Ugaglia ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 2016 Monica Ugaglia Aristotle and the Mathematical Tradition on diastēma and logos: An Analysis of Physics 3.3, 202a18–21 Monica Ugaglia RISTOTLE’S PHYSICS 3.3 contains interesting evidence of an open debate in mathematics, concerning the inter- changeability of the notions of diastēma and logos in the theory of harmonics. Because of the standard interpretation of the passage, however, this reference to harmonics has gone un- noticed: a slightly different understanding is proposed in this paper, which restores the relevance of the passage and its place in the contemporary debate. In §1 Aristotle’s text is analysed. The context is that of mo- tion: in order to justify the apparently counterfactual assertion that a single motion corresponds to two different objects, for it is the actuality of the movable (by the mover) but also of the mover (in the movable), Aristotle resorts to a simile: likewise a single diastēma, namely the interval between two notes, cor- responds to two different logoi; both the ratio (logos) of the higher note to the lower and the ratio of the lower note to the higher can be used to describe the same interval. The historical relevance of the diastēma/logos example is dis- cussed in §2: although both concepts are employed in har- monics, the diastēma is a purely harmonic-theoretical concept, while the strictly connected notion of logos primarily pertains to proportion theory, so that the way of relating them is far from straightforward. In the so-called Pythagorean tradition, the notion of diastēma is not defined, but the interval between two notes is identified with their ratio (logos). The further unspoken convention of taking the notes in a precise order—the ratio is of the higher note to the lower and not vice versa—leads to a one- A 50 DIASTĒMA AND LOGOS ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 to-one correspondence between diastēma and logos, leaving no room for ambiguities. But this correspondence is a result of Pythagorean conven- tions, and different traditions rely on different assumptions, so much so that in the Aristoxenian theory, for instance, the two notions of diastēma and logos become completely separate. It is reasonable to assume that when a text from the Pythagorean tradition, where the two terms are used interchangeably, is read according to a more Aristoxenian perspective, or, more generally, according to any philosophical rather than strictly mathematical perspective, ambiguities like the one mentioned by Aristotle will arise. The emergence of a discussion on this subject is particularly evident in the well-known passage about the harmony of the world-soul in the Timaeus: where Plato comfortably employs diastēma as a synonym for logos, interpreters like Theon or Era- tosthenes feel uncomfortable, and try to assess the question within a harmonic-theoretical context, asking for example why the ratio of the higher note to the lower will be preferable to the ratio of the lower note to the higher in order to single out an interval. In fact, there is no reason other than convention for pre- ferring one logos to the other, and once the situation is clear, the ambiguity is perfectly acceptable, even if the acceptance needs some sort of discussion. My claim is that Aristotle’s passage must be read in this perspective, as evidence of a wider debate, some parallel testimonies for which I list and discuss in §3. 1. Aristotle’s text Chapters 1–3 of Physics Book 3 deal with motion (kinēsis). At the very beginning motion is defined as “the actuality of what potentially is, qua such.”1 The definition is then restated as “the actuality of the movable, qua movable,” and the long discussion that follows is concerned with the problems that this raises. In particular, the whole of 3.3 is devoted to illustrating the seem- 1 ἡ τοῦ δυνάµει ὄντος ἐντελέχεια, ᾗ τοιοῦτον (201a10–11). MONICA UGAGLIA 51 ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 ingly paradoxical consequence that motion is a shared actu- ality: since ‘movable’ is a relative term (πρός τι) it cannot be defined, qua movable, without its correlative term, namely the mover. But if the movable is always a movable-by-a-mover, motion is the actuality of the movable, but also of its mover. But what does it mean that motion, while being a single ac- tuality, is the actuality of two objects? To make the statement acceptable to his audience, Aristotle sets forth an analogy between motion and diastēma (202a18–21):2 T1: ὥστε ὁµοίως µία ἡ ἀµφοῖν ἐνέργεια ὥσπερ τὸ αὐτὸ διά- στηµα ἓν πρὸς δύο καὶ δύο πρὸς ἕν, καὶ τὸ ἄναντες καὶ τὸ κάταντες· ταῦτα γὰρ ἓν µέν ἐστιν, ὁ µέντοι λόγος οὐχ εἷς· ὁµοίως δὲ καὶ ἐπὶ τοῦ κινοῦντος καὶ κινουµένου. So that one is the actuality of both, similarly to the way in which it is the same interval “one to two” and “two to one,” and the ascending and the descending: for they are one but the logos is not one: and similarly about the moving thing and the moved.3 In short, we have the following proportion (P): P) diastēma : two logoi :: kinēsis : two logoi The current interpretation of passage T1, ultimately trace- able to Themistius,4 reads logos as an abbreviation of “logos of 2 Unless otherwise stated, translations are Hussey’s, slightly modified. 