This study presents a new Timaeus analysis demonstrating how the Timaeus articulates a musical paradigm of the cosmos. It is a complete, rigorous analysis of each element of the Timaeus text from 35Â Aâ36Â D. It both discovers the primary musical scale dwelling hidden within Platoâs recipe for constructing the world soul and breaks new ground, inter alia, by finding the exact âfabricâ that Plato cut to form the Ï (âchiâ) figure instrumental to making the world soul. It characterizes the fabric as a particular matrix of numbers intimately connected with Platoâs musical cosmology and far more extensive than ancient or modern commentators have allowed. It is open to the possibility that the ancient commentators knew what they were about in their attempts to find a âscaleâ in the text, over against the efforts of some modern philosophers to discount a musical interpretation of the work.
The position of Francis Cornford, Luc Brisson, and Walter Meyerstein that Plato could not primarily have been constructing a musical scale, and so, a musical cosmology, in the Timaeus, seems ultimately untenable.1 It too easily discounts the existing musicological scholarship on Platoâs Timaeus and some of his other works, such as the Myth of Er, in the Republic, and sections of Book V of the Laws from 737 Eâ741 A and 745 Bâ745 E. It also largely neglects the ancient evidence documenting the emergence of the octave scale in the West; the cosmological significance of music in Platoâs time; and the place of music education as a vehicle for the formation of the soul in the ancient Greece of Platoâs day.2 These factors are important to a proper evaluation of the Timaeus because they reveal the musical issues and ethos extant in Platoâs time.
Although Luc Brisson has given some interesting alternative interpretations of the construction of the world soul that are not without validity for a multivalent text, they do not rule out a musical interpretation.3 In addition, although Cornford raised an interesting issue for the long-established view that Platoâs text elaborates a musical cosmology, it is hardly an insurmountable one. He claimed that, if Plato had really meant to articulate a harmonic conception of the cosmos, he would not have stopped at four octaves and a major sixth, the limitation that commentators universally apply to Platoâs harmonic construction of the world soul. Such a limitation, he argued, does not reflect a closed system and seems arbitrary from a musical standpoint.4 It is, indeed, an odd limitation; but Timaeus warns one from the outset that his likely account of the origin and structure of the universe may not be wholly self-consistent (Timaeus 29Â C). Accordingly, the lack of complete consistency in a musical metaphor can hardly be taken to argue against Platoâs intent to articulate a musical cosmology.
Taking the strangeness of the limitation as hard and steadfast proof that the whole body of ancient tradition acknowledging a musical cosmology in Platoâs text is wrong in its fundamental instincts, as though the ancient commentators were not themselves very educated men much more deeply steeped in a living tradition than any modern observer of Plato, also seems misplaced. It appears more appropriate to assess that, being only human, they did not see the whole picture.
Perhaps some modern scholarship, also, simply does not see the whole picture. Maybe, as Handschin suggested, Plato primarily furnished the principles of a musical scale and only secondarily articulated a particular one.5 As long as all relevant principles are articulated in the text, the musical interpretation remains viable and cosmologically important.
Alternatively, Plato may not, in fact, have limited the harmonics to four octaves and a major sixth, despite the universal assumption, otherwise. The current study indicates that such a limitation depends upon an unnecessary assumption concerning Platoâs termination of the numerical sequence at twenty-seven. It demonstrates how that termination possibly has a different meaning than commentators have previously given it that is, nonetheless, exactly in line with the Pythagorean mode of thinking that many persons, ancient and modern, have attributed to Plato in the Timaeus. Previous commentators have missed the possibility suggested by this study at least partly because of the onerous investment of time that the discovery requires and the particular mathematics required to reveal it.
