These two books are very different, the one a work of imagination at least as much as of learning, the other not only learned in itself, but demanding erudition in the reader. The late Flora Levin enters the lists to champion Aristoxenus in his quarrel with the Pythagoreans; we may examine how well she does so before turning to Stefan Hagel's account of musical practice insofar as it can be known over the course of Greek antiquity.

If there is one thing that persons even slightly acquainted with ancient music theory know, it is that the Pythagoreans understood intervals as ratios capable of multiplication and division, Aristoxenus as measurements capable of addition and subtraction. For both parties, the only simple concords are the octave, the fifth and the fourth (only perfect intervals come into consideration). The Pythagoreans, measuring string-lengths, declared that the respective ratios were 2 : 1, 3 : 2, 4 : 3; since these ratios were either multiple or superparticular, they concluded that only multiple and superparticular ratios were capable of generating concords,Footnote ¹ and therefore the eleventh, though concordant to the ear, was a priori a discord as being generated by the superpartient ratio 8 : 3. Further calculation showed that the fourth exceeded the ditone (9² : 8² = 81 : 64) by a minor semitone or λεῖμμα (‘leaving’) of 256 : 243, whose complement to the tone was the major semitone or ἀποτομή (‘offcut’) of 2187 : 2048, the difference between them being the comma (κόμμα, ‘chip’) of 531441 : 524288. Aristoxenus would have none of this: any concord expanded by an octave yielded another concord, and the tone that separated the fifth from the fourth was divisible into two equal semitones, five of which constituted the fourth.

Although Aristoxenus had many followers among Greek handbook-writers, his approach failed to satisfy serious intellectuals; even Ptolemy, who gives him best over the eleventh, rejects his additive conception and his equal semitones. The opprobrium he thus incurred has lasted into modern times, even though equal semitones are now in use, and reckoning in cents, determined by a logarithmic procedure that neither party could have understood, has transcended the conceptual opposition between them by reconciling the additive principle with ratios.

It is this opprobrium from which Levin aims to vindicate him, claiming (p. xvii) that Aristoxenus – properly understood, it will appear, only by Ptolemais of Cyrene and Levin herself – employed a ‘profoundly dialectic’ method from which he ‘obtained a fixed constant of measurement’, practising analytic number theory centuries before Lejeune Dirichlet, Riemann, Cantor, Kronecker and Weierstrass (listed thus against chronology, and never mentioned again). I do not pretend to make sense of this assertion; but over and over again she writes with more engagement than exactness; suffice it to cite from p. 206, n. 4: ‘When the geometry of the line, which encompasses the harmonic series, is integrated with a melodic topos that has been equally tempered, a most mysterious element is produced: melodic gravity.’ Even though topos is used in its Aristoxenian sense of musical space, nothing could be less Aristoxenian than this sentence.

Levin also writes con amore, a fact that must endear her to many readers but entails certain problems. On p. xiv she is impressed by the ancient Greeks' ability to distinguish words according to their pitch-accent: the meanings of writers' words ‘depended in the fullest sense on the semantic nature of their accompanying pitch-accents’, a phenomenon that bespoke ‘an acutely sensitive and highly developed auditory sense’, perhaps extending even to absolute pitch. What is this but that idealisation of classical Hellas which we associate with Winkelmann, Goethe and Karl Marx? There are many other tonal languages in the world whose speakers must presumably have, in Levin's phrase, an at least equally sensitive sense; far fewer words in ancient Greek are distinguished by pitch alone than in Chinese or Vietnamese or Thai.Footnote ²

That, however, is a venial fault compared with the errors in mathematics of one who can proclaim that the ratio of the octave ‘to its internal twelve semitones’ (p. 120), ‘when merged with the ratios of its innermost interval’ (p. 184), is 0.6 recurring, which she absurdly regards as a form of infinity.Footnote ³ More seriously, she flings around the concepts of incommensurability and irrationality without knowing what they are, applying them to the perfectly rational fractions 3/2, 4/3 and 9/8 because their square roots are irrational; as a result her discussion of Aristoxenus' disagreement with the Pythagoreans is in large part nonsense.

This supposed incommensurability was (we read on p. 9) resolved by the allegedly brilliant invention of equal temperament; in other words, by the imposition of genuine irrationality on all intervals below the octave and the contrivance of semitones governed by the ratio 2^1/12 : 1. So deep is this pianistic love of equal temperament that Levin can declare (p. 20, n. 39): ‘modulation, to work at all, requires an equally-tempered attunement’, which leaves one wondering not only how the modern violinist manages to modulate, but how the medieval singer mutated between hexachords when even just intonation lay in the future. Like many others since the sixteenth century, she deems equal temperament to have been Aristoxenus' invention.

