Pars prima

PRET PRIMA THE RITUS CANENDI VETUSTISSIMUS ET NOVUS OF JOHANNES LEGRENSE A CRITICAL EDITION WITH TRANSLATION, INTRODUCTION AND NOTES ON THE TEXT by RICHARD VAUGHAN HUGHES IN TWO VOLUMES SUBMITTED TO GLASGOW UNIVERSITY, THE FACULTY OF ARTS, IN FULFILMENT OF THE REQUIREMENTS OF THE DEGREE OF DOCTOR OF PHILOSOPHY MARCH 1996 VOLUME TWO (©) RICHARD HUGHES 1996

CONTENTS Luca VOLUME TWO THE TREATISE 115 117 118 120 123 Its Structure The Manuscripts Manuscript Characteristics and Relationships Editorial Practice ABBREVIATIONS USED IN THE APPARATUS CRITICUS 125 ABBREVIATIONS USED IN THE ENGLISH TEXT 126 THE TEXT AND TRANSLATION 129 | Praefatio Libelli Musicalis de Ritu Canendi Vetustissimo 130 et Novo. Preface to the music treatise which aims to deal with the 131 old and new methods of singing. Pars Prima The First Part —> Liber Primus The First Book I. Quis hominum primo cecinerit? Who was the first man to sing? IL. 136 137 Quid sit aut a quo dicatur musica, quodque sit universalis linguis omnibus, ac de quatuor mathematicis una. The nature of music, and the derivation of the word. 140 The fact that it is the universal language amongst all languages, and how it comes to be one of the four Il. mathematical disciplines. Quid sit sonus, quid phthongus, quid tonus, quid 141 ‘semitonium minus, cum his quae redundant ex illis. 144 The definitions of sonus, phthongus, tonus and semitonium minus. In connection with these topics, the things which emanate from them. -

De primo et antiquissimo tetrachordo: qualefuerit ac unde venerit. Concerning the first and oldest tetrachord: its nature 156 and its source. 157 Addidisse philosophos suprascriptis chordis et alias undecim per intervalla temporum nec unquam illos excessisse quindecim numerum. 160 The philosophers added eleven other pitches at different times to the ones mentioned above: they did not add any VI. VIL. beyond fifteen. 161 Harum quindecim vocabula chordarum ac interpretationes earum. The terms used for these fifteen pitches, with their meanings. 166 167 Has quindecim chordas hic more Graeco per solum genus diatonicum divisas. These fifteen pitches at this point are classified in the Greek fashion only by the diatonic genus. 172 173 VI. Alia duo tetrachorda primis duobus simillima; cur sint ab illis per unum tonum disiuncta. 178 Two other tetrachords, which had the same structure as the first two: why they were separated from them by the interval of a tone. 179 Cur quintum sit inventum tetrachordum et cum chorda mese ligatum. 184 Why the fifth tetrachord was invented, and why it was joined to the mese. | Ad sonitum malleorum Tubal-Cain; Jubal concepisse totam 185 in numeris musicam consistere. 190 Regarding the sound of the hammers of Tubal-Cain; Jubal discovers that music consists entirely of numbers. XI. 191 Quibus proportionibus numerorum Jubal adaptari voluit consonantias vocum atque sonorum. 194 With which numerical ratios Jubal wished to relate the XII. consonances produced by sounds and pitches. 195 Numeros eiusdem esse naturae voces et sonos. 200 That pitches or sounds have the same natural proportions as numbers,

2Capitulum primum: 3Quis hominum primo cecinerit? 4Miror viros nostri temporis, doctos atque peritos, maxime tamen ecclesiasticos, adhibere posse fidem his qui tradunt modos musicos sub petris, ut supra tractatum est, fuisse repertos aut in guttis aquarum et terrae cavernis, nisi forte putent Jubal organa tantum aut citharas fabricasse, quod stulti cogitatus est, nec illum prorsus aut quempiam alterum ante diluvium cecinisse. SQuod si verum est ut post diluvium Graecus, Latinus aut barbarus dulces prior modulari sonos inceperit, nemo necesse est ad illa usque tempora canendi formam habuit, et si nullus ante diluvium huius rei notitiam perceperunt, profecto sacra pagina, quae non mentitur, nobis verum non tradidit. 6Scripsit enim Moyses de praefato Jubal qui, ni fallar, extitit ab Adam septimuse stirpe Cain utpote generatus, quod paterfuit canentium in organis et citharis, cuius frater, Tubal-Cain, artem eo tempore fabrorum invenit. 7Refert quoque Josephus, grandis auctoritatis apud Hebraeos, Graecos et Latinos historiographus, hunc Jubal adeo tenuisse caram sonorum quam exquisiverat artem ut illam in duabus columnis, verens diluvium, sculperet. AN~A AlvH lv hiis pro his A sil supra lin A praefacto A ex pro e À a Deo pro adeo A

