every diminished seventh a feeling of despair. Aesthetically speaking, these rudimentary associations vanish when subordinated to other organizing principles. [In a work of art], the basic intrinsic relations do not count as expression or representation. These elements are "symbolic" because their subject is not directly represented. They represent something essentially different. If yellow presents jealousy, if the key of G major is cheerful, it the cypress tree mourns, it is because these interpretations are based on a definite psycho-physiological relation to definite feelings -- but these meanings exist only in our interpretation, not in the color, the sound, or the plant themselves. Isolated chord never represent a determinate feeling, and much less so when it is hear within the musical work as a whole. Music has no means for fulfilling its alleged mission of conveying feeling than through the analogy of motion and the symbolism of tones.
  17. Seeing, then, that the nature of tonal elements demonstrates the inability of music to represent definite feelings, it seems almost incredible that this fact is not generally accepted.  Let all whose heartstrings resonate when listening to some piece of instrumental music explain which feeling is the content of that music. This test is essential. Listen, for example, to Beethoven's "Prometheus" Overture. The attentive musical ear successively hears, in this sequence, the following: The tones of the first bar, then following a fall into the lower fourth, a gentle upward motion in rapid succession, then the same movement repeated in the second bar. The third and fourth bars continue further upward. Water droplets propelled by a fountain trickle down and rise once more, only to repeat, as the following four bars repeat the figure of the preceding four. So the listener perceives that melodic symmetry between the first and second bars, then again between these two bars and the succeeding two, and finally there is the wider arc of the first four bars and the corresponding arc of the following four. The bass which indicates the rhythm marks the beginning of each of the first three bars with a single beat, of the fourth with two beats, and the same rotation is observed in the next four bars. The fourth bar, then, is different from the first three, and, this point of difference establishes a symmetry with the following four bars, agreeably impressing the ear with this equilibrium between the old and the new. The harmony of the melody exhibits the same correspondence of one large and two small arches: the C major of the first four bars corresponds to the four-two chord in the fifth and sixth bars, then to the six-five chord of the seventh and eighth bars. This correspondence of melody, rhythm, and harmony generates a symmetrical yet varied structure, enriched by additional gradations of light and shade through the timbres of the various instruments and the variations in volume:
  18. Beyond this, we cannot point to anything additional in the theme, at least not anything that counts as the feeling it represents or necessarily arouses in the listener. Admittedly, any analysis of this kind makes a mere skeleton out of a living thing, and it destroys its beauty. But it also destroys all unjustified interpretations.
  19. No other instrumental music fares any better than this randomly selected passage. Many music lovers think that it is an exclusive characteristic of the older, "classical" music that it does not represent feelings, and it is immediately admitted that nobody seeks a feeling as the content of any of the forty-eight preludes and fugues of J. S. Bach's Well-Tempered Keyboard. However, this distinction [between older and more recent "classical" music] is both simplistic and arbitrary. It is explained by the fact that the older music more obviously affords aims at nothing beyond itself, so that interpretations of the kind mentioned would, in this case, prove too difficult to merit the effort. The existence of such music is sufficient to prove that music does not necessarily awaken feelings and present feelings as its content. Otherwise the whole domain of figural counterpoint must be ignored. But if huge domains of aesthetically and historically important art have to be ignored for the sake of a theory, then that theory is false. A single leak will sink a ship, but those who find that to be too slow are at liberty to knock out the whole bottom. Let them play the theme of a symphony by Mozart or Haydn, an adagio by Beethoven, a scherzo by Mendelssohn, a piano piece by Schumann or Chopin. Choose anything, in short, from our substantial music. Or select a more popular theme from the overtures of Auber, Donizetti, Flotow. Who is bold enough to identify a definite feeling as the content of any of these themes? One person will say "love." He might be right. Another thinks "longing." Perhaps. A third feels "religious devotion." Who can contradict of any them? How can we call any music a representation of a definite feeling when nobody agrees on what is represented? Probably all will agree about the beauty or beauties of the composition, whereas all disagree concerning its content. To represent something requires exhibiting it clearly and distinctly. But how can we specify the content that is represented by art when we find that the vaguest and most ambiguous elements of that art are forever the basis of our disagreement?
