The “Elements” of Musical Composition.

A neo-Aristotelean ontology of musical works.

By Adam Voight.

Abstract

The current work defines an Aristotelean approach to the ontology of musical works and other related abstracta. The theory would satisfy multiple conditions: 1) It would provide a workable theory of abstract artifacts. 2) It would be consistent with modern scientific naturalism (broadly defined), and 3) It is at least a possible reading of what Aristotle has to say as well as what he should say if he were to answer the questions concerning the coming-to-be of musical compositions.

Table of Contents

2

I.1. Introduction 2

I.2 The Ontological Strangeness of Musical Works 2

II. The Idea of “Musical ‘Physics’ ” 3

II. The General Sense of “Elements” in Aristotle 4

II.A. “Elements” vs. “Matter” 4

II.B. “Elements” defined. 4

III. Musical Elements in Aristotle. 5

III. A. Musical “Units” In the Metaphysics. 5

III. B. Music and other Elements in De Sensu. 6

IV. The Genealogy of Musical Elements. 7

Proximate / Ultimate Elements of Music. 9

V. Music and Ontology. 11

V. A. Music and Substance. 11

V.A.1. Pythagoreanism – Numerical Substance. 11

V.A.2. Aristotle: Music as Mathematical Science. 12

V.A.3. The Aristoxenian Paradigm Shift in Music Theory. 13

V.B.1 Science, Music, and Substance in Aristotle. 15

V.B.2. Aristotle’s ‘substance’. 15

V.B.3. The distinction between Substantial and Analogous sciences. 16

V.B.4 Analogical Sciences in Book Lambda. 17

VI. An Possible Objection from Final Causes. 18

VII. Conclusion. 19

I.1. Introduction

A surprisingly hot subfield of analytic philosophy in the past generation has been aesthetics. Of late, one of the more active topics has been the ontology of music, especially the problem of the individuation of musical works. Thus far, no one approach to defining the being of musical works seems to ‘save the phenomena’ to the satisfaction of those involved. A perusal of the ideas on offer is daunting, but thus far there is no distinctly Aristotelean perspective available.¹ Two extreme positions include “Musical Platonism”² and “musical fictionalism”.³ The former claims that musical works are eternal ideas and the latter claims that “music” does not really refer. Aristotle’s general approach was intended to chart a middle path between two similar extremes – Platonism and materialism, and so we might benefit from something similar tailored to today.⁴ However, in order to make a proposal, I have questioned one thesis that is often taken to be essential to Aristoteleanism: that unlike matter, forms are unchangeable. In my view, we should explore the possibility for an Aristotelean conception of changes of forms and thus essences. If this could be done, it is likely that such a view would be superior to those who ignore essences on the one hand and those who deny their changeability on the other.

My goal being to define a hylomorphic theory of said “abstract artifacts”, I must first defend the claim that there are abstract “elements” or “matter” from which abstract artifacts are made. It is the latter more limited goal with which the current work shall attempt.

I.2 The Ontological Strangeness of Musical Works

Musical works (songs, symphonies, concertos, etc.) are ontologically unique for many reasons:

1) They are abstracta , which ‘are’ in a radically different way from concreta. I will assume that Platonic Ideas are “abstract” in the modern sense assumed here.⁵

2) They are created. While numbers are not generally thought to be created or invented, it seems much more intuitive to say that composers create their works in some way, while numbers are simply “discovered”. Perhaps transfinite numbers or imaginary numbers might be “invented”, but in general the natural numbers are often though of as “discovered”. Of course, Musical Platonists have differed on this point Kivy (1987) famously claims that they are eternal.

3) They are arbitrary or contingent. – Likewise, abstract artifacts are in most cases far more contingent that the numbers. While there is no room for creativity in the integers, it seem that there is a lot about musical works that is radically contingent or arbitrary. For example, Beethoven could have transposed the Ninth Symphony up a whole step and it would still be the Ninth Symphony, whereas it seems that numbers are pretty much unchangeable. People may debate whether zero, negative numbers, irrational numbers etc. are invented, but they seem much less contingent than musical works.

4) Another difference is that musical works are “perishable”, but not in the same way as an apple or table. Musical works are “lost” when we can no longer know or learn how to perform them and this is something that applies to many other classes of abstract products from biological species to poems or inventions.

In order to handle these difficulties, we shall treat music as a part of Aristotle’s “physics”; a goal-oriented behavior of an organism that takes place in space and time and which is causally efficacious.

II. The Idea of “Musical Physics”

Metaphysics and physics both have as part of their mission the description and explanation of change. Things are not created ex nihilo but from existing matter. In many cases, this matter must be made, as when bricks must be made first for a house, or plants must be grown first for animal’s food. If there were poesis of abstract products, such as a prose, poetic or musical compositions, it would also operate on existing matter. It is typical that this matter would be different matter from concrete products. All products would have their own forms specific to the matter that they are, just as bricks have their own matter and form, and the houses made from brick their own matter and form. For musical composition, this matter is not made of material but rather abstract elements. I will not make too much about the details of how we construe abstractions; perhaps it would have been better to call them “virtual elements”. Here we shall focus on the material cause of music, a.k.a. the classical “Elements of Harmony”. But first we shall have a close look at what “elements” are in Aristotle’s philosophy in the broader non-musical sense.

II. The General Sense of “Elements” in Aristotle

II.A. “Elements” vs. “Matter”

In Aristotle there are two words with similar meaning that might well refer to the “that-from-which” of abstracta: “elements” [stoicheia] and ”matter” [hyle]. I am using the term “elements” rather than “matter” for the following reasons:

1. It is a term with slightly wider meaning. In other words, all “matter” are “elements”, but not vice versa. For example, the while the “matter” of geometry is space or alternatively, the genus of space, the elements of geometry include in addition to space itself, points, lines, shapes, axioms, theorems et cetera. ‘Elements’ has a much wider applicability.
2. “Intelligible matter” is only mentioned three times in all of Aristotle’s corpus, whereas “elements” is much better explained at length in many different contexts.
3. Intelligible matter is only ever related to arithmetic and geometry, while “elements” are mentioned with respect to grammar, logic, and many other sciences.
4. “Elements”, has its own entry in Aristotle’s glossary (Book Delta, see below.), while “matter” does not.

For these reasons, I will use the term “elements”, except where said elements are spoken of as a material cause.

II.B. “Elements” defined.