However, though not the most modest of authors, nor the most sparing of his opponents' faults, nowhere does Aristoxenus claim to have reformed tuning, or accuse others of making their intervals the wrong size; rather, he claims that the semitone is (not merely ought or needs to be) the exact half of the tone – that is to say, of the difference between a fourth and a fifth – which he purports to demonstrate by the experiment of tuning in fourths and fifths till an original fourth has been expanded by a semitone in each direction, producing an interval that he maintains is found concordant by the ear and therefore must be a fifth.Footnote ⁴ We need not follow the remainder of the reasoning, for if the fourths and fifths are the pure intervals that he has not disowned, the outcome of his experiment is not a fifth (3 : 2) but a fourth plus two leimmata (262144 : 177147), which is less than the pure interval by a comma, or 23.460010 cents. If he really could not perceive the difference, which according to Boethius is the smallest audible,Footnote ⁵ gone is that enviable fineness of ear with which Levin credits not only him but all Greeks. It is all the more amusing to read at p. 184, n. 71 that Aristoxenus ‘seemed to feel that such an interval as a comma and such an unmanageable [sic] interval as the fifth were nature's – that is, musical nature's – way of informing us that we are doing something wrong’.Footnote ⁶

In order to succeed, the experiment requires equal temperament: if, for instance, our fourth is the descending tetrachord a–e, the resulting interval of b♭–d♯ must become its (in the modern sense) enharmonic equivalent b♭–e♭Footnote ⁷ (for ancient Greek notation modifies the lower, not the higher note). But Aristoxenus did not stipulate its use in the experiment, and could not have done so without being seen to load the dice in his favour; nor was he the first theorist to posit it, for it is presupposed by the very division of the old Dorian ἁρμονίαFootnote ⁸ into twenty-eight quartertones that he despises. That is not to say that it was already the norm for practical music, since that of Greece in his day did not need it, and that of modern Europe, which did, was slow enough to devise it.

However, the entire discussion is based on a premiss already known to be false in antiquity,Footnote ⁹ that tuning in real life was as precise as in theory. In fact, the experiment would have been a matter of mere chance: as we have seen, fourths and fifths as barely distinguishable as the equal-tempered (500 and 700 cents) and the Pythagorean (498.044999 and 701.955001 cents) yield a noticeably different outcome. Rather, the great musician has decided a priori what must be, and bluffed an experiment to prove it; we are not so very far removed from the fable of Pythagoras in the smithy.

From numbers to words. All too often, Levin makes Aristoxenus reach across the millennia to modern musicians by equivocation of language or of thought, for example equating his tensions and relaxations of the voice, which produce respectively higher and lower notes, with the tensions and relaxations of melodic progression (pp. 110–11); Aristoxenus is explicitly teaching elementary theory, not music appreciation or composition like the nineteenth book of Aristotelian Problems.Footnote ¹⁰ Even when he speaks of a note's or an interval's δύναμις, ‘force’, meaning its position in the scale, or as we say its function, he does not tell us how the good composer will deploy it; even when he discusses the succession of intervals, he is distinguishing correct from incorrect, not beautiful from ugly or interesting from dull.

The inadequacies of musical notation have become a commonplace; but they have nothing to do with Aristoxenus' statement (in her translation, p. 84), ‘It is not necessary that one who has written down a Phrygian melody know exactly what a Phrygian melody is’: the point is not that notation might in any way fall short of recording the melody as performed, but that ability to employ it does not entail ability to define, or even recognise, the τόνος. Nor, when the Platonic Socrates' unfavourable view of artistic creation as the outcome of divine madness is favourably restated by Peter Kivy as a draught from the depths of the unconscious, does this echo Aristoxenus' assertion that critical understanding (ξύνεσις), attained through perception and memory, has sunk down somewhere into the ψυχή and is not close to hand or obvious to the many (p. 85); for ψυχή excludes neither the conscious nor the rational.Footnote ¹¹

For Levin, however, critical understanding is not enough: she requires the inspiration of genius, known as intuition, which she reads into ξύνεσις.Footnote ¹² Accordingly, on pp. 45–6 her heroine Ptolemais is made to describe the extraction of musical meanings as an intuitive act, ‘a zopyron, a “flash” of knowledge that defies all analysis’. But ζώπυρον means not ‘flash’ but ‘spark’; to quote Levin's own translation on p. 258, Ptolemais says that ‘Pythagoras and his followers are pleased to accept perception to begin with as a guide for reason, so as to offer reason some sparks, as it were; but reason, having been stimulated by those sparks, is treated by the Pythagoreans as something that works independently, standing quite apart from perception’ – and if reason's conclusions conflict with perception, then perception is declared to be in error. Not a flash of intuition, then; perception is summoned to light the lamp of reason and having done so is sent back to the servants' quarters. Of musical meanings, not a trace.Footnote ¹³

There is another feature of Levin's work that requires comment, though some readers will not consider it a fault. Interpretation of a text entirely in accordance with its author's historical situation, without regard to subsequent events or works not known, has in recent years appeared to critics an unwarrantable confinement of the mind; the fashion is rather to pursue thoughts stimulated by the text wherever they may lead, even to considerations that neither the author nor the first readers can have entertained. However, such criticism is often presented, not as the critic's own reaction to the text, but as a drawing-out of the author's hidden thoughts; in this philosophers have long shown the way, attributing to Plato or Aristotle inferences that they never drew and answers to objections that they never heard. Levin's book should have been entitled, not Greek Reflections on the Nature of Music, but Reflections on the Nature of Music, Stimulated by the Reading of Greek Authors.

