Paul C. Kuntz is professor of philosophy at Emory University.
The following article appeared in Process Studies, pp.43-65, Vol. 29, Number 1, Spring-Summer, 2000. Process Studies is published quarterly by the Center for Process Studies, 1325 N. College Ave., Claremont, CA 91711. Used by permission. This material was prepared for Religion Online by Ted and Winnie Brock.
The author writes that Whitehead put forth a kind of correlation of simple harmony as faith in an order of things and in an order of nature.
I. The Pythagorean traditions of four kinds of harmony: musical harmony the root metaphor
"Harmony" is a word now used only in metaphorical senses.1 In the long and complicated history of this word, which we cannot here trace, the literal sense has been forgotten. The ancient Greek a r m o z . meant joint, and a r m o z e i n meant to arrange. The physician Galen used the abstract noun a r m o n i n in a way defining his orthopedic meaning "the union of two bones by mere apposition" (New 96).2
Joining or arranging presupposes logically that there are different things together. Joining parts of the body allows functioning together, as parts cannot function separately All the metaphorical extensions seem to have this in common: they are cases of the many that become one. The most important metaphorical extension indicates the very essence of music: that there is "agreement [or] concord of sounds" (New, V, 96-98). If music is the chief metaphorical meaning, and the others are controlled by it, then to think of the world as a harmony is to regard everything as if all things sound well together.
In the tradition of both the ancient and the modern world, musical harmony is only one of four kinds. There is, according to the Pythagoreans, a harmony of strings, but also a harmony of body and soul, a harmony of the state, and a harmony of the starry sky In a doctrine of universal harmony; the metaphors go back and forth (Spitzer 8-9). According to Jean Bodin, an eminent Renaissance thinker whose thought in this regard has only recently been recovered from obscurity, held that there is harmony in numbers, harmony in geometric progression, harmony in musical systems, and harmony in nature. Bodin extended the latter harmony in a doctrine that toleration facilitated harmony among religions. Harmony then functions to cure, as in the case of Leibniz and Whitehead, one of the ills of civilization, religious war.
The world then is made up of many harmonies. This seems the essence of the tradition; that there should be four and only four seems somewhat accidental and arbitrary (see various works by Kuntz). The lists of four, as can be seen above, do not coincide. We may indeed regard the starry sky as an aspect of nature, and harmony of strings is one application of a musical system. Should we then conflate the two lists and conclude that there are six fundamental harmonies of the universe? The question of how many harmonies there are should be deferred because it is a function of prior decisions. Bodin distinguished harmony in numbers from harmony in geometric progressions. Yet another philosopher might prefer to group both under the head "mathematical" harmony.
II. Whitehead’s reduction into two basic types of harmony: logical and aesthetic
The simplification of kinds of harmony into a simple correlation is found in a twentieth-century son of Pythagoras, whose thought about harmony deserves exploration. In Science and the Modern World, Whitehead expounds the central idea of Western civilization as faith in an order of things and in an order of nature.
Faith in reason is the trust that the ultimate natures of things lie together in a harmony which excludes mere arbitrariness [. . . .] The faith in the order of nature which has made possible the growth of science is a particular example of a deeper faith. This faith cannot be justified by any inductive generalization. It springs from our direct inspection of the nature of things as disclosed in our own immediate present experience [. . . .] To experience this faith is to know that in being ourselves we are more than ourselves: to know that our experience, dim and fragmentary as it is, yet sounds the utmost depths of reality: to know that detached details, merely in order to be themselves demand that they should find themselves in a system of things: to know that this system includes the harmony of logical rationality, and the harmony of aesthetic achievement: to know that, while the harmony of logic lies upon the universe as an iron necessity, the aesthetic harmony stands before it as a living ideal molding the general flux in its broken progress towards finer, subtler issues. (18)
There is, Whitehead suggests, a harmony of the abstract possibilities with which formal sciences deal, and then another harmony of the changing process that we study in the concrete. Our study of the latter, as in history; is based upon sense.
Whitehead’s principle of moving from the abstract to the concrete seems to be the principle implicit in both the Pythagorean list of four harmonies and in Bodin’s list of four. The musical and mathematical harmonies are cited first, then the mind moves to the harmony of body and soul and the harmony of human society, until finally we think of the richest harmony of the all-embracing world in which we live and think. Let us rather begin with the greatest and most concrete order, and use the principle in inverse form, to end with the least tangible.
III. Nine kinds of harmony: all required in Whitehead’s illustrations
Since, as we have illustrated, "harmony" is a metaphorical term, the meaning must change from one level to another. Things of different sorts are related by different principles. Especially because the following list is based on a richer selection of instances, we need to suggest a sharper clarification of differences (New 96-99).
I propose that we consider the kinds of harmony we commonly speak of under nine heads. I arrived at these nine empirically by thinking of the kinds represented in the English language. It may be more than a coincidence that John Milton celebrated a "ninefold harmony."3 Even Whitehead, who in the above significant passage reduces all harmony to "logical" and "aesthetic," says only that the "system of things includes" these, and elsewhere, as we shall see, he refers to others. At one point my list was of twenty harmonies, but some of these distinctions are in this present list reduced to subtypes.
Type I: Cosmic Harmony
There is a harmony of heavenly bodies. The principle is the regularity and therefore the predictability of motion, as developed in astronomy
Type II: Natural Harmony
There is a harmony of plants and animals living in their environment. The harmonic principle is balance, as in ecology.
Type III: Human Harmony.
There is a harmony between humans in families, businesses, schools, churches, armies, states, etc. The harmonic principle is agreement of interest, as studied in ethics, history, and the social sciences.
Type IV: Harmony of Action
There is a harmony between thought and behavior, or speaking traditionally of substances, between mind and body; The harmonic principle is correspondence. One sort of correspondence is Leibniz’s "pre-established harmony"
Type V: Spiritual Harmony
There is a harmony of individual consciousness. The principle is self-identity as we find it in rational psychology, some forms of philosophy and psychiatry.
Type VI: Aesthetic Harmony
There is a harmony of sensed objects such as sounds, colored things, etc. The principle is aesthetic fittingness of various kinds, as studied in aesthetics.
Type VII: Doxic Harmony
There is a harmony of stories, propositions, doctrines, and theories. Beliefs or statements that are claimed to be true or false are subject to principles of logical laws, such as identity, contradiction and excluded middle. There are also certain epistemological principles of relevance and coherence.
Type VIII: Mathematical Harmony
There is a harmony of numbers, of geometrical shapes such as lines, etc. The principle is proportion that can be stated, as an algebraic formula.
Type IX: Metaphysical Harmony
There is a principle of harmony underlying the unity of things and of process. The principle is teleology, as in metaphysics, that what can be actualized ought to be, and that the fulfillment of possibilities is good.