3 Insofar as motion is concerned, Aristotle can speak indifferently of the movable/mover relation or of the moved thing/moving one relation. For when the movable (κινητόν, what potentially is moved) is in actuality, it is a moved thing (κινούµενον: present participle, meaning it is actually moved), but then its correlative term is no longer a mover (κινητικόν, what poten- tially moves), but a moving thing (κινοῦν, that actually moves). As Aristotle says at the beginning of 3.3, “the mover is such for its power, but is moving for its actuality, and it is actualizing the movable” (202a16–18; cf. Metaph. Δ 15, 1021a14–21). 4 Themistius’ rendering of T1 is as follows: “The actuality of the moving thing and the moved is one whenever the former moves and the latter is moved, and they are one in their underlying subject, since in their definition and in essence they are two. Compare uphill and downhill: in their case the interval is the same but if you start from here it is uphill, from there down- hill, and because of this it is two in definition. Likewise with motion, the 52 DIASTĒMA AND LOGOS ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 the essence” (ὁ λόγος ὁ τοῦ τί ἦν εἶναι), namely the definition, and opposes it to diastēma, namely the generic interval, or dis- tance, between the two terms, understood as the substratum of the definition.5 To sum up, the proportion may be read as fol- lows: P1) diastēma (generic interval) : two logoi (definitions) :: kinēsis (actuality) : two logoi (definitions) Undoubtedly this is an admissible reading, perfectly consistent with the conclusion Aristotle will reach at the end of his anal- ysis: motion as a substratum is one, but it can be defined in two ways: as the actuality of the movable (by the mover) but also as the actuality of the mover (in the movable) (202b19–22): ___ actuality of the moving thing and the moved one is one with respect to the underlying subject”: µία µὲν ἀµφοῖν ἡ ἐνέργεια τοῦ κινοῦν τος καὶ τοῦ κι- νουµένου, ὅταν τὸ µὲν κινῇ τὸ δὲ κινῆται, µία δὲ τῷ ὑποκειµένῳ. τῷ γὰρ λόγῳ δύο καὶ τὸ τί ἦν εἶναι, ὥσπερ τὸ ἄναντες καὶ τὸ κάταντες. ὡς γὰρ ἐπὶ τούτων τὸ µὲν αὐτὸ διάστηµα, ἀλλ’ ἔνθεν µὲν ἀρχοµένῳ ἄναντες, ἔνθεν δὲ κάταντες, καὶ διὰ τοῦτο τῷ λόγῳ δύο· οὕτως καὶ ἐπὶ τῆς κινήσεως κατὰ µὲν τὸ ὑποκείµενόν ἐστι µία ἡ ἐνέργεια τοῦ κινοῦντος καὶ τοῦ κινουµένου (Themist. In Phys. p.76.15–21 Schenkl; transl. Todd, modified). In turn, Themistius’ reading seems to follow the Aristotelian compiler of Metaphysics K, who in his simplified version of the passage directly replaces λόγος with εἶναι (Metaph. K 9, 1066a31–34; on the question of the inauthenticity of Book Κ see for example P. Aubenque, “Sur l’inauthenticité du livre K de la Métaphysique,” in P. Moraux and J. Wiesner [eds.], Zweifelhaftes im Corpus Aristotelicum: Studien zu einigen Dubia [Berlin/New York 1983] 318–344. 5 In the most recent editions, we find the following translations: “As the same interval is the interval between one and two and between two and one. It is a case of there being one thing definable in two ways” (W. D. Ross, Aristotle, Physics [Oxford 1936] 361); “As it is the same interval from 1 to 2 and from 2 to 1, and as the uphill and the downhill. These are one, yet the definition is not one” (E. Hussey, Aristotle Physics. Books III and IV [Ox- ford 1983] 4); “Just as there is the same interval from 1 to 2 as there is from 2 to 1, just as uphill and downhill are identical (for these things are the same, although their definitions are different)” (R. Waterfield, Aristotle, Physics [Oxford 1996] 60); “à la manière dont c’est le même intervalle qu’il y a entre un et deux et entre deux et un, et à la manière de la montée et de la descente. Car toutes ces choses sont une, bien que leurs définitions ne soient pas une” (P. Pellegrin, Aristote, Physique [Paris 2000] 167–168). MONICA UGAGLIA 53 ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 T2: ὅλως δ’ εἰπεῖν οὐδ’ ἡ δίδαξις τῇ µαθήσει οὐδ’ ἡ ποίησις τῇ παθήσει τὸ αὐτὸ κυρίως, ἀλλ’ ᾧ ὑπάρχει ταῦτα, ἡ κίνησις· τὸ γὰρ τοῦδε ἐν τῷδε καὶ τὸ τοῦδε ὑπὸ τοῦδε ἐνέργειαν εἶναι ἕτε- ρον τῷ λόγῳ. Generally speaking, teaching is not the same as learning, in the primary sense, nor acting as suffering, but that to which they be- long, namely motion. To be the actuality of this one in this one, and to be the actuality of this one by this one is different in logos. However, this is not the only admissible reading of T1. First of all, it is not only consistent but actually coincident with the conclusion T2, where it is reasonable to expect that T1 con- tains more information: indeed, within the framework of chap- ter 3.3, T1 does not play the role of a simple anticipation of the conclusion, but is intended as a first step in the argument that will lead to the conclusion T2 itself. As Aristotle makes clear, motion can be the act of two differ- ent “things” only because these “things” are relations:6 motion is the act of the “movable by the mover” but also of the “mover in the movable.” To understand not only that we have two different definitions, but also that these definitions are con- nected with two different relations, is indispensable for un- derstanding Aristotle’s argument. Indeed, Aristotle not only explains why the movable and the mover can share the same actuality, but also why this does not mean that the mover, and more generally the acting object, also suffers the same passion that it acts, when it acts. Both conclusions are reached by re- sorting to the properties of relatives. 