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Re-Stringing the Daphnē Harp?

In: Greek and Roman Musical Studies
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Stefan Hagel Austrian Academy of Sciences Vienna Austria
University of Vienna Vienna Austria

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Abstract

D. Najock has criticised a tentatively proposed diatonic-chromatic tuning for the triangular frame harp found at Daphnē and suggested a different enharmonic-diatonic stringing. While his criticism may well be justified, it is argued that his alternative proposal is physically problematic and historically implausible. A possible compromise is discussed.

After the remains of a harp had been discovered in an Athenian grave from the Classical period and the structure of the instrument had been reconstructed by Chrḗstos Terzḗs, I discussed the question of how it might have been strung and tuned.1 I argued that the geometry of the instrument rules out some possibilities such as a continuous diatonic or chromatic scale in the modern sense and tentatively proposed a comparatively slack modulating tuning which replicates the tonality of contemporary lyres in the lower range while doubling and partially tripling the same pitches in the higher octaves. A few years later, this suggestion faced a powerful attack by Dietmar Najock:2 only a significantly higher tuning would yield a satisfactory sound. Najock argued mostly on the basis of modern harp lore, combining string physics with the rules of thumb of modern instrument makers, his experience with early modern harps and his inspection of a reconstruction of an ancient Egyptian instrument.

There is more at stake than Najock gave away. If his particular tuning proposition is correct, the reconstructed late Classical system of harmoníai is not, as will become clear below. But we need to address the differences as two separate questions: on the one hand, the range and general pitch of the tuning, and on the other, the intervallic steps into which this range was articulated.

The aim of this contribution is, above all, to clarify the differences between the two approaches, and make explicit the assumptions which they share and on which they disagree. In this way, it is hoped, the reader may find it easier to pass judgement on the merits and disadvantages of each. Currently this task is needlessly encumbered because Najock and I not only failed to agree about at least one fundamental notion, we also talked in different languages. Whereas Najock, in pursuit of delicate sound, focuses on various string diameters and consequently defines the boundaries of the physically possible in terms of ‘breaking loads’, my fundamental concept was that of ‘breaking pitch’; while I define pitch in terms of frequencies and their meaning in terms of ancient music and its notation, Najock liberally employs modern note names. Similar note names, in the context of my argument, however, designated relative pitches, related to a tuning standard of Lydian mésē = ‘a’ = 490 Hz/2. Najock, in contrast, implicitly appears to talk in terms of 20th-century a′ = 440 Hz. Such a lack of compatible conceptual frameworks is all too well known in ancient musical studies, and it is extremely tedious to resolve.

I will therefore try to clarify the pitch ranges in question. My original proposal placed the lowest string at about 160 Hz, which I transcribe as ‘D’ relative to Lydian mésē as ‘a’ a fifth higher, while modern music would find the same pitch between its D3 and its E3 – a difference of three quartertones, as long as we don’t quibble about how to notate different octaves. Najock, in contrast, suggests “a pitch near modern d′” (10). His argument leaves no doubt that this means indeed D4 in scientific notation, at a frequency of 293.7 Hz. The difference is enormous, just three quartertones short of a full octave. Evidently at least one of our two divergent approaches must be way off the mark.

Things get slightly murkier when Najock addresses the relation of his pitches with those of ancient music. In his proposed tuning, the lowest string sounds a whole tone below Hypolydian hypátē mesôn , which he transcribes as relative ‘e’ “without respect to real (modern) pitch or to historical continuity” (15 n. 41). The bass note thus becomes relative ‘d’, and Najock is pleased that this gives the same letter as the absolute d′ he had derived as its optimal pitch: “this method yields a system just an octave below the real pitch of the harp, certainly a didactic advantage” (ibid.). But what about real (ancient) pitch? According to all available sources, Hypolydian relative ‘d’, i.e. the note , had a pitch of approximately 123 Hz. Najock’s ‘Hypolydian’ places the same note at 146.8 Hz, a minor third higher. One might discount this significant difference on the grounds that the evidence for ancient pitch largely stems from a later period. This would however make the choice of Hypolydian arbitrary, unless ‘Hypolydian’ notation could serve as a general scale that was used without regard to an absolute pitch. Najock appears to hint at such a stance when referring to “the Hypolydian scale, which was often used for didactic purposes” (14 n. 41). No further reference is given, and I am afraid I cannot see which passages in which ancient musical authors Najock may have had in mind. Certainly Bellermann’s Anonymi, whose current standard edition was prepared by Najock himself, uses the Lydian scale instead, and so do all other sources I am aware of.3