The study claims that the commentators of the ancient Academy did not unpack Platoâs riddle at Timaeus 35Â Aâ36Â D, completely, at least in any form that survives, though it appears that at least one commentator of Augustine of Hippoâs day, Favonius Eulogius, knew how the riddle ended. The study heeds the work of Jacques Handschin, a prominent musicologist fascinated with the problems that an effort to find a Timaeus scale poses; it takes seriously John Dillonâs suspicion, deriving from Speusippus, that the Decad may be the âAll Perfect Animalâ; it follows-up Mitchell Millerâs intuition about a particular âGod-given methodâ at work in the Timaeus; and it gives significant weight to Platoâs link to the Pythagorean community at Tarentum.6
The new interpretation argues that Plato delineated the proposed musical paradigm for the Timaean cosmos against a more fundamental harmonic conception of the universe. It also hypothesizes a structure of human perception governed isomorphically by the proportions proper to the movement of the whole cosmos. It demonstrates how the pattern of the Decad not only describes the ordo of human perception in the Timaeus but also constitutes the pattern according to which all things are generated, including the form of the world soul and the primary diatonic musical scale that is its symbol. It establishes the primary diatonic musical scale against a large but finite set of competing, secondary scale possibilities.
The analysis is compelling in its articulation of the matrix comprising Platoâs fabric and in its illumination of a number of other problems of textual interpretation. The latter include the precise identity of the primary Timaeus scale; the manner of forming the âchiâ from the fabric and the soul sphere from the âchiâ; the creation of bands from the soul sphere; the meaning of the differences among the subsets of three and four bands resulting from the splitting of the band of difference at Timaeus 36Â D; the reasons for the differing motions and speeds of the band of same (36Â C) and the bands of difference, as well as the significance of those differences; the relationship of the motions of all bands to the motions of the cosmos and the planets, in their orbits; the reasons for the appearances of âchromaticâ elements among the numbers corresponding to the dominant musical scale emerging from Platoâs text; the specific order of their emergence; and their relevance to identifying the band of âsameâ at 36Â C.
The study illustrates that the chromatic distortions critical to identifying the band of âsameâ at 36Â C arise, as the octave sequences of the primary Timaeus scale repeat, because Platoâs original world soul divisions at Timaeus 35Â Bâ36Â A created continuously overlapping octaves and musical twelfths (octave plus fifth) in line with the predictions of both Jacques Handschin and Ernest McClain.7 This continuous overlapping causes a corresponding overlapping and interference among the fourths included within those two series of intervals as they continue running together, with the result that numbers properly belonging to one fourth interval among the musical twelfths begin to show up as extraneous additional numbers in octave sequences and vice versa. The two series are out of synchronization with each other, so to speak. Handschin explained that âif one follows Plato strictly,â one will, therefore, soon âtransgress the limits of diatonicism.â8
In the dominant diatonic Timaeus scale found by this analysis, the chromatic distortions arise in an orderly sequence following the same logic of fourfold progression pertaining to every other scheme of generation in Platoâs text. They also arise in an orderly fashion in all of the secondary diatonic Timaeus chains.
The invasion of the octave scale by chromatic elements is exactly what Jacques Handschin predicted would happen to octave periodicity with continued iterations of the diatonic scale, under the particular conditions for the scale set by Plato in the Timaeus. Handschin also predicted that the octave periodicity would eventually give way to fifth periodicity but expressed doubt that Plato understood the implications attached to his text.9 Handschin did not prove from the text his predictions about the degeneration of octave periodicity to fifth periodicity. The analysis offered here, however, accomplishes exactly that proof. It arrives at its conclusions independently, based upon a detailed and rigorous examination of Platoâs text, departing from Handschin only in his suggestion that Plato was clueless about the degeneration of octave periodicity to fifth periodicity.