6 Aristotle never speaks in term of relations, but only of properties like being relative-to-something (πρός τι). In this way he makes clear that no ad- ditional entities must be introduced beyond the relative and its correlative. For the sake of simplicity and in accordance with current practice, I will notwithstanding use the noun “relation” and the corresponding adjective “relational.” By saying that the movable and the mover are in relation, or that there is a relation between the movable and the mover, I simply com- bine the two assertions that the movable is a relative to a mover and a mover is a relative to a movable. 54 DIASTĒMA AND LOGOS ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 For this reason, I suggest first of all recovering the interpreta- tion of those commentators who, like Simplicius or Philoponus, maintain the polysemy of logos, which in T1 does mean both definition and ratio.7 In addition, I propose keeping in mind that the definitions at issue are definitions expressed in terms of relatives, and finally I suggest identifying the particular diastēma 7 See in particular Simplicius: “As the interval from one ‘to two and from two to one is one and upwards and downwards’ are one in substrate, but two in definition, the ratio from one to two being a half and from two to one double, so is the relation of change in the case of the changer and the changed, since they are one in substrate, but is viewed one way in the changer, another way in the changed,” ὡς οὖν τὸ διάστηµα τοῦ ἑνὸς “πρὸς δύο καὶ δύο πρὸς ἓν καὶ τὸ ἄναντες καὶ τὸ κάταντες” ἓν µέν ἐστι τῷ ὑποκειµένῳ, δύο δὲ κατὰ τὸν λόγον, ἀπὸ µὲν τοῦ ἑνὸς πρὸς τὰ δύο ἥµισυ, ἀπὸ δὲ τῶν δύο πρὸς τὸ ἓν διπλάσιον, οὕτως ἔχει καὶ ἐπὶ τοῦ κινοῦντος καὶ κινουµένου ἡ κίνησις, µία µὲν οὖσα τῷ ὑποκειµένῳ, ἄλλως δὲ ὡς ἀπὸ τοῦ κινοῦντος καὶ ἄλλως ὡς ἐν τῷ κινουµένῳ θεωρουµένη (In Phys. pp.439.33– 440.1 Diels; transl. Urmson; cf. p.439.5–8); and Philoponus: “That it is not surprising for there to be one and the same actuation of two things which differ only in relation he confirms by the examples. ‘Just as’, he says, ‘there is the same interval between one and two as between two and one’. For this is one and the same thing in respect of the substratum. For what is the difference between a change from two to one and a change from one to two? That is, to say what ratio two has to one does not differ from saying what ratio one has to two, excepting only in the relation, while as to the substratum they are one and the same. But if there is one interval, it is the relations then that differ. For two has to one the ratio of being double, while one has to two the ratio of being half. Likewise, the upward and downward are different relations, while the distance is single,” καὶ ὅτι οὐ θαυµαστὸν µίαν καὶ τὴν αὐτὴν ἐνέργειαν δύο εἶναι πραγµάτων τῇ σχέσει µόνον διαφέ- ρουσαν, τοῖς παραδείγµασι πιστοῦται. “ὥσπερ,” φησί, “τὸ αὐτὸ διάστηµα ἓν πρὸς δύο καὶ δύο πρὸς ἕν”· ἓν γὰρ τοῦτο καὶ ταὐτὸν κατὰ τὸ ὑποκείµενον. τί γὰρ διαφέρει ἐκ τῶν δύο πρὸς τὸ ἓν κινηθῆναι, καὶ ἐκ τοῦ ἑνὸς πρὸς τὰ δύο; οἷον τὸ λέγειν ὅτι ὃν λόγον ἔχει τὰ δύο πρὸς τὸ ἕν, οὐδὲν διαφέρει τοῦ εἰπεῖν ὃν λόγον ἔχει τὸ ἓν πρὸς τὰ δύο, εἰ µὴ µόνον τῇ σχέσει, τῷ ὑπο- κειµένῳ δὲ ταὐτὸν καὶ ἕν. ἀλλ’ εἰ καὶ µία ἡ διάστασις, ἀλλ’ οὖν αἱ σχέσεις διάφοροι· τὰ µὲν γὰρ δύο πρὸς τὸ ἓν διπλασίονα λόγον ἔχει, τὸ δὲ ἓν πρὸς τὰ δύο τὸν ἥµισυν. ὁµοίως δὲ καὶ τὸ ἄναντες καὶ τὸ κάταντες σχέσεις µὲν διάφοροι, διάστασις δὲ µία (In Phys. pp.375.25–376.5 Vitelli; transl. Ed- wards). MONICA UGAGLIA 55 ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 that Aristotle is referring to with the harmonic interval, avoid- ing to equate it with a generic substratum. To sum up, I suggest summarizing P in the following way: P2) diastēma (musical interval) : two logoi (numerical ratios) :: kinēsis (actuality) : two logoi (definitions in term of relatives) On the left horn diastēma is the musical interval and the two logoi are the two possible ratios between the values of the two terms, which define the diastēma: namely the pitches of a higher and a lower note, one pros two and two pros one. It is worth noting that it is not by chance that this particular example was chosen, for the interval two-one is the interval of an octave, or diapason, namely the most perfect accord.8 On the right horn kinēsis is the actuality and the two logoi are the two ways of re- lating the two terms, which define the kinēsis: namely the mover pros the movable, and the movable pros the mover (or—which amounts to the same thing—the moving thing pros the moved one and the moved thing pros the moving one). Of course, this reading too is compatible with Aristotle’s con- clusion T2, although it does not coincide with it: in this case the conclusion is not immediate, but is reached in two steps. The first step is to set the philosophical opposition between kinēsis and logos in analogy with the more handleable math- ematical opposition between diastēma and logos. The second step is to observe that both the mathematical logos of the last example and the philosophical logos of the first one are in fact logoi also as far as the definition is concerned.9 8 Even those commentators who, like Simplicius and Philoponus, recog- nise the ambiguity of logos, fail to recognise the harmonic content of the example (see n.7): to them one and two have nothing to do with any pitch, but are simply the end points of a generic diastēma, in the sense of spatial distance. On the Aristotelian tradition of reading of T1 see M. Ugaglia, “Alcune osservazioni sull’uso di λόγος in Aristotele,” in F. Franco Repellini and G. Micheli (eds.), La scienza antica e la sua tradizione (Milan 2011) 81–117. 