For example, the first chapter, ‘All Deep Things Are Song’, begins with Bacchius the Old's definition of music, ‘the εἴδησις of melody and the things pertaining to melody’; Levin translates εἴδησις not as ‘knowledge’ but as ‘conceptual knowledge’, which would cast a strange light on the heading in a treatise on horse-medicine, Εἴδησις περὶ κόλλης, ‘Knowledge about Glue’. She then exalts his first definition of melody, ‘the fall and rise generated by melodious notes’, as deliberately circular, so that he may anticipate Wittgenstein in declaring melody a closed system, but find logic in melody and not vice versa. This apparently sets him in a somewhat alarming agreement with Schopenhauer, but in opposition not only to the Pythagoreans, whose notions of cosmic music are still entertained in certain quarters, but to Deryck Cooke, who elevated his own emotional responses to musical devices into objective meanings as in a dictionary. Having quoted Hindemith and Stravinsky against such equations of music and language, Levin brings in Philodemus, for whom music is a source of aural pleasure and nothing else. More engagement with Cooke follows, leaving him barely upright.

There is much food for thought here (for a start, could Cooke respond with the appropriate mood to an Indian rāga heard without explanation?);Footnote ¹⁴ but with the best will in the world I cannot extract any of it from Bacchius' first definition of melody, which resembles less the pronouncements of Schopenhauer and Wittgenstein than the famous definition of an archdeacon as one who performs archidiaconal functions. Bacchius was manifestly no logician: his second definition of melody, ‘That which is composed of notes and intervals and durations’, is little more satisfactory. The thing was done much better in the Excerpta Neapolitana: ‘The use, affording pleasure to those who hear it, of an intervallic, inflected voice.’Footnote ¹⁵

But is music in fact sufficient unto itself? On p. 86 Levin asserts that ‘with Aristoxenus, we are … among the musicians of the world – performers and composers – for whom the modes of music represent all the states of the human soul’. I do not ask which among the many musical senses of ‘modes’ she intends, but I do ask where she found such a notion in her author, and how a music that represents the states of the soul is any more a closed system than one that represents the movements of the cosmos. In fact, despite her praise of Aristoxenus, Levin is sufficiently taken by this latter, Pythagorean and Ptolemaic, notion to find it in supersymmetric string theory, according to which the universe is made up of tiny strings vibrating at resonant frequencies. Should that, in any of its variants, happen to be right, it will stand in the same relation to the cosmic music of the ancients as modern particle physics to ancient atomism; but the suggestion suffices for Levin to declare (p. 169) that if scientists can find music in the universe, then musicians may find the universe in music. In particular, causality is there, as she argues from Roger Scruton (pp. 169, 171, n. 39), combining memory with expectation, and therefore locating the past and the future within the present as if in answer to St Augustine's quest for them. That is an interesting notion: remembering how a composition has developed, and perceiving how it is developing, one forms an expectation of how it will develop; that expectation is sometimes fulfilled and sometimes frustrated, but even in the latter case one may in retrospect recognise causality. (Indeed, the relative frequency of fulfilment and frustration might be made into a test of merit in both composer and listener.) But despite Levin the notion is not entertained by Aristoxenus, who speaks only of remembering the past and perceiving the present.

Although Levin more than once finds in Aristoxenus the opinions of his teacher Aristotle, she does not always make him philosophical: ‘being the complete musician he indeed was, [he] evidently had little patience with the subtleties of arguments such as that of Theophrastus’ (p. 109). She then imagines that it was Theophrastus who raised the question, whether the voice really moves, that Aristoxenus dismissed as belonging to a different inquiry; that is not a sign of impatience with other considerations, for music and acoustics are different disciplines even now. Yet Andrew Barker has recently suggested that Theophrastus, so far from attacking Aristoxenus, in fact borrowed from him in order to demolish a theory cited in Plato's Republic.Footnote ¹⁶

Barker's book may have been published too late for Levin, but she does not always engage with scholarship available to her. In chapter 4, ‘Magnitudes and Multitudes’, she represents the Division of the Canon, rightly as she thinks ascribed to Euclid, as the summa of Pythagorean theory, maintaining it is all of a piece, including the proem, which some scholars have found problematic, and in which she finds an anticipation of Fourier worthy of Euclid's genius. Yet although she cites Barbera's edition,Footnote ¹⁷ nowhere does she consider his arguments to show that the treatise originally expounded an arithmetic view of intervals before being revised in the Pythagorean interest, even though it is on that basis that he sets out the text.