The present interpretation suggests that Plato could plausibly have known about the decay of octave periodicity to fifth periodicity, entailed by his text, and, perhaps, even entertained a pedagogical purpose for allowing it. One sees fifths fall out, in proper order, too, as the fifth periodicity replacing octave periodicity itself disappears with the completion of Platoâs divisions at Timaeus 36Â B. It is possible that Plato deliberately intended to set up conditions in the Timaeus under which octave periodicity would give way to a fifth periodicity which would, then, itself give out, just so that he could show his reader the ordo according to which the fifths fill out the octave. One might at least so hypothesize in connection with the shadowy figures whose tradition the Timaeus reflects.
The primary scale emerging and degenerating in the Timaeus is an ascending Lydian diatonic scale, if one interprets the numbers resulting from Platoâs divisions of the world soul stuff as numbers indexed to string vibration or impacts on air, or, alternatively, a descending Dorian scale, if one interprets them as string lengths (384 as an index to string vibration or impacts on air is a low pitch given the set of mostly larger numbers comprising Platoâs set; but as a string length, it is a high pitch, representing a relatively short length in comparison with those other numbers).10 Plutarch elected for the Lydian, rather than Dorian option, and that is my preference as well.11 The results of this study are valid, in either case, because the two scales are exact reciprocals of each other. An ascending Lydian scale, that is, has the same tone/diesis pattern as a descending Dorian scale.12 Tone/diesis sequences are inverse for the ascending Lydian and ascending Dorian and for the descending Lydian and descending Dorian. Plato or those whose tradition the Timaeus reflects may have meant for ambiguity to remain pertaining to an ascending or descending interpretation of the scale just to emphasize that the octave scale emerges according to the same principles whether it ascends or descends. One simply interprets the scale in reverse order, for a rising and declining scale, identifying the initial number as the lowest pitch for the ascending Lydian scale and as the highest one for the descending Dorian.13
The numbers resulting from Platoâs divisions of the world soul are sufficient to account for the following phenomena in relation to the primary diatonic scale built in the text. They show the rise to the scale from its elements; then they articulate a perfect disdiapason (two-octave sequence) followed by four chromatically distorted disdiapasons for a total of a decad of octaves; then they exhibit a decad of incomplete octave sequences that accomplish the replacement of octave periodicity by fifth periodicity and end in the demise of fifth periodicity, as well. The decad of octaves stands to the decad of incomplete sequences as monad stands to duad; and so, this monad/duad relation in Platoâs text reproduces the octave proportion vis-Ã -vis its members.
At the same time, each of the decads, inasmuch as each is structurally a decad of elements, marks the fourfold progression (monad to tetrad 4:1) normally associated with the disdiapason (two octave sequence); so the duad of decads marks the eightfold progression proper to a duad of disdiapasons. A duad of disdiapasons, however, is a tetrad of diapasons (octaves). A tetrad is never achieved in Platoâs text except through a fourfold progression in the pattern of the Decad; so the tetrad of diapasons points to a higher order decad standing behind the monad/duad pair made out by octave periodicity/other periodicity. The higher order decad governing the pair works like any other decad. The fourfold progression marked by the pattern of the Decad terminates in a tetrad just as it gives rise to a disdiapason. Accordingly, imitating the Decad constituting the exemplary pattern and âAll Perfect Animal,â the higher order decad standing behind the primary monad/duad pair of the Timaeus text (octave periodicity/other periodicity) structures the cosmos as a disdiapason, just as Favonius Eulogius had written much later in his In somnium Scipionis.14
In addition to revealing phenomena pertinent to the primary scale and structure of the cosmos, identified above, the analysis demonstrates how an entire system of ancient Greek musicâseven dominant octave species in all three genera, diatonic, chromatic, and enharmonicâcan be derived from the matrix of numbers hidden within the Timaeus text. This system recalls the fifth century Eratocles, in its demonstration of cyclical variations to achieve different octave species, as well as the spirit of fourth century experimentation with perfect systems against fifth century developments, otherwise. The study sheds new light on possible avenues of development of the Greater Perfect System (âGPSâ), Lesser Perfect System (âLPSâ), and Unmodulating Perfect System (âUPSâ) in the fourth century.