But why fuss about Hypolydian at all, if absolute pitch is not of interest? After all, the keys (tónoi) of the developed system of ancient Greek notation differ merely by pitch. Actually the Hypolydian key enters the argument through the way in which Aristides Quintilianus chose to notate the scale structures of the ancient ‘harmoníai’, which Najock implements in his tuning proposal. All the notes here may be regarded as taken from the Hypolydian, if one allows for a mixture of enharmonic and diatonic as well as the inclusion of the modulating synēmménon tetrachord. Such a purely Hypolydian interpretation is certainly correct from a rigidly technical viewpoint; but it is doubtful whether it does justice to all the scales in question. In particular, an exclusively Hypolydian viewpoint attributes the modulating tetrachord to the Dorian and Phrygian harmonía. This tetrachord is however part of the regular Lydian key. Taken on their own, one would inevitably read the said two harmoníai as notated in the Lydian key, where the functions of the notes correctly reflect the internal structure of the two harmoníai (cf. Figure 1). Much the same is true of the Mixolydian harmonía, and ultimately all of Aristides’ harmoníai might just as well be regarded as notated in the Lydian key, with the sole exception of – confusingly – the Lydian harmonía, which features distinctly Hypolydian notes. The putative reason for this stunning mismatch needs not concern us here, nor is there any reason to resolve the question of Aristides’ own view about the involved keys for our present purposes. In fact, all this is relevant mostly for the side issue of how best to transcribe the ancient notes with modern note letters: I would prefer to label the Lydian mésē as ‘a’ instead of the Hypolydian mésē .

Figure 1
Figure 1

The notation of the harmoníai in Aristides Quintilianus 1.9, pp. 19–20 Winnintgon-Ingram

Citation: Greek and Roman Musical Studies 10, 1 (2022) ; 10.1163/22129758-bja10041

Another issue, however, is of the greatest consequence: Najock’s proposal depends on the assumption that Aristides’ way of notating the harmoníai does not merely illustrate the intervallic structure of each of them individually, but also reflects their original interrelation in terms of pitch (cf. Figure 1). This notion tacitly contradicts my reconstruction of the historical embedding of these harmoníai within a modulating scheme, related to the endeavour of creating modulating versions of the aulos.4 I had originally proposed it solely on the basis of the literary testimonies, but later found it confirmed, inter alia, by the extant melody fragments from the Hellenistic period; meanwhile hard archaeological evidence has been surfacing as well.5 Of course I cannot expect that Najock accepts this theory; but, as far as I know, the basic concept has not been challenged in over twenty years since its first publication. As long as nobody has proposed another explanation for the various strands of concurring evidence, it should perhaps not be neglected without any discussion. If the theory is accepted, on the other hand, the harmoníai were originally aligned at their upper ends, quite unlike their apparent relation in Aristides. This would remove the basis for Najock’s proposal, which simply adds up all the notes that are found in Aristides’ lists (but, as we have seen above, does not interpret them in terms of historical pitch) and adjoins a whole tone at the lower end without further justification (“may be added” 15).