The above schema is no more than a sketch and could be fully expounded and defended only with a chapter expounding each meaning. Yet I wish to press several distinct advantages over all previous theories of harmony that have come to my attention. Previous lists of kinds of harmony seem to me to be over-simple, and a richer and more complete summation is required. This is a metaphysical advantage: A variety of distinct definitions is what we have in the case of Being and Existence. There are very diverse sorts of relations called "harmonies" and no conventional definition using "agreement" or "correspondence" is helpful in defining such diverse cases. Greater precision is needed. Each of the nine kinds can be further specified into subtypes and each admits of degrees of specificity.
IV All harmonies are relations analyzable in terms of the logic of relations
The table of nine kinds accepts the plurality of meanings and hence does not beg the question whether "harmony" has an identical meaning in every instance. Harmony may be a general name for a family of terms. If there is a common essence, that remains to be demonstrated.
All we can assert at this stage is that harmony has many meanings, yet sufficiently analogous that a common metaphor applies to all. The analogy or sameness in difference remains to be discerned.
Just as we began by saying of any harmony that there are "different things together," so we may now say that all harmonies require relation of a definite sort. All harmonies are orders (see Kuntz, Concept).
If it is true that all harmonies are relations between terms, then we can ask, How many terms are required or allowed? We can also ask, What are the formal properties of the relations? These two leading questions are couched in the familiar language of the logic of relations.
Can there be a one-term relation, or a relation that is reflexive? Stated formally; is there a harmony of the form aRa? The only likely case is form V. A person is said to be in harmony with himself/herself. Is there here really a many that is one? Surely more than one term is required. Are there many feelings, thoughts and actions that are unified by one design or purpose?
The relation of harmony; where there are two terms, aRb, has the formal property of symmetry. Under Type III we speak of the harmony between labor and capital. We might just as well say the harmony between capital and labor, or formally aRb = bRa. Type IV is also a two-termed relation of harmony; Type II, balance, is also two-termed. If a balances b exactly; then we must also say b balances a exactly.
Yet other harmonies require three terms. Type VIII, mathematical harmony; includes "harmonical proportion." Here we have a relation that is an ordering relation in the strict sense of a series, or a relation that is asymmetrical, transitive and connected. "Harmonical proportion" was stated by Morley in 1597: "Harmonical proportion is [. . .]when the greatest of three terms is to the least as the difference of the greatest and the middle terms is to the difference of the middle and least," (New 96-99). This is clearer if stated in terms of a, b and c a : c :: (a-b) : (b-c). This proportion is illustrated by the progression 30, 15, 10. For 30 :10 :15 : 5. The transitivity of the relationship is that the same relation that holds between the first and second, and the second and the third, holds also between the first and the third. The relations are asymmetrical because we may not reverse them without changing the sense or direction. And given the numbers in such a relation -- there must be three but may be any number -- we can find one and only one place for each within the order.
Harmonies of several types are not limited to one, two, or even three terms. Notably Types I, III, VI, VII, and VIII have common examples of many-termed relations. The balance of nature, Type II, although commonly stated in a metaphor of two terms, obviously involves many terms such as animals, waste, water, oxygen, algae, industries, governments, conservation groups, etc. Many-termed harmonies may contain certain relations that are disharmonious. Discordant aspects may be resolved in the totality of harmony, as in the most interesting and complex musical compositions.
Not all harmonies are so clear; few have an ear that can catch the music of the spheres. Our musical instrumentals can scarcely reproduce musica mundana. According to Johannes Kepler the basic musical intervals are 4/5, the major third of Saturn, 5/6, the minor third of Jupiter, 2/3, the fifth of Mars, 15/ 16, the half-tone of Earth, 14/15 of Mercury, 5/12, the octave plus minor third of Mercury (Apel 375).
But how are we judging whether relations are harmonious or disharmonious? Are we limited to listening to musical notes sounding together? We are indeed often pleased by sweet sounding notes, but also by colors that vivify one another and shapes that together excite our vision. Yet there are mathematical relations associated with sweet sounding notes or underlying auditory pleasure. The great thrust of Pythagoreanism is to regard Type VIII, Mathematical Harmony; as fundamental to all other types. Mathematical relations as such are not sensed, hence Pythagoreans claim that the unseen harmony explains the seen, or the unheard the heard. Is there then ultimately only the conceptual standard of harmony?
Surely the most interesting abstract principle of musical harmony that would explain all heard consonances and dissonances is that of the Pythagoreans. One statement of harmonic relations begins with the octave (1/2) as simple, goes on to 1/3, (the fifth), 1/4, (the double octave), 1/5 (the third above the double octave), etc. But the disharmonic relations are complex, such as 7/8, etc.4
Concord of any sort can be compared to music, as discord can be compared to noise. If mathematical formal relations underlie song and instrumental compositions, do they also underlie harmonies of Type III, Human Harmony? I should rather leave the question as did Milton: "Harmonie to behold in wedded pair/ more grateful then harmonious sound to the eare" (Paradise Lost VIII, 605, Adam of Eve: "in us both one soul").
V. Is harmony dead? An historicist challenge to any attempt, as Whitehead’s, to reconceive the Category of Harmony
The great philological historian of ideas, Leo Spitzer, wrote his account of Classical and Christian Ideas of World Harmony with nostalgia. The present state is only one of loss of faith. René Wellek says in his "Preface":
While Spitzer’s regret or half regret for the destruction of the old belief in world harmony faded, as no illusion could long keep his allegiance, he surely preserved his aesthetic admiration for the old world-picture, his historical interest in understanding it and his feeling for its survivals in our time and in our languages. Stimmung is such a survival as he shows convincingly, and so is ‘mood’ as moods preserve ancient feelings. The musica mundana, the well-tempered man, was or tried at least to be in harmony with the great musica mundana, while musica instrumentalists was a means of reconciling microcosm and macrocosm, man with nature -- the work or even the composition of God. Without succumbing to superstition or sentimentality we may feel this even today in our deepest experience of music and poetry. (viii-ix)
The story of the idea of world harmony ends then with rejection, not with the acceptance stated so vigorously by Whitehead in Science and the Modern World (18). We cannot logically agree with Spitzer-Wellek and with Whitehead. Perhaps nothing can help us towards deciding as a closer examination of Whitehead’s theory of world order.