ÎTHE FIRST BOOK 2Chapter 1: 3Who was the first man to sing? 4] am surprised that men of the present day, who are themselves learned and experienced people-in particular men of the Church-can put their trust in those who claim that musical modes were discovered, as we have discussed above, under stones, in drops of water, or in caverns under the earth, unless by chance they think that Jubal merely built organs and lyres-this stems from foolish thinking-and that he did not sing at all nor indeed did anyone else, before the time of the Flood. 5But if this is true, that after the time of the Flood, the Greeks, Latins and barbarians were the first to begin to sing sweet sounds, then it must follow that no-one until then had formed the habit of singing; further, if no-one before the Flood had any knowledge of this subject, then assuredly Holy Scripture, which does not lie, has not given us a true account. 6For Moses wrote about the above-mentioned Jubal who, unless I am mistaken, was the seventh generation after Adam from the stem of Cain, that he was the father of all who make music on organs and lyres.$ His brother, Tubal-Cain, discovered during that time the art of metalwork.? Moreover, Josephus, who enjoyed great respect among the Jews, the Greeks and the Latins as a historian, refers to the fact that this man Jubal held so dear the art of sounds that he had discovered that, for fear of a flood, he had carved out details of it on two columns.8 6Gn 4,21: Iubal ipse fuit pater canentium cithara et organo. 7Gn 4,22. 8Cf, Peter Comestor, Historia Scholastica (PL 198, p. 1079). Concerning association of this story with Josephus, see Judith Cohen, Jubal in the Middle Ages, Dissertation, University of Tel-Aviv, 1975.*

8Quarum unam postea gens renovata vidit, prophetarat etenim illis Adam quod mundus cito periret aut per ignem aut per aquam, ob quod Jubal unam de columnis illis fecit latericiam ne solveretur ab ignibus, alteram vero marmoream, ne putrefieret in fluctibus. 9Alibi quoque legitur prius illum pro tedio pastorali mitigando cecinisse, dein ad sonitum sui fratris malleorum causam rei nimis subtiliter exquisisse. Quod si quis noscere cupit qualiter, legat prius egregii doctoris Boetii musicam, et postea decimum, si placet, huius libelli nostri capitulum, quamvis imitatus Graecorum Boetius fabulas et iactantiam, huius rei philosopho Pythagorae totam ascribat gloriam. !!Mei namque propositi non est theoricam huius artis velle post tam eximium virum, nisi forsan raro coactus tractare, quin potius veram priscorum Ecclesiae Christi practicam, quae tota nihilominus ab illo fonte procedit, si possim renovare. 9. 10. 11. legat in marg A prius supra lin H si placet in marg H raro in marg H

8A later people-renewed after the flood-witnessed one of these, and indeed Adam had prophesied to them that the world would quickly come to an end either by fire or by water.? It was because of this that Jubal had made one of these columns out of bricks, to prevent its being consumed by the flames, and the other out of marble, to avoid its crumbling away in the waters. 9Elsewhere, we read that he had first begun to sing in order to allay the boredom of country life, and then had researched the reason for this phenomenon, according to the sound of his brother's hammers.10 10]f anyone wishes to familiarise himself with the details of this, let him first read the De Musica of Boethius, that distinguished scholar, and then, if he wishes, the tenth chapter of this treatise of mine.!! Though Boethius, taking as his models the stories and the claims of the Greeks, ascribes the whole of the credit for this idea to the philosopher Pythagoras.!2 11It is not my intention to deal with the theoretical aspects of this art in the steps of such a distinguished person, unless this I am compelled to do so on rare occasions. Rather would I discuss the true practices of the early Christian fathers; the practical side emanates from them after all, and I hope to be able to breathe fresh life into them. 9 osephus Antig. lud. 1,70. 10Cf, Peter Comestor, Historia Scholastica (PL 198, p. 1079) 11See below Pars prima 1.10.4. 12De inst. mus. 1,10.