Book V of the Metaphysics consists in a series of definitions of Aristotle’s philosophical terms, including and section three is as follows:

“ ‘Element’ [Greek stoicheion] means (1)[⁶] the primary component immanent in a thing, and indivisible in kind into other kinds; For example, grammar- the elements of speech are the parts of which speech consists and into which it is ultimately divided, while they are no longer divided into other forms of speech different in kind from them. If they are divided, their parts are of the same kind, as a part of water is water (while a part of the syllable is not a syllable).” “Those who speak of the elements of bodies mean the things into which bodies are ultimately divided, while they are no longer divided into other things differing in kind; and whether the things of this sort are one or more, they call these elements.” “The so-called elements of geometrical proofs, and in general the elements of demonstrations, have a similar character; for the primary demonstrations, each of which is implied in many demonstrations, are called elements of demonstrations; and the primary syllogisms, which have three terms and proceed by means of one middle, are of this nature. (1014a26 – b5)⁷ 

Notice that we have examples of elements from three separate sciences: grammar, nature, and logic. Of these three, it seems that the one most similar to music is that of speech, for the following reasons: 1)The elements are neither wholly physical nor exclusively intelligible. For grammar, the elements something like letters, syllables, and words, with letters being the elements of syllables, which are in turn the elements of words. For logic, the elements are terms, operators, and quantifiers which are the elements of propositions. 2) For both grammar and logic, the elements are made into utterances in much the same way that musical elements are made into musical works. 3) For all three sciences, they deal with objects which are abstract or universal by virtue of the “one above many” argument. In logic, it is possible to give the same argument on different occasions. For grammar, one may also make the same utterance on various occasions, and in music, one may perform the same musical work on various occasions. And in all of these sciences, one cannot say, argue, or perform the same thing with out the thing said/argued/performed first coming to exist in the first place. For these and other reasons, we can see that the works of Aristotle are filled with the exact sorts of “elements” of which we speak here. But that is not all; he often refers to specifically musical elements in many contexts.

III. Musical Elements in Aristotle.

III. A. Musical “Units” In the Metaphysics.

The concept of a distinctly and explicitly musical element is common in the Aristotelean corpus. In the following passage from Metaphysics V, he is defining “one” or “unity”.

“The essence of what is one is to be some kind of beginning of number; for the first measure is the beginning, since that by which we first know each class is the first measure of the class; the one, then, it the beginning of the knowable regarding each class. But the one is not the same in all classes. For here it is the quarter-tone, and there it is the vowel of the consonant; and there is another unit of weight and another of movement. But everywhere the one is indivisible either in quantity or in kind.” (1016b18-24)

 

“Units” here are the most fundamental parts or elements, in that letters make up words just as dieses (meaning a smallest interval in music) make up melodies.

“We have said previously… that ‘one’ has several meanings…..In music the measure is the diesis, since it is the smallest, and in speech it is the letter … but the measure is not always numerically one. Sometimes there are several, as for instance there are two dieses – not those given by the ear, but those found in ratios – and several articulate sounds that we use for measuring [in phonetics].” (Metaphysics X.1 1053a12-17)

Here we see a musical element compared with others in grammar and units of measurement for weight alongside various physical elements. The elements of each fields are the fundamental units of which those beings are composed. Again, the same comparison is made for letters in grammar and the smallest musical interval. Elements or fundamental units are different in nature for different fields of study. Some fields of study elements that are substances, but others do not, including music.

III. B. Music and other Elements in De Sensu.

In the physical treatises, Aristotle considers the elements of music to be analogous to those of other sciences of sensibles. In the following passage, he treats them in conjunction with color. His discussion assumes an analysis where notes are the elements of chords in (analogically) the same way that black and white pixels can combine into a gray field:

We must now speak of the other colours, reviewing the number of ways in which it is possible for them to arise. It is possible, first, that the white and the black are laid side by side in such a way that while each of them is invisible because of its smallness, the combination of the two becomes visible. This cannot appear as either white or as black, but since it must necessarily have some colour, and can have neither of those, it must be something mixed, a different kind of colour. In this way then, it is possible to accept that there are more colors than just white and black, and that they are many in ratio: for they may lie side by side in the ratio of three to two or that of three to four or in other relations of numbers. (Some may be in no ratio whatsoever, but in some incommensurable relation of excess and deficiency.) Thus they may be in the same condition as concords [symphoniai]: the colors that depend on well ratioed numbers, like concords in their domain, are taken to be the pleasantest of colors (purple and red and a few others of that kind – few for the same reason that concords are few), while those that are not in numbers are the other colours.⁸ (439b19-440a4)

Aristotle is here anticipating some very modern ideas: primary colors which combine in order to produce secondary colors as well as what we now call “pixels” (the smallest visible unit of visibility). His hypothesis is that the underlying mechanism behind concords and color-wheel aesthetics are based on an underlying unity of principle, which was taken up by Johannes Itten in modern color theory. This is far ahead of his time, since the analogy between them is based on wave-phenomena – one of sound, the other of light. In De Anima, he expands this to taste:

If a concord is a sound, and if a sound and the hearing of it are in a way one, while a concord is a ratio, then the hearing must necessarily be a ratio. For this reason either element in excess – either the high or the low – destroys the hearing : similarly in flavors such excess destroys the taste, in colours what is exceedingly bright or shadowy destroys the sight, and in smelling the same applies to a powerful smell, whether sweet or biter, since the perception is a ratio. That is why, while things are pleasant when they are brought pure and unmixed into the ratio (things such as the high-pitched or the sweet or the salty: for they are pleasant in such circumstances) nevertheless what is mixed, concord, is more pleasant than the high or the low. The perception is a ratio, and things in excess dissolve or destroy it.” (426a27-b7)

So clearly Aristotle’s work is filled with “elements” of many sorts, not all of which are substances in the strict sense. Grammar and music treat of relations among substances: animals and air are substances, but they are not the per se focus of music theory, rather these substances are only “musical” insofar as they contribute to the composition and performance of musical works. The principles of music are not those of a substance per se, but rather emerge from the interactions of many substances, in much the same way as the principles of grammar and strategy. In the next section, we shall treat in detail the process of such emergence of analogous (nonsubstantial) per se objects from the relations among substances.

IV. Elemental “Genealogies” for Houses and Music.

On the view defended here, a neo-Aristotelean theory of music will start with some kind(s) of concrete substance and tell how some quantity, relation, affection, etc. thereof relates to the science in question. The following is a simple but modern description of how the phenomenon of music comes from the relations, qualities or affectations among substances. To clarify this process in true Stagirite fashion, we shall use the analogy with house building.