VI. Whitehead, a person with deep Pythagorean feeling
When Whitehead wanted most of all to state his deepest feeling of hope in the midst of anguish, his words took a distinctly Pythagorean form. The dedication of The Principles of Natural Knowledge is the best case in evidence:
To ERIC ALFRED WHITEHEAD
Royal Flying Corps
November 27, 1898 to March 13, 1918
Killed in action over the Foret de Gobain giving himself that the city of his vision may not perish. The music of his life was without discord, perfect in its beauty
My work on harmony in history reveals that several historians have erred in counting harmony dead before it was expressed significantly, as by Whitehead. An historian must keep his own beliefs from coloring the texts of past thinkers, and there is no personal cost in writing an obituary of the ideas of others.
Whitehead is writing from within the living tradition of Pythagoras. The belief is fervent, and a conviction without which he could not go on living and working. He is not an historian of ideas writing coolly of others, even with homesickness for a lost faith.
VII. Harmony, according to Whitehead, essential to civilization and therein vindicated
Whitehead counts Harmony one of the seven main doctrines of Plato. But it is not merely of historic interest. Harmony is "as important for us now, as [it was] then at the dawn of the modern world, when civilizations of the old type were dying" (Adventures 147). The doctrine couldn’t be, as Wellek judges it, an "illusion," or a "survival," a "superstition or sentimentality." Whitehead concludes: "The Greek doctrine of Composition and Harmony has been vindicated by the progress of thought" (Adventures 154).
Unfortunately Spitzer did not study Whitehead, and Wellek ignored him. Therefore we have no direct and explicit statement of exactly the respects in which the idea of world harmony is a failure in Wellek’s opinion. Nor is Whitehead altogether clear about the respects in which the idea of world harmony is a success. Whitehead certainly regarded faith in an order of nature (Type I and Type II) as a vindicated faith. I doubt if Wellek would deny that mathematics has proved the key used in discovering the regularity of planetary motion or that humanity has now extended its knowledge to the delicate interrelations of life in an environment (See Whitehead, Science Chapters 1,11 and IV). Perhaps Wellek’s case for failure is based on the prevalence of strife rather than harmony in natural and human life. That the "vision" of community of interests did not prove a total success is illustrated by the tragic death of Whitehead’s son. Is the doctrine of world harmony as naive a form of optimism as Voltaire in Candide mocks Leibniz’s doctrine of pre-established moral harmony? is Whitehead’s doctrine of world harmony some Platonic optimism that insofar as anything is true, it is also good and beautiful, and insofar as anything is true, good and beautiful, it is also most real? Lastly, does Wellek mean by failure of the doctrine of world harmony that philosophy has not demonstrated or proved in a strong sense the necessity of an ultimate ground of order?
VIII. Whitehead’s category couples disharmony with harmony
Whitehead grants the existence of much strife, of great tragedy, of failure of the actual world to conform to the ideal, and of lack of finality in matters of ultimate trust. Indeed, as I read Whitehead on harmony; he seems to be building into his theory these supposed refutations in order to render it more adequate and defensible.
Whitehead’s theory of harmony does not deny strife in nature and human society. A theory of harmony-disharmony accepts the partial truth of applying the "struggle for existence" to economic life. In short, in Adventures of Ideas, Whitehead does not dismiss Social Darwinism as wholly false:
What the notions of ‘form’ and ‘harmony’ were to Plato, that the notions of ‘individuality’ and ‘competition’ were to the nineteenth century. God had placed his bow in the skies as a symbol; and the strip of colours, tightly read, spelt ‘competition.’ The prize to he competed for was ‘life.’ Unsuccessful competitors died; and thus, by a beautiful provision of nature, ceased from constituting a social problem.
Now it is quite obvious that a much needed corrective to an unqualified, sentimental humanitarianism is here being supplied. Strife is at least as real a fact in the world as Harmony. If you side with Francis Bacon and concentrate on the efficient causes, you can interpret large features of the growth of structure in terms of ‘strife.’ If, with Plato, you fix attention on the end, rationally worthy, you can interpret large features in terms of ‘harmony.’ But until some outline of understanding has been reached which elucidates the interfusion of strife and harmony, the intellectual driving force of successive generations will sway uneasily between the two. (31-33)
Whitehead grants that in much recent history strife was fundamental and "harmony was a secondary effect, merely Romance gilding Strife" (Adventures 31-33).
IX. Whitehead’s theory of harmony-disharmony is not to be confused with the liberal Nineteenth Century compromise
What of the reconciliation of Strife with Harmony?
The political, liberal faith of the nineteenth century was a compromise between the individualistic, competitive doctrine of strife and the optimistic doctrine of harmony. It was believed that the laws of the Universe were such that the strife of individuals issued in the progressive realization of a harmonious society. In this way, it was possible to cherish the emotional belief in the Brotherhood of Man, while engaging in relentless competition with all individual men (Whitehead, Adventures 33).
There is no logical contradiction in the theory of liberalism. The repugnancy is between the misery of workers in factory and mine, when competition is unregulated, and the profession of humanitarian faith. In terms of our table of types of harmony; Type VII may be satisfied while clearly there is a lack of Type III, Human Harmony, and probably also of Types IV, Harmony of Action, and Type V, Spiritual Harmony.
X. The Ancient Greek success in discerning the mathematical harmony in music inspires hope that we can likewise discover a basis for coherent policies
If recent Western civilization is moral failure in achieving harmony in society, it is to be counted an intellectual success in comprehending the harmony of nature. Whitehead agrees with Plato that "any selections [of Ideas] are either compatible for joint exemplification, or are incompatible. It thus follows, as he notes, that the determination of compatibilities and incompatibilities is the key to coherent thought, and to the understanding of the world in its function as the theatre for the temporal realization of ideas" (Adventures 147). The most signal intellectual success in the search for harmony lies, says Whitehead in Adventures of Ideas, in the application of Mathematical Harmony (Type VIII) to Aesthetic Harmony (Type VI). This may be the clue to why Pythagoreans chose musical harmony as the root metaphor.
In respect to Harmony, the Greeks made a discovery which is a landmark in the history of thought. They found out that exact Mathematical Relationships, as they exist in Geometry and in the numerical proportions of measurements, are realized in various outstanding examples of beautiful composition. For instance Archytas discovered that, other circumstances being equal, the note given Out of a stretched string depends on the length of the strings [. . .] Also they investigated the dependence of the beauty of architecture upon the preservation of the proper proportions in the various dimensions. This was an immense discovery, the dependence of the qualitative elements in the world upon mathematical relations [. . . .] But the Greeks, with their power of generalization, grasped the full law of the interweaving of qualitative fact with geometrical and quantitative composition. (149)
XI. Harmony as a category applicable to numerous sets
Whitehead meets an objection that even if Pythagoras contributed a theory toward understanding proportions of music and architecture, these only are the fine arts: Whitehead argues that the category applies to chemistry, physiology; and economics.