9 Phys. 2.3, 194b26–29: “According to another ‘way of speaking’ the form or model is a cause, and this is the definition (the logos of the essence, ὁ λόγος ὁ τοῦ τί ἦν εἶναι) and its genera, thus the cause of an octave is the two 56 DIASTĒMA AND LOGOS ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 Just as two ratios give rise to the same musical interval—for example the quoted interval of an octave can be defined by either the ratio “two pros one” or the ratio “one pros two”—so two ways of putting the movable and the mover in relation give rise to the same motion. Indeed, motion can be defined either in terms of the moved pros the moving thing or of the moving thing pros the moved one, depending on whether one says that it is the actuality of the “movable by the mover” or that it is the actuality of the “mover in the movable.” As requested for proving that the mover does not suffer the same passion that it acts, when it acts, this happens without the two relations neces- sarily coinciding: for just as it is not the same to state that “one is half of two” and that “two is the double of one,” it is not the same to state that “the moved is being moved by the moving” and that “the moving is moving the moved.” However—and this is the crucial point—the two statements are not independent. On the contrary, they entail each other: if the one holds, the other necessarily holds as well, and it is impossible to think of them separately. In this case the terms ἄναντες and κάταντες too acquire a quasi-technical meaning in T1: aside from their more evident affinity with Heraclitus’ roads,10 they find a natural—albeit less philosophically evocative—counterpart in the notions of ascending and descending intervals in harmonic theory.11 ___ to one (τὰ δύο πρὸς ἕν), and more generally number.” 10 Heraclitus 22 B 60 D.-K. Cf. Aristotle’s reference to the road from Thebes to Athens, set in contrast to the road from Athens to Thebes, in Phys. 3.3, 202b10–16. 11 See in particular Aristoxenus Harm., 10.11–2 Meibom.: “Stretching is the continuous motion of the voice from a deeper position to a higher one, tightening from a higher position to a deeper one,” ἡ µὲν οὖν ἐπίτασίς ἐστι κίνησις τῆς φωνῆς συνεχὴς ἐκ βαρυτέρου τόπου εἰς ὀξύτερον, ἡ δ’ ἄνεσις ἐξ ὀξυτέρου τόπου εἰς βαρύτερον. The terminology was traditional, the ascending motion being called ἐπίτασις, which literally denotes stretching (usually that of a string), the descending motion being called ἄνεσις, literally loosening. Aristoxenus redefined these terms, freeing them from any mechanical connotation. MONICA UGAGLIA 57 ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 A reason for preferring the second reading is that it does justice to Aristotle’s philosophical argument, restoring the original complex articulation of the analogy, impoverished by the standard interpretation. In the standard interpretation, in order to clarify the way in which a single subject, namely mo- tion, has two possible definitions, Aristotle would have simply produced an additional example, the one of diastēma, standing on the same footing as the one of movement. In other words, we have nothing but two perfectly equivalent instances of the same relationship, and one may wonder why Aristotle thinks that it would be simpler to understand that the logoi (= defini- tions) of a single interval can be two different ones than to posit that the logoi (= definitions) of a single state of motion can be two different ones. On the contrary, in the interpretation I suggest the parallel case of the diastēma has a stronger cognitive function: it moves onto a new level, introducing a technical mathematical simile, which is actually different, but also—and most importantly— simpler, than the philosophical counterpart that it is intended to clarify. Indeed, Aristotle frequently resorts to mathematics when he needs to illustrate a particularly difficult philosophical point, as though mathematics were an unproblematic realm that can help clarify more complex states of affair.12 In this case, in addition to Aristotle’s general penchant for mathematical sim- 12 See for instance the use of the construction of the right angle in a semicircle in order to explain the procedure of knowledge in Metaph. Θ 9 (1051a27–29) and the notion of epagoge in An.Post. 1.1 (71a17–b5), or the mention of the line cutting a plane figure parallel to its side, used for clarifying the role of the middle term as a “reason” in a syllogism in Top. 8.3 (158b29–34). More generally, some twenty-odd mathematical examples can be found in Metaphysics (excluding the “mathematical” books M and N), roughly the same number in Physics, a dozen or so in De anima, and a few in the biological works. For an analysis of Aristotle’s philosophical use of math- ematical examples see M. Ugaglia, “Knowing by Doing: the Role of Geo- metrical Practice in Aristotle’s Theory of Knowledge,” Elenchos 36 (2015) 45–88. 