IV. A. The Genealogy of the Elements of Houses.

House building is a “science”, and its per se object is the production of houses. Pace Plato, the knowledge of a house-builder will include the Form of the House, but following Aristotle, it must also include the matter of the house (wood, stone, bricks), the efficient causes (the different workers and tools available) and the final causes. It is not enough to know the overall purpose of a house (“to live in”), but also the lower-level purposes such as “create a level foundation”, and “make sure the walls are square”. A house builder will not only know the form of level and square, but also why houses need to be level and square in the first place.

Houses are not substances in the strict sense and exist by convention. Their “forms” are not natural but emerge from the skillful interaction of humans and nature. The Form of the House cannot be found in a dictionary or even in a building code, but can only be in the mind of a qualified architect. This is the main difference between a productive science and a theoretical science in Aristotle: a theoretical science knows about a substance such as an atom, a cell or a plant, while a productive science knows about something which is not a substance but whose essence is primarily in the mind of the maker. The principles of housebuilding include axioms that are not the essence of a substance and might not be deduced therefrom. For instance “always make all floors and walls level, plumb, and square” cannot be deduced from the essence of any substance, neither from the essence of the house’s matter, nor from the definition of “house”. While the definitions of “level”, “plumb”, and “square” refer to abstract geometry, the presence of these terms in the definition of the essence of “house” is not rigorously demonstrated but rather emerges from the interaction of builders with material over many generations. This being the case, in place of a demonstration, we need a causal story which I will call a “genealogy”. Such a genealogy will be implicit in the principles and causes of all sciences whose per se objects are not “substances” in the strict sense. The genealogy of the principles and elements of housebuilding are as such:

1. Substances – First we have atoms, molecules, energy and living things.
2. Other categories. – Some living things need “shelter” from other things.
3. Some materials have been found useful to “construct” said shelter.
4. There are a lot of useful rules to follow that make building such shelter more effective, including some with arithmetic and geometry. Contra Pythagoras, such elements are not being used qua geometrical but are used qua useful for a specific purpose.
5. Once construction is finished, then living things can “live in” the shelter.

Contra Plato, the builder’s tekne cannot be deduced a priori but are rather learned by those who cooperate to build houses and discuss the pros and cons of different ways of building. So with this in mind, let us look at a similar genealogy for the science of music.

 

IV. B. The Genealogy of the Elements of Music.

As with house-building, so with music, we need to start from some set of commonly-accepted sumbstances and construct our nonsubstantial elements therefrom.

1. Substances – First we have atoms, molecules, and living things.
2. Other categories. – The energy imparts motion to the atoms and molecules.
3. Some forms of this motion are made or perceived as “sound” by some living things.
4. Sound is used by living creatures for the following purposes: sensation (mere hearing), communication, or music.
5. There are a lot of useful rules for making musical sound, including many that involve some arithmetic. Contra Pythagoras and kata Aristoxenus, such rules are not being used qua geometrical but rather qua musical.

According to this framework, music is a science somewhat like phonetics, house building, computer science, or military strategy. In all of these fields, there is a physical substrate or set of elements which can take on various forms imposed on it by rational agents for various purposes. Thus while “music” has no Aristotelean substance as its per se focus, it can define its focus as a certain set of activities that assume a certain physical substrate, principles, purposes, and rational agency of those involved. With that in mind, let us give a full catalogue of the elements of music, from the most fundamental to the most final:

1. Atoms⁹
2. Molecules
3. Sound
4. Musical Sound – sound made of notes, intervals, and rhythm.
5. Melody – Musical sounds in a dynamic sequence.
6. Harmony – Melodies arranged simultaneously.
7. Works – Songs, Concertos, Operas, Musicals, etc.
8. Performances – Social events.
9. Culture (Ethos) of a People.
10. The Final Final Cause – There may be some higher telos for music than contributing to the life of a people who have a certain culture.

 

IV. C. Proximate and Ultimate Elements of Music.

Art and sciences take matter from some more fundamental art: the house builder takes his material and tools from the makers of tools and bricks. Music is similar in this respect. Notice that many of the above elements are not part of music per se:

1. Elements 1-3 pertain to physics.
2. Elements 4-10 pertain “music theory” in the widest sense, which might study the ultimate basis for the smallest intervals and scales.
3. Elements 4-8 are the proper study of musical artists.
4. Element 9 is in political philosophy.
5. Element 10 is theology.

The distinctively musical elements (4-8) in this “scala musica” are not substances, but derived by cognitions concerning substances “in a certain respect” – those relations which are musically relevant. For music to be a science, we must know:

1. What are the per se phenomena that are the focus of music. – Musical sound.
2. What it is about the focus that makes it music. – The sound exhibiting proportions and patterns of a certain type.
3. How the elements are defined. – The elements are those most useful for defining said proportions and patterns that define music.
4. Other causes: formal causes, final causes, etc.

Something like this will be the the most simple version of our theory: There are various substances, including atoms, molecules and living things. The atoms and molecules collect in “atmospheres”; layers of gas surrounding some planets. Atmospheres transmit sound, which animals find useful for hearing events in their environment. Some animals also use sound for “music”, whose purpose is unclear, and it may have multiple uses. However it seems clear that communication is a large part of it, because we find that musical sound has been split into distinguishable elements rather similar to the elements of codes or languages.

This last line is where we come to the fundamental principles of music: in other words, we begin to find the ultimate causes and principles that underly the distinction between normal sound and music. Music exhibits its distinctive character by having all pitches and beats limited to one of a few selected our of many. So the fundamental elements of music are both melodic and rhythmic, but in the following, I shall focus on melodic units or elements, which are intervals. But why is this the case? Because of communication – each unit (pitch or note) must be distinguished from the others so that patterns are easier to recognize.¹⁰ This is the origin of the “diesis” or smallest interval. In Greek music, it was a quarter tone, but later on it was dropped and the diesis was made the semitone, perhaps due to the increasing importance of harmony over melody in Western music. In almost all Greek music, harmonies were sung in unison. With the later increase in polyphony, however, quarter tones perhaps seemed too cluttered. Since complex polyphony provided a great many more possibilities than single melodies, Western composers dropped the quarter-tone.