Not only does Whitehead call attention to Type VIII as a key to Type VI, that is Mathematics as a theory applicable to Beauty; but also to Mathematics as a key to Cosmic and Natural Harmony "Plato drew the conclusion that the key to the understanding of the natural world, and in particular of the physical elements, was the study of mathematics" (Adventures 149).
He adds, also in Adventures of Ideas:
An intense belief that a knowledge of mathematical relations would prove the key to unlock the mysteries of the relatedness within Nature was ever at the back of Plato’s cosmological speculations. In one passage he reprobates the swinish ignorance of those who have failed to study the doctrine of proportions incapable of expression as numerical ratios. He evidently feels that the chance of some subtle elucidation of the nature of Harmony is being crassly lost. (151)
And in the same pages:
The Platonic doctrine of the interweaving of Harmony with mathematical relations has been triumphantly vindicated. The Aristotelian classification based upon qualitative predicates have a very restricted application apart from the introduction of mathematical formulae. Indeed, Aristotelian Logic, by its neglect of mathematical notions, has done almost as much harm as good for the advancement of science. We can never get away from the questions -- How much, -- In what proportions, -- and In what pattern of arrangement with other things. The exact laws of chemical proportions make all the difference; CO will kill you, when CO2 will only give you a headache. Also CC2, is a necessary element for the dilution of oxygen in the atmosphere; but too much or too little is equally harmful. Arsenic deals our either health or death, according to its proportions amid a pattern of circumstances. Also when the health-giving proportion of CO2 to free oxygen has been obtained, a rearrangement of these proportional quantities of carbon and oxygen into carbon monoxide and free oxygen will provide a poisonous mixture. In Political Economy, the Law of Diminishing Returns points to the conditions for the maximum efficiency of a dose of capital. In fact, there is hardly a question to be asked which should n(it be fenced round with qualifications as to how much, and as to what pattern of circumstances (153).
XII. As the Aristotelian Categories of Quality and Quantity are required in understanding chemical substances, so Whitehead suggests we must grasp the relations between Harmony and other categories
Harmony is one of the fundamental notions concerning the factors of fact. Very often Whitehead couples "The Harmony" with "The Mathematical Relations." Exactly how Harmony interweaves with Ideas, in contrast to The Physical Elements, to The Psyche, to The Eros, to The Receptacle, is a task of a philosophical system. Whitehead allows many ways of understanding this interweaving (see Adventures 158). It would be most helpful to know in detail how Harmony changes its role as we go from system to system. Whitehead is here suggestive of further study.
XIII. Harmony sometimes describes processes of nature and sometimes prescribes social policies. Is there a basis in fact for what ought to be? Harmony may be a norm of logical coherence or of the aesthetic and moral good
Just as above we made the contrast between the failure to achieve social harmony while people gained knowledge of natural harmony; we must now go a step further in recognizing that Whitehead’s Harmony is an ideal as well as a factor of the actual world, though this contrast is not as sharp in Whitehead as in many philosophers. Given such a concept of harmony as regularity, a given situation approximates to the ideal. Harmony is a logically normative concept. Harmony is also an aesthetic and moral concept.
XIV Bridging the gap between the natural and the ideal also connects the True with the Beautiful and the Good. There are gradations both of Beauty and of social harmony
There is in nature "a tendency to be in tune." This is a suggestion in the form of a question, and it leads to another form of the question. How is the natural harmony connected with the ideal harmony? The explanation by way of conformal feeling is too complex to be considered in detail; suffice it to say here that Whitehead does believe he has bridged the gap between the natural and the ideal:
The attainment of such conformation would belong to the perfection of nature in respect to the higher types of its animal life. There is no necessity about it. Evidently there is failure, interference and only partial adjustment. But we have to ask whether nature does not contain within itself a tendency to be in rune, an Eros urging towards perfection. This question cannot be discussed without passing beyond the narrow ground of the truth-relation. (Adventures 251)
There are gradations of aesthetic harmony Beauty, which is "defined as being the perfection of Harmony;" comes in two forms, minor and major. The minor form, Whitehead says, is "the absence of mutual inhibition" (Adventures 252). In terms of harmony, this is the mere absence of discord. But the major form is a synthesis of new contrasts with the content. It is the latter kind of harmony that Whitehead says is strong: "In the sense here meant, Strength has two factors, namely; variety of detail with effective contrast, which is Massiveness, and Intensity Proper which is comparative magnitude without reference to qualitative variety. But the maximum of intensity proper is finally dependent upon massiveness" (Adventures 253).
There are comparable gradations of social harmony Civilization as well as art, Whitehead says, is "nothing other than the unremitting aim at the major perfections of harmony" (Adventures 271). One excellent harmony of society is the exclusion of destruction. But this is a lower form of perfection. The failure to realize a higher form of perfection is preferable for Whitehead: "Progress is founded upon the experience of discordant feelings. The social value of liberty lies in its production of discords" (Adventures 257).
The argument of analogical resemblance of degrees of aesthetic harmony and degrees of social harmony; raises the question of a highest degree of perfection.
XV Is all harmony finite or can there be an infinite harmony? Can all harmonies be harmonized together in perfection Itself
"There are perfections beyond perfections" (Whitehead, Adventures 257-58). Degrees, as steps of a ladder, suggest a topmost rung. If there is more perfect, must there not also be a most perfect? Traditional hierarchical thought culminates in Perfection Itself, sometimes including a Summum Verum, Truth Itself, a Summum Pulchrum, Beauty Itself, in addition to more familiar Summum Bonum, the Good. Whitehead clearly rejects this version of the Great Chain of Being:
all realization is finite, and there is no perfection which is the infinitude of all perfections. Perfections of diverse types are among themselves discordant. Thus the contribution to Beauty which can be supplied by Discord -- in itself destructive and evil -- is the positive feeling of a quick shift of aim from the tameness of outworn perfection to some other ideal with its freshness still upon it. Thus the value of Discord is a tribute to the merits of Imperfection" (Adventures 257) - Whitehead goes on to judge the stale perfection of Hellenism in contrast to the freshness of most religions and barbarism. (Adventures 257)
XVI. Can there be bad harmony and good disharmony? Can there be too much harmony and too little disharmony? Can a person or a civilization or work of art be too harmonious?