58 DIASTĒMA AND LOGOS ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 iles, the choice of the specific example has a particular raison d’être, for the opposition diastēma/logos explicitly refers to a well- known lexical controversy in ancient Greek mathematics, which presumably would have been familiar to an educated reader in Aristotle’s time. As we show in the next section, the extant Greek sources on harmonic theory bear witness to the emergence of a termino- logical discussion concerning the notion of diastēma, a purely harmonic-theoretical notion, and the strictly connected notion of logos, which primarily pertains to proportion theory. In short, the question concerns the equivalence of the two concepts, which is warranted in the so-called Pythagorean tradition but denied in the Aristoxenian one. Aristotle’s statements, chronologically intermediate between these two poles, testify to the difficulties connected with the confusion which inevitably goes with the existence of such op- posite points of view, and of the discussions accompanying it. 2. Harmonic background It is customary to gather under the improper label of ‘Pythagoreans’ a wide family of Greek musical theorists who, though belonging to different philosophical schools and per- iods, share a mathematical approach which can in some way be traced back to the work of Pythagoreans like Archytas and Philolaus.13 According to the Pythagorean tradition, harmonics is not a self-standing discipline, and its objects of study— namely, the elements out of which a melody is composed—are not independent entities. More specifically, sounds are viewed as the result of particular motions of physical instruments, so that notes, which are sounds with a particular pitch, are as- 13 On this basis, the musical theories that are implicitly or explicitly in the background of the expositions of Plato, Theon of Smyrna, Nicomachus, Ptolemy, Aristides Quintilianus, and also Euclid’s Sectio Canonis are con- veniently labelled as ‘Pythagorean’. On this question see, for example, the introduction and accompanying bibliography in A. Barker, Greek Musical Writings II Harmonic and Acoustic Theory (Cambridge 1989). MONICA UGAGLIA 59 ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 sociated with the physical properties of the instrument (the diameter, length or tension of the strings). In their turn, physical properties can be quantified and associated with arith- metical objects, so that any harmonic notion can be read in mathematical terms. In particular, the intervals between notes, understood as the relation between the corresponding pitches, are represented by the ratios between their numerical values: an interval is a consonance when it is represented by a ratio of simple numbers.14 This ultimate reduction to arithmetical terms makes it un- necessary to define harmonic notions in an independent way. In particular, we can note the absence of a Pythagorean defini- tion of interval, the term diastēma being employed merely as the harmonic equivalent of the arithmetical logos (= ratio).15 By convention, the ratio is always the ratio of the higher note to the lower one, and not vice versa. In contrast, the Aristoxenian tradition considers harmonics an independent discipline, grounded on sensory data and ac- countable in terms of purely harmonic entities.16 Although the elements forming a melody are the objects of study of har- monics, they are not the basic entities of the theory, the main 14 A systematic account of such an approach is presented, for example, in the opening sections of the first book of Ptolemy’s Harmonics. 15 In the Euclidean Sectio Canonis, for example, which chiefly deals with harmonic ratios, we find only six occurrences of the term logos, the har- monic ratios always being designated by the term diastēma. Leaving aside the three occurrences of logos in the Introduction, whose authenticity is doubtful, and the one at the end of Prop. 12, where it is part of a sentence whose spurious character is evident, the remaining two occurrences are in Prop. 3, and they are induced by a change in the paradigm of reference. The second occurrence is contained in a verbatim quotation of Elements 8.8 —i.e., a proposition from the arithmetic books (a fact that should in any case raise some doubts as to the authenticity of the quotation): as a con- sequence, one must resort to the terminology of arithmetic. The first occur- rence paves the way to the second. 16 See in particular the beginning of Book 2 of Aristox. Elem.Harm., pp. 32.31–34.32 Meibom. 60 DIASTĒMA AND LOGOS ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 object of interest being the so-called musical motion. This is a kind of Aristotelian local kinēsis, where the contrary places be- tween which motion arises are not up and down, but high and low (pitch). In the case of voice, for instance, the subject of motion makes a continuous motion in speaking, but a dis- continuous one in singing. During this discontinuous motion, which Aristoxenus calls διαστηµατική, “intervallic,”17 the voice again and again breaks this process of ascending or descending motion by settling on a note, which is then read as a point of discontinuity of motion. In such a picture, an interval is that which separates two notes. As a consequence, it is a part of motion, in the same way as, for Aristotle, what separates two ‘nows’ in time is still a part of time.18 It is clear that within this perspective there is no need to relate intervals to any kind of ratio;19 therefore, an explicit definition of interval must be pro- vided. Aristoxenus defines it as “what is determined by two notes not having the same tension.”20 Similar definitions are offered in later musical writings.21 In sum, there is a crucial difference between the Pythagorean point of view and the Aristoxenian: the tacit convention of reading a musical diastēma as an ordered relation builds a one- to-one correspondence between the notions of diastēma and logos, so that although the two terms apply to two different theoretical fields, in a purely Pythagorean context there is no room for ambiguities. 