ICapitulum secundum: 2Quid sit aut a quo dicatur musica, quodque sit universalis linguis omnibus, ac de quatuor mathematicis una. 3Ars igitur musica Deo placens ac hominibus, omne quod canitur discernens et diiudicans, ac de cunctis quae fiunt, non solum intendendo voculas atque remittendo, sed etiam tempus metiendo, veram inquirens rationem. 4Nam et a verbo Graeco, quod inquirere significat, musa descendere dicitur, et musicus, apud Boetium, est cui de modis atque rhythmis deque generibus cantilenarum adest secundum speculationem et rationem facultas. SOmnis enim ars sive disciplina honorabiliorem habet naturaliter rationem quam artificium, quod manu artificis atque opere exercetur. $Non parum igitur errant qui musicam aliud esse putant in Gallis, aliud in Italia et in Graecia, seu aliud in singulis nationibus, cum omnium utique sit communis linguarum ac universalis, non aliter quam caeterae tres mathematicae sunt artes. 7Sicut enim arithmetica de numeris, geometria de terrae mensuris, astrologia de stellis et de earum motibus, ita quidem musica de sonis scientia est ac vocibus. 8Is ergo qui penes nos par habetur numerus, apud quosdam fortassis populos dispar erit et contrarius, aut quadrum hic per geometriam in quatuor triangulos resolutum, id non erit apud gentes omnium nationum? NAYRS AlvH2r de pro deque A (ars) aut (sive disciplina) HA cum sit omnium utique A deomA

IChapter II: The nature of music, and the derivation of the word. The fact that it is the universal language amongst all languages, and how it comes to be one of the four mathematical disciplines. 3Music then is an art which gives pleasure both to God and to man.13 It exercises discernment and evaluation in all that is sung, and searches for true reason in all things-not only in matters concerning the raising and lowering of sounds, but also in those concerned with the measurement of time. 4Musa is said to come from a Greek word meaning ‘inquire’ !4; ‘a musician’, according to Boethius, ‘is a person who has a particular gift for melody and rhythm, and indeed for different styles of composition, according to speculative and rational guidelines.15 5Every art or discipline, by its very nature, possesses a system commanding more respect than a craft, which is practised by the manual efforts of the artisan.'16 Those who believe that there is one kind of music amongst the Gauls, another in Italy or Greece, and others in other individual nations, are therefore making a serious mistake; for music is in every way a common | universal language above all other languages, and this it has in common with the other three mathematical arts.17 7Just as arithmetic deals with number, geometry with the measurement of the earth, and astrology with the stars and their movements, so music is the science of sound and pitch.!8 8A number which in our society is considered equal to another-would this number perhaps be considered unequal and opposite in other social groups? Or the fact that a rectangle, according to the rules of geometry here is made up of four triangles!9—will this not be true among all nations of the earth? 13Cf, Burtius Florum Libellus p. 59. 14Cf, Isidore Ety. 3.15.1; Marchetto Lucidarium 6,2-3; see N. Swerdlow, “Musica Dicitur A Moys, Quod Est Aqua,” JAMS 20 (1967): 3-9. 15De inst. mus. 1,34 (225,11-15). 16De inst. mus. 1,23 (223,28-224,1). 17Cf below Pars secunda 3.10.4. 18This reference to the disciplines of the quadrivium is notable for its use of the word astrologia rather than astronomia. Medieval definitions of the quadrivium ultimately relate to De inst. arith. 1,1. 19Cf. De inst. arith. 2,6 (91,18): quadratus in quattuor triangulos divisus.

9Quod namque sol unus oriatur cunctis gentibus per diem, ac una luna per noctem, non est qui ambigat. 10Sic de musica sentiendum, o cantores, quoniam, etsi ritus modulandi varii sint varias in nationes atque varius ad docendum usus, non poterit tamen ullus homo canens vocem sursum intendere seu inflectere deorsum quin tonum proferat aut minus semitonium, ditonum aut semiditonum, tritonum aut diatesseron, perfectum diapente seu imperfectum, tonum cum diapente vel semitonium, ditonum cum diapente vel semiditonum, diapason perfectum aut imperfectum, seu ex his quippiam cum illa compositum; circa quae procul dubio tota versari debet contemplatio musicorum. 11Nam et ipse Jubal qui, sicut audistis, pater fuit, hoc est, primus et princeps cantorum, quid valuit, dictante natura, canere nisi quoddam ex illis de tono semitonioque compactum? 12Sicut enim de littera, quae pars est compositae vocis minima, syllabae fiunt, ac de syllabis dictiones, et de dictionibus constructiones in grammatica, sic et phthongi quidem sive soni musici canore vocis originem habent, e quibus ortae syllabae musicales, tonus ac semitonium minus, iunctae simul in varias huius artis concrescunt vocum resonantias, quae tamen in ipsis resolvi queant omnia phthongis. 10. 12. sic bis HA in varias nationes A flectere A pars compositae est A vocum in marg H