Modern scholarship affirms that Greek writers were agreed on the basic shape of GPS by the late fourth century; and certainly the Sectio Canonis, a late fourth century Pythagorean text attributed to Euclid, provides direct evidence of GPS, LPS, and UPS.15 In relation to the Timaeus, one might place the Sectio Canonis at the end of the experimental period for UPS. It explicitly discusses and demonstrates the immutable system and reflects all of the concerns that one might expect of a Pythagorean thinker for the importance to music of superparticular ratios in the mathematical sense.16
Stefan Hagel has assigned the beginning of interest in modulating music to the second half of the fifth century, remarking upon Pythagoras of Zacynthusâ early success in tuning an entire set of strings to the Dorian, Phrygian, and Lydian harmoniai in such a way that one could proceed from one to another without interruption. He has also mentioned Pronomosâ invention of the modulating aulos before 400, integrating at least the Dorian, Phrygian, and Lydian harmoniai, stating that the incorporation of several scales on one instrument was indubitably a milestone in the evolution of a modal system.17
Both Andrew Barker and Stefan Hagel have discussed the importance of the fifth century Eratoclesâ efforts to the development of UPS in the fourth century.18 Barker noted that octaves can be organized without a change in genus in seven different ways. In his opinion, compelling evidence shows that Eratocles recast the harmoniai under which fifth and early fourth century melodies were classified as the seven octave species: Dorian, Phrygian, Lydian, Mixolydian, Hypodorian, Hypophrygian, and Hypolydian. He observed that, for Eratocles, each of the seven harmoniai or species of octave is generated by removing the interval at the top of its predecessor and replacing it at the bottom.19 Hagel and Gustave Reese have affirmed the existence of the octave species identified, above, by the fourth century.20
Evidence that the seven octave types identified were the primary octave types recognized in the fourth century is derived from the writings of Plato himself and Aristoxenus, Platoâs younger fourth century contemporary, who commented on earlier musical practice.21 At Republic 398Â Eâ399Â A, Plato expressly mentioned the Mixolydian, various other âLydianâ modes, including âtenseâ and âlax (Hypolydian?)â varieties, the Dorian, the Phrygian, and the âIonianâ but protested, from Socratesâ mouth, that he did not know the modes. At Laches 188 D, he named the âIonian,â Phrygian, Dorian, and Lydian. Barker has drawn particular attention to the Mixolydian, Lydian, Dorian, and Phrygian in Plato, while Reese has clarified that the label âIonian,â was sometimes a label for the Phrygian.22
Stefan Hagel acknowledges the Hypophrygian, Hypodorian, Dorian, Phrygian, Lydian, Mixolydian, and (probably also) the Hypolydian as having predated Aristoxenus, whose writings imply their earlier use, and opines that the âIonianâ had no roots in traditional pre-Aristoxenian musical practice comparable to those of the others mentioned.23 Thomas Mathiesen mentions all modes except the Hypolydian as having existed among the âharmonicistsâ predating Aristoxenus, but recognized Aristoxenusâ knowledge of all seven octave species preceding his own experimentation in association with the tonoi.24
Barker has observed that Aristoxenus worked from the foundation that Eratocles laid, noting that Aristoxenusâ notion of the role of the tonoi was essentially tied up with their relation to the octave species.25 Mathiesen has also mentioned both Aristoxenusâ knowledge of Eratoclesâ efforts and his understanding that Eratoclesâ primary interest was the possible cyclic orderings of octave intervals.26 It may interest some readers that Ptolemy, a much later musical theorist, far from following Aristoxenus, essentially reverted to an approach stressing seven tonoi in accord with the seven diatonic tunings for octave species.27
Hagel has opined that Eratoclesâ tuning diagrams certainly represent a system of tonoi reflecting a highly abstract conception of tonal relations. Like Barker, he has observed that Eratocles analyzed different scales as successions of intervals which can be transferred from one end to the other; thus his seven octave species are certainly possible sources for the âcanonicalâ seven tonoi widely acknowledged by the time of Aristoxenus or not long after.28