How should we judge harmony and disharmony? How should we deal with disharmony? Whitehead’s chief novelty in his theory of harmony comes form considering the respects in which disharmony can be good and should be respected. It is fair to say that Whitehead has no simple theory of harmony but, just as he has a theory of order-disorder, so also a theory of harmony-disharmony. If major beauty requires contrast, and the basic contrast rendering harmony vivid and fresh is disharmony, then it follows that not all harmony is good nor is all disharmony bad. Whitehead’s case is best made in Adventures of Ideas with the example of Hellenistic, as contrasted to Hellenic culture:
Perfection was attained, and with the attainment inspiration withered. With repetition in successive generations, freshness gradually vanished. Learning and learned taste replaced the ardour of adventure. Hellenism was replaced by the Hellenistic epoch in which genius was stifled by repetition. We can imagine the fate of the Mediterranean civilization if it had been spared the irruption of Barbarians and the rise of two new religions. Christianity and Mohametanism: -- For two thousand years the Greek art-forms lifelessly repeated: The Greek schools of philosophy, Stoic, Epicurean, Aristotelian, Neo-Platonic, arguing with barren formulae: Conventional histories: A stabilized Government with the sanctity of ancient ceremony, supported by habitual pieties: literature without depth: Science elaborating details by deduction from unquestioned premises: Delicacies of feeling without robustness of adventure. (257-58)
XVII. Whitehead’s warning against harmony used to stifle individuality, spontaneity, and originality
Progress requires therefore that the case for harmony be not used to stifle individuality. The individual (exemplified by "each actual occasion," as Whitehead says) displays "Spontaneity, originality of decision [...] Freshness, zest, and the extra keenness of intensity arise from it [individuality] [. . .] Thus the wise advice is, not to rest too completely in any continued realization of any perfection of type" (Adventures 258).
XVIII. Warning against disharmony that is destructive, uncoordinated, unsubordinated, and therefore evil
It is not the "basic disharmony of the actual world" that is evil, rather it is destruction that is evil. Disharmony is relatively good for Whitehead because it prevents mere repetition (Adventures 259). Rather than block out disharmony or absorbing it in the generally positive apprehension, Whitehead recommends what he calls "readjustment" and what we may call synthesis in terms "relevant to both the inharmonious systems." At this point Whitehead is very abstract and fails to use examples from the history of music that many times shows that a new combination of sounds, once rejected as cacophonous, is taken up into a richer system of sounds. In western culture, as contrasted to Byzantium, history of music is made through revolutions, and musical education must therefore proceed by the wisdom of accepting shock:
The third way depends on another principle, that a readjustment of the relative intensities of incompatible feelings can in some cases reduce them to compatibilities. This possibility arises when the clash in affective tones is a clash of intensities, and is not a sheer logical incompatibility of qualities. Thus two systems of prehensions may each be internally harmonious; hut the two systems in the unity of one experience may he discordant, when the two intensities of their subjective forms are comparable in magnitude. There may he discordance in feeling this as much as that or in feeling that as much as this. But if one be kept at a lower intensity in the penumbra of feeling, it may act as background to the other, providing a sense of massiveness and variety. This is the habitual state of human experience, a vast undiscriminated, or dimly discriminated background of low intensity, and a clear foreground. This third way of eliminating discordance may be termed the method of ‘reduction to a background.’ Alternatively, it can equally well he termed the method of ‘raising to a foreground’ (Adventures 260).
XIX Including the disharmonious in the harmonious
The disharmonious can be an asset to the harmony of a whole when there is no clash of equal intensities. That is, when one is put in the background of the other. This is sub-ordination, and there is also coordination.
Whitehead’s explanation of how the inharmonious systems are included in an harmonious system is "a fourth way": "This novel system is such as radically to alter the distribution of intensities throughout the two given systems, and to change the importance of both in the final intensive experience" (Adventures 260).
The beneficial disharmony of individuality is illustrated both by strong individuality in society and strong individuality in the arts. "The problem of social life is to make possible a harmony of strong individuals. This ‘is the problem of the co-ordination of [various grades of activities], including the limits of such co-ordination."6 As in civilization, so in art, individuals appear immortal: "The very details of [. . .] compositions live supremely in their own right. They make their own claim to individuality, and yet contribute to the whole. Each such detail receives an access of grandeur from the whole, and yet manifests an individuality claiming attention in its own right" (Whitehead, Adventures 282)
Whitehead’s example in Adventures of Ideas is here needed to make the point effective:
The sculpture and tracery in a Gothic cathedral -- Chartres for instance -- subserve the harmony. They lead the eye upward to the vaulting above, and they lead the eye onward horizontally to the supreme symbolism of the altar. They claim attention by their beauty of detail. Yet they shun attention by guiding the eye to grasp the significance of the whole. Yet the sculpture and the tracery could not perform this service apart from their supreme individuality, evoking a wealth of feeling in their own right. Each detail claims a permanent existence for its own sake, and then surrenders it for the sake of the whole composition.
Again, the value of discord arises from this importance of the forceful individuality of the details. The discord enhances the whole, when it serves to substantiate the individuality of the parts. It brings into emphatic feeling their claim to existence in their own right. It rescues the whole from the tameness of a merely qualitative harmony (282-83).
XX. As followers of Pythagoras rejected the tendency to reduce all harmonies to simple ratios, so Whitehead rejects the reduction of all harmonies to logical compatibility
Just as Aristoxenus7 in The Harmonics transcended the Pythagorean reduction of all problems of harmony-disharmony to ratios, so Whitehead likewise in Adventures of Ideas: "The Harmony is felt as such, and so is the Discord. Now Harmony is more than logical compatibili5; and Discord is more than logical incompatibility. Logicians are not called in to advise artists"(261-62). The reason for this reinforces what Whitehead said about individuality in society and the arts. The artist pays attention to the individuality of an ‘It.’ The logician ignores any individual ‘It’ (Adventures 262). It would follow that the harmony of logic is abstract in contrast to social harmony or aesthetic harmony; which are richer, or in a word, concrete.
