H
V
Ru
Image: sig4
Image: sig5
Image: sig7
εἴδη συμφωνιῶν ἐν τῷ τελείῳ συστήματι ἕξ ςʹ διὰ τεσσάρων δʹ διὰ πέντε εʹ διὰ πασῶν διὰ πασῶν καὶ διὰ τεσσάρων δʹ διὰ πασῶν καὶ διὰ πέντε εʹ δὶς διὰ πασῶν ὅτι τρεῖς τόποι τρόποι φωνῆς ὀξύς μέσος βαρύς ὅτι πάθη τῆς μελῳδίας τέσσαρα δʹ ἄνεσις ἐπίτασις μονή στάσις Ἄνεσίς ἐστι κίνησις μελῶν ἀπὸ τοῦ ὀξυτέρου φθόγγου ἐπὶ τὸ βαρύτερον ἐπίτασις δέ ἐστι κίνησις μελῶν ἀπὸ τοῦ βαρυτέρου φθόγγου ἐπὶ τὸ ὀξύτερον μονὴ δέ ἐστιν ὅταν ὅτ᾽ ἂν ἐπὶ τοῦ αὐτοῦ φθόγγου πλείονες λέξεις μελῳδῶνται μελωδοῦνται στάσις δέ ἐστιν ὕπαρξις ἐμμελοῦς φθόγγου οἱ τοὺς τρεῖς τρόπους τόπους ᾄδοντες ᾄδουσι λύδιον φρύγιον δώριον οἱ δὲ τοὺς ἑπτά μιξολύδιον λύδιον φρύγιον φρύγιον, λύδιον, δώριον ὑπολύδιον ὑποφρύγιον ὑποφρύγιον, ὑπολύδιον, ὑποδώριον ὑπερφρύγιον ὑπερλύδιον ὑπερδώριον μιξολύδιον ἢ μιξολύδιον]. μιζολύδιον Τούτων ὁ μιξολύδιος ὀξύτερος ὀξύτατος τούτου ἐχόμενος ὁ λύδιος ποσῶς exp. βαρύτερος ἡμιτονίῳ τοῦ δὲ λυδίου βαρύτερος ὁ φρύγιος τόνῳ τοῦ δὲ μιξολυδίου τριημιτονίῳ τριημιτόνῳ βαρύτερος δὲ ὁ δώριος τοῦ μὲν φρυγίου τόνῳ τοῦ δὲ λυδίου διτόνῳ διατόνῳ τοῦ δὲ μιξολυδίου τῷ διὰ τεσσάρων δʹ τούτου δὲ βαρύτερος ὁ ὑπολύδιος ἡμιτονίῳ τοῦ δὲ φρυγίου τριημιτονίῳ τοῦ δὲ λυδίου τῷ διὰ τεσσάρων δʹ τοῦ δὲ μιξολυδίου τριτόνῳ τοῦ δὲ ὑπολυδίου βαρύτερος ὁ ὑποφρύγιος τόνῳ τοῦ δὲ δωρίου τριημιτονίῳ τοῦ δὲ φρυγίου τῷ διὰ τεσσάρων δʹ τοῦ δὲ λυδίου τῷ διὰ πέντε εʹ τοῦ δὲ μιξολυδίου τετρατόνῳ τοῦ δὲ ὑποφρυγίου βαρύτερος ὁ ὑποδώριος τόνῳ τοῦ δὲ ὑπολυδίου διτόνῳ διατόνῳ τοῦ δὲ δωρίου τῷ διὰ τεσσάρων δʹ τοῦ δὲ φρυγίου τῷ διὰ πέντε εʹ τοῦ δὲ λυδίου τετρατόνῳ καὶ ἡμιτονίῳ τοῦ δὲ μιξολυδίου πεντατόνῳ τρόπος δέ ἐστι πλοκῆς ἐμμελοῦς σχῆμα πλοκὴ δὲ μέλος τι τί ἐστὶ ἐστιν ὃ διὰ τῶν ἔγγιστα φθόγγων μελῳδεῖται ὁτὲ ὅτε μὲν ἀνιεμένης τῆς μελῳδίας ὁτὲ ὅτε δὲ ἐπιτεινομένης δηλοῦσι δὲ φθόγγοι τὰ ἓξ ⲋʹ εἴδη τῶν συμφωνιῶν