The “whole tone” is another intervallic element derived from the space between the two concords of the fourth and fifth. In both ancient Greek and modern Western scales, we find that the middle of each octave is taken up with the whole step that divides the fourth from the fifth degrees. Below the fourth and above the fifth, we always find a mix of whole tones and smaller intervals depending on the tonality needed for the occasion. Dieses could in theory be defined in many ways, but in order to be more compatible with the structure defined by the concords, it should be some whole number fraction of a major fourth. In modern Western music, we have five semitones below the fourth degree which can be broken up into either the major scale (whole, whole, semitone) or the minor scale (whole, semitone, whole). If you were to try to divide the fourth into three equal units, they would be slightly larger than the whole tone and not so much larger that they would be readily distinguishable nor mathematically proportionate with the other intervals. The three whole tone interval falls directly between the two concords and is the most discordant interval, rarely used for most serious music, but in blues and other blues-influenced styles it is prominent. However, the harmonic structure of such music has been simplified to the extent that it is not too cluttered. If Bach were to try a fugue on the theme containing a tritone, it would not work, but some popular music can get away with it.

This is what we might expect to find as the essence of musical elements – a mix of nature and convention, not so different from grammar and logic. In none of these sciences are the elements substances in the strict sense, but instead they define their elements based on a mix of natural and pragmatic considerations. Once we have the fundamental melodic elements defined as the octave, concords, semitone and whole tone we can add them together to make melodies, which melodies must then obey the rules of “dynamics”. These rules are generalizations of what sorts of rising or falling series of notes or chords “make melodic sense”. Said melodies must at the very least must seem like a unified entity and be complex enought to hold interest but not be too complex to exhibit perceivable order..

In order to accomplish this, composers will follow certain principles:

1. Define a “motif” or “theme” by the compostition of lower level elements such as notes and rhythyms.
2. Repeat the motif.– the motif can be used over and over again in the same way that many of the same type of brick are needed to make a house.
3. The motif undergoes “development”, “variation”, “restatement” – the elements of the motif are slightly re-arranged into a related motif or variation.
4. Then “resolution”, other dynamic patterns … and so on and so forth.

Thus we have various sets of principles that are not reducible to those of lower levels but which build on them to further the same purpose. Thus far, I have only given a superficial look at the physics of musical poetics or composition; next we shall explore the deeper metaphysics and philosophy of science involved.

V. Music and Ontology.

V. A. Music and Substance.

Music was a prominent topic in classical Greek metaphysics starting from the Pythagorean school, which influenced Plato and Aristotle’s ideas concerning music as mathematical science. Even as late as Aristotle Metaphysics Books I and VII. After that Aristotle’s student Aristoxenus continued the same trend to be even more empirical than the Stagirite, and our views are very much in this latter vein.

V.A.1. Pythagoreanism – Numerical Substance.

On my Aristotelean reading, Pythagorean “substance” is ultimately numerical, so Pythagorean substances are non-sensible ideal beings. My interpretation of them here is based solely on the assumption that their numbers are ideal or abstract beings, their placement “in” those things of which they are the substance notwithstanding. Sensible beings may not seem numerical at first glance, however according to our reading of Pythagoreanism the substance of these beings must be numerical in some way. One way this could be seen is where there is some unlimited substrate, which substrate then takes form through numerical proportionality. On this reading, music is seen to be an example of a sensible phenomena whose essence / substance has been shown to be mathematical ratios that underly rhythms and melody. So while music is not substance per se, it is shown to be more substantial than many other things whose mathematical essence is less clear and which are therefore less beautiful. On this view, the closer to the numerical substance a phenomenon is, the more beautiful it will be. On this view music is far more substantial than other sensible beings, and contrary to our position, it would be one of the substantial sciences, as it was under Platonically-inclined thinkers.

Since Pythagorean metaphysics makes the substance of beings numerical, Pythagorean science should be somewhat “numerological”. In Pythagoreanism, it is of the essence of planets that there are a certain number of them. Which number it is is up for debate, but most numerological astronomers counted seven. The fact that there were seven planets was taken to be a clue to their essence, and their research consisted in looking for other sets of sevens, such as the seven “metals of antiquity”, days of the week, and the number of notes in the diatonic scale. On this view, the discovery of Uranus would throw the “numerological” astronomy into crisis, because through the change of number there would be a corresponding change of planetary essence. However, for either modern or Aristotelean science the number of planets is not essential to the nature of planets. On both of these views, planets are natural concreta whose number is accidental to their nature. Other planets in other solar systems may be fewer than in ours and they will still be essentially the same as our own.

V.A.2. Aristotle: Music as Mathematical Science.

Aristotlean substances are natural concreta that are not mere aggregates but are a separate “this”: in modern terms (which for conveniences’ sake I will use in this work), the following are what he would call Aristotelean substances: atoms, molecules, cells, organs, organisms, planets, and stars. (Whether the inclusion of atoms in this list undermines my entire approach is something best left for a separate work.¹¹)

Aristotle differs from Pythagoras in claiming that numbers are not themselves substance; instead, numbers are properties of concrete physical substances. Thus, because the of the nature of reality, there happen to be planets (for example); the fact that there are a certain number of planets is not really essential to their nature. Pythagoreans, on the contrary, tend to think that the number of planets is essential to their planetary natures, whether these are the number of planets counted, their number in order from the center of the solar system outward, or their periods of revolution. For Pythagoreans, these quantities are the very essence of substance of what the planets are.¹² Aristotle is having none of this; for him, there are material beings of such and such type who move in a certain way based on their physical nature, and the number of these beings is accidental. As a result, the number of planets is of no more consequence for astronomy than the number of continents is for geology; in other words, the discovery of a new one (changing its number) does not change its substantial essence.

However, Aristotle has taken up the conception common to his idealistic predecessors that mathematical sciences are more scientific than their non-mathematical counterparts. Some empirical sciences, such as music or astronomy are essentially mathematical while other branches of ‘physics’ are not. Strangely enough, this would include the field of study that we call “modern physics”. In Posterior Analytics, he makes this assumption without any argument:

…[i]t is the task of those who use perception to know the fact that, and that of the mathematical scientists to know the reason why: for the latter possess the demonstrations of the causes, and often do not know the fact that, just as people who study the universal often do not know some of the particular instances because they have not observed them. (78b34 – 79a6)

It is difficult to see how this could be under the more naturalistic approach of Aristotle, where mathematical entities are not substance, nor essence, but rather the mere definition of the essence. In the following, we see where he went wrong with this approach. Through the examples of astronomy and music he seeks to show how mathematical sciences can define the essence of sensibles.