Hagel has pointed out that a demonstration of Eratoclesâ seven octave species, assigning the same pitch to similar functional notes, required a structure of two octaves such as GPS.29 He has opined that such a two octave system was probably already known to Plato and was important, practically, for the boring of the single-holed aulos.30 He states also that Plato treated the Pythagorean diatonic as a given (though the Eratoclean school was concerned only with the enharmonic), notes that the Timaeus presupposes the fixed notes skeleton of a musical scale, and observes that it is but a minor step from the Timaean framework of fixed notes to the conception of a perfect system.31
This study indicates that Platoâs Timaeus may provide indirect evidence of GPS, LPS, and UPS, perhaps while there was still some controversy concerning their structures. The text may also demonstrate a concerted interest in Eratoclesâ seven octave species, noted above, and the means by which they can be generated from each other by interval transference. It proceeds from a Pythagorean rather than Aristoxenian standpoint, in its approach to previous developments, and stresses the diatonic, rather than the enharmonic, in contrast to the Eratoclean school, interested solely in the enharmonic.32
The study adopts use of the word tonos, in connection with the musical preoccupations of the Timaeus, as appropriate. Reese clarified the legitimacy of using the word âtonosâ in association with the thinking of various ancient Greek writers, including Plato, as follows:
In many of the Greek treatises, the word harmonia (tonal structure) appears ⦠[W]hen high tuning came into use and when the eleven- and twelve-string kitharas yielded, as we have seen, a range of two octaves, âharmoniaâ was employed in at least two senses: as a synonym for tonos and as a name for the individual octave-species as projected upon the Greater Perfect System. It is often difficult to determine which meaning is intended, but it seems fairly certain that such writers as Plato, Aristotle, and Herakleides used it in the sense of tonos.33
The study claims, in addition, that, although Plato was primarily interested in diatonic phenomena in the Timaeusâcertainly, as James Haar has noted, all ancient writers agreed that the Timaeus scale is in the diatonic genus,âthe system arising from the Timaeus numbers accommodates the chromatic and enharmonic octave genera, as well, also within a UPS structure.34 Diatonic, chromatic, and enharmonic interests would certainly all have been possible for Plato.
Andrew Barker, for example, points to Philolaus fragment 6A as evidence for the provenance of the diatonic genus before Plato; and he marks the fourth century Archytasâ diatonic, chromatic, and enharmonic divisions of the fourth.35 Hagel notes Aristoxenusâ specific acknowledgment, in the fourth century, that the diatonic genus was older than the other two genera, evidencing knowledge of all three by his time. In relevance to the plausibility of Platoâs diatonic interests, Hagel relates that the diatonic genus considerably predates Hellenic culture, probably deriving from Sumerian music. He marks its witness in Old Babylonian cuneiform tablets; observes that Near Eastern music had codified the complete cyclic system of seven diatonic tunings in the second millennium B.C. and probably earlier; notes that most or all of these tunings were actually used in musical practice; and states that, in the Sumerian tradition, they were exemplarily constructed and construed on a stringed instrument. He opines that the diatonic division of tonal space similar to scales known for the ancient Near East was, in some respect, the basis for Greek lyre music.36
Hagel further makes reference to an early preference, in pre-Aristoxenian musical practice, for enharmonic music, that gave way in the fourth century to a chromatic preference and, even later, to a diatonic preference; and Mathiesen specifically notes Aristoxenusâ awareness, in the fourth century, of the diatonic, enharmonic, and chromatic genera.37 Surely other sophisticated fourth century thinkers were also aware of all three genera.
Although it is somewhat speculative to suggest that Plato or the persons whose work informed the Timaeus sought to experiment with accommodating all three genera within a unified UPS framework, the study supports the suggestion and points scholars to further researches in that direction.