This fundamental divergence about how to interpret the evidence from the ancient harmoníai makes minor differences moot. I will nevertheless mention one point, because it is of broader concern and affects other possible reconstructions of the harp’s tuning. Following the transmitted structures of the ancient scales, Najock embraces an enharmonic tuning, i.e. one including triplets of adjacent strings that are separated by quartertones, reckoning that “before Agathon’s victory in 416 BC only diatonic plus enharmonic were used in tragedy and that these were probably the dominant genera in the time of the Daphnē harp” (14). There can be little doubt that this is correct for the kind of music accompanied by the aulos. However, the cithara is never associated with the enharmonic, but with the chromatic and its evolution, which Aristoxenus supposes to predate tragedy.6 Such an organological division of the genera, building on a common ground of diatonic, appears natural enough, because the enharmonic was evidently produced by partially opening finger holes on the wind instrument, while the chromatic emerges naturally by extending the basic tuning procedure of an instrument with open strings, such as the lyre or the harp, beyond the completion of a diatonic octave: a fourth beneath paramésē b’, one obtains khrōmatikḗ mesônf ♯’, a note that seems to have been presupposed by Archytas,7 who may have been born around the time when the Daphnē harpist died; a fifth above again, khrōmatikḗ diezeugménōn c♯’ is established.8 A priori, we would therefore expect a closer connection between the harp and the cithara, not the aulos, and consequently a chromatic rather than enharmonic tuning.9 Thanks to its greater coherence in terms of concords, a chromatic tuning would furthermore benefit the internal sympathetic resonance of any instrument with open strings.

So much for the complicated and inevitably disputable issues of ancient music-theoretical sources and their potential relation to the musical practice of the fifth century. Let us now turn to physics, which might hopefully provide firmer ground, and the question of musical acceptability, which might not. The latter is pivotal in assessing the lowest possible pitch for each of the 26 strings involved; the former, in turn, sets the upper limits. Very slack strings have two disadvantages: they sound dull, and they produce a more noticeable pitch shift when plucked, from an initial higher pitch down to a longer-lasting, slightly lower sound. The latter may be a welcome effect in certain musical cultures; but we have no reason to assume that an ancient Greek would have subscribed to one of these. Quite to the contrary, ancient theorists consistently defined a musical note as a stable pitch.10 But how dull is too dull and how much pitch shift would have been considered too much? While I had relied on published formulae that take into account plucking position, initial displacement of the string at the plucking position and a maximally allowed shift that I defined as an eighth of a tone,11 Najock employs a general rule of not slackening any string below 10% of its breaking tension (9). With suitable strings, he finds, my proposal may be just acceptable; but he points out that it is close to the lower limit throughout, falling short of the tone quality that might be achieved with tauter stringing and consequently a higher pitch.

On a general level, one might express doubts whether rules that derive from experience with curved harps and wooden soundboards necessarily apply to instruments where the strings are connected to a straight resonator via a soundboard made of stretched hide; but such doubts may be voiced regarding both approaches. Najock has even gone to some lengths to possibly accommodate my low values, against the stricter limits normally respected by modern makers. Indeed I suspect he might have been too lenient in this regard: when expressing the minimal allowed tensile forces as a percentage of the breaking point, it does not seem to make sense to apply the same rule to strings of different quality. If some gut strings of similar twist snap more easily than do others, I doubt that these weaker strings would consequently sound better in the bass region; but this is apparently what Najock’s application of a fixed percentage implies. In any case, if Najock is right on this point, our disagreement about the lower limits is comparatively small.

On balance, a lower limit is always vague. I have tested my bass string tension on a tortoise-shell lyre with similar string length and diameter and found the sound adequate; but it may have been far from acceptable in Classical Athens. As far as I know, neither Najock nor I have presented replicas of the Daphnē instrument with our proposed tunings to a modern audience – nor would this necessarily settle the question.

On the other hand, strings of a given length and material also have an upper limit, defined by the breaking pitch. Fortunately, this pitch limit does not depend on the thickness of a string. As Najock correctly insists, we only know the upper limit of the string diameter used on the Daphnē harp, which is defined by the 0.8 mm holes in the string holder. Najock dismisses my argument for equal diameters of ancient harp strings, which was based on a literary source,12 as inconclusive, because I had implicitly presumed that the term tásis, ‘tension’, would mean tensile force and definitely not “tensile force divided by cross-sectional area” (5). I must admit that I fail to see how a Peripatetic author would have come up with the latter notion, which I think has little to no ground in the physical experience of tuning the instrument and was certainly not accessible to ancient experimentation.13 In contrast, if anybody in anti-quity really cared, it would have been straightforward to hang similar weights from a couple of suitable strings and determine the actual truth of what the ancient author states as a truth. Nonetheless, Najock is certainly right that thinner strings towards the treble region would sound better, even though it is difficult to assess whether the ancients would have exploited the advantages (as they did on lyres).