…in all these examples it is clear the nature of the thing and the reason of the fact are identical: the question ‘What is an eclipse?’ and its answer ‘The privation of the moon’s light by the imposition of the earth’ are identical with the question ‘What is the reason of the eclipse?’ or ‘Why does the moon suffer eclipse?’ and the reply ‘Because of the failure of the light through the earth’s shutting it out’. Again, for ‘What is a concord? A commensurate ratio of a high and a low note’, we may substitute ‘What reason makes a high and low note concordant? Their relation according to commensurate numerical ratio.’ ‘Are the high and low note concordant?’ is equivalent to ‘Is their ration commensurate?’; and when we find that it is commensurate, we ask ‘What then, is their ratio?’ (90a15-24)

In the former example, we see that clearly geometrical analysis is essential to predicting and explaining eclipses; however this should not be taken too far: the assumption that Euclidean geometry is axiomatic for physics has recently been disproven and discarded under relativity. However, Euclid will suffice for the solar system’s orbital dynamics as known to Aristotle and Newton. In a sense, modern physics’ recourse to non-Euclidean geometry undermines Aristotle’s argument. Admittedly it is still geometry with different axioms, but there are so many different ways to do non-Euclidean geometry. How does one choose how many spatial dimensions and what topology to use? This can only be derived from the study of cosmology. Thus rather than geometry ruling over astronomy as under the ancien regime, modern astronomy uses whichever version of geometry suits its purpose. Of course, Euclid is still interesting form most mid-scale phenomena, but it no longer exerts the sort of absolute authority we find in ancient science. In my view, this same dethroning of the exact sciences over the empirical in modern astronomy is implied in Aristoxenus’ criticism of dogmatically mathematical music theories.

V.A.3. The Aristoxenian Paradigm Shift in Music Theory.

Aristoxenus (fl. 335 BC), a student of Aristotle, wrote the first major work of music theory, the “Elements of Harmony”. While a student of Pythagoreanism in his native Italy, he converted to Aristoteleanism and eventually created a theory of music that was even less Pythagorean and more ‘physical’ than his teacher’s. Aristoxenus was more faithful to the naturalistic spirit of Aristotle and disregarded the above-criticized assumption that music is an essentially mathematical substance. Because he pursued a science of music theory and because his innovation required a change in how music itself was defined, I call it a “paradigm shift”.

While Aristotle still saw mathematical ratios as being radically essential to music, Aristoxenus’ claim that mathematics was less essential than a species of aesthetic sensation. Thus the essence of music is not Pythagorean substance nor sound qua mathematically rationalized, but rather sound qua sensibly proportioned, by which we mean that which appears properly proportioned rather than that which conforms most exactly to mathematical proportions. In Aristotle’s view, concords just are numerical ratios and nothing else besides. (90a30) But with Aristoxenus, concords have a curious relation with numerical ratios without being identical, almost like the relation between the astronomical solar calendar and paper calendars. Just as there needs to be days added onto leap years to keep our yearly tally of days in line with the revolutions of the earth, so also do we need to adjust the arithmetical proportions of pure Pythagorean temperment to keep it in line with our musical perceptions.

“Through hearing we assess the magnitudes of intervals, and through reason we apprehend their functions. … While it is usual in dealing with geometrical diagrams to say ‘let this be straight line’, we must not be satisfied with similar remarks in relation to intervals. The geometer makes no use of the faculty of perception; he does not train his eyesight to assess the straight or the circular or anything else of that kind either well or badly: it is rather the carpenter, the wood turner, and some of the other crafts that concern themselves with this. But for the student of music accuracy of perception stands just about first in order of importance, since if he perceives badly it is impossible for him to give a good account of the things which he does not perceive at all.” (Barker 150)

This means that contra Aristotle, musical proportion is not a species of mathematical proportion. However since we are retaining an Aristotelean conception of science, we have to say that musical proportion is not a species of arithmetical proportion and is defined separately.

Likewise, as befits the author of the “Elements of Harmony”, Aristoxenus also believes in elements that are essentially musical, but which are analogous to other sorts of virtual or abstract elements:

“… the order which relates the melodic and unmelodic is similar to that concerned with the combination of letters in speech: for from a given set of letters a syllable is not generated in just any way, but in some ways and not others.” (Barker 153)

He also adheres to a rigorous distinction between arithmetic and musical elements. On the one hand “… we accept that from a purely abstract point of view there is no least interval.”(Barker 160), but on the other

“The claim that there is no least interval by which we divide ad infinitum in melody is one that commands assent: there is some greatest number of parts into which melody divides each of the intervals.” (Barker 160)

What prevents a contradiction with the one before is the qualification “in melody”; once we assume that we are speaking of musical intervals and not mere differences in merely physical frequencies, which is what he is taking about “from a purely abstract view.”

Furthermore, there is also found in Aristoxenus the view that musical composition is the placing the musical elements in a certain arrangement:

However, there is a major hurdle in this conception of music; how to explain the presence of numerical ratio in pre-rational sensation without recourse to an abstract conception of substance or subordination of music to mathematics. In my view, this is done by giving an account something like that given above for the ultimate basis for whole tones, semitones, and how they are pieced together to make scales.

“The last part of the science is that concerned with melodic composition itself. Since many forms of melody, of all sorts, come into existence in notes which are themselves the same and unchanging, it is clear that this variety depends on the use to which the notes are put: and this is what we call melodic composition.” (Barker 155)

Here we find that the Musician also has our own conception of Universal Hylomorphism: the idea that there are changes where units of matter are arranged into a form without themselves undergoing any change. Just as bronze is not changed by being made a sphere, so also are notes not changes by being composed into a song. The fact that said “matter” is neither wood nor molecule does not change the fact that songs are made from notes in the way a sphere is made from bronze.

V.B.1 Science, Music, and Substance in Aristotle.

Under the idealistic systems of Plato and Pythagoras, one of the main arguments that substance is the “argument from the sciences”. On this view, the sciences of the ideal were the most rigorous and certain and thus the most suitable per se objects are ideal beings. If substance is prior in definition, knowledge and time, (as in Aristotle 1028a30) then the idealists argue that ideal objects are “substance” in the strict sense. This is an objection which Aristotle went to great pains to answer, devoting not only significant portions of books I and VII but all of books XIII and XIV to this and related problems. In the following, I will try to explain a plausible Aristotelean way to rebut the argument from the sciences, which, if successful would undermine Aristotle and boost Plato.

V.B.2. Aristotle’s ‘substance’.