9For no-one doubts the fact that it is one sun that rises over all nations, tribes and languages by day, as does a single moon by night. 100 you singers! This is the view one should adopt about music-since, even if styles of singing and teaching methods vary from country to country, no man will be able, while he is singing, to raise the pitch of his voice or lower it without making use of the following intervals: the tone, the minor semitone, the ditone or semiditone, the tritone, the diatessaron, the perfect or imperfect diapente, the diapente plus tone or semitone, the diapente plus ditone or semiditone, the perfect or imperfect diapason, or any interval which is a combination of these. Surely it is to these topics that all the thoughts of musicians should be directed. !!For even Jubal himself, who, as you have heard, was the father, that is, the first and chief among singers, what could he sing at nature’s bidding except some combination of tone and semitone? !2Just as syllables are made up of individual letters, which are the smallest units of a compound sound, and words from syllables, and grammatical constructions from words, so sounds or musical pitches have their origin in the inflections of the voice. From these pitches develop ‘syllables’-the tone and the minor semitone. These syllables in combination develop various ‘melodic shapes’ which we associate with the art of music.20 Nevertheless, all these structures can be broken down into their individual pitches. 20Cf. Mus. ench. 1 (ed. Schmid p. 1, II. 1-5); see also Quintilian Inst. Orat. 1,4,6 and Calcidius Comm. in Timaeum Platonis 1,44. A similar analogy occurs in Lucidarium 9.1.2. where the ditones are called 'words'.*

1Capitulum tertium: 2Quid sit sonus, quid phthongus, quid tonus, quid semitonium minus, cum his quae redundant ex illis. 3Ecce liquet quoniam Jubal, quem natura primum canere docuit, sicut suum idioma proprium absque vocalibus et consonantibus enuntiare nequibat, ita nec ullatenus cantare, si non aliquas de suprascriptis melorum speciebus, aut, ut magis proprie loquar, e toni semitoniique compositionibus proferret. 4Ob quod absurdum non estimo, non aliter quam alphabeti nostri litteras in vocales partiri solemus et consonantes ac iterum in mutas, semi-vocales ac liquidas, huiusmodi quoque musicales syllabas et sonoras quonammodo dictiones hic primum suis in partibus dividere, quo necnon vocabulo quaelibet per se vocitetur quove modo diffiniatur declarare. SQuoniam, ut dixi, Jubal haec omnia prius humana voce discrevit, ac forsitan in organo, liris et citharis, sola iuvante natura, multum exercuit, ac ubi vero rerum causas inquirendo multa naturae secreta reserasset, normam docendi coaevos invenit. 6Diffinitio soni generalis: sonus ergo, iuxta Boetium, est percussio aeris indissoluta usque ad auditum. 7Sonus autem non ille generalis, sed quem Graeci phthongon appellant, est, ut ait isdem Boetius, vocis casus emmeles, id est, aptus melo, in unam intentionem. 8Intervallum vero, dicit adhuc, soni est acuti gravisque distantia. 9Phthongi ergo soni sunt, sed proprie musici qui scilicet legitimis ab invicem distant spatiis, et sunt ad cantandum aptissimi. no A2rH2v hisom A quem om A 4, loquitur pro loquar A proferrent H exstimo pro estimo A vero in marg H phthongi pro phthongon A

IChapter II: The definitions of sonus, phthongus, tonus, and semitonium minus. In connection with these topics, the things which emanate from them. 3It is clear then that Jubal, the first whom nature taught to sing, could not speak in his native language without using vowels and consonants. Similarly, he could not sing without producing at least some of the melodic formulae mentioned above, or, to speak more precisely, the structures made up of the tone and semitone. 4As our custom is to divide the letters of our alphabet into vowels and consonants, and to sub-divide them into mutes, semi-vowels and liquids, similarly, I think it a good idea first to divide musical syllables and ‘sound utterances’ into their basic units, and then to explain what terminology is . adopted for each one, and how it is defined.2! 5Further, as I have already pointed out, Jubal had already made these distinctions in the field of the human voice, and possibly had employed them greatly in playing the organ, the lyre and the cithara, solely with the help of nature. When he had unlocked her many secrets by researching into the meaning of things, he found a way of teaching his discoveries to his contemporaries. 6The definition of general sound: sound, according to Boethius, consists of a disturbance of the air which remains intact until it reaches the ear.22 71 do not mean sound in general, but that which the Greeks call phthongos, which, as Boethius again points out, is an emmeles, that is to say, musical, resolution of the voice onto a particular pitch.23 8He also points out that an interval is the distance between a high and a low pitch.24 9Therefore, phthongi are sounds which are particularly musical ones; they are separated from each other by established distances and are particularly suitable for singing. 21Cf, Quintilian Inst. orat. 1,4,6.and above note 20. 22 De inst. mus. 1,3 (189, 22-23).* 23De inst. mus. 1,8 (195, 2-3).* 24 De inst. mus. 1,8 (195, 6).* 25Cf. Mus. ench. 1 (3,7-8): Ptongi autem non quicumque dicuntur soni, sed qui legitimis ab invicem spaciis melo sunt apti.