However, we need not resolve the question of string diameter here, because it does not affect the upper pitch limits and consequently has no bearing on the playable range (after all, there is agreement on the bass strings being of 0.8 mm diameter or only slightly thinner). So what is the breaking pitch of such gut strings? Curiously, this seemingly simple question leads us to the paramount difference between Najock’s solution and mine. The physical properties of sheep gut, one might assume, would stay quite constant over time, and so would the optimal techniques of twisting these to uniform, flexible as well as robust, if delicate, strings. Consequently, as Najock’s figures show, he works with more or less modern values, assuming that a normal (i.e. low-to-medium-twist) string of 1m length breaks at approximately 250 Hz or even more.14 In contrast, I have assumed a significantly lower value, following historical data from the writings of Marin Mersenne (1588–1648) before the establishment of chemical treatment:

The strength we calculate for his gut is less than that of modern gut. This is to be expected, since the sulphur treatment of gut which cross-links the collagen molecules, thus conferring greater strength, seems to have been a more recent innovation. […] It is questionable whether Mersenne used […] a selected and expensive string for the experiment he reported. If he used an inferior string the strength given here is an underestimate.

Abbott and Segerman 1974, 49

The respective breaking index for the gut string Mersenne had used amounts to merely 185 mHz, while Abbott and Segerman give an extraordinarily high value for modern strings of 275–295 mHz (58, Table 1). The former is the figure I have used, and it is more than a fourth lower than Najock’s. My reason for settling on potentially not a first-class string from Mersenne’s time was originally grounded in the observation that cithara and lyra strings were apparently of similar length, so that the two types of lyre almost certainly played in the same range. This would have meant that suitable strings must have been available for anybody owning a lyre, not only for a few professional players, leading to a substantial demand in cities where lyre-playing belonged to basic education. Consequently I concluded that the proper range must have been realisable with lower-quality material as well.15

My original considerations then appeared corroborated when I interpreted the inferred vibrating lengths of ancient lyre strings in terms of Mersenne’s value: the obtained range coincides precisely with the pitch standard that was deduced from the vocal scores as well as from the design of woodwind instruments in the archaeological record.16 If these calculations hold, there would be little reason to assume that first-quality gut was instead used throughout for harps, which enjoyed comparatively smaller reputation but required more string material.17 Moreover, on a lyre it would suffice to select one or two strings of excellent material for the treble strings, whereas on a triangular harp of the Daphnē design, at least seven strings would suffer similar top stresses.

Najock’s breaking pitch is closer to a well-known standard that was deduced from data provided by Mersenne’s contemporary Michael Praetorius (1571–1621). If described in the most convenient way as the product of string length and frequency, which may for our purpose be regarded as a constant, Praetorius’ highest usable tensions appear to measure around f L = 210 mHz.18 These would have been tuned at least a tone below breaking pitch, which thus cannot have been lower than 235 mHz; assuming a safer minor third would take us to Najock’s 250 mHz.

However, we ought to bear in mind that, among Praetorius’ instruments, such high values are only obtained for lutes, both plucked and bowed.19 These have comparatively few strings, all of which are typically of the same length, so that the required extreme stress resistance concerns only a single string (or a single course), which to select carefully as well as replace regularly did not involve excessive inconvenience or costs. The corresponding values for the many-stringed harps are much lower (as are those of other lutes), between 105/122 (Groß Doppel-Harff) and 171 mHz (Gemeine Harff). This is just in the range that is also suggested by Mersenne’s string. Since a large part of the pitch differences on a harp (especially a curved harp) is effected by different string lengths, string tensions are more even on these instruments, and particularly high tensions would consequently affect several strings, which would all have needed regular replacement – lest any of them should snap during performance.