17 Elem.Harm. pp.8.14–10.10 Meibom. 18 See for example Phys. 4.10 (218a16–21), 11 (219a22–30). 19 In fact, the notes themselves, which as dimensionless points are devoid of any quality, cannot be considered in any reciprocal relation, the interval itself aside. On this point see for instance Ptolemy’s critique in Harm. 1.9 (20.3–9 Düring). 20 τὸ ὑπὸ φθόγγων ὡρισµένον µὴ τὴν αὐτὴν τάσιν ἐχόντων (Elem.Harm. p. 15.22–23 Meibom). 21 See Cleonides Harm. p.179.11–12, Nicomachus Harm. p.261.8, Bac- chius Harm. p.292.20–21, Gaudentius Harm. p.329.23–24 (page numbers of C. Jan, Musici scriptores graeci [Leipzig 1899]). MONICA UGAGLIA 61 ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 On the contrary, no mention of any privileged direction is to be found in the Aristoxenian definition of diastēma as “what is determined by two notes.” In this perspective, any attempt to relate a diastēma to a logos necessarily gives rise to ambiguities: even assuming that a logos can be associated with the two notes defining an interval, why would the logos associated with the relation of the higher note to the lower be preferable to the logos associated with the relation of the lower note to the higher? As a consequence of all of this, ambiguities may arise in texts such as those of the Pythagorean tradition, where diastēma and logos are used interchangeably, if the two terms are read in an Aristoxenian perspective, or, more generally, in any philo- sophical rather than strictly mathematical perspective, thereby breaking the one-to-one correspondence between diastēma and logos. The ambiguity arising from an Aristoxenian reading of Pythagorean texts is exactly what Aristotle alludes to in T1. 3. Parallel texts The remark in T1 can be found, expressed in the very same terms, in other passages in the Greek musical corpus. In the first passage, extracted from Nicomachus’ Enchiridion, we find the example of the one-two interval, as in Aristotle’s T1, together with an attempt to clarify the diastēma/logos am- biguity by referring it to the διαφορά/σχέσις pair, where a diaphora is defined as the excess or defect of two notes, and a schesis, or logos metrētikos, as its (double) measure (Harm.Ench. 12 [p.261.8–17 Jan]): T3: διάστηµα δ’ ἐστὶ δυοῖν φθόγγων µεταξύτης. σχέσις δὲ λόγος ἐν ἑκάστῳ διαστήµατι µετρητικὸς τῆς ἀποστάσεως· δια- φορὰ δὲ ὑπερβολὴ ἢ ἔλλειψις φθόγγων πρὸς ἀλλήλους. κακῶς γὰρ οἴονται οἱ νοµίζοντες διαφορὰν καὶ σχέσιν τὸ αὐτὸ εἶναι. ἰδοὺ γὰρ τὰ δύο πρὸς τὸ ἓν διαφορὰν µὲν ἔχει τὴν αὐτὴν, ἣν ἓν πρὸς δύο, σχέσιν δὲ οὐ τὴν αὐτήν. τὰ µὲν γὰρ δύο διπλάσια, τὸ δὲ ἓν ἥµισυ. καὶ πάλιν ἐν πᾶσιν ἀριθµητικῆς µεσότητος ὅροις τρισὶν ἢ καὶ πλείοσι διαφορὰ µὲν ἡ αὐτὴ ἐν πᾶσι, σχέσις δὲ ἄλλη καὶ ἄλλη. 62 DIASTĒMA AND LOGOS ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 An interval is what lies between two notes. A relation is the ratio which measures the distance in any interval: and a difference is the excess or deficiency of one note with respect to another. Those who think that relation and difference are the same are wrong: for as you will understand, the difference between 2 and 1 is the same as that between 1 and 2, whereas their relation is not the same. For 2 is double 1, while 1 is half of 2. Or again, the difference between three or more terms in arithmetic pro- portion is the same in each case, but the relation is different in each. (transl. Barker) The same example of the two-one interval can be found in the Expositio rerum mathematicarum ad legendum Platonem utilium by Theon of Smyrna,22 where the notions of diastēma and logos are contrasted (Exp. p.81.6–16 Hiller): T4: διαφέρει δὲ διάστηµα καὶ λόγος, ἐπειδὴ διάστηµα µέν ἐστι τὸ µεταξὺ τῶν ὁµογενῶν τε καὶ ἀνίσων ὅρων, λόγος δὲ ἁπλῶς ἡ τῶν ὁµογενῶν ὅρων πρὸς ἀλλήλους σχέσις. διὸ καὶ τῶν ἴσων ὅρων διάστηµα µὲν οὐδέν ἐστι µεταξύ, λόγος δὲ πρὸς ἀλ- λήλους εἷς καὶ ὁ αὐτὸς ὁ τῆς ἰσότητος· τῶν δὲ ἀνίσων διάστηµα µὲν ἓν καὶ τὸ αὐτὸ ἀφ’ ἑκατέρου πρὸς ἑκάτερον, λόγος δὲ ἕτερος καὶ ἐναντίος ἑκατέρου πρὸς ἑκάτερον· οἷον ἀπὸ τῶν βʹ πρὸς τὸ ἓν καὶ ἀπὸ τοῦ ἑνὸς πρὸς τὰ βʹ διάστηµα ἓν καὶ τὸ αὐτό, λόγος δὲ ἕτερος, τῶν µὲν δύο πρὸς τὸ ἓν διπλάσιος, τοῦ δὲ ἑνὸς πρὸς τὰ βʹ ἥµισυς. An interval and a ratio are different, for an interval is what lies between homogeneous and unequal terms, while a ratio is quite generally the reciprocal relation of homogeneous terms. For this reason, while there is no interval between equal terms, their re- ciprocal ratio is one and the same, namely, that of equality; on the other hand, the interval between unequal terms, from any of them to the other, is one and the same, but the ratio of any of them to the other is different and the opposite: for example, from 2 to 1 and from 1 to 2 there is one and the same interval, while the ratio is different: double that of 2 to 1, one half that of 22 On the sources of the passage see the Introduction in F. M. Petrucci, Teone di Smirne: Expositio