XXI. Harmony of truth based on a narrow relation; harmony of beauty based on a broad relation. Truth-relation not necessarily good or evil Beauty-relation self-justifying
Compatible with the contrast between the abstract harmony of logic and the concrete harmony of society and the arts is the place given by Whitehead to the harmony of truth in contrast to the harmony of beauty; Truth "is the conformation of Appearance to Reality [. . . .] The notion of ‘conformation’ in the case of Truth is narrower than that in the case of Beauty. For the truth-relation requires that the two relata have some factor in common" (Adventures 265). Beyond the limitation of the truth-relation in contrast to the beauty-relation is the service of the former to other ends. In Adventures of Ideas, Whitehead explains:
A truth-relation is not necessarily beautiful. It may not even be neutral. It may be evil. Thus Beauty is left as the one aim which by its very’ nature is self-justifying. The Discord in the Universe arises from the fact that modes of Beauty are various, and not of necessity compatible. And yet some admixture of Discord is a necessary factor in the transition from mode to mode. The objective life of the past and future in the present is an inevitable element of disturbance. Discord may take the form of freshness or hope, or it may be horror or pain. (266)
The general importance of Truth for the promotion of Beauty is overwhelming. After all that has been said, yet the truth-relation remains the simple, direct mode of realizing Harmony. Other ways are indirect, the indirectness is at the mercy of the environment. There is a blunt force about Truth, which [. . .] is akin to cleanliness -- namely, the removal of dirt which is unwanted irrelevance. The sense of directness which it carries with it, sustains the upstanding individualities so necessary for the beauty of a complex. Falsehood is corrosive. (266)
XXII. Truth and beauty combine into what has "rightness in the deepest harmony." Goodness belongs "to the constitution of reality"
A major Platonic step towards "a Harmony of Harmonies" is the combination of Truth and Beauty. The kind of Truth involves penetrating insight, perhaps what some call "revelation." In Adventures of Ideas, Whitehead notes:
The type of Truth required for the final stretch of Beauty is a discovery and not a recapitulation. The Truth that for such extremity of Beauty is wanted is [. . .] a Truth of feeling, and not a Truth of verbalization. The relata in Reality must lie below the stale presuppositions of verbal thought. The Truth of supreme Beauty lies beyond the dictionary meanings of words [. . .] Truth in the service of Beauty achieves security and promotes Beauty of feeling [….] The element of anticipation under the influence of Truth is in a deep sense satisfied, and thus adds a factor to the immediate Harmony. Thus Truth, in itself and apart from special reasons to the contrary, becomes self-justifying. It is accompanied by a sense of tightness in the deepest Harmony. But Truth derives this self-justifying power from its services in the promotion of Beauty. Apart from Beauty, Truth is neither good nor bad. (267)
XXIII. The role of Creative Advance is to doom mere harmony of repetition to decay while fostering harmony that is freshened by discord and spontaneity
With Truth and Beauty interrelated, Whitehead’s next step is to bring in" Good ness [as] the third member of the trinity." Goodness is not an aim of art, but belongs "to the constitution of reality; which in any of its individual actualizations is better or worse. Good and evil lie in depths and distances below and beyond appearance [....]The real world is good when it is beautiful" (Adventures 268).
The search for Truth and Beauty, Science and Art, are able to tap "the infinite fecundity of nature," and thus become "sources of Harmony" Whitehead, Adventures 272). This point is stressed, in that nature supports the harmonies of man’s efforts but nature is not supportive of mere harmony, that is the stale repetition of old perfections lacking the freshness of discord. Whitehead writes:
Decay, Transition, Loss, Displacement belong to the essence of the Creative Advance. The new direction of aim is initiated by Spontaneity, an element of confusion. The enduring Societies with their rise, culmination, and decay, are devices to combine the necessities of Harmony and Freshness. There is a deep underlying Harmony of Nature, as it were a fluid, flexible support; and on its surface the ripples of social efforts, harmonizing and clashing in their aims at ways of satisfaction. (Adventures 286)
Thus Adventure is added to the great regulative properties in addition to the qualities of Truth, Beauty and Art.
XXIV What is the ultimate harmony?
The "harmony of harmonies" is called "peace" because it hinds together distinct kinds of harmony Yet the "harmony of harmonies" is not infinite, for all harmony must be limited. In politics, as in art, evil comes from trying to join what cannot be conjoined. Not even God can surmount impossibility.
Regarded as aspects of civilization in search of Harmony; we have been given many more or less distinct kinds or types of harmony. In Adventures of Ideas, Whitehead notes: "We require the concept of some more general quality [. . .] the notion of a Harmony of Harmonies, which shall bind together the other four qualities. [. . .] I choose the term ‘Peace’ for that Harmony of Harmonies which calms destructive turbulence and completes civilization.
Thus a society is to be termed civilized whose members participate in the five qualities – "Truth, Beauty, Adventure, Art, Peace" (284-85).
Yet Harmony remains finite and not infinite. According to Whitehead, every
actuality is in its own nature finite. There is no totality which is the harmony of all perfections. Whatever is realized in any one occasion of experience necessarily excludes the unbounded welter of contrary possibilities. There are always ‘others’, which might have been and are not. This finiteness is not the result of evil, or of imperfection. It results from the fact that there are possibilities of harmony which either produce evil in joint realization, or are incapable of such conjunction. This doctrine is a commonplace in the fine arts. It also is -- or should be -- a commonplace of political philosophy. History can only be understood by seeing it as the theatre of diverse groups of idealists respectively urging ideals incompatible for conjoint realization. You cannot form any historical judgment of right or wrong by considering each group separately. The evil lies in the attempted conjunction. (Adventures 276-77)
What follows with regard to the affirmation of God as infinite Harmony? Again, in Adventures of Ideas, Whitehead says:
This principle of intrinsic incompatibility has an important bearing upon our conception of the nature of God. The concept of impossibility such that God himself cannot surmount it, has been for centuries quite familiar to theologians. Indeed, apart from it there would be difficulty in conceiving any determinate divine nature. But curiously enough, so far as I know, this notion of incompatibility has never been applied to ideals in the Divine realization. We must conceive the Divine Eros as the active entertainment of all ideals, with the urge to their finite realization, each in its due season. Thus a process must be inherent in God’s nature, whereby his infinity is acquiring realization. (277)
Whitehead’s theory of harmony-disharmony is a great success. This is particularly remarkable because the learned historians who know the literature as Whitehead did not, had pronounced it dead.
Whitehead knew enough history of the idea to know why it had been given a nostalgic obituary. With regard to order, which is only a general name for such particular types or families as harmony, Whitehead writes, "there is not just one ideal ‘harmony’ which all actual entities should attain and fail to attain." (Process 84) (This passage also mentions the failure of Plato, and could also cite the best known philosophy of Leibniz, or Augustine’s theology I have inserted "harmony" for "order").
Several criteria for a successful philosophy of harmony are listed, and Whitehead’s satisfies all of them. Does consideration of harmony take the correlative disharmony into account? Does the theory assume one and only one meaning? Does the theory assume one and only one principle of harmony applying to all actual occasions? Does the theory assume final perfect satisfaction of the one ideal? By these four tests, all past theories of harmony fail, but Whitehead’s avoids these fatal errors.
Whitehead’s theory is not the only one in our century passed over. One other one that I have studied and written about is George Santayana’s. A comparison reveals some remarkable agreements, yet some disagreements also.
What I have presented as Whitehead’s category of harmony which should more carefully be harmony-disharmony; has been drawn exclusively from Adventures of Ideas. But in that text harmony is nowhere called a "category Such designation comes from Process and Reality, where, among nine Categoreal Obligations, we meet the seventh, "The Category of Subjective Harmony" (1. II. III. with subsequent commentary in III. III. VII, 27 and 254-55. In the latter we find "The Category of Aesthetic Harmony")
In Adventures of Ideas Whitehead found analogous meanings of harmony in almost every realm of being and culture. But one conspicuously missing in the text of 1933 was harmony of mind and body. Since that had already been dealt with in the text of 1929, we may think of the two, Process and Reality and Adventures of Ideas, as supplementary accounts of harmony.