οὗτοι τὴν μὲν διὰ τεσσάρων δʹ 𝈒𝈩 Ζ𝈪 𝈒 καὶ Φ𝈓 Φ𝈸 Φ τὴν δὲ διὰ πέντε εʹ 𝈒𝈩 καὶ ϹϹ τὴν δὲ διὰ πασῶν 𝈒𝈩 καὶ I𝈶 τὴν δὲ διὰ πασῶν καὶ διὰ τεσσάρων δʹ 𝈒𝈩 καὶ Ʊ𝈼 τὴν δὲ διὰ πασῶν καὶ διὰ πέντε 𝈒𝈩 𝈒 καὶ 𝈚𝈿 𝈚 τὴν δὲ δὶς διὰ πασῶν 𝈒𝈩 Ζb καὶ Ιʹ𝈶ʹ 𝈩Ι𝈶 𝈪Ι𝈶 ἡ οὖν διὰ τεσσάρων ἐκ τόνων συνέστηκε β ϐ̄ καὶ  ἡ δὲ διὰ πέντε ἐκ τόνων γ γ̄ ἡ δὲ διὰ πασῶν ἐκ τόνων ⲋʹ ⲋ̄ ἡ δὲ διὰ πασῶν καὶ διὰ τεσσάρων δʹ ἐκ τόνων η η̄ ἡ δὲ διὰ πασῶν καὶ διὰ πέντε εʹ ἐκ τόνων θ ἡ δὲ δὶς διὰ πασῶν ἐκ τόνων ιβʹ ιϐ̄
ω: Archetypus
codd: codices
H: Heidelbergensis Pallatinus gr.
V: Vaticanus gr. 192.
Ba: Bakcheios, Introduction to the Art of Music 44-49, 11-12; Jan 1895:302.14-304.5, 293.11-294.16
Vpc: Vaticanus gr. 192. (corr.)
Ru: Ruelle Ch. E. (1975).
Jan: Jan 1895 sugestions
codd Vpc Ru JanH V Vpc Ru Jan : inc.
111 ‌εἴδη συμφωνιῶν ἐν τῷ 22τελείῳ συστήματι,
codd RuH V Ru : ςʹ
ἕξ· διὰ 33
VV : δʹ
τεσσάρων, διὰ
codd H V : εʹ
πέντε, διὰ 44πασῶν, διὰ πασῶν καὶ διὰ 55
codd H V : δʹ
τεσσάρων, διὰ πασῶν καὶ 66διὰ
codd H V : εʹ
πέντε, δὶς διὰ πασῶν.
722 ‌ὅτι τρεῖς
ω H V Ba : τρόποι
τόποι φωνῆς· 28ὀξύς, μέ­σος, βαρύς.
933 ‌ὅτι πάθη τῆς μελῳδίας, 210
ω Ru JanH V Ba Ru Jan : δʹ
τέσ­σαρα· ἄνεσις, ἐπίτασις, 311μονή, στά­σις. Ἄνεσίς ἐστι 412κίνησις μελῶν ἀπὸ τοῦ 513ὀξυτέρου φθόγγου ἐπὶ τὸ 614βαρύτερον· ἐπίτασις δέ 715ἐστι κί­νησις μελῶν ἀπὸ τοῦ 816βαρυτέρου φθόγ­γου ἐπὶ τὸ 917ὀξύτερον· μονὴ δέ ἐστιν, 1018
HH : ὅτ᾽ ἂν
ὅταν ἐπὶ τοῦ αὐτοῦ 1119φθόγγου πλεί­ο­νες λέξεις 1220
VV : μελωδοῦνται
μελῳδῶνται· στά­σις δέ 1321ἐστιν ὕπαρξις ἐμμελοῦς 1422φθόγγου.
3144 ‌οἱ τοὺς τρεῖς
RuRu : τό­πους
τρόπους 22ᾄδοντες, ᾄδουσι
codd RuH V Ru : om.
λύδιον, 33φρύγιον, δώριον· οἱ δὲ 44τοὺς ἑπτά, μιξολύδιον, 55
codd RuH V Ru : φρύ­γι­ον, λύδιον,
λύδιον, φρύ­γι­ον, δώριον, 66
codd RuH V Ru : ὑπο­φρύγι­ον, ὑπο­λύδιον,
ὑπο­λύδιον, ὑπο­φρύγι­ον, 77ὑποδώριον,
BaBa : om.