Similarly, Praetorius’ instruments give us an idea about the lowest notes that his contemporaries found acceptable on gut strings of a given length. For a few lutes, the respective values even drop below 40 mHz. For the mentioned harps, they have been calculated as 57 and 73 mHz, respectively.

In comparison, my proposed tuning for the Daphnē harp moves between 66 mHz for the bass string and 109 mHz in part of the upper half (dropping once more to 97 mHz on the treble strings), which appears well in line with the standards of Praetorius’ period. Najock’s tensions, in contrast, appear too high for a harp (cf. Figure 2).

Figure 2
Figure 2

Ranges of f L product in historical harps, ancient standard-size lyres (kithára, tortoise-shell lýra) including hyperypátē () and Daphnē tuning proposals

Citation: Greek and Roman Musical Studies 10, 1 (2022) ; 10.1163/22129758-bja10041

Indeed, strings of Mersenne quality would appear unable to support Najock’s proposed tuning range. As shown by the broken grey line in Figure 3 and Figure 4, about seven strings in the higher range would be expected to snap outright before they reach the required pitch; others, being stretched beyond the customary upper limit of a minor third below breaking, would hardly survive being plucked.

Figure 3
Figure 3

Najock’s proposed pitches on the Daphnē harp in relation to the respective breaking pitches of Mersenne’s gut string

Citation: Greek and Roman Musical Studies 10, 1 (2022) ; 10.1163/22129758-bja10041

Figure 4
Figure 4

Percentages of Mersenne-string breaking pitch for each of Najock’s proposed notes on the Daphnē harp

Citation: Greek and Roman Musical Studies 10, 1 (2022) ; 10.1163/22129758-bja10041

Would such strings perhaps support Najock’s concept of a ‘Hypolydian’ range instead? After all, if his proposed lowest note is not interpreted as a modern d′, but, in accord with Najock’s reasoning, as the octave of the pitch a whole tone beneath Hypolydian hypátē mesôn, it emerges as the octave of , a note that also forms the Lydian proslambanómenos and as such becomes a quite plausible candidate for a bass note. However, even such a downward-‘corrected’ tuning may in some cases become too high for Mersenne-type strings, as transpires from the continuous grey lines in the figures. On the other hand, it would be compatible with the harp strings used on the instruments described by Praetorius (Figure 2, rightmost column). From a purely physical viewpoint, therefore, a range starting from the bottom of the Lydian double octave is certainly viable.

Here we have arrived at the heart of the disagreement. Najock maintains that my tuning would not sound well enough, and that we would expect ancient instrument makers to take better advantage of their material. I concede this possibility, though similar objections might then be raised against some early Baroque instruments. On the other hand, I argue that Najock’s proposal would inconvenience the musician by frequently breaking strings, unless first-class material is used throughout – better material, at any rate, than Mersenne had at hand. Do I overlook new insights concerning historical gut strings? Unfortunately Najock does not explicitly correct or even discuss my diverging breaking pitch. Ultimately this part of our dispute thus boils down to potentially conflicting assumptions about the minimal quality of fifth-century BC strings: did the harpists of Classical Greece have recourse to invariably better material than did those of the early seventeenth century? Or might strings like Mersenne’s suffice for the sweet polychordal instrument of antiquity? My experience with cithara and lyra stringing suggests they might, since such strings not only predict the required pitch ranges accurately but also produce usable instruments when the obtained ranges are implemented in practice.20