Assuredly, the interpretation of the Timaeus offered by this study depends upon a much larger set of numbers than is usually attributed to Platoâs divisions of the world soul. Accordingly much of the analysis is devoted to exhibiting just how one arrives at the set. It makes manifest that the smaller sets of numbers achieved by commentators to date is partially due to a misstep that they make at Timaeus 36Â B. That passage directs the reader to put sesquioctave intervals into all sesquitertian parts produced when harmonic and arithmetic means are inserted between members of the original set of numbers defined at 35Â Bâ36Â A. Commentators err, as the text shows, by failing to consider all of the possibilities for textual interpretation of the sesquitertian parts that Plato intended his reader to fill. Indeed, they miss the ambiguity in the text that gives rise to possibilities beyond the most obvious. Plato gave his reader a veiled hint at the very beginning of the Timaeus that correctly identifying the sesquitertian parts is the key to his whole riddle. The sesquitertian part, correctly elaborated, could certainly be Platoâs missing fourth guest mentioned at the beginning of the dialogue.38
Some modern commentators have moved in particularly promising directions. Luc Brisson, for example, argued in Le même et lââ¯autre that the Timaeus articulates a mathematical model for the universe, but he did not consider that the text itself could be a kind of number generating machine.39 That the text is a number generator makes sense, as a good working hypothesis, from the standpoint of Platoâs close friendship with the Pythagorean community at Tarentum. Certainly, cosmogony, cosmology, and number generation would have walked hand-in-hand for them.
The beauty of the present analysis is its beginning from a theory of text as number generator and not from musical presuppositions. Its method is simply to find the maximum set of numbers that Platoâs divisions legitimately allow one to construct on the basis of some principle already implicit in the Timaeus text. Only after the set is identified is there an attempt at an interpretation. The entire set of numbers is a completely nonarbitrary set, yielding the results skeletally described above. The best argument for the method and the set of numbers it generates is that they yield an interpretation making sense of Platoâs Timaeus cosmogony down to the fabric and the âchiâ figure, in line with the best instincts of the ancients; the formation of the sphere from the âchiâ figure; and the further divisions from the soul sphere of the bands of same and difference. The study also yields a new bridge to musical material in the Laws. In the end, the studyâs best justification for itself is its delivery of a complete, cogent exposition of Timaeus 35Â Aâ36 D.
Francis Cornford, Platoâs Cosmology, The Timaeus of Plato Translated with a Running Commentary (London: Routledge and Kegan-Paul, 1951), 68â70; Luc Brisson and Walter Meyerstein, Inventing the Universe, Platoâs Timaeus, The Big Bang, and the Problem of Scientific Knowledge (Albany: State University of New York, 1995), 35; Luc Brisson, Le même et lââ¯autre dans la structure ontologique du Timée de Platon, un commentaire systématique du Timée de Platon, International Plato Studies, eds. Luc Brisson, Tomás Calvo, Livio Rossetti, Christopher J. Rowe, and Thomas A. Szlezák, no. 2 (Sankt Augustin: Academia Verlag, 1994), 314â332.
François Lasserre, âLââ¯Education musicale dans la Grèce antique,â in De la musique par Plutarque, texte traduction commentaire précédés dââ¯une étude sur lââ¯Ã©ducation musicale dans la Gréce antique (Olten & Lausanne: Urs Graf-Verlag, 1954) 53â74.
Luc Brisson, Le même et lââ¯autre dans la structure ontologique du Timée de Platon, un commentaire systématique du Timée de Platon, International Plato Studies, eds.Luc Brisson, Tomás Calvo, Livio Rossetti, Christopher J. Rowe, and Thomasa. Szlezék, no. 2 (Sankt Augustine: Academia Verlag, 1994), 33â54.
Cornford, Platoâs Cosmology, 68â69.