10fllud autem spatium, quod est inter phthongon et phthongon, appellatur intervallum. 11De tono: tonus est duorum phthongorum legitimis spatiis ab invicem distantium quam integra modulatio; diciturque tonus a tonando quod de phthongo perfecte tonet in phthongon, nec habere valet ultra solum intervallum. 12Omnes siquidem huiusmodi canorae coniunctiones unum semper minus habent intervallum quam habeant voces. 13De minori semitonio: semitonium minus est duorum phthongorum legitimis spatiis ab invicem distantium non integra modulatio. !4Dictumque semitonium non a semi, quod sit medium, immo sicut dicitur semivir aut semivivus imperfectum, non enim medius tonus est, sed de duabus toni partibus pars minima, quod perfecto demonstratur in Boetii musica. 15De ditono: ditonus est trium phthongorum ac duorum tonorum adunatio, dictus a duo Graece quod est duo Latine, nam etsi tres sonos sive voces habeat, duos tantummodo tonos in suis spatiis occupat. 16De semiditono: semiditonus quaedam est trium similiter phthongorum sed toni tantum ac semitonii minoris copulatio, dictus a semi sicut semitonium et ditonus, quasi non perfectus ditonus. 11. 14, de supra lin A phthongon scripsi phthongom A phthongum H pars in marg Hom A

10That distance between one pitch and another is called an interval. ll’The tone: a tone is an absolutely complete progression between two pitches which are separated from each other by established distances. The word tonus is derived from the verb tono, because it produces a perfect tone between one pitch and the next; it cannot contain more than one interval. 12Certainly, all harmonious pitch combinations of this kind always have one interval less than they do pitches. 13The minor semitone: the minor semitone is not a complete progression. It is made up of two pitches separated from each other by established distances. 14The word 'semitone' is not derived from semi in the sense of ‘half’, but in the sense of ‘imperfect’ as it is used in the words semivir or semivivus.26 It does not represent the halfway point in a whole tone; rather, it refers to the smaller of the two divisions of the whole tone. This is clearly explained in the De Musica of Boethius.27 I5The ditone: this is made up of three pitches, and is a combination of two whole tones. It is derived from the Greek övo, which is the Latin word duo. For even though it contains three sounds or pitches, it contains only two tones within its range. 16The semiditone: this is similarly made up of three pitches, but is a combination of a whole tone and a minor semitone, The word is made up from semi, as in semitone, and ditonus. It is, as it were, an imperfect ditone. 26For ‘semi’ meaning ‘imperfect’, cf Lucidarium 2.5.18, Micrologus 4,5 (p. 103), and Johannes Afflig. Musica in GS 2, p. 238. For use of 'semivir', see Summa musicae in GS 3 . 210.* Der, e.g, De inst. mus. 1,16 (203,8-9): Sed utraque semitonia nuncupantur, non quod omnino semitonia ex aequo sint media, sed quod semum dici solet, quod ad integritatem usque non pervenit. See also, De inst. mus. 1,7; 2,29; 3,1.

17De tritono: tritonus quatuor phthongorum est ac trium tonorum valde discors aggregatio, sic a tris atque tonus dicta, cum sit ex tribus tonis contiguis tota confecta. 18De diatessaron, prima consonantia: diatessaron quatuor est phthongorum ac duorum tonorum uniusque semitonii minoris aggregatio, dicta quidem a dia quod est de vel per et tessara quatuor, eo quod sit de quatuor sonis effecta. 19Haec est prima trium perfectarum consonantiarum atque simplicium; sicut enim in alphabeto, demptis quinque vocalibus, aliae litterae consonantes sunt, ita quidem, separatis hic tribus perfectis consonantiis, aliae sunt omnes dissonantiae, quamquam ditonus ac semiditonus, tonus cum diapente sive semitonium et huiusmodi sint compassibiles. 20De quibus loco suo tractabitur, ut spero, diligentius. 21De diapente perfecto: diapente perfectum est quinque phthongorum atque trium cum uno semitonio minori tonorum connectio. 22Dicta siquidem a dia quod est per aut de, et pente quinque, nam de quinque sonis haec tota conficitur secunda simplex et perfecta consonantia. 23Quadruplex est etiam, sicut et diatessaron triplex, tot etenim species habere potest omnis coniunctio vocum quot intervalla possidet, de quo tractandum est diligenter in sequentibus. 24De diapente imperfecto: diapente non perfectum est etiam quinque phthongorum, sed duorum dumtaxat tonorum ac duorum minorum semitoniorum discors quaedam compositio. 19. (hic) se (tribus) dele H 20. supero pro spero A