  20. We have intentionally selected instrumental music for the examples, because only what is true of instrumental music is true of all music as such. If we wish to examine music's own nature and properties, to determine its limits and purposes, then should consider instrumental music. What instrumental music can never do, music can never do, because it alone is music purely and absolutely. We may regard vocal music as superior to or more effective than instrumental music, or the reverse--an unscientific preference, by the way, which is prompted by one-sided dilettantism--one must admit that the term "music" does not strictly apply to music composed for words. In vocal or operatic music it is impossible to separate the effect of the music from that of the words so that an exact assignment of the poetical and the musical can be assigned.  An account of musical "content" must also ignore compositions with descriptive titles as well as so-called program music. Its union with poetry enhances its power without widening its boundaries. 
  21. Vocal music is a fused compound, and it is impossible to determine the relative importance of each of its constituents. If we were discussing the nature and function of poetry, it would never occur to anyone to cite opera as an example. Although it requires a greater effort, it requires no deeper insight to do the same when discussing the fundamental principles of musical aesthetics.
  22. With vocal music, the music provides colors, as it were, to the black-and-white of the poetic drawing. In the musical elements we find both splendid and delicate hues, with associated symbolic meanings. Thus music might transform second-rate poetry into a passionate expression of feeling. However, it is not the music but the text which determines the subject. It is not the coloring but the drawing which determines the subject of the picture. 
  23. We appeal to the listener's capacity for abstraction. Choose a dramatically effective melody. Imagine the melody apart from the context of the words. A dramatic melody, for instance, which effectively represents anger, will now possess only a quick, impulsive motion. The same melody can be effectively combined with words expressing passionate love, diametrically opposed in meaning. When thousands (including Jean Jacques Rousseau) were moved to tears by the air from Orpheus:
  24. J'ai perdu mon Euridice, Rien n'égale mon malheur! 
  25. Boyé, a contemporary of Gluck, observed that precisely the same melody would work equally well, if not better, with words conveying exactly the reverse, thus:
  26. J'ai. trouvé mon Eurydice, Rien n'égale mon bonheur! 
  27. We are not sure in this case that the composer is quite free from blame, inasmuch as music certainly possesses other tones which more truly express a feeling of profound sorrow. If, however, we select this example from among innumerable instances, we have done so because, first, it concerns the composer [Gluck] who is credited with the greatest accuracy in dramatic expression; and, second, because several generations have praised this very melody for its rendering of the intense grief which it expresses in its words.
  28. But even far more definite and expressive passages from vocal music, when considered apart from the text, enable us at best to guess the feeling they are intended to convey. They resemble a silhouette, the original of which we recognize only after being told whose likeness it is.
  29. What is true of isolated passages is true also in a wider application. There are many cases where an entirely new text has been employed for a complete musical work.  . . .  Has the reader never heard the fugato from the overture to Mozart's The Magic Flute changed into a vocal quartet of quarreling Jewish peddlers? Mozart's music, though not altered in a single note, fits the low text appallingly well, and the enjoyment we derive from the gravity of the music in the opera can be no heartier than our laugh at the humor of the parody. Countless examples are available to demonstrate the adaptability of every musical theme and human emotion. The feeling of religious devotion is rightly considered to be the least liable to musical misconstruction. Yet in countless German village and country churches, the organ plays pieces like Proch's "Alphorn" or the finale from the [the opera] La Somnambula (with the leap of a tenth for "into my arms") during Eucharist. Germans who visit churches in Italy are astonished to hear the most popular themes from operas by Rossini, Bellini, Donizetti, and Verdi. Pieces like these and of a more secular character, provided they are played with a quality of sobriety, do not disturb the devotions of the congregation, who, on the contrary, appear to be greatly edified by them. However, if music in itself were capable of representing the feeling of piety, this kind of quid pro quo would be impossible; it would be like having the preacher reciting from the pulpit a novel by Tieck or an act of Parliament in place of the sermon. 
  30. The greatest masters of sacred music offer abundant examples in support of our proposition. Handel, in particular, went to work in composing with the greatest nonchalance. Winterfeld has shown that many of the most famous melodies in The Messiah, including some most admired as expressions of religious feeling, are transcribed from secular duets (mostly erotic) which Handel originally composed in the years 1711-1712, when he composed madrigals from texts by Mauro Ortensio for the Electoral Princess Caroline of Hanover. The music of the second duet has this text: 
  31. "No, I will not trust you/Blind love, cruel beauty,/You are too deceitful,/flattering deity!" 