[ὑπερφρύ­γι­ον, 88ὑπερλύ­διον, ὑπερ­δώριον, 99
RuRu : ἢ μιξο­λύδι­ον].
VV : μιζολύδιον
μιξο­λύδι­ον]. Τού­των ὁ 1010μιξολύδιος
RuRu : ὀξύτατος
ὀξύ­τερος· 1111τούτου ἐχόμενος ὁ λύδι­ος 1212
RuRu : exp.
[ποσῶς], βαρύτε­ρος 1313ἡμιτονίῳ· τοῦ δὲ λυδίου 1414βαρύτε­ρος
codd RuH V Ru : om.
ὁ φρύγι­ος τόνῳ, 1515τοῦ δὲ μιξολυδί­ου
codd H V : τρι­ημι­τόνῳ
τρι­ημι­1616τονίῳ· βαρύτερος δὲ ὁ 1717δώρι­ος, τοῦ μὲν φρυγίου 1818τόνῳ, τοῦ δὲ λυ­δίου 1919
codd H V : διατόνῳ
διτόνῳ, τοῦ δὲ μιξο­λυδίου 2020
RuRu : om.
ω JanH V Ba Jan : om.
VV : om.
τῷ διὰ
codd H V : δʹ
τεσ­σάρων·
VV : om.
τού­του 2121δὲ βαρύτερος
codd RuH V Ru : om.
ὁ ὑπολύδιος 2222ἡμιτονίῳ, τοῦ δὲ φρυγί­ου 2351τριημιτονίῳ, τοῦ δὲ λυδίου 242
ω JanH V Ba Jan : om.
τῷ διὰ
codd H V : δʹ
τεσσά­ρων, τοῦ δὲ 253μι­ξολυδίου τριτό­νῳ· τοῦ δὲ 264ὑπολυ­δίου βαρύ­τερος
codd RuH V Ru : om.
ὁ 275ὑποφρύγιος τόνῳ, τοῦ δὲ 286δωρίου τρι­ημιτονίῳ, τοῦ δὲ 297φρυ­γί­ου τῷ διὰ
codd H V : δʹ
τεσ­σά­ρων, 308τοῦ δὲ λυδί­ου τῷ διὰ 319
codd H V : εʹ
πέντε, τοῦ δὲ μιξολυ­δίου 3210
RuRu : om.
τετρα­τόνῳ· τοῦ δὲ 3311ὑποφρυγίου βα­ρύτε­ρος
ω Vpc JanH V Ba Vpc Jan : om.
ὁ 3412ὑποδώριος τόνῳ, τοῦ δὲ 3513ὑπο­λυδίου
codd H V : διατόνῳ
διτόνῳ, τοῦ δὲ 3614δωρίου
codd JanH V Jan : om.
τῷ διὰ
codd H V : δʹ
τεσσάρων, 3715τοῦ δὲ φρυγίου
codd JanH V Jan : om.
τῷ διὰ 3816
codd H V : εʹ
πέντε, τοῦ δὲ λυδίου 3917τετρα­τόνῳ καὶ ἡμιτονίῳ, 4018τοῦ δὲ μιξο­λυ­δίου πεντα­4119τόνῳ.
2055 ‌τρόπος δέ ἐστι πλοκῆς 221ἐμμε­λοῦς σχῆμα·
BaBa : πλοκῆς δὲ μέλος τί ἐστι;
πλοκὴ δὲ 371μέλος
ω RuH V Ru Ba : τί
τι
V RuV Ru : ἐστιν
ἐστὶ ὃ διὰ τῶν 42
VV : om.
ἔγγι­στα φθόγγων μελῳδεῖ­53ται,
H RuH Ru : ὅτε
ὁτὲ μὲν ἀνιεμένης τῆς 64μελῳδί­ας,
H RuH Ru : ὅτε
ὁτὲ δὲ 75ἐπιτεινομένης.
666 ‌δηλοῦσι δὲ φθόγγοι τὰ
codd RuH V Ru : ⲋʹ
ἓξ 27εἴδη τῶν συμφωνιῶν 38οὗτοι· τὴν μὲν διὰ
VV : δʹ
τεσ­49σάρων
VV : Ζ𝈪
RuRu : 𝈒
𝈒𝈩 καὶ
codd H V : Φ𝈸
RuRu : Φ
Φ𝈓,
RuRu : om.
τὴν δὲ 510διὰ
codd H V : εʹ
πέντε 𝈒𝈩 καὶ ϹϹ , τὴν 611δὲ διὰ πα­σῶν 𝈒𝈩
codd H V : om.