Aristotle uses many of the same words for various related or “analogous” senses. The most famous is “’Being’ is said in many senses.”. As a result, many other substances have analogous senses of the various “be” verbs. Key to the argument of Book Zeta, there is a distinction between two senses of “substance” which we shall discuss on the way to our presnet conclusion. For the sake of distinguishing them in this chapter, we will call them “substance₁” and “substance₂”. For Aristotle, the following are true:

1. A substance₁ is a compound of matter and form.
2. Substance₂ is the form of a substance_1.
3. The essence of a substance₁ is a substance_2. (2 and 3 are equivalent statements.)
4. A substance₁ is anything that has substance₂.
5. Conversely, substance₂ belongs most properly to substance_1..
6. Substance₂ cannot exist separately.
7. Only substance₁ can exist separately.
8. When substance₂ is spoken of as if it were separate from substance₁, it is being spoken of “abstractly”.
9. A substance₁ cannot be artificial since artificial beings do not have substance₂ in the full and proper sense.¹³ (The formal cause belongs to it only extrinsically, and the efficient and final causes even less so.)

Such are the basic assumptions concerning substance in what follows.

V.B.3. The distinction between Substantial and Analogous sciences.

It is substance₁, the concrete substance₁, that is most real. The latter formal substance₂ is the content of science, while substance₁ is the object of science (in our modern sense of “objective”). I say an object of science, because sciences do not only learn the form of the substance, but the other causes as well, a fact which further tells against the idealist “argument from the sciences”.¹⁴ But what I call a “substantial science” does have a substance as its per se object, but these substances are concrete, and the science studies the form as form of the concrete: examples of this include chemistry (the study of atoms and molecules), biology, botany, zoology, medicine, astronomy, geology.

These sciences deal with substances, meaning that members of a particular genus are individuated into concrete units which cannot be divided into smaller units of the same kind. So if you divide an atom, you do not get another atom, but rather an other type of substance. When you divide a molecule, you do not get molecules, but rather atoms. When you divide a cell, you do not get another cell, but rather parts of a cell which cannot come to be nor survive separately. When you divide an organ such as a heart, you do not get another heart, but rather tissue, a mere aggregate of cells of a certain type. Organisms, planets, and stars also exhibit a similar unity, and the fundamental principles of the science include the following the study of atoms and molecules as substances.

1. The form of the genus – what all atoms share qua atoms.
2. The elements of matter of the genus – protons, neutron, electrons, etc.
3. The formal causes of the substance. For atoms, this includes

• • Genus – the essential form shared by all atoms as well as
  • Differentia – the various ways that atoms differ based on the different arrangements of the elements of the genus.

4. Fourth, other causes as applicable, including efficient and final causes.

A conception of atoms as a certain kind of substance might provide the fundamental principles of a natural science that studies atoms. Today we would call such a science “physics” or “chemistry”, which, for the sake of convenience, would include as well the study of molecules. However, given that our current topic music concerns how matter reacts to certain sorts of sonic energy, we can call this science “physics”. It is exemplary for how a science can be defined by its primary concern with a particular type of substance. In addition, biology is defined by its concern with another type of substance, the organism, which forms its natural ‘unit’ in the same way that atoms and molecules do for our sense of “physics”. The fact that molecules are a different kind of substance only means that its inclusion in the same science is only due to their ontic proximity or pragmatic concerns. It is not so different from how biologists not only study complete organisms, but also their organs and cells. Whether some biologists find it better to specialize in cells of organs is contingent on the usefulness of such a strategy w.r.t. epistemology or application rather than ontology.

So now that we have a preliminary conception of substantial sciences, we also need to see how an analogical science, even those of logic and arithmetic, can find their rigour without having a per se focus on a primary substance.

V.B.4 Analogical Sciences in Book Lambda.

The idealist can respond to the above by pointing out that on Aristotle’s view, the most rigorous sciences paradoxically have the least substantial objects. If mathematics and geometry are not sciences of substance, then what is? Aristotle gives many examples of rigorous sciences that do not focus on per se substances – arithmetic, logic, grammar, and music among others. How would such a science work if it did not have a substantial per se object? The answer may be found in the following passage:

The causes and the principles of different things are in a sense different, but in a sense, if one speaks universally and analogically, they are the same for all. For one might raise the question whether the principles and elements are different or the same for substances and for relative terms, and similarly in the case of each of the categories. But it would be paradoxical if they were the same for all. For then from the same elements will proceed relative terms and substances. (1070a31ff)

Elders (1972) reads this and other nearby related passages as referring solely to Aristotle’s criticism of Platonism where different substances are not univocal in the senses of their categories.¹⁵ In that reading, each of the “different things” in line 31 are the different substances whose various categories and predicates are analogically but not univocally “the same” as they are for other classes of substance. In other words, the “different things” refers to different members of the category “substance” – for instance stars, organisms, and atoms. But there are two reasons why we might not limit the reading of “different things” to the category of substance, and they include the following:

1. In these passages, it seems that the primary difference being discussed is between substance and other categories:

• • “different or the same for substances and for relative terms” (1070a34-35)
  • “for then from the same elements will proceed relative terms and substances” (1070a37-b01)
  • “[T]here is nothing common to and distinct from substance and the other categories….” (1070b01)
  • “Substance is not an element in relative terms, nor is any of these an element in substance.” (1070b02-3)
  • “None of the elements, then, will be neither a substance or a relative term; but it must be one or the other.”(1070b7-9)

2. There is independent reason to think that for many sciences, we are forced to speak of non-substances as being “substantial” in a derivative or loose sense. It is these sciences that we speak of here as being “analogical” (as in 1070a31), and the independent grounds for this assumption will be the primary topic of this chapter.
   

There are two ways that one might argue for such a reading: first, one might claim that this is what Aristotle meant in his texts, or secondly, Aristotle must argue something like this in order to claim that rigorous sciences can have per se objects which are not substance in the strict sense. In the following, I shall pursue the latter thesis, that something like this is needed for a science of music, not to mention logic, grammar, rhetoric, strategy, geometry, arithmetic, and many others. So from this point, I shall argue under this assumption, that the science of “music” grants its objects with a what I shall call “virtual substantiality”, and as such they are the sort of thing that are composed of ‘virtual elements’ or ‘abstract elements’.