  32. Handel used the same music, unaltered both in key and melody, for the chorus in the first part of The Messiah, "For unto us a Child is born." The third part of the same duet, "So per prova i vostri a`nganni," has the same tune as the chorus of "All we like sheep" the second part of The Messiah. The music of the sixteenth madrigal (a duet for soprano and alto) is essentially the same music as the duet from the third part of The Messiah, "O Death, where is thy sting?" But the words of the madrigal express a very different feeling. 
  33. There are many similar instances in the music of J. S. Bach, but we need only refer here to the series of pieces in the Christmas Oratorio which, as everyone knows, were naively taken from dissimilar secular cantatas. And Gluck, whose music, we are taught, attained the pinnacle of dramatic accuracy only by fitting every note scrupulously to the situation, indeed, by deriving the melodies from the very rhythm of the syllables of the verse--Gluck has transferred to Armida no fewer than five pieces from his earlier Italian operas (see my The Modern Opera, p. 16). We see that vocal music, which in theory can never determine the essence of music itself, is in practice unable to challenge the principles we derive from instrumental music.
  34. Furthermore, the proposition we are attacking has become so integrated into the current view of musical aesthetics, that any theory deriving from it or similar to it are likewise treated as invulnerable. The prevailing view includes the theory that music can reproduce visual and auditory impressions of a nonmusical nature. Whenever the question of the representation of objects by musical means (Tone-painting) is debated, we are told over and over that although music is unable to portray phenomena which are foreign to its province, it can picture the feelings which the objects excite. However, the very reverse is true. Music can attempt to imitate objective phenomena, not the specific feelings they arouse. The falling of snow, the fluttering of birds, the rising of the sun -- these can be painted musically only by analogy, by producing auditory impressions that are dynamically related to those phenomena. In pitch, intensity, tempo, and rhythm, sounds present to the ear a figure that has an analogy to specific visual impressions which sensations of various kinds bear to one another. Just as there is, physiologically, such a thing as a vicarious substitution of one sense for another (up to a certain point), so there is, aesthetically, some substitution of one sense for another. There is a well-founded analogy between motion in space and motion in time; between the color, texture, and size of an object and the pitch, timbre, and intensity of a tone. So one can portray an external object musically. However, it is ridiculous to portray by musical means the "feeling" which falling snow, a crowing rooster, or a flash of lightning produces in us. 
  35. Although, as far as I can remember, all musical theorists implicitly accept and build on the assumption that music can represent definite feelings, their better judgment has kept many of them from openly endorsing it. The absence of definite ideas in music troubles them and prompts them to modify their principle. They say that music arouses and represent indefinite feelings, not definite ones. A more rational proposal is that music can only reproduce what we termed the dynamic element of a feeling, apart from what is felt, and it is this capacity which we have unhesitatingly conceded to music. But this aspect of music does not enable it to represent indefinite feelings. The terms "represent" and "indefinite" are contradictory terms. Mental motion, considered as motion apart from the content it involves, can never become the subject of a work of art, because there is no way of making them specific without answering questions about what moves or what does the moving. What is true in the proposition--namely, that music does not portray a definite feeling--is only a negative concession. But what is positive and creative in musical art? An unspecified feeling cannot supply content. To employ an unspecified subject, an art would have to face the problem: How should it be formed? But all art individualizes, in making something definite out of what is indefinite, the particular out of the general. The theory that music offers "indefinite feelings" proposes the exact opposite. It offers more difficulties than the earlier theory that music represents something, though it is impossible to know what. From here we are only a step removed from the recognition that music never represents feelings, either specific or indefinite. Yet what musician would surrender this immemorial piece of his art's established domain? 
  36. Our conclusion might allow us to embrace the view that the representation of a specific feeling is a musical ideal that is never wholly realizable in practice. But, as an ideal, is necessary to constantly strive for it. The many high-sounding phrases about music's power to shatter its lack of specificity and to become actual speech, no less than the popularity of compositions aiming--or supposed to be aiming--at this, demonstrate the popularity of this theory. 
  37. Having absolutely denied the possibility of representing feelings by musical means, we must be even more emphatic in attacking the view that it is an ideal that serves as an aesthetic principle of music. 