rerum mathematicarum ad legendum Platonem utilium (Sankt Augustin 2012), with a detailed analysis of the passage at 388–389. MONICA UGAGLIA 63 ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 the 1 to 2. Here the context is more general, but the fact that the two-one example comes from harmonics is suggested by the possible origin of Theon’s analysis. Some very similar remarks are re- peated a few lines later, but now they are explicitly ascribed to Eratosthenes (Exp. pp.81.17–82.5): 23 T5: Ἐρατοσθένης δὲ ἐν τῷ Πλατωνικῷ φησι, µὴ ταὐτὸν εἶναι διάστηµα καὶ λόγον, ἐπειδὴ λόγος µέν ἐστι δύο µεγεθῶν ἡ πρὸς ἄλληλα ποιὰ σχέσις· γίνεται δ’ αὕτη καὶ ἐν διαφόροις <καὶ ἐν ἀδιαφόροις>.24 οἷον ἐν ᾧ λόγῳ ἐστὶ τὸ αἰσθητὸν πρὸς τὸ νοητόν, ἐν τούτῳ δόξα πρὸς ἐπιστήµην, καὶ διαφέρει καὶ τὸ νοητὸν τοῦ ἐπιστητοῦ ᾧ καὶ ἡ δόξα τοῦ αἰσθητοῦ. διάστηµα δὲ ἐν διαφέ- ρουσι µόνον, ἢ κατὰ τὸ µέγεθος ἢ κατὰ ποιότητα ἢ κατὰ θέσιν ἢ ἄλλως ὁπωσοῦν. δῆλον δὲ καὶ ἐντεῦθεν, ὅτι λόγος διαστήµατος ἕτερον· τὸ γὰρ ἥµισυ πρὸς τὸ διπλάσιον <καὶ τὸ διπλάσιον πρὸς τὸ ἥµισυ>25 λόγον µὲν οὐ τὸν αὐτὸν ἔχει, διάστηµα δὲ τὸ αὐτό. In his Platonicus, Eratosthenes says that an interval and a ratio are not the same, for a ratio is a qualified reciprocal relation be- tween two magnitudes. It comes to be between different ; for instance, the sensible is to the intelligible in the same ratio as opinion is to science, and the intelligible differs from the knowable as opinion differs from the sensible. An inter- val, instead, is given only between terms that are different either in magnitude, in quality, in position, or in any other way. It is clear that a ratio is also different from an interval from this point of view: the half to its double do not have the same ratio, but do have the same interval. 23 This is one of the only two surviving passages containing an explicit reference to the Πλατωνικός, and it has proven crucial for the attempt to re- construct the contents of that lost work of Eratosthenes. The other passage, again in Theon of Smyrna (Exp. p.2.3–12), concerns the problem of the duplication of the cube. On these passages and on the content of Era- tosthenes’ work see for example B. Vitrac, “Eratosthène et la théorie des médiétés,” in Ch. Cusset and H. Frangoulis (eds.), Eratosthène: un athlète du savoir (Saint-Etienne 2008) 77–103. 24 Add. Hiller; ἀδιαφόροις perperam corr. Düring. 25 Add. Hiller. 64 DIASTĒMA AND LOGOS ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 Without going into the general problem of the reliability of Theon as a witness to Eratosthenes’ mathematical achieve- ments, one must observe that the compilatory nature of the Expositio, together with its philosophical rather than mathemati- cal aims, generally does not allow us to precisely distinguish Eratosthenes’ statements from material coming from other sources. In some cases, however, it is possible to clarify the situ- ation by comparing Theon’s report and analogous accounts by other authors. Concerning the passage at issue, a happy coinci- dence provides a parallel text dealing with the same topic and referring to the same examples. The argument, now framed within a harmonic context, is explicitly ascribed to Eratosthe- nes by Porphyry in his Commentary on Ptolemy’s Harmonics (In Harm. pp.90.24–91.9 and 94.2–4 Düring): T6: “λόγος δὲ λέγεται δύο µεγεθῶν ὁµογενῶν ἡ κατὰ πηλικό- τητα ποιὰ σχέσις,” κατὰ δὲ τοὺς Ἀριστοξενείους “τὸ περιεχό- µενον ὑπὸ δύο φθόγγων ἀνοµοίων τῇ τάσει”· καὶ ἄλλοι ἄλλως ἐδόξασαν περὶ τοῦ διαστήµατος. Ἐρατοσθένης µὲν οὖν φησιν ἕτερον εἶναι διάστηµα λόγου· ἐν γὰρ ἑνὶ διαστήµατι δύο λόγοι γίνονται. ὁ δὲ λόγος δὶς φέρεται, ὅ τε τοῦ µείζονος πρὸς τὸ ἔλαττον καὶ τοῦ ἐλάττονος πρὸς τὸ µεῖζον καὶ κοινὴ διαφορὰ ὑπεροχῆς καὶ ἐλλείψεως ὡς τῆς διαφορᾶς δηλονότι τὸ διάστηµα ποιούσης. διπλασίου τε γάρ φησι πρὸς ἥµισυ καὶ ἡµίσεος πρὸς διπλάσιον ὁ µὲν λόγος ἕτερος, τὸ αὐτὸ δὲ διάστηµα […] ὅτι µὲν τοίνυν ὁ λόγος ἐν διαφόροις γίνεται ὅροις, ὁµογενέσι δὲ πάντως, καὶ ἐν ἀδιαφόροις, ὡς Εὐκλείδῃ δοκεῖ, δειχθήσεται· διάστηµα δ’ ἐν τοῖς διαφέρουσι µόνον, φανερόν. “Ratio is said to be a qualified relation in respect of size between two homogeneous magnitudes,” and following the Aristoxenians “what is contained by two notes dissimilar in pitch”; others have held different opinions on intervals. Eratosthenes says that an in- terval is different from a ratio, for in a single interval two ratios come to be. Ratio presents itself in two ways: that of the greater to the smaller and that of the smaller to the greater, and the interval of excess and deficiency is common, for, obviously, it is the difference that makes the interval. Indeed, he says that the ratio of the double to its half and of the half to its double is different, but the interval is the same. […] It will be shown that a ratio comes to be between different yet still totally homo- MONICA UGAGLIA 65 ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 geneous terms, or between similar terms, as Euclid thinks. It is clear, instead, that an interval comes to be only between differ- ent terms. What clearly emerges here is Eratosthenes’ intention to stress the non-coincidence between the notions of diastēma and logos. As to the meaning of logos, both Theon and Porphyry have no doubts: insomuch as they make Eratosthenes quote the defini- tion in terms of the schesis found in Euclid Elem. 5.def.3, a logos is a sort of relation. As for the diastēma, whether