17The tritone: this is made up of four pitches, and three whole tones. The interval produced by these sounds is very dissonant. The word is made up from tris and tonus because its total range is made up of three adjacent whole tones. 18The diatessaron, the first consonance: this is made up of four pitches, and is a combination of two whole tones and a minor semitone. The term is derived from dia which (in Latin) means de (from) or per (through), and tessera, which (in Latin) is quatuor (four).This is because the interval is made up of four sounds. 19This is the first of the three perfect simple consonances. Just as in the alphabet, if we take away the five vowels, we are left with consonants, so if we take away the three perfect consonances, all the rest are dissonances; though the ditone and semiditone, the diapente plus tone, and the diapente plus semitone and others of this type, are compatible.28 201 shall discuss these in greater detail, I trust, in the appropriate place. 21The perfect diapente: this is made up of five pitches-a combination of three whole tones and a minor semitone. 22This term is derived from dia, which translates as de or per, and nevte which is the Latin quinque. The whole of this second consonance is made up of five sounds, and is a simple, perfect consonance. 23It is made up of four intervals, in the same way as the diatesseron is made up of three; each combination of pitches is able to contain as many species as it has intervals, butI shall treat this matter in greater detail in the following pages. 24The imperfect diapente: there is also an imperfect diapente consisting of five pitches, but it is a dissonant combination of only two tones and two minor semitones. 28 Concerning compassibilis, cf. below Pars secunda 3.2.. For Marchetto’s coinage of the term, see Introduction, p. 75.

25De tono cum diapente: tonus cum diapente sex phthongorum est et unius semitonii minoris cum quatuor tonis quaedam auditui compassibilis copulatio. 26De semitonio cum diapente: semitonium cum diapente sex etiam phthongorum est, sed duorum minorum semitoniorum cum tribus tonis integris quaedam quoque non tota discors coacervatio. 27De ditono cum diapente: ditonus cum diapente septem est phthongorum et cum uno minori semitonio discors tota quinque tonorum associatio. 28De semiditono cum diapente: semiditonus cum diapente septem est etiam phthongorum, sed cum duobus semitoniis minoribus quatuor tonorum quae tota discordat collectio. 29De diapason, perfectissima consonantiarum ac tertia simplicium: diapason est octo phthongorum legitimis ab invicem spatiis distantium ac quinque tonorum cum duobus semitoniis minoribus dulcissima modulatio, dicta videlicet a dia quod est de vel per, et pan, omne vel totum, eo quod omnes huiuscemodi vocum aggregationes ipsa contineat et, ut pia mater, in sinu suo foveat ac enutriat. 30Haec tertia consonantia simplex atque perfecta quae, iuxta datam regulam, octo voces habet, intervalla septem et septem species, de quibus disserendum est dum tempus venerit per singula. 3!Quam profecto, si sibi diapente copules, iam duplicem effectam diapason diapente vocabis, et si diapason duplices, bisdiapason erit; sicque bisdiapason cum diapente et terdiapason replicare potes in infinitum. 25. auditui compassibilis in marg H 26. quaedam quoque non tota discors in marg H 29. nutriat A

25The tone plus diapente: this interval contains six pitches-four whole tones plus a minor semitone: it is a sound not incompatible to the ear. 26The semitone plus diapente: this interval also contains six pitches, but made up of two minor semitones and three whole tones. It too produces a combination of sounds which is not totally dissonant. 27The ditone plus diapente: this interval contains seven pitches; its combination of five whole tones and a minor semitone produces an absolute dissonance. 28The semiditone plus diapente: this interval also contains seven pitches, and its combination of four tones and two minor semitones is also an absolute dissonance. 29The diapason is the most perfect of the consonances and the third of the simple ones. The diapason contains eight pitches separated from each other by established distances. It is made up of five whole tones and two minor semitones, and is a very pleasing combination of sounds. The term is derived, of course, from Sua (in Latin de or per) and nov, which translates into Latin as omne or totum. This is because it contains within itself all the combinations of such sounds and, like a devoted mother, cherishes them in its embrace and gives them nourishment. 3°This third consonance is simple and perfect. It contains eight pitches, and, according to the established rule, seven intervals and seven species within it, which I must discuss individually when the time comes. 31If one couples the diapente to this interval, you will then refer to this as a compound interval, the diapason diapente. If one doubles the diapason, the interval thus formed will be the bisdiapason. Thus, one can continue doubling to form the bisdiapason diapente and so on up to the terdiapason.