There are good reasons both for coupling Process and Reality and Adventures of Ideas and for contrasting them. There is much more in Adventures of Ideas on harmony than in Process and Reality, but many of the essential points were laid down in Process and Reality. Adventures of Ideas on harmony may be read as an expansion. First, Whitehead uses Leibniz’s term in Process and Reality "preestablished harmony"; he did not in Adventures of Ideas, but affirms the same principle: in the nature of things incompatible data cannot be conjoined or united in one feeling. In the relation of organism to environment the harmonious relation may be narrow or wide (Process III) In Adventures of Ideas this is the contrast between the truth-relation and the beauty-relation.
In Adventures of Ideas harmony may require exclusion of irrelevance, corrosion, or dirt. In Process and Reality "the right chaos, and the right vagueness are, jointly required for any effective harmony" (112).
In Process and Reality Whitehead makes a clear use, as in Adventures of Ideas, of analogy as that between the structure of body and that of mind. The mind "is only one more example of the general principle on which the body is constructed" (Process 109). The most daring extension of the category of subjective harmony is to God, whose role is harmonization such that there is no final loss of value. All is "woven by rightness of feeling into the harmony of universal feeling" (Process 346).
But on the other hand, is there not a deep difference between the categoreal scheme of Process and Reality and such a category as harmony in Adventures of Ideas? Certainly the concept of category in Process and Reality is more Aristotelian, while that in Adventures of Ideas is more Platonic. Each of Aristotle’s ten -- substance, and nine predicables -- is complex. As quality and quantity are sharply contrasted, so we are not to confuse an actual entity with an eternal object. But the whole strategy of Pythagoras is to find a correspondence between the mathematical and the sensed sounds, or colors, etc. But when in Process and Reality we come to Categories of Obligation, dealing with binding relations, we find them interwoven. Unity and harmony "jointly express a pre-established harmony in the process of concrescence of any one subject" (27).
My hypothesis is that Whitehead began his Categorical Scheme in Process and Reality with Aristotle’s in mind, but found kinds of relations that were more complex. Hence he turns to Plato’s interwoven modes of order to be true to process. By modes of order, I mean not merely series, but rhythm, balance, hierarchy; as well as harmony, even analogy and dialectic.8
I have extended the logic of relations beyond series, and in this I follow Josiah Royce, rather than Whitehead or Russell. It is thoroughly in keeping with Whitehead’s practice to begin with such an order as ABC, or 1, 2, 3, "a relation which is asymmetrical, transitive, connected." But if order is aesthetic as well as logical, simplicity, once found, is to be finally escaped. What we find in Whitehead on harmony is that it cannot be analyzed in the complex patterns of events without dealing also with balance and with hierarchy. We may suppose that Whitehead had Pythagorean harmonies in mind when he wrote: the Platonic doctrine of interweaving of Harmony with mathematical relations has been triumphantly vindicated" (Adventures 149-50)9 But it is of great philosophic importance to note the interweaving of modes of order.
The modes of order are interwoven. This, I believe, is found in other discussions of rhythm or periodicity as well as of harmony, but I would now wish to explore further what I suggested in my Alfred North Whitehead in Twayne’s English Author’s Series, and to make out the case, as I did in the case of Voegelin, that the cosmos cannot be comprehended without hierarchy and balance.10 This way of stating the categories can employ the logic of relations, among which analogy is necessary, and a superior way of stating the categories.
1. The author expresses thanks to Professor George J. Allan, the assessors of the manuscript, the editor of Process Studies for editing the essay, Ms. Ivia Cofresí and Ms. Lynda McMorris, and the librarians of Woodruff Research Library of Emory University.
2. So Galen used the term a r m o n i a .
3. This "ninefold harmony" is in context of the spheres. This recalls the myth of Er in Plato’s Republic, X. On each wheel an intelligence sings one tone and together form a harmony. Poets commonly invoked Nine Muses, and Christian poets thought of the ninefold hierarchical order of angels, from those on the moon to the seraphs nearest to God. This is crucial, as we shall see, to Whitehead’s position that modes of order are interwoven: harmony involves hierarchy.
4. Cf.: Theodore Presser, "Harmonics," Grove’s Dictionary of Music and Musicians. Also Donald Francis Tovey, The Forms of Music and Joseph Otten, "Harmony," The Catholic Encyclopedia; A. Lalande, Vocabulaire Technique et Critique de la Philosophie; Rudolf Eisler, Handwörterbuch der Philosophie; José Ferrater-Mora, Diccionario de Filosofia; Encyclopedia Filosofica; Paul Edwards, Encyclopedia of Philosophy has no article "Harmony" but interesting aspects are discussed under "Monad and Monadology," "Utopias and Utopianism," "Chinese Philosophy," "Hungarian Philosophy"; A Syntopicon of Great Books of the Western World discusses harmony under five heads: "Beauty," "Justice," "Man," "Medicine, and "Relation."
5. Fragments of his writing are translated, Gutarie, pp. 177-201. Archytas was the general of the Pythagoreans in Tarenum, then "Magna Graeca," now Traranto on the western side of the heel of Italy. Not only did he contribute to mathematics, geometry, and harmonic theory, but also as a ruler attracted Plato who visited him in 388 BC. "It is possible that Archytas was Plato’s model for the [. . .] ‘Philosopher King."’
6. See Victor Lowe’s Understanding Whitehead. See also Whitehead, Adventures 34. Another book from the same period by another one who defends more explicitly the interweaving of modes of order is John B. Cobb, Jr., A Christian Natural Theology. To aim at ideal harmony is also to aim at balance (101, 112, 128) and involves "graded relevance" (154, 201, etc.). Dorothy Emmet notes that harmony is associated with rhythm in Whitehead and suggests a Platonic text, Timaeus 90d. Whitehead’s Philosophy of Organism, 240. This passage (Timaeus 90d) occurs in 109, cf. also 179, 215, 247.
7. Aristoxenus, as Archytas, also of Tarentum, a student of Aristotle whose On the Pythagoreans does not survive, continued both to contribute to the theory and to tell the story of the movement. His life of Pythagoras is the source for Diogenes Laertius and the Neo-Platonists Porphyry and Iamblichus. (Guthrie: Sourcebook, 38-39). His Rules of Education advised, in educating sons, to chose that they be born and grow up in a city ruled by good laws (Sourcebook, 145). Aristoxenus was a prolific author, credited with four hundred books, survives only in three books the Elements of Harmony, to which Whitehead refers, and also part of book two of Elements of Rhythm. The latter book is an important example of the interweaving of two modes of order, rhythm and harmony. ‘Without rhythm melody was felt to be incomplete and formless. Melody was sometimes regarded as female, rhythm as the male partner in bringing music to life" (xxiii).