If it is accepted that Najocks proposition is physically problematic, on the one hand, because it puts a strain on the strings that significantly exceeds that known for early modern harps, and historically implausible, on the other, because it mistakes the Roman-period exempli-gratia notation of the harmoníai for their original context, the question of harp tuning is open again. Najock effectively agrees with me that some kind of scale is required that has more than seven but less than twelve notes to the octave, and we have both settled on ten as an optimal value. If the auletic harmoníai cannot give us those ten, my chromatic-tetrachords solution may still be considered as most closely matching the available evidence regarding strings in the Classical Period. These may of course start at a higher pitch than I have originally proposed, in order to achieve a better sound. Mersenne’s data appear to allow tuning a similar sequence up to about 550 cents higher, i.e. more than a fourth. This would place the bass string somewhere between Lydian hyperypátē (my original suggestion) and Lydian diátonos mesôn , which is also Phrygian mésē. The evidence from Praetorius suggests we might even go a tone higher, up to Lydian mésē . From our limited understanding of ancient music-making we can hardly pinpoint a universally convincing solution. One might consider suggesting octave-doubling either the Dorian, Phrygian or Lydian Unmodulating System, starting an octave above the respective proslambanómenos: at a pitch comfortably in the middle of the plausible range in the case of the Dorian (about 195 Hz), at its upper end for Lydian (about 245 Hz), or in between for Phrygian (about 219 Hz). However, I regard it as highly unlikely that the two-octave Unmodulating System was conceived earlier than the triangular harp, let alone its application to a system of interrelated keys whose development may only just have started when our instrument was interred. My personal image of the Greek musical world at the start of the second half of the fifth century is dominated by a diversity of lyre tunings (but not an exuberant one) including modulation, which had engendered the emergent chromatic, and various kinds of auloi playing a mixture of diatonic and (a relatively wide) enharmonic. Consequently I still somehow fancy my original lyre-based interpretation, if it turns out to be in any way feasible. After all, the harp’s bass string length is just so seductively similar to that of the lyre strings. On the other hand, bass lyre strings were probably thicker than the harp’s maximum of 0.8mm. If the sound of such a thin string appears unacceptable at lyre pitch, a Lydian Perfect System at the higher octave might be a promising competitor. We would not even need to presume that the two-octave system had already been developed at the period in question. Instead, we would posit an instrument that could play together with the lyre, play music that was also played on the lyre, and/or accompany songs that were also accompanied on the lyre. Its principal range would then lie an octave above the lyre, naturally making it the women’s instrument that it appears to have been. In the fifth century, such a tuning would not yet have been associated with the key name ‘Lydian’. It is only the proposed modulating scale, combining the lyre’s parypátē and khrōmatikḗ, which makes its pitches predominantly ‘Lydian’, though only in hindsight, from the viewpoint of the developed system of modulating keys. In order to maintain a range of two octaves and a fifth with 26 strings it would be required not to include a modulating b♭ in the bass region, in conformity with the structure of the Perfect System that has this note only in its central octave (cf. Figure 5).

Figure 5
Figure 5

An alternative higher tuning for the Daphnē harp

Citation: Greek and Roman Musical Studies 10, 1 (2022) ; 10.1163/22129758-bja10041

The very nature of the harp would thus have encouraged its original designers to extend the lyre range not only upwards, enabling the famous playing in octave parallels,21 but also downwards, quite reasonably right to the note an octave beyond pivotal mésē. In this way, the triangular harp would not at all depend on the conception of the Perfect System; contrarily, under the present hypothesis the latter might have been conceived with the experience of harps in mind. At present, however, this is only one of at least two hypotheses with some claim to historical informedness.

Acknowledgements

This publication is based on research funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 787522). The views presented here, however, reflect only those of the author; the ERCEA is not responsible for any use that may be made of the information contained.

Bibliography

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  • Terzḗs, C. (2013). The Daphnē Harp. GRMS 1, pp. 123149. DOI: https://doi.org/10.1163/22129758-12341240.

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1

Hagel 2013; Terzḗs 2013.

2

Najock 2015.

3

Cf. Hagel 2009, 97–102. It is true that the notation tables in Gaudentius preserve only Hypolydian, Hyperlydian, Aeolian and part of Hypoaeolian, but this is an accident of the scribal tradition, and I do not think anyone believes that the tables did not originally start with Lydian, precisely as do those of Alypius.