καὶ I𝈶 , 712τὴν δὲ διὰ πασῶν καὶ διὰ 813
codd H V : δʹ
τεσσάρων 𝈒𝈩 καὶ Ʊ𝈼 , τὴν 914δὲ διὰ πασῶν καὶ διὰ πέντε 1015
RuRu : 𝈒
codd H V : om.
𝈒𝈩 καὶ
RuRu : 𝈚
𝈚𝈿 ,
RuRu : om.
τὴν δὲ δὶς διὰ 1116πασῶν
VV : Ζb
𝈒𝈩 καὶ
HH : 𝈩Ι𝈶
VV : 𝈪Ι𝈶
Ιʹ𝈶ʹ.
1777 ‌ἡ οὖν διὰ τεσσάρων ἐκ 218τόνων συνέστηκε
RuRu : ϐ̄ καὶ
β ἡ δὲ 319διὰ πέντε, ἐκ τόνων
RuRu : γ̄
γ ἡ 420δὲ διὰ πασῶν, ἐκ τόνων
RuRu : ⲋ̄
ⲋʹ· 521ἡ δὲ διὰ πασῶν καὶ διὰ 622
VV : δʹ
τεσσά­ρων, ἐκ τόνων
RuRu : η̄
η
RuRu : om.
ἡ 791δὲ διὰ πασῶν καὶ διὰ 82
VV : εʹ
πέντε, ἐκ τόνων θ ἡ δὲ δὶς 93διὰ πασῶν, ἐκ τόνων
RuRu : ιϐ̄
ιβʹ.
100
111‌#In the first section, the author refers to the six species of consonances that are found in the Perfect System, in other words, within the double octave range. Undoubtedly the passage draws on the Aristoxenian school, as the eleventh is included among the exposed concords. However, by bounding the available voice spectrum within the limits of the double octave, the double octave plus fourth and the double octave plus fifth are excluded, even though Aristoxenus counts them among the consonances. On the other hand, the author refrains from alluding to the ratios defining concords, thus setting the framework within which provided material can be perceived by students keen on music practice. The fragment’s collator drew on the Eisagōgē only as regards the passage of the six species of consonances, neither expanding on their number per species nor enumerating the consonances in the Lydian tonos as appearing in the Eisagōgē.  The species of consonances 22within the perfect system are 33the following six: the fourth, the 44fifth, the octave, the octa­ve plus 55fourth, the octave plus fifth and 66the double octave.
722‌  That there are three regions 28of the voice: high, intermediate 39and low.
1033‌  That there are four melodic 211modifications: relaxation, 312tension, persistence and 413standing still. Relaxation is the 514melody movement from the 615higher note towards lower pitch; 716tension is the melody movement 817from the lower note towards 921higher pitch; persi­stence is 102whenever more (than one) 113words are melodically produced 124on one and the same note; and 135standing still is the existence of a 146melo­dically produced note.
744‌#Ruelle emended the word τρόπους preserved in the Spanish manuscript to τόπους, even though his conjecture cannot be verified either by the rest of the examined manuscripts or the available parallel texts. Alypius, from the outset, conceives terms τόνος and τρόπος as similar in meaning between each other and subsequently substitutes τόνος by τρόπος, suggesting the modern sense of ‘key signature’. Bacchius, on the other hand, does not provide the structure of his ‘seven tropoi’ unless the relative intervals he exposes between the proslambanomenoi of each tropos are implied ‒ and consequently expanded to all the steps of each tropos in the Perfect System. Should it be so, Bacchius’ tropoi can be identified as keys. On the other hand, an underlying concept of modality cannot be excluded: not only because he speaks of seven tropoi ‒ and not thirteen or fifteen ‒ but more importantly, on the grounds of his definition of tropos, provided in paragraph five, as it were the form of the musical structure comprising ascending or descending series of adjacent pitches. The fragment’s manuscript tradition is corrupted at the outset of the fourth paragraph: although the author declares the presence of three tropoi, he nevertheless exposes ten. Moreover, the order of their presentation is deprived of reason since the lower-pitched Phrygian precedes the higher Lydian in the manuscripts. Therefore, I restore here the fragment’s text on the grounds of the corresponding text of Eisagōgē, discerning the three early tropoi from the later seven and fixing their original order. Moreover, manuscript readings enable me to emend Ruelle’s haplographic omission τοῦ δὲ ὑποφρυγίου ... τετρατόνῳ (4.24‒30) owed to the homoioteleuton τετρατόνῳ, and to provide the corrected intervallic magnitudes between hypophrygian and the rest of the tropoi.  