32Inter quas etiam cadunt ditonus cum diapason aut semiditonus cum eodem, et tonus cum diapason, diapente vel semitonium identidem. 33Quae quidem dissonantiae compositae sunt, sed compassibiles ut in simplicibus. 34Non haec tamen rerum innovatio, sed praecedentium eiusdem naturae vocum replicatio. 35Quod totum utique simul in una figura collectum, si paucis subiectis litteris demonstretur, puto visus humanus in ea satis delectabitur et sensus capacior erit. 36Sit igitur A b tonus, b C semitonium minus, CD tonus, DE tonus, EF semitonium minus, FG tonus, G et iterum A tonus, A autemet b sic quadratum tonus, sed A et b sic rotundum sit semitonium minus. 37Tunc quod AC non sit semiditonus quis velle dicere praesumat et AD diatessaron, AE diapente, AF semitonium cum diapente, AG semiditonus cum diapente, et Aa diapason perfectum? 38Nam et CE ditonus est, et BF diapente non perfectum, F autem ad b quadrum tritonus, et D & quadrum tonus cum diapente, C b quadrum ditonus cum diapente, verum b primum ac b rotundum inter se gignunt diapason imperfectum. 39Octo namque phthongos habet sicut et illud optimum, sed quia cum tribus semitoniis minoribus tonos quatuor tantummodo colligit, ad illius veri diapason dulcissimam concordiam non pertingit. 32. in ter pro inter A 35. capatior A 36. si pro sit A 37. 39. minus in marg H et om À etom À

32Between these intervals fall the ditone plus diapason, the semiditone plus diapason, the tone plus diapason, the semitone plus diapason, the semitone or tone plus diapente diapason, and so on. 33These compound intervals are, strictly speaking, dissonant, but are compatible, like their simple counterparts. 34They are not new intervals, but a doubling up of intervals of the same nature which we have mentioned previously. 35AIl this material, in any case, can be collated onto one diagram; if it can be explained by the use of a few basic letters appended, I consider that the human eye will take pleasure from it, and our understanding be improved. 36Let us therefore establish the following: the distance between A and b is a whole tone; from È to C a minor semitone; from C to D a whole tone; from D to E a whole tone; from E to F a minor semitone; from F to G a whole tone; from G to A again a whole tone; from A to square b is a whole tone, but the distance between A and roundb is a minor semitone. 37Then no-one will dare deny that A to C forms a semiditone, A to D a diatessaron, A to E a diapente, A to F a semitone plus diapente, A to G a semiditone plus diapente, and A to aa perfect diapason. 38C to E forms a ditone, b to F an imperfect diapente, F to square b a tritone, D to b a tone plus diapente, and C to square b a ditone plus diapente. However, the first square b and the round b produce between them an imperfect diapason. 39This interval contains eight pitches, as does its perfect counterpart, but since it is formed from three semitones and four whole tones only, it does not achieve the perfect harmony of the true diapason.

40FIGURA VOCUM OMNIUM CONIUNCTARUM Quicquid Jubal cecinerit seu canendo protulerit; quicquid Graecus aut Italicus quidve Gallus aut barbarus. Aliud nemo cecinit neque potest aut potuit modulari viriliter quam quod hic annotavimus. -—————————— Tonus cum diapente | Diapason perfectum Semiditonus cum diapente Diapente perfectum171 Semiditonus q — Dita | Tritonus A Tonus b Semi-aTonus I Tonus E Semi- F Tonus G Tonus A Semi-b Semi b I" tonium je" Pen imperfectum Semitonium culo Litmus cum diapente Diapason imperfectum tonium| tonium

404 DIAGRAM SHOWING ALL THE COMBINATIONS OF SOUNDS Whatever Jubal sang, or produced by singing; whatever the Greek, the Italian, the Frenchman or the barbarian sang. No-one ever sang anything else; no-one is able, or ever has been able to sing properly other than by using what we have drawn here. —— Whole tone plus diapente Perec diapason | Semiditone plus diapente Perfect se] Semiditone — Ditone —| Tritone A Toneb Semi-C Tone D Tone E Semi- F Tone G Tone A Semi-b Semi-b — Dar _] tone tone I Imperfect diapente Semitone plus diapente | I _ Ditone plus diapente Imperfect diapason tone tone