  38. Even if the representation of specific feelings were possible, the degree of beauty in music would not depend on the degree to which it accurately represents those feelings. Let us, for argument's sake, suppose that it is possible to represent specific feelings with music. Let us examine its practical consequences. 
  39. It is impossible to test this fiction with instrumental music, since I do not recognize the possibility of its representing specific feelings. We must, therefore, make the experiment using examples of vocal music in which the music emphasizes states of mind clearly delineated in the text. 
  40. Here the words determine what the composer is to represent. The music may animate it and provide it with a more or less distinct individuality. The music does so by utilizing characteristics of motion and by exploiting the symbols associated with various tones. When composers keep in mind the prominence of the words and do not strive for independent musical beauty, the music can produce a high degree of individuality of feeling. The delusion may even arise that the music alone expresses the emotion which, though heightened by the music, was already inherent in the words. This tendency [to attribute what is in the words to the music alone] is in its consequences very similar to the alleged representation of a feeling as the content of a particular piece of music. Suppose, hypothetically, that the genuine and assumed power of music co-exist, so that it was possible to represent specific feelings by musical means, as the content of musical compositions. If this assumption is granted, the logical consequence would be to evaluate as best those compositions which perform the task most perfectly. Yet who does not all know works of exquisite beauty without any such content? We need but remember Bach's preludes and fugues. On the other hand, there are vocal compositions which try to portray certain emotions with great accuracy, within the limits just explained, in which case the fundamental goal is success in this portrayal. On close examination of actual cases we find that the most rigorous fitting of music to feeling is generally successful in an inverse ratio to the independent beauty of the music. Rhetorical-dramatic precision and musical success go together only halfway, after which they part company. 
  41. Operatic recitative clearly demonstrates this truth, since it is that type of music which best accommodates rhetorical requirements, down to the very accents appropriate to each individual word, attempting only to be a faithful copy of specific, rapidly changing states of mind. Therefore, accordance to the feeling-theory before us, recitative should be the highest and most perfect music. But in reality, the recitative reduces music to a mere helper and relinquishes its musical individuality. This proves that the representing of specific mental states is contrary to the purpose of music, and that in the final analysis the two purposes are opposed to one another? Play a long recitative, leaving out the words, and ask the audience about its musical merit and meaning. All music alleged to be capable of portraying feeling should be able to stand this test. 
  42. Recitative is not the only culprit. In the highest and most advanced arts, musical beauty tends to disappear in proportion as the expression of some specific feeling is attempted; for the former demands its autonomy, while the latter relegates music to a subservient role. 
  43. [Hanslick give an extended analysis of opera as a sort mixed marriage, the many compromises of which produce no principles that are useful in the other arts.] 
  44. For our purposes, it is enough to denounce Wagner's main idea as stated in the first volume of Opera and Drama: "The mistake respecting opera, viewed as art, consists in the fact that the means (music) is regarded as the end, and the end (drama) is made the means." But this is false, because an opera in which the music is really and truly employed solely as a means for dramatic expression is a musical monstrosity. 
  45. One of the many inferences to be drawn from Wagner's claim (respecting means and ends) is that all composers who combine mediocre librettos [i.e., words] with anything better than mediocre music are guilty of a serious error, as we ourselves must be if we admire such music. 
  46. The union of poetry with music and with the opera is a sort of morganatic marriage [whose children receive no inheritance], and the more closely we examine this morganatic union of musical beauty and pre-existing content, the more skeptical we become about the fate of the marriage. 
  47. How is it that in every song we can make minor alterations that, without in the least detracting from the accuracy of its expression of feeling, immediately destroy the beauty of the musical theme? This would be impossible if the latter [the musical success] were dependent on the former [the accuracy of expression]. Again, how is it that many songs that successfully express the drift of the text are nevertheless intolerable as music? The theory that music is capable of expressing emotions offers us no explanation of these phenomena. 
  48. So what accounts for the beautiful in music, now that we have rejected the accurate expression of feeling as the guiding principle? A completely different, autonomous factor, which we shall now examine.


This version of chapter II is the Gustav Cohen translation of 1891, altered in many places after consulting both the German text and the 1986 translation by Geoffrey Payzant. The Cohen translation is in the public domain and may be freely reproduced. This adaptation is my own work and should not be reproduced without due credit, including this statement.


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