On this view, Aristotle could answer the idealists thusly: the rigour of the exact sciences comes not from the substantiality of their per se objects, but rather the fact that they limit their investigation to some dependent category which has well-defined objects. On this view, math investigates substances but not qua substance but rather qua quantifiable being. This places math in a secondary class of sciences that do not deal with a substance as their per se object, but only treat substance qua some other category. If Aristotle is to answer the idealist’s challenge, each accepted science must have some account that defines how it relates to substances in the full concrete sense. So for math, he claims that it deals with substances, if at all, solely in the category of quantity and that this limitation of focus is what gives it its rigour. Other sciences limit themselves in other ways and other categories: logic deals with propositions insofar as they are true or false, grammar deals with sound insofar as it is articulate and meaningful, and music deals with sound insofar as it forms the ‘essence’ of musical works. While there former exact sciences are simply the sciences of the category of quantity, the others are the sciences of something is by nature in the overlap of the ta phusika and the ta pragmata: grammar, for example is the science of articulate sound, meaning that it looks at a particular physical phenomenon – sound, but sound only insofar as it is used by animals for language. Military science, for example, looks at men, horses, weather and terrain – but only insofar as these elements are related to the need for armed groups to control territory. Musical theory also has a similar account that it must give for how it treats sound- sound in so far as it relates to the need for certain living creatures to make sound that is musically structured.

 

VI. An Possible Objection from Final Causes.

In performance, and existing form is applied to existing matter. In composition, a form is created from abstract elements. Only once this form is created can it then go on to be the form of a musical performance. Thus we have a explanation of a change that occurs. However, there is more than matter and form iin Aristotle’s physics, there is also the final cause. It seems that the analogy between performance and composition might break down down due to the lack of an existing form as final cause. Since there is no form as final cause, how can the change happen? The performer knows what they are after in a performance; how does the composer know?

My initial view, which will be postponed for a future work, is that composition is more akin in this respect to praxis than to tekne. While the content of the science of composition has a lot of overlap with the tekne of performance, in terms of teleology. In this respect they are similar to the relations between military praxis and military science. Praxis is that form of goal-oriented behavior which has no clearly defined form as its telos. If we say that that the goal of praxis is the “Form of the Good”, this is in a much looser sense than with the form of a house. It is highly unlikely that the Good has a form in the same sense as other concreta. When the composer composes, they are seeking to implement a certain specific way of being “good” in a way that we find in other goal-oriented processes that create forms rather than instantiate them:

1. Praxis – Political action which seeks to maximize the Good.
2. Invention – Technical action which seems to create a form in matter that can acheive a goal.
3. Rhetoric – Technical action which seeks to maximize the persuasiveness of speech.
4. Poetic composition – Technical action which seeks to create a form of poetic speech that is poetically Good.
5. Musical composition – Technical action which seeks to create a form of musical sound that is musically Good.
6. Legislation – Political action that seeks to define laws of such a form as to achieve the Good for a people.

In all of these sciences, poesis is at the service of a Good rather than a Form. In each the Form is the product rather than the telos as it is with productive arts or nature. In virtual poesis, the form is created by the maker according to the process given above.

 

VII. Conclusion.

In contemporary ontology of musical works, there are extreme views who we have been influenced by and we hope that we have saved the relevant phenomena using an Aristotelean “middle path”.

To the fictionalists who deny that compositions are real¹⁶ we say that there are many ways of saying “real”, and each differs by virtue of the essence of what is spoken of. For nonsubstances like musical works, we have a conventional or derivative sort of “reality”, but it is its own reality nonetheless, a reality suited to the being of music. Our above “genealogy” of musical elements details the difference between the substantiality of material elements and living things and analogous reality of musical elements of works.

To the Platonists who say that musical works are substances, we claim that such a view is subject to the same objections given by Aristotle so long ago, chief among them being the following: 1) The objection from lack of causality. “Above all, one might discuss the question what on earth the Forms contribute to sensible things, either to those that are eternal or to those that come into being and cease to be.” (991a7-8) How do eternal forms cause composers to reveal them to us in the Plato’s Cave? What is the relationship between the two? We might be satisfied with leaving it open for future inquiry if only there were not a superior option in Aristotle’s immanent forms. 2) The point that no universal is a substance, given that universals are predicated of concreta (1038b15) and cannot exist apart from them.

This is not to say that neither of these views is lacking in value, but we hope that something like our view will seem plausible both for the issue of artificial abstracta but natural ones as well, including biological essences and natural languages. In our view each of these beings has virtual elements specific to the sorts of beings they are: genes¹⁷, phonemes, memes or others as needed to save the phenomena in those domains.

End notes

1. The one who seemed me as the most Aristotelean among them, Nicolas Wolterstorff (1980) is called a sort of a “Platonic (eternal) norm-kind/norm types” in Killin (2018.) 272

2. Peter Kivy and Julian Dodds are the most respected such “Platonists” of whom neither actually ever cites Plato, something they share with mathematical Platonists since Frege. The best introduction is Kivy (1987) and Ostertag (2012) In the present work, I will refer to “Platonism” with respect to musical works to refer to Kivy’s position in the above-cited piece.

3.  Killin (2018).

4.  I admit that we have modern theories that explain music in terms of set theory, qualia, and others which are not part of classical metaphysics and which do not clearly resemble anything that he dealt with. However, the factors adduced in favor of these more recent approaches may be even more amenable to an Aristotelean analysis.

5.  Rosen 2018

6.  In the first line, the “(1)” means that these are the primary meanings of ‘element’, the next section (“2”) (not quoted here) begins “people also transfer the word ‘element’ from this meaning and apply it to” another secondary meaning. But the following are the primary meanings of ‘element’.

7.  All Aristotle quotes, unless otherwise specified, are from McKeon (1941).

8.  Barker (1990) Page 47.

9.  Aristotle’s relationship with the atomism of his time is contentious. However, I will simply assume that atoms qualify as Aristotelean substances on the following basis: They are the smallest unit of material bodies which is a “this” and not merely an aggregate of such-and-such. I think that this topic deserves its own full treatment in another piece, but here I shall assume that modern atoms and molecules are substances.

10.  This is rather similar to how languages separate certain sounds to make “letters” while other sounds are not excepted because they would muddy up the code. For example, letters are often dropped from languages, especially when phonemes from other languages are introduced. For example, when the Francophile Normans conquered England, that introduced a great many new sounds into Old English which led to the elimination of the Scandinavian “ð”, which is midway between a “d” and a “th” or “t”. With the more crowded list of elements to choose from, the Scandinavian “ð” just muddied things up.