Those who make music (in) 28the three modes, sing Lydian, 331Phrygian, #Μονή and στάσις are both used by Aristoxenus, Aristides Quintilianus, Porphyry, Gaudentius, and Anonymi in describing the meaning of τάσις. Cleonides introduces and defines τονήν (‘sustained note’) by using both terms. However, in contrast to Bacchius, the aforementioned authors do not incorporate the terms in the framework of melodic modifications (πάθη τῆς μελῳδίας). According to Aristides and Ptolemy, relaxation (ἄνεσις) and tension (ἐπίτασις) constitute species of voice (φωνῆς εἴδη), not modifications of melody (πάθη τοῦ μέλους). Nevertheless, their definitions are almost similar to those of Bacchius. Persistence (μονή), that is, according to Bacchius, singing more than one syllables on the same pitch, perhaps can be related to πεττεία (cf. n. Error: Reference source not found), as described by Cleonides, although the latter’s definition appears more applicable to instrumental music. At first glance, the use of ἄνεσις, ἐπίτασις, and μονή ‒ if one demands Aristoxenian precision ‒ should be considered ‘species of voice-tension’, even though the author has declared four ‘melodic modifications’ (πάθη τῆς μελῳδίας) at the beginning of the section. However, if the loose use of Aristoxenian terms slips between the cracks, Bacchius presents here four possible ‘states’ of melody: the segmented up‒down‒persistence (ἄνεσις‒ἐπίτασις‒μονή) and the non-segmented standing still (στάσις), echoing perhaps the views of another valid old parallel tradition. Under this aspect, the author can dispense with the confusion of the ‘voice species’ with the ‘melodic modifications’. Ultimately, even if the Aristoxenian ‘species of voice-tension’ have been conveyed here as ‘melodic modifications’, the more liberal use of the Aristoxenian nomenclature in introducing youths to the basics of music playing or singing may be considered legitimately accepted within the musical schooling from the Roman Imperial Period onwards.Dorian; (in) #Through τόπος φωνῆς, the author delivers the notion of the Aristoxenian voice range resulting from the ascending or descending voice movement within certain boundaries. The concept of τόπος here, departs from the technical meaning assigned to the τόπος of the movable notes of the tetrachord, an abstract conception deriving from the sum of each of the accurate positions of lichanos and parhypatē in the three genera (diatonic, chromatic, enharmonic). Bacchius discusses the three voice ranges (high, intermediate and low), alluding to the distinct tetrachordal ranges of the Perfect System (Hypatōn, Mesōn, Diezeugmenōn, Hyperbolaiōn). The term is similarly treated by the Anonymi Bellermann, who, however, comment on four voice ranges, assigning particular names to each of them: ὑπατοειδής, μεσοειδής, νητοειδής, ὑπερβολοειδής, thus echoing the tetrachordal divisions of the ‘Perfect System’ too. Moreover, the Anonymi provide the number of tetrachords on each of the fifteen tonoi that fall within each of the four voice ranges. The term φωνῆς τόπος is also used with the general meaning of ambitus in other sources.the seven 441(modes), they sing Mixolydian, 52Lydian, Phrygian, Dorian, 63Hypolydian, Hypo­phrygian, 74Hypodo­rian [Hyperphry­gi­an, 85Hyper­lydi­an, Hyper­do­rian, 96Mixoly­dian]. Of these, the Mixo­107lydian is highest; the Lydian 118follows [quanti­tively], being 129lower by semi­tone; the Phrygian 1310is lo­wer than the Lydian by tone, 1411and (lo­wer than) the Mixoly­dian 1512by three semitones; the Dorian, 1613then, is lower than the Phrygian 1714by a tone, (lower than) the 1815Lydian by a di­tone, and (lower 1916than) the Mixo­lydian by a 2017fourth; the Hypolydian, then, is 2118again lo­wer by a semitone, (lo­w­2219er than) the Phrygian by three 2320semi­tones, (lower than) the 2421Lydian by the fourth, and (lower 2522than) the Mixolydian by a tri­2623tone; the Hypo­phrygian, then, 2751stands lower than the 282Hypolydian by a tone, (lower 293than) the Dorian by three se­mi­304tones, (low­er than) the Phrygian 315by a fo­urth, (low­er than) the 326Lydian by a fifth, and (lo­wer 337than) the Mixolydian by four 348tones; and, (finally), the Hypo­359dorian stands lower than the 3610Hypo­phrygian by a tone, (lower 3711than) the Hypolydian by a 3812ditone, (lower than) the Dorian 3913by a fourth, (low­er than) the 4014Phrygian by a fifth, (lower than) 4115the Lydian by four tones plus 4216semito­ne, and (lower than) the 4317Mixo­lydian by five tones.