If in Levin's eyes Aristoxenus is too little known except to scholars, there is another author whom she rightly considers too little known even to scholars, Ptolemais of Cyrene, whom Porphyry – our sole witness to her existence – adduces as the source of Ptolemy's source Didymus and whom he honoured by not mentioning her sex (p. 230). He did not need to do so, since not only the name but the gender of the descriptor ἡ Κυρηναία made that clear, but it is true that, as one learned person quoting another, he made no comment on the fact; nor did Aulus Gellius, not the ancient world's chief feminist, when he cited the Alexandrian miscellanist Pamphile. The title Ptolemais of Cyrene's Pythagorean Elements of Music sets Levin romancing: an author identified by her ethnic must (heaven knows why) have been of exalted social status, even a princess, born in the mid-third century bc to Demetrius the Fair, king of Cyrene, and Apame, mother of his intended bride.Footnote ¹⁸ This is but one more testimony to scholars' weakness for gentle birth; royal names regularly make their way down the social scale, and hers is found in the most insalubrious contexts. In any case it is not her blood but her brain that matters.

Furthermore, as we shall see, Levin's theory entails the existence of a furious and ongoing polemic between Pythagoreans and Aristoxenians, all scribbling away in the century or so between the appearance of his work and Ptolemais' treatise; one wonders whether the topic was really so hot, or the Pythagorean school so lively. All we know about her date is that Ptolemais wrote before Didymus, who wrote before Ptolemy, and that Porphyry classes her with the ‘ancient’ writers, a very vague term, which he uses with a great variety of reference; but since at p. 4, l. 11 Düring it denotes all writers before Ptolemy and at p. 92, l. 28 it includes Dionysius of Halicarnassus from the late first century bc, the very time in which Pythagoreanism was revived, there is no reason to place Ptolemais any earlier.

From the writings of Ptolemais, Ptolemy and Porphyry, Levin attractively creates on pp. 254–62 a dialogue on perception and reason. Whereas for Ptolemy the Pythagoreans relied exclusively on the latter and the Aristoxenians no less one-sidedly on the former, Ptolemais makes distinctions. Some people relied exclusively on one or the other: Pythagoreans zealous against the (presumably Aristoxenian) ‘musicians’ referred everything to reason (despite starting out from perception), instrumentalists cared only for the ear. Others accepted both, some on equal terms, others giving the preference to one over the other. Aristoxenus accorded equal importance to both, requiring reason both to start from perception and to reach conclusions conformable to it. The transmitted text then makes Pythagoras and his successors also accept them on equal terms, even though as Ptolemais herself observes they accepted perception only as giving initial sparks to reason; at Porphyry p. 25, ll. 26–7 in Düring's edition we should read οὐχ ὁμοίως, ‘not equally’, or ἀνομοίως, ‘unequally’.Footnote ¹⁹ Opposed to them are certain Aristoxenian musicians, who, coming from an instrumental background, give the priority to perception and admit reason only at need, allowing that ‘the rational postulates [λογικαὶ ὑποθέσεις] of the canon’ may not agree with perception. It is evident that Ptolemais was extremely well read in the literature of the subject; we can but wish that we had access to her sources.

Levin understands the canon to be the monochord, not explaining how an inanimate object can entertain or propound postulates. Yet Ptolemais herself has insisted in other extracts that the canon is not the monochord but the ὑποθέσεις of musicians and those adopted from mathematics. If, as Levin suggests on p. 276, the musicians' postulates are the consonance of certain intervals and the dissonance of the rest, the mathematical postulates those relating to ratios, then the postulates within the canon rejected by these Aristoxenians will be those relating to ratios, λόγοι. But it remains unclear how they differed from Aristoxenus himself, and what subordinate role they nevertheless allocated to reason, λόγος.

These uncertainties, however, do not inhibit Levin from representing Ptolemais, despite the title of her treatise, as a champion of Aristoxenus who uses the phrase under discussion ‘in open disapproval’ (p. 273); that would make her a champion not of Aristoxenus but of the Aristoxenian instrumentalists. In fact this open disapproval is reserved for the Pythagorean zealots who affect to treat reason as entirely self-sufficient but ‘are completely refuted by their accepting something that is perceived by the senses at the very beginning and then disregarding it altogether’ (p. 261); by contrast the true Pythagoreans acknowledged the initial contribution of sense. In all likelihood Ptolemais dismissed the instrumentalists as mere empirical practitioners unworthy of intellectual consideration, found the vulgar Aristoxenians who flouted the canon as much wanting as the vulgar Pythagoreans who disowned perception, and respectfully compared Aristoxenus himself with Pythagoras and his successors before finally siding with the latter. But no: Levin's Ptolemais must be the exponent of Levin's Aristoxenus.