4

Hagel 2000, 165–90; Hagel 2009b; Hagel 2009a, 366–93.

5

Cf. Terzḗs and Hagel 2022 in this volume.

6

Aristox. ap. [Plut.] Mus. 1137d–e: τῷ γὰρ χρωματικῷ γένει […] τραγῳδία μὲν οὐδέπω καὶ τήμερον κέχρηται, κιθάρα δέ, πολλαῖς γενεαῖς πρεσβυτέρα τραγῳδίας οὖσα, ἐξ ἀρχῆς ἐχρήσατο “tragedy makes no use of the chromatic genus […] even today, while the kithára, being many generations older than tragedy, has been using it from the beginning”. Cf. Philochorus ap. Ath. 638a, on Lysander of Sicyon: χρώματά τε εὔχροα πρῶτος ἐκιθάρισε “he was the first to play well-coloured colours on the kithára”. The close relation between diatonic and chromatic emerges from PHibeh 13; cf. Hagel 2009, 446–8.

7

Ptol. Harm. 1.13, 31.2–6 Düring.

8

In addition, a chromatic structure was established whenever a tuning incorporated synēmménon modulation, which introduced ‘b♭’ along mésēa’ and paramésēb’.

9

Najock’s most intriguing argument for an enharmonic harp – that Euripides reportedly owned some kind of plucked instrument and that this instrument would have had to be enharmonic if he used it in the process of composing the music for his tragedies (14 n. 40) – faces the problem that the psaltḗrion named in the source (Schol. Eurip., vita 5 Schwartz) may well have been a different instrument; the Daphnē harp was more probably called a trígōnos.

10

Various wordings go back to Aristoxenus’ definition of pitch (tásis: Harm. 1.12, 17.2–4 Da Rios: τοιοῦτον οἷον μονή τις καὶ στάσις τῆς φωνῆς; 1.13, 18.3: ἐρημίαν φωνῆς) and note (phthóngos: 20.16–19: φωνῆς πτῶσις ἐπὶ μίαν τάσιν ὁ φθόγγος ἐστί).

11

Abbott and Segerman 1974.

12

[Aristot.] Pr. 19.23.

13

Najock is aware of the problem: “This is not to say that the author of the Problems used the term epitasis in the modern technical sense of tension, but probably the concepts of tension and tensile force were not clearly separated” (5). But can one eat their cookies and have them? Even in order to not distinguish a Newtonian idea of tension one needs to have developed some respective concept (albeit unconscious); in the present case it must furthermore be explicit enough to warrant the assertion of ‘equal tásis’, in the first place.

14

A value of 250 mHz is most straightforwardly derived from the remark “a string of normal material with a diameter of 0.40 mm and a length of 24 cm at its breaking tension (i.e. stretched with c. 44 N) would produce the note c′″” (8): c′″ corresponds to a frequency of 1046.5 Hz; 1046.5 Hz × 0.24 m = 251.16 mHz.

15

Hagel 2009a 89f.

16

Hagel 2009b, 90–4; Hagel 2012, 105; Hagel 2019, 185.

17

The Daphnē harp features 6.7 m of vibrating string length, plus 26 times the additional lengths used for fastening these at both ends. A contemporary seven-stringed lyre needed about 3 m plus 7 times somewhat larger additional lengths. The amount of required gut additionally depends on string diameters, which presumably were on average larger on the lyre.

18

Segerman 1998; Segerman 1999; Segerman 2011.

19

Segerman 1998, 37; Segerman 1999, 58, Table C: Chorlaute: 211; Paduanische Theorba: 209; Mandörgen (“Mandoraen”): 208. Bas-Geig de bracio: 207 (233); Viola bastarda: 209; Italianische Lyra de bracio: 215; Italianische Lyra de Gamba 220.

20

This reasoning must hold a fortiori even though I necessarily use modern commercial strings: since such modern strings are stronger, Najock’s method implies that their lowest usable note would also be higher. If my bass lyre notes are still acceptable, they would need to sound even better with strings of the Mersenne type.

21

Cf. Barker 1988.

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