6155‌#Tropos, according to Bacchius, is defined as the form of melodic interweaving (πλοκή). Although the term appears to be a substitute for the word τόνος (‘key’) as it is commonly transmitted in later musical writings, the introduction of the word plokē within the definition blurs our understanding, allowing possible interpretations of tropos in the context of melody making (μελοποιΐα). Cleonides and Aristides Quintilianus present plokē as a particular sort of melody which moves in leaps ‒ avoiding the use of adjacent notes ‒ either ascending or descending. In contrast, when the melody flows in continuous steps using adjacent notes, they introduce the term agōgē (ἀγωγή). Solomon (1980:360) perceives plokē as the successive set of non-adjacent intervals, while Barker (1989, 431 n. 144) argues that the term probably describes a sequence of regularly rotating melodic steps, in ascending or in descending order. Despite the possibly corrupted text, I am inclined to think that Bacchius' description accords with Barker’s interpretation. However, the probability that part of the original text has perished in the course of the manuscript tradition cannot be excluded. Jan (304.3) in his printed text has adopted πλοκῆς δὲ μέλος, as preserved in the manuscripts of Bacchius’ Eisagōgē; although the phrase πλοκῆς (...) μέλος does not appear in other sources it may convey a correction in the archetype of the copying error πλοκῆς δὲ μέλους, either influenced by the word ἐμμελοῦς in the line above or implying a more extended corruption, which involves the omission of the definition of agōgē. Interestingly, the fragment’s manuscripts transmit the correct «πλοκὴ δὲ μέλος τι ἐστὶ» revealing a possibly earlier version of the corresponding question of the Eisagōgē, «πλοκὴ δὲ μέλος τί ἐστι;», should we counter-adapt the indefinite, to the interrogative τί.   Tropos (mode) is a form of 22melodic interweaving; and plokē 33is a melody that proceeds 44through adjacent notes, moving 55sometimes towards lower pitch 66and sometimes towards higher 77pitch.
7166‌#‡Ruelle’s sixth paragraph misses a substantial part of its content, owed to textual corruption in Es, his singular source. Ruelle prints Es’ text unchanged without providing possible improvements. From the other manuscripts however, the text can be completely restored. Here the fragment’s compiler has returned to the topic of consonances. Setting out from Lydian proslambanomenos, the six species of consonances are exemplified.  And these are notes that 22exemplify the six spe­cies of 33consonances: The fourth, 𝈒𝈩 (A) 44and Φ𝈓 (D); the fifth, z𝈩 (A) and 55SϹ (e); the octave, z𝈩 (A) and 66I𝈶 (a); the octave plus fourth, 77z𝈩 (A) and Ö𝈼 (d); the octave 88plus fifth, z𝈩 (A) and Å𝈿 (e’); 99and the double octave, z𝈩 (A) 1010and Ιʹ𝈶ʹ (a’).
1177‌#The seventh paragraph details the sizes of the six abovementioned consonances. Ruelle’s text reflects the corrupted text of Escorialensis, which omits the size of the twelfth; the text can be restored from the other witnesses. The Aristoxenian school is still followed since the consonance sizes are expressed as tones and semitones, not ratios. The content of the final two sections (6, 7) added to the content of first section of the fragment, concludes the eleventh chapter of Bacchius’ Eisagōgē. The reason why the compiler of the fragment split and scattered the section on consonances at the outset and the end of the fragment is inexplicable at first glance. Nevertheless, it is not improbable that the Bacchius fragment, in its present form, has drawn on an earlier, more complete text of the same prose style now lost.  Now the fourth consists of 212two and a half tones, the fifth, of 381three and a half tones, the 42octave, of six tones, the octave 53plus fourth, of eight and a half 64tones, the octave plus fifth, of 75nine and a half tones, and the 86double octave, of twelve tones.
9631