For all her enthusiastic imprecision, Levin does vindicate Aristoxenus' understanding of real music against the devotees of Pythagorean mathematics, insofar as he recognised melody as the movement of the voice through a sonic range or space. Those who fly to the stratosphere when thinking about music as well as listening to it will find a great deal in her book to admire; but far less of it will be related to the ancient Greeks than the title professes.Footnote ²⁰

By contrast, Hagel's book is soundly rational; it is also, as its author said in an email of 29 January 2010 to the Moisa mailing list for students of ancient music, ‘severely technical’, and ‘may be the ideal gift to your academic foes’. Indeed, it cannot be described as easy reading, the less so because staff notation is eschewed for a reason that will seem strange and anti-historical to readers of Early Music History: that ‘its visual focus on thirds obscures the inherent structures [of ancient music], which divide the octave mostly into fourths and fifths’ (p. xviii); a better, though not irrefragable reason would have been that from its medieval invention onwards staff notation has presupposed the diatonic genus, which did not prevail before Roman times.

The book is also aimed at readers with some knowledge of the subject; it will not suffice, to quote Martin West's accessibility statement in his Ancient Greek Music, ‘[to know] roughly what an octave is’. Hagel's intended reader already knows better than to give such names as Dorian, Phrygian and Lydian the meaning that Ciconia and Gaffurius and Glareanus gave them: they were not modes, beginning on D, E and F respectively, and exhibiting different octave species – though in Plato's day they had belonged to irregular modes or ἁρμονίαι set out by Aristeides Quintilian 1. 9 – but either keys, whose internal structure varied according to genus only and whose double octaves began (in the conventional transcription, to be discussed below) on B♭,Footnote ²¹c and d,Footnote ²² or octave species in the three genera with intervals corresponding to those ascending respectively from hypate meson, lichanos hypaton and parhypate hypaton, so that the diatonic Dorian species has the intervallic structure of our Phrygian and vice versa.Footnote ²³ Deceptively familiar too will be the Greek note-names: being accustomed by our medieval predecessors to equate mese with Alamire in acutis,Footnote ²⁴ we need to learn that the Greek name may be used not just positionally (κατὰ θέσιν) within the basic (Hypolydian) scale for A in acutis, but functionally (κατὰ δύναμιν) within the key, as la or mi or re or indeed any other syllable.Footnote ²⁵ Hagel does not make this point till later, and then not in those terms, since he transcribes mese in the basic scale as modern a (p. 6) while accepting the common opinion that it was closer to f or f♯ (p. 71).Footnote ²⁶ Readers to whom all this is familiar will profit greatly from his book, at least if they can read Greek; those in search of elementary instruction had better look elsewhere.

Hagel begins with ancient Greek notation, which (p. 2) ‘combines two ideas: that of the regular model scale (sýstēma), and those of keys (tónoi or trópoi)’. Several pages pass before this notation is set out, and even then in a table in which:

Another author might have seen fit to give an example, but reader-friendliness is evidently not an object; it is not even stated that the vocal notation is given in the leftmost column and the instrumental in the rightmost.

The use of Helmholtz notation in this book interposes another barrier to understanding. It is disconcerting enough to find on pp. 20–1 that two notes an octave apart are both labelled e; on p. 24 and elsewhere the notes below e′ and e are given as respectively d and D; even if this is because the lowest Greek notated pitch is E, it is still misleading. (The notation is correctly used in a modern context at pp. 221–2.) But on p. xviii we read: ‘When modern note names refer to absolute pitch and when they merely indicate relative pitch relations will become clear from the context.’ As we read further, we find that Hagel has very often equated hypate, mese and nete in the central Lydian octachord with e, a and e′ respectively instead of the conventional a, d′ and a′; at this point we suppose, bearing in mind repeated references to relative pitch, that he wishes to reproduce in his transcription the correct intervallic structure, which in diatonic ascends STTTSTT. However, at the very end of his book, on pp. 452–3, he offers different arguments: the conventional transcription was established by Bellermann when less was known about ancient music, it is acknowledged to be too high, and although it matches the fully developed notation, it wrongly privileges Hypolydian above Lydian, the central τόνος round which the notation evolved; therefore, we should equate a, not with the Hypolydian mese but with the Lydian mese ,Footnote ²⁷ with a true value not of f or f♯, but of b♭ or b. Even the disadvantage that the Hypolydian paramese must be represented as f♯ instead of b is punningly mitigated by the observation that this note, chromatic in the modern sense, is also chromatic in the ancient, being the Lydian chromatic lichanos meson or χρωματικἡ for short;Footnote ²⁸ unfortunately, Hagel does not tell us how to transcribe Hypolydian trite and paranete diezeugmenon in enharmonic. He has a case, though it hardly seems unreasonable to base transcriptions on the final form of the notation, rather than its beginnings, and performers will still have to transpose, even though upwards by a semitone or tone rather than downwards by a major or minor third; but however readers may judge his plea, for clarity's sake in this review I shall stand upon the ancient ways.