Jacques Handschin, âThe Timaeus Scale,â Musica Disciplina 4 (1950):31â35.
See Handschin, âTimaeus Scale,â 3â41; John Dillon, âThe Timaeus in the Old Academy,â in Platoâs Timaeus as Cultural Icon, ed. Gretchen J. Reydams-Schils (Notre Dame: University of Notre Dame Press, 2003), 82â84; Mitchell Miller, âThe Timaeus and the âLonger Wayâ: God-Given Method and the Constitution of Elements and Animals,â in Reydam-Schils, Cultural Icon, 23â50; Carl A. Huffman, Philolaus of Croton, Pythagorean and Presocratic, A Commentary on the fragments and Testimonia with Intepretative Essays (Cambridge: Cambridge University Press, 1993), 21â25 (discussing the dialogue of Plato and the Old Academy with Pythagoreans); Dominic OâMeara, Pythagoras Revisited: Mathematics and Philosophy in Late Antiquity (Oxford: Clarendon Press, 1989), 146â149, 179â183 (discussing Proclusâ attribution of Pythagorean inspiration to Platoâs science of the divine, as expressed in Timaeus and other texts). Speusippus was Platoâs sisterâs son, a member of the Academy, and Platoâs immediate successor as its head upon his death. Leo Tarán, Speusippus of Athens, A Critical Study with a Collection of the Related Texts and Commentary (Leiden: Brill, 1981), 5. Speusippus, however, conceived of numbers only as a collection of units, according to Tarán, and spurned the idea of ideal number. Ibid., 16. A similar contempt is not to be found in the Timaeus. See Sarah Klitenic Wear trans., The Teachings of Syrianus on Platoâs Timaeus and Parmenides, Studies in Platonism, Neoplatonism, and the Platonic Tradition, eds. Robert Berchman and John Finamore, vol. 10, Ancient Mediterranean and Medieval Texts and Contexts (Leiden: Brill, 2011), 40 (for the notion that the dyad contained the intelligible Decad of forms in the much later neoplatonic tradition).
Handschin, âTimaeus Scale,â 21; Ernest McClain, Pythagorean Plato, Prelude to the Song Itself (York Beach, Maine: Nicolas Hys, Inc. 1978), 61 and 62.
Handschin, âTimaeus Scale,â 24; see also, McClain, Pythagorean Plato, 59 (identifying the triple intervals with musical twelfths).
Handschin, âTimaeus Scale,â 23â24.
Ibid., 24 and 26.
Ibid.; see also, Plut., Moralia, 13.1.2.13.1018D, 13.1.2.18.1021E (manifesting that Plutarch assigned smaller numbers to lower notes and higher numbers to higher notes; the implication is that Plutarch interpreted Plato as having defined an ascending Lydian scale); see James Haar, âMusica mundana: Variations on a Pythagorean Themeâ (Ph.D. diss., Harvard University, 1960), 17â21, for a good discussion of the ancient and modern debates on the ascending or descending scale question and 1â70, more generally, for an excellent account of the history of interpretation of the Timaeus scale from ancient times.
Gustave Reese presented the seven different possible diatonic arrangements of tones and semitones in the Greek octave interval, thus, in descending order of pitch: TTTSTTS (Mixolydian, B or D); STTTSTT (Lydian, C or C#); TSTTTST (Phrygian, D or B); TTSTTTS (Dorian, E or A); STTSTTT (Hypolydian, F or G#); TSTTSTT (Hypophrygian, G or F#); TTSTTST (Hypodorian, A or E). Gustave Reese, Music in the Middle Ages, with an introduction on music of ancient times (New York: W.W. Norton & Co., 1940), 28 and 30.
Handschin stated that it would have been characteristic of Platoâs style to have left the question open. Handschin, âTimaeus Scale,â 15. He observed that Pseudo-Timaeus, Proclus, and Psellus, did, in fact, leave the choice between the Dorian or Lydian tonos open for the Timaeus scale, just as this study has done. Ibid., 21.