it is understood as an excess or a difference, as that which separates two terms or as that which puts them in relation,26 the problem is that a single diastēma corresponds to two logoi, or to two ways of read- ing it, and this is what Eratosthenes, like Aristotle before him, points out. The main difference is that while Aristotle accepts the am- biguity and simply turns the one-to-one correspondence into a one-to-two correspondence—the same diastēma corresponds to two logoi, which both can be used to define it—Theon, Nicoma- chus, and Eratosthenes are more interested in the negative side of the ambiguity, that is, in the fact that the two terms refer to two distinct objects. Aristoxenus will carry the situation to the extreme consequence of regarding only one of them, the dia- stēma, as a true harmonic notion. 4. Conclusions: Plato’s original sin According to Porphyry, who claims to be following here the criticism of “Demetrius and Panaetius the mathematicians,”27 many authors too hastily conflated the meanings of the terms diastēma and logos. Besides the Pythagoreans in general, he lists 26 According to Porphyry (In Harm. pp.91.10–92.13), Eratosthenes did not give an explicit definition of diastēma, even though he set it in contrast to logos. It is perhaps for this reason that interpreters, both before and after him, defined the interval as an excess (ὑπεροχή), or a difference between notes, as did Philolaus and Thrasyllus. 27 In Harm. p.92.19–21. 66 DIASTĒMA AND LOGOS ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 Plato,28 and in particular Plato’s discussion about the division of the world-soul by means of harmonic intervals (Ti. 35C–36B): T7: µετὰ δὲ ταῦτα συνεπληροῦτο τά τε διπλάσια καὶ τριπλά- σια διαστήµατα, µοίρας ἔτι ἐκεῖθεν ἀποτέµνων καὶ τιθεὶς εἰς τὸ µεταξὺ τούτων, ὥστε ἐν ἑκάστῳ διαστήµατι δύο εἶναι µεσότη- τας, τὴν µὲν ταὐτῷ µέρει τῶν ἄκρων αὐτῶν ὑπερέχουσαν καὶ ὑπερεχοµένην, τὴν δὲ ἴσῳ µὲν κατ’ ἀριθµὸν ὑπερέχουσαν, ἴσῳ δὲ ὑπερεχοµένην. ἡµιολίων δὲ διαστάσεων καὶ ἐπιτρίτων καὶ ἐπογδόων γενοµένων ἐκ τούτων τῶν δεσµῶν ἐν ταῖς πρόσθεν διαστάσεσιν, τῷ τοῦ ἐπογδόου διαστήµατι τὰ ἐπίτριτα πάντα συνεπληροῦτο, λείπων αὐτῶν ἑκάστου µόριον, τῆς τοῦ µορίου ταύτης διαστάσεως λειφθείσης ἀριθµοῦ πρὸς ἀριθµὸν ἐχούσης τοὺς ὅρους ἓξ καὶ πεντήκοντα καὶ διακοσίων πρὸς τρία καὶ τετ- ταράκοντα καὶ διακόσια. After this he went on to fill the double and triple intervals by cutting off still more portions from the mixture and placing these between them, in such a way that in each interval there were two middle terms, one exceeding the first extreme by the same fraction of the extremes by which it was exceeded by the second, and the other exceeding the first extreme by a number equal to that by which it was exceeded by the second. These connections produced intervals of 3/2, 4/3, and 9/8 within the previous in- tervals. He then proceeded to fill all the 4/3 intervals with the 9/8 interval, leaving a small portion over every time. The terms of this interval of the portion left over are as the numbers 256/ 243. (transl. Zeyl, modified) If one reads Plato’s words in a purely Pythagorean perspective, the text does not leave any room for misinterpretation: the interval between two terms is the ratio of the greater to the lesser: the hemiolic interval is the ratio of 3 to 2, the epitritic of 4 to 3 and so on, until the minimal interval, or lemma, which is 28 In Harm. p.92.13–18. The other quoted authors are Dionysius of Hali- carnassus (pp.92.28 and 94.25), Archytas (92.29, 93.4–19, 94.24), Euclid (92.29–93.4, 94.25), and Eratosthenes himself (92.24–25). On Porphyry’s passage and his sources see M. Raffa, “The Debate on logos and diastēma �in Porphyry’s Commentary on Ptolemy’s Harmonics,” Greek and Roman Musical Studies 1 (2013) 243–252. MONICA UGAGLIA 67 ————— Greek, Roman, and Byzantine Studies 56 (2016) 49–67 the ratio of 256 to 243. On the other hand, texts like T4–6 show a certain unease on the part of the interpreters, who try to eliminate an ambiguity which they evidently perceive. Something has changed, between Plato’s writing and Eratos- thenes’ reading: something that has to do with the emergence of a different way to approaching harmonic notions. Eratosthe- nes’ testimony, just like Aristotle’s before it,29 bears witness to the strengthening of this approach, which will eventually take the concrete shape of the Aristoxenian theory of music. October, 2015 Scuola Normale Superiore Pisa, Italy monica.ugaglia@sns.it 29 The hypothesis that Eratosthenes’ exposition of the problem depends on Aristotle, as the extreme similarity between the texts suggests, is corrob- orated by another fact. Theon’s quotation is part of a longer text extracted from Eratosthenes’ Πλατωνικός. In addition to the quotation, the text contains another terminological discussion (p.82.6–21 Hiller), aimed at avoiding the confusion between the notions of ἀναλογία (proportion) and µεσότης (mean): as in the case of diastēma and logos in Phys. 3.3, a textual comparison shows a strong similarity with an Aristotelian passage in Eth.Nic. 5.3: here too, Aristotle’s intention is to use a mathematical example to clar- ify a philosophical concept, in this case the notion of justice and its forms. For a comparison between Eratosthenes’ work and the passage in Aristotle’s Nicomachean Ethics see B. Vitrac, “La Définition V. 8 des Eléments d’Euclide,” Centaurus 38 (1996) 97–121.