Hagel is convinced that the history of Greek notation, as well as of Greek music, is not to be understood without reference to Greek instruments, in particular the aulos; so much for Aristoxenus' abstractly intellectualising indifference to notation and instruments alike. In the very first chapter he argues that the notation was originally devised for the enharmonic, not the diatonic, genus, and specifically for the ancient Lydian ἁρμονία. The reader is puzzled why the scale in question has been transposed by a fourth between pp. 18 and 20; eventually it emerges on p. 34 that it is an intruder, the true Lydian being Aristeides' Syntonolydian, notated a fourth too low in his table.

Comparing the two Delphic Paeans, the First, by the singer Athenaios son of Athenaios, in the expected vocal notation and the Phrygian and Hyperphrygian τόνοι, and the Second, by the citharistFootnote ²⁹ Limenios son of Thoinos, in instrumental notation although texted and in the Lydian, Hypolydian and Hyperlydian τόνοι, Hagel not only explains the latter's notation as the preference of an instrumentalist, but also suggests that the primacy of Lydian and its side-keys was too firmly established in the instrumental signs, which distinguished uninflected from inflected pitches by their shape, to accommodate the notation by τόνος that the vocal signs, which put all pitches on a par, could more easily adopt.

At this point the reader will object that the hypothesis would be stronger if Limenios' tonal material had not been Lydian, for Hagel at this point is trying to prove, not that instrumentalists were reluctant to compose in other τόνοι, but that they did not care to notate in them. However, at the beginning of the next chapter Bellermann's Second Anonymous is cited to show that citharists did indeed confine themselves to Lydian until they added Iastian (B–b′); next, the aberrant teachings of Ptolemy and Porphyry on the τόνοι are analysed to show on the one hand that whereas Ptolemy's Dorian is what any normal person called Lydian, Porphyry took it to be Hypolydian, on the other that whereas nearly everything in the music known to have been current when Ptolemy wrote could be played in a Lydian enneachord from hyperhypateFootnote ³⁰ (g) to nete diezeugmenon (a′), by Porphyry's day the Hypolydian octachord e–e′ accounts nearly as well for the tonal material and better than Lydian for the actual melodies. Hagel passes then to absolute pitch, whose existence he is inclined to doubt except for the aulos; indeed, he contends that it was for the aulos that the system of τόνοι was elaborated.

On p. 71 Hagel is puzzled by the statement in the Third Anonymous (§94) that ‘The human voice is naturally measured by the interval of the triple octave’; he suggests that it refers to the entire range of human voices, not to any individual,Footnote ³¹ even though the Anonymous goes on to say that we compress the outer octaves and sing only the Lydian double octave in the middle,Footnote ³² because the lowest notes are indistinguishable and the highest hard to sing.Footnote ³³ All this would have surprised the Renaissance writer who expected the bass to have a range of twenty-two notes ‘con eguale tondezza di tuba’.Footnote ³⁴

Having contrasted the lyre, whose fifteen strings were played by the fingers and allowed no scope for modulation, with the plectrum-played modulating cithara, Hagel wrestles with Boethius' chapter on the successive addition of strings to the latter instrument with their δυνάμεις, conjecturally interpreted as diatonic; comparing it with a preserved fragment of the latter's source Nicomachus, he demonstrates that what was still familiar from practice in the second century ad was misunderstood in the sixth. He then attempts to establish from measurement of vase-paintings the actual pitch of the Lydian nete diezeugmenon, conventionally transcribed a′ like the pitch (misprinted a at p. 90, l. 9) whose frequency is now set at 440 Hz. Since (he argues) this lies just above the breaking-point for gut strings, and hence about a minor third above the highest usable note, the modern belief is confirmed that the true pitch was somewhere between f′ and f♯′.Footnote ³⁵ Perhaps it is rather the result that validates the method, but as Hagel points out, all his data except the vase-measurements have been based on other scholars' work, mostly published beforehand. Likewise, his findings on the primacy of the Lydian τόνος meshes well with West's emendation of the vocal registers listed by the Third Anonymous.