Eulogius somnio Scipionis 25.
For the development of UPS, including GPS and LPS, in the fourth century, see Andrew Barker, The Science of Harmonics in Classical Greece (Cambridge: Cambridge University Press, 2007), 12â17 (noting, inter alia, that by the late fourth century, all Greek writers on harmonics were agreed on the basic shape of the system); Donald Creese, The Monochord in Ancient Greek Harmonic Science (Cambridge: Cambridge University Press, 2010), 21 (GPS was a fourth century construct). Stefan Hagel, Ancient Greek Music, A New Technical History (Cambridge: Cambridge University Press, 2010), 5â6 (defining âUnmodulating Systemâ).
See André Barbera, ed. and trans., The Euclidean Division of the Canon: Greek and Latin Sources, Greek and Latin Music Theory, eds. Thomas J. Mathiesen and Jon Solomon (Lincoln: University of Nebraska Press, 1991), 21, 129â131, 134â135, 150â159, 170â171, 174â175, 178â179, 186â187; see also, Andrew Barker, âEarly Timaeus Commentaries and Hellenistic Musicology,â in Ancient Approaches to Platoâs Timaeus, eds. Robert W. Sharples and Anne Sheppard, Bulletin of the Institute of Classical Studies Supplement, ed. Geoffrey Waywell, no. 78 (London: Institute of Classical Studies, School of Advanced Study of the University of London, 2003), 76 (for dating of the text).
Hagel, Ancient Greek Music, 378â379.
Barker, Science of Harmonics, 83, 224; Hagel, Ancient Greek Music, 373â374, 378, 387.
Barker, Science of Harmonics, 83, 224.
Hagel, Ancient Greek Music, 4â5 (mentioning all but the Mixolydian, but referring to a seven tonoi system predating Aristoxenus); Reese, Music, 35 (dating the âhighâ Mixolydian to 475â¯B.C.).
See Thomas J. Mathiesen, âGreek Music Theory,â in The Cambridge History of Western Music Theory, ed. Thomas Christensen (Cambridge: Cambridge University Press, 2002; reprint, Cambridge: Cambridge University Press, 2004), 113 (establishing Aristoxenus as one of Platoâs younger fourth century contemporaries).
Barker, Science of Harmonics, 309; Reese, Music, 32; see also, Hagel, Ancient Greek Music, 430.
Hagel, Ancient Greek Music, 4â5, 8.
Mathiesen, âGreek Music Theory,â 119, 125.
Barker, Science of Harmonics 224, 227.
Mathiesen, âGreek Music Theory,â 119.
Hagel, Ancient Greek Music, 5, 387.
Ibid., 8, 378, 387, 430â431; see, also, Barker, Science of Harmonics, 297 (for fourth century placement of Aristoxenus).
Hagel, Ancient Greek Music, 387â388.
Ibid., 387â388.
Ibid., 430â431, 448â449.
Hagel, Ancient Greek Music, 431.
Reese, Music, 44 (relying upon Bonaventura Meyer, âARMONIA,â Bedeutungsgeschichte des Wortes von Homer bis Aristoteles, 1931).
Haar, âMusica Mundana,â 15.
Barker, Science of Harmonics, 38, 264, 292, 293â307; see, also, Hagel, Ancient Greek Music, 135 (noting that diatonic heptatony, associated with stringed instruments has origins beyond the second millennium B.C.) and 143 (recognizing Philolaus as evidence of the diatonic genus among Greeks prior to Plato).
Hagel, Ancient Greek Music, 10 (n. 35), 106, 414, 436, 442.
Ibid., 44â52; Mathiesen, âGreek Music Theory,â 123.
Plato Tim. 17 A.
See Brisson, Le même et lââ¯autre, generally, for Brissonâs explanation how the Timaeus posits a mathematical model of the universe.