11. I do this to simplify my exposition, to make this work more readable and relevant for non-Aristoteleans. I am thereby choosing to make my thesis primarily a “neo-Aristotelean” theory as opposed to an interpretation of Aristotle’s work. However, my goal is an argument that resembles something he might support if he were alive today.
12.  The fact that none of these numbers works out to exact ratios could, in a Pythagorean research program, be either explained away or be the goal of future work. For example, the desire to square the numerical messiness of the heavens with the beauty of whole numbers was a major impetus behind Mesoamerican astronomy, and the Pythagoreans could undertake such a project of their own. One might also claim that this mathematical inelegance is empirical “noise” as opposed to the pure signal of the mathematical “music of the spheres”.
13. Note that the demotion of products of skill from substantiality is especially crucial in the anti-Platonism the motivates the theory of abstract artifacts. Any further treatment of substance will be given when we treat of natural abstract or virtual products, such as biological essences and perhaps natural languages.
14.  Physics II.2 194a21- 27
15.  Elders (1972) pg. 114ff.
16.  Killin 2018
17.  For biological essences, the distinction of composition and performance is exactly analogous to that of phylogeny and ontogeny, with phylogeny being the manipulation of genes through the efficacy of natural selection.

    Bibliography

 

Barker, Andrew, ed. 1990. Greek Musical Writings Volume 2: Harmonic and Acoustic Theory. Cambridge UP.

Elders, Leo. Aristotle’s Theology. 1972. Van Gorcum and Co. N.V., Assen, The Netherlands.

Killin, Anton, 2018. “Fictionalism about musical works.” Canadian Journal of Philosophy. Vol. 48, No. 2, 266-291.

Kivy, Peter, 1987 “Platonism in Music: Another Kind of Defense.” American Philosophical Quarterly Vol. 24, Number 3, July 1987.

McKeon, Richard, ed. 1941 The Basic Works of Aristotle. Random House. New York.

Ostertag, Gary. 2012. “Critical Study: Julian Dodd. ‘Works of Music: An Essay in Ontology.’” Nous. 46:2 (2012) 355-374.

Rosen, Gideon, “Abstract Objects”, The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/fall2018/entries/abstract- objects/>.

Witt, Charlotte. 1989. Substance and Essence in Aristotle. Ithaca and London. Cornell U.P.

Wolterstorff, Nicolas. 1980. Works and Worlds of Art. Clarendon Press, Oxford UK.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Bibliography

 

Barker, Andrew, ed. 1990. Greek Musical Writings Volume 2: Harmonic and Acoustic Theory. Cambridge UP.

Elders, Leo. Aristotle’s Theology. 1972. Van Gorcum and Co. N.V., Assen, The Netherlands.

Killin, Anton, 2018. “Fictionalism about musical works.” Canadian Journal of Philosophy. Vol. 48, No. 2, 266-291.

Kivy, Peter, 1987 “Platonism in Music: Another Kind of Defense.” American Philosophical Quarterly Vol. 24, Number 3, July 1987.

McKeon, Richard, ed. 1941 The Basic Works of Aristotle. Random House. New York.

Ostertag, Gary. 2012. “Critical Study: Julian Dodd. ‘Works of Music: An Essay in Ontology.’” Nous. 46:2 (2012) 355-374.

Rosen, Gideon, “Abstract Objects”, The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/fall2018/entries/abstract- objects/>.

Witt, Charlotte. 1989. Substance and Essence in Aristotle. Ithaca and London. Cornell U.P.

Wolterstorff, Nicolas. 1980. Works and Worlds of Art. Clarendon Press, Oxford UK.

1The one who seemed me as the most Aristotelean among them, Nicolas Wolterstorff (1980) is called a sort of a “Platonic (eternal) norm-kind/norm types” in Killin (2018.) 272

2Peter Kivy and Julian Dodds are the most respected such “Platonists” of whom neither actually ever cites Plato, something they share with mathematical Platonists since Frege. The best introduction is Kivy (1987) and Ostertag (2012) In the present work, I will refer to “Platonism” with respect to musical works to refer to Kivy’s position in the above-cited piece.

3Killin (2018).

4 I admit that we have modern theories that explain music in terms of set theory, qualia, and others which are not part of classical metaphysics and which do not clearly resemble anything that he dealt with. However, the factors adduced in favor of these more recent approaches may be even more amenable to an Aristotelean analysis.

5 (Rosen 2018)

6 In the first line, the “(1)” means that these are the primary meanings of ‘element’, the next section (“2”) (not quoted here) begins “people also transfer the word ‘element’ from this meaning and apply it to” another secondary meaning. But the following are the primary meanings of ‘element’.

7All Aristotle quotes, unless otherwise specified, are from McKeon (1941).

8Barker (1990) Page 47.

9Aristotle’s relationship with the atomism of his time is contentious. However, I will simply assume that atoms qualify as Aristotelean substances on the following basis: They are the smallest unit of material bodies which is a “this” and not merely an aggregate of such-and-such. I think that this topic deserves its own full treatment in another piece, but here I shall assume that atoms and molecules are substances.

10This is rather similar to how languages separate certain sounds to make “letters” while other sounds are not excepted because they would muddy up the code. For example, letters are often dropped from languages, especially when phonemes from other languages are introduced. For example, when the Francophile Normans conquered England, that introduced a great many new sounds into Old English which led to the elimination of the Scandinavian “ð”, which is midway between a “d” and a “th” or “t”. With the more crowded list of elements to choose from, the Scandinavian “ð” just muddied things up.

11. I do this to simplify my exposition, to make this work more readable and relevant for non-Aristoteleans. I am thereby choosing to make my thesis primarily a “neo-Aristotelean” theory as opposed to an interpretation of Aristotle’s work. However, my goal is an argument that resembles something he might support if he were alive today.

• 12 The fact that none of these numbers works out to exact ratios could, in a Pythagorean research program, be either explained away or be the goal of future work. For example, the desire to square the numerical messiness of the heavens with the beauty of whole numbers was a major impetus behind Mesoamerican astronomy, and the Pythagoreans could undertake such a project of their own. One might also claim that this mathematical inelegance is empirical “noise” as opposed to the pure signal of the mathematical “music of the spheres”.

13 Note that the demotion of products of skill from substantiality is especially crucial in the anti-Platonism the motivates the theory of abstract artifacts. Any further treatment of substance will be given when we treat of natural abstract or virtual products, such as biological essences and perhaps natural languages.

14Physics II.2 194a21- 27

15 Elders (1972) pg. 114ff.

16Killin 2018

17 For biological essences, the distinction of composition and performance is exactly analogous to that of phylogeny and ontogeny, with phylogeny being the manipulation of genes through the efficacy of natural selection.

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