This primacy, as Hagel observes in a very short third chapter, was recognised by Boethius, whose system of eight τόνοι he finds to preserve one known to and scorned by Ptolemy, but also to cast light on the latter's working methods: basing himself exclusively on the seven octave species, he had reconciled lyre music with theory, but by departing from notation, which indeed he never uses, he had condemned himself to be misunderstood (p. 102). We have already seen Ptolemy misunderstood by Porphyry; in chapter 4 he has confused ‘most modern readers’ (p. 108) by his ‘positional’ and ‘functional’ terminology. Be that as it may, Hagel extracts from Ptolemy's tunings evidence for continuity of citharistic practice over several centuries, not least in the use of positional and not functional terminology; he is thus able to demonstrate that in Aristotle and the Problems mese is indeed the middle string of the lyre or cithara and not the middle pitch of the τόνος. He then turns to the table variously headed in manuscripts κοινὴ ὁρμασιά,Footnote ³⁶ understood to mean ‘common tuning’, or κατὰ κιθαρῳδίαν, ‘according to citharody’, which as Jan saw does indeed comprise instructions for tuning the cithara; the case is carefully proved, no doubt for the refutation of those who delight in doubting the obvious. His further examination, without solving all the problems that had defeated previous researchers, advances understanding far beyond what they had achieved.

In chapter 5, ‘Fine Tuning’, Hagel examines what ancient authors had to say on how the tetrachord was, or ought to be, divided. He remarks indeed on Aristoxenus' presumption of equal temperament (pp. 153–4, n. 43), but pays more attention to his conflation of ‘the versatile auletic pyknón and the rigid tunings of the lyre’ (p. 152) and his preference for that kind of enharmonic (truly beautiful and classical, but rarely practised in his degenerate day) which comprised two quartertones and an interval twice that between fourth and fifth – the ‘Pythagorean’ ditone.Footnote ³⁷ Pythagorean writers are then surveyed: the obsession with superparticularity documented in all its mathematical ingenuity and practical unreality. As elsewhere, Hagel emphasises that the aulos and not the lyre or cithara is often enough the instrument of reference; it was Ptolemy who concentrated on strings to the exclusion of wind. His various tunings are examined along with their alleged experimental derivation, which in two cases is shown to have been guided by the desired results, though we learn in the next chapter that they seem to be correct.

One theoretical statement, however, is missing, as it is in every other book I have read on this subject; it was made by the great Jewish Platonist Philo of Alexandria, expatiating on the number seven, that 6 : 1, 5 : 2, and 4 : 3 are all of musical significance (On Creation 96). The last of these we know; the first, he says, gives the distance from the lowest to the highest note, which he thus agrees with Aristoxenus in making two octaves and a fifth;Footnote ³⁸ but the second ratio, which generates the major tenth,Footnote ³⁹ ‘displays the greatest power in music (ἁρμονία), almost rivalling the octave, as is very clearly set out in music theory (ἁρμονική)’.

In chapter 6, ‘Going beyond Ptolemy?’, Hagel ingeniously explains Ptolemy's diatonic tunings, which exclude our familiar thirds, as using a different segment of the harmonic series from that underlying our major and minor tonalities. He then attempts to locate their focal pitches, first a priori, then from the evidence of actual music; his theories are better borne out in those pertaining to Lydian and Iastian (indicating that Ptolemy's septimal thirds were indeed in use), but not the less favoured Hypolydian and Hyperiastian. (No support whatever is found for Didymus' diatonic, with its pure thirds.) Further investigation suggests that by that time, in the tuning known as λύδια, the Lydian framework had shifted a tone downwards, from hypate, mese, paramese and nete (a, d′, e′, a′)Footnote ⁴⁰ to hyperhypate, diatonos, mese and synemmeneFootnote ⁴¹ (g, c′, d′, g′), with a resonant third between diatonos and paramese; that is to say, the disjunctive tone between mese and paramese had, in defiance of Pythagoras, Aristoxenus and right reason, been reduced to 10 : 9. Ptolemy discreetly looked the other way.

In consequence, the fifth from hypate to paramese fell out of use in Lydian compositions; in the Mesomedean hymns it is found only in the invocation to the Muse, which may be earlier, and between vv. 10 and 11 in the hymn to Nemesis (Hagel wrongly says Helios), but between verses ‘even a (clearly dissonant) tritone is found’ (Hymn to Helios, vv. 15–16). Readers of Early Music History will agree that tritones are clearly dissonant, but then they will say the same of fourths; Gaudentius, however (§8, p. 338, ll. 3–7) classed the tritone with the third from diatonos to paramese as ‘between consonant and dissonant, but appearing consonant in the accompaniment’,Footnote ⁴² and Aristoxenus claims that Olympus found beauty in the ethos of a fall from paramese to diatonic parhypate. Just such a descending tritone, in Hypolydian, from b to f occurs in Limenios’ Paean, at the very point when the Maenads bang down their thyrsi; the effect is marvellously imitative, but it is not clear what ethos Aristoxenus had in mind.