8. It is fruitful to think of the closeness of Whitehead to Aristotle in thinking of categories as adverbial interrogatives, accompanying verbs, rather than as nouns and adjectives. See Marion Leathers Kuntz and Paul G. Kuntz, "Naming the Categories; Back to Aristotle by Way of Whitehead."
9. Josiah Royce, "Order," Encyclopaedia of Religion and Ethics, ed. James Hastings, Edinburgh: T. and T. Clark, 1917, Vol. 9, 553-40. I have included among mathematical relations the mean, and other modes of balance between extremes. I have assumed, if harmony is a "category" so also is rhythm. By the "interweaving" of modes of order the most obvious example is Whitehead’s An Enquiry Concerning the Principles of Natural Knowledge, 195-200. We see that rhythm is a series of states, in which we recognize alternation between in higher and lower, which is gradation or hierarchy. There is reversal of process to attain some balance. One concept of balance between extremes which Whitehead used was, not too structured, as a crystal, nor too unstructured as fog or gas. Because rhythm is as obvious as beating of the heart and the alternation of light and darkness, I begin my book Alfred North Whitehead, with "The Discovery of Rhythm." This mode of order, I argue, is most prominent in Whitehead on his life and theory of learning. Balance is most commonly now used in stating the problems of the environment. I deal with this in "American Wilderness: Too Much or Too Little? Clarifying the Concept of Balance with the Help of Aristotle’s Mean and Cannon’s Homeostasis."
10. In my article on Voegelin, there is another example of a living philosophy of interwoven modes. I put hierarchical order first (135-44) and at far greater length than would be appropriate to Whitehead, and also polarity or balance between opposites (147-56). I believe much more can now be understood about balance through more careful use of Aristotle and the discovery in our century of homeostasis (see note 9).
A New English Dictionary on Historical Principles, Oxford: Clarendon Press, 1901, Vol. V.
Adler, Mortimer J. Ed. A Syntopicon of Great Books of the Western World. Chicago: Encyclopedia Britannica, 1952.
Apel, William. Harvard Dictionary of Music. 2nd Edition. Cambridge: Harvard UP, 1969.
Bodin, Jean. Colloquium of The Seven about Secrets of the Sublime (Colequium Heptaplomeres de Rerum Sublimium Arcanis Abditis), Translation, with Introduction, Annotations, and Critical Readings by Marion Leathers Daniels Kuntz. Princeton: Princeton UP, 1975.
Bodin, Jean. The Six Books of a Commonweal, Facsimile of 1606 translation, Ed. Kenneth Douglas McRae, Cambridge: Harvard UP, 1962, 455-67.
Cobb, John B. Jr. A Christian Natural Theology. Philadelphia: Westminister, 1965.
Edwards, Paul. Encyclopedia of Philosophy. New York: Macmillan, 1967. Eisler, Rudolf. Handwörterbuch der Philosophie. 2te Auflage, Berlin: E. S. Mittler & Sohn, 1922.
-- Kritische Untersuchung des Begriffs der Weltharmonie und seiner Anwendung bei Leibniz, Berlin: S. Calvary. 1895.
Emmet, Dorothy. Whitehead’s Philosophy of Organism.1932 London: Macmillan, 1966.
Enciclopedia Filosofica, Venezia-Roma: Centro di Studi Fiosofici di Gallarte, 1957, Vol. I.
Ferrater-José, Mora. Diccionaria de Filosofia, 3d ed., 1951. Guthrie, Kenneth Sylvan. The Pythagorean Sourcebook and Library, Grand Rapids, Mich.: Phanes, 1987.
Kuntz, Marion Leathers and Paul G. Kuntz. "Naming the Categories: Back to Aristotle by Way of Whitehead." Journal of Speculative Philosophy 1.2 (1987): 30-47.
Kuntz, Marion Leathers Daniels. "Harmony and the Heptaplomeres of Jean Bodin." Journal of the History of Ideas 12.1 (1974): 31-41.
Kuntz, Paul G. Alfred North Whitehead. Boston: Twayne, 1984.
-- "American Wilderness: Too Much or Too Little? Clarifying the Concept of Balance with the Help of Aristotle’s Mean and Cannon’s Homeostasis." Contemporary Philosophy 12 (1989): 1-8.
-- The Concept of Order. Seattle: U of Washington P, 1968.
-- "The Continuity of the Pythagorean Tradition." (with "Select Bibliography on Harmony") Unpublished manuscript.
-- "Harmony According to Santayana: A Pythagorean Theme in the Cosmos, Art, Psyche and Spirit, Society and State, A Metaphysical and Theological Principle." Unpublished essay.
"Leibniz’s Theory of Order." Acta Conventus Neo-Latinus Goelpherbytani: Proceedings of the Sixth International Congress of Neo-Latin Studies, Ed. Stella Revard, Fidel Riidle, Mario A. DiCesare, Binghamton: (1988): 625-34.
-- "Voegelin’s Experiences of Disorder Out of Order and Vision of Order Out of Disorder: A Philosophic Meditation on His Theory of Order-Disorder." Eric Voegelin’s Significance for the Modern Mind. Ed. Ellis Sandoz, Baton Rouge. Louisiana: Louisiana State UP, 1991. 111-173.
-- "Weiss’s Search for Adequacy." The Modern Schoolman 66 (1969): 251-70.
Lalande, A. Vocabulaire Technique et Critique de la Philosophie, 8th ed., Paris: Presses Universitalres de France, 1960. 401.
Lowe, Victor. Understanding Whitehead, Baltimore: Johns Hopkins UP, 1962.
Milton, John. "On the Morning of Christ’s Nativity" 1629. -- Paradise Lost.
Orten, Joseph. "Harmony." The Catholic Encyclopedia, New York: Robert Appleton, 1910.
Presser, Theodore. "Harmonics." Groves Dictionary of Music and Musicians. 1926.
Plato. R.G. Bury’s translation. Cambridge: Harvard UP, 1957. Spitzer, Leo. Classical and Christian Ideas of World Harmony. Ed. Anna Granville Hatcher, Baltimore: Johns Hopkins UP, 1963.
Tovey, Donald Francis. "Harmony." The Forms of Music. New York: Meridian Books, 1956.
Whitehead, Alfred North. An Enquiry Concerning the Principles of Natural Knowledge. Cambridge: Cambridge UP, 1919.
-- Science and the Modern World. 1925. New York: Free Press, 1967.