After a brief chapter on resonators, which finds the Vitruvian theatre design to suit ‘the general requirements of Roman Imperial music’ (p. 255), even if not necessarily the citharodic refinements previously discussed, Hagel turns in chapter 8 to the extant musical documents, with special attention to scalar and modal questions. In discussing the settings from Iphigenia in Aulis, while uncertain whether anything at all can be read, he discounts the Hyperaeolian interpretation on the ground that that τόνος (and proposed alternatives) did not exist in Euripides' day, or even Aristoxenus', but was created (he assumes) long after the papyrus; he therefore falls back on a theory of extreme modulation, only to dismiss it as unlikely. In other words, we are left nowhere.

We may readily grant that no τόνος was called Hyperaeolian even when the papyrus was written in the middle of the third century bc, but that is not the point: was there really none that used the notes that Pöhlmann and West claim to read? If scholars are wrong to doubt whether those passages of the play were written in 405 bc,Footnote ⁴³ then since on Hagel's own showing (p. 392) Aristeides' list of ἁρμονίαι is not complete, but presents only those discussed by Plato, it is necessary to show that those notes could not have been used in any scale known in the late fifth century; if they were later additions, the same applies to music (perhaps too avant-garde for Aristoxenus) composed in 340 (the year in which the actor Neoptolemus revived one of Euripides' Iphigenia plays) or at least before c. 250.Footnote ⁴⁴

Next comes chapter 9, on aulos types and pitches. Examination of vase-painting showing male auletes (the exclusion of female players is not explained) suggests a median pitch of f♯, a quartertone or so higher than the hypate of the cithara as determined by the same method; moreover, extant auloi are said to furnish a central Lydian octave matching that of the cithara, from conventional a to a′ and actual f♯ to f♯′. (It is actual pitch-equivalents that are used for the diagram on p. 331; but on p. 339 we are back to Hagel's reformed transcription.) The historical development of the aulos is then traced from classical Greek through Hellenistic to Roman times; the most important development was the invention of the modulating aulos on which Pronomos could play Dorian, Phrygian or Lydian at will. The belief that the accompaniment was always higher than the melody is exploded. Finally, Hagel remarks that the common understanding of the Aquincum water-organ is archaeologically insecure, ‘even though the presumed ranges are in best accord with the rest of the evidence’ (p. 364); arguably then, although it does not confirm his findings, it is confirmed by them.

In chapter 10, ‘Before Aristoxenus’, Hagel begins by revising his own reconstruction of notational history, suggesting that the system began with functional, not positional values, and that the Lydian transcription of the old ἁρμονίαι that we find in Aristeides Quintilian is due to a later renotation. He then turns to pre-Aristoxenian harmonic theory, attempting to reconstruct the diagrams that reportedly mapped the entire melodic space without their enharmonically fixated creators' realising the fact. Detailed discussion of the two divergent sequences proposed by Aristoxenus' predecessors for the old ἁρμονίαι finds them to have been arranged by the location of the functional mese, which differed according as the disjunctive tone in Mixolydian was in the middle, as in the older view, or on top, as Lamprocles had stated,Footnote ⁴⁵ and furthermore to imply a πυκνόν of respectively two quartertones and one three-quartertone, the latter disposition being in Aristoxenus' view devised for the aulos. It is this second system that Hagel takes to be the earlier, despite the novel analysis of Mixolydian, on which the first system retained or revived the older theory; indeed, he maintains that the semitone πυκνόν beloved of Aristoxenus was (had he but known) an auletic innovation made shortly before his birth, as the precondition of that very New Musical modulatory exuberance which he deplored.Footnote ⁴⁶ Few other ears ever loved the resultant ditone, favouring instead the new fashion for the chromatic genus.

Likewise, Hagel rebuts the common opinion that the undivided semitone antedated the division into quartertones, pointing out the difficulties in Aristoxenus' account of Olympus' invention.Footnote ⁴⁷ Accordingly, having demolished the notion of early Greek pentatony in its anhemitonic form, he casts serious doubt (not least because diatony already existed in the Near East) on the hemitonic variety; he sees enharmonic trichords as pseudarchaic, based either directly or more probably through theoretical writings on the single case of Olympos' auletic σπονδεῖον.Footnote ⁴⁸ Hence, although readers impatient with musical moralism will have relished the remark (p. 406, n. 98): ‘The technical advancements especially of the aulos might have obscured from a late-fourth-century mind how often the cherished self-restriction of old composers was in fact mere physical necessity’, this may not have been the case on the cithara (p. 441); however, melodic restraint, as Hagel does not note, was compensated by the rhythmical variety and inventiveness of classical lyric poetry. Nevertheless, improvements to instruments came at a price, as Hagel observes in the historical overview that concludes his book: no longer could the amateur entertain a party with a tolerable imitation of professional performance.

There are many more riches in this book, which cannot be neglected by any serious student of the subject. It is also handsomely produced, with an attractive Greek music fount of Hagel's own devising. Our understanding is much enlarged by his work.Footnote ⁴⁹