1Capitulum quartum: 2De primo et antiquissimo tetrachordo: quale fuerit ac unde venerit. 3Praesto nunc, o cantores, qui naturam vocum et sonorum his nostris temporibus ignoratis, nunquam vidistis viros a natura tam mirabiliter modulari voces ut iam non dicam melius, sed certe multo lascivius quam vos illi canant, et nihilominus regulas vestras, ciferas, litteras et characteres, notas atque mensuras non intelligunt? 4Quis in civitate Mantua citharoedum illum quem appellabant 'passerem' non vidit, qui novas in harpa sua saepe fabricavit, me teste, cantiones, quas nunquam per se tamen describere scisset? 5Alius etiam civis extitit illis in diebus in eadem Italica civitate qui tumultuarias quoque - componens cantilenas, atramento vel carbone pinxit aliquando magis quam scripserit, quas nemo quidem canere praesumpsisset unquam nisi quidam cantor mihi notus more communi prius illas descripsisset. "Nunc autem quis dubitet quoniam Jubal a principio sic egerit? 7Instigante siquidem natura primo cecinit, ac postea paulatim discrevit sonorum differentias, fecitque fortassis organum et citharas, dulces cantiones etiam composuit, deinde aetas longaeva rerum causas investigat et exercitat sensus. 8Quo tamen ritu coaevos suos modulari docuerit, aut quibus ad docendum et scribendum usus sit notis, litteris, ciferis, characteribus, aut quibus canendo nescitur, etsi credendum sit Noe filios post diluvium suaves iterum hominibus tradidisse canendi modulos, 1. 5. A 4r et hoc capitulum om in toto H tumultuarias scripsi tumulas A notus scripsi noctus A exercitat scripsi exercitati A

IChapter IV: 2Concerning the first and oldest tetrachord: its nature and its source. 3Give some attention to this, you singers who are not conversant with the nature of pitches and sounds during these times. Have you never encountered men who pitch their voices so instinctively and so beautifully that they sing not better admittedly, but certainly more uninhibitedly than you, but do not understand your rules, your ciphers, your letters, characters,29 pitches and metres? 4There can be no-one in the city of Mantua who has not seen the lyrist whom they called 'The SparrowTM in my presence , he often composed original melodies on his harp, but he would never have known how to write them down by himself. 5In fact, there was another inhabitant of the same city at that time who also improvised melodies: he would sometimes paint them, rather then notate them, with black ink or carbon. As a result, no-one would ever have dared to sing them if a certain singer of my acquaintance had not written them out, using the familiar notation. "Now who can be in any doubt that Jubal would initially have acted in this way?30 7Certainly at the beginning he sang instinctively, but then gradually he discovered the differences between sounds; he probably built an organ and some lyres, and composed sweet melodies. Then his long life gave him the opportunity to delve into the nature of things and to cultivate his sensitivity. 8No-one knows however how he taught his contemporaries to sing, or what notes, letters, cyphers or characters he used for teaching and writing, or which he used for singing. However, we must believe that after the Flood, the sons of Noah passed on to their fellow men in turn beautiful melodies to sing. 29Concerning the term character, see F. Reckow in Handwörterbuch de musikalischen Terminologie (Wiesbaden, 1970). * For an analagous useage of the term, see Micrologus 5, 30Cf, above nn. 4 and 6.

maxime Japhet qui totus, ut legitur, virtuti deditus erat, et a quo nobis illa philosophorum emanavit progenies, totaque Graecorum et aliarum omnium fere nationum gentilitas. 9Nam et primas illas quatuor chordas, quarum Graeci iactitant inventorem extitisse Mercurium, opinor magis ante diluvium ab ipso Iubal exquisitas ac inventas, sed et ad docendum in ordine dispositas atque regulatas, deinde post diluvium a praefatis Noe filiis novae genti novisque populis ita sicut prius erant fuisse traditas. 10In ea namque musica, quam totiens allegatus Boetius de Graeco vertit in Latinum, legitur illam a principio fuisse simplicissimam, adeo quod quatuor nervis ipsa tota constaret. !!Inde tetrachordum a quatuor chordis appellatum est. 12Primus autem nervus et quartus diapason invicem resonabant consonantiam, medii vero simul tonum habentes, ad extremos diatessaron ac diapente reddebant, quod totum his quatuor litteris sequens figura demonstrat. 13Si ergo diatessaron AB, tonus autem BC, quisque AC diapente sit negare poterit? 14Iterumque si BD diapente sit et diatessaron AB resonabit, et AD diapason perfectissimum erit. —_ _——T———__r_________— nationum scripsi nationem A

This is particularly true of Japhet, who was totally committed to goodness, so we read.3! It is from him that the noble line of philosophers sprang, together with the entire pagan learning of the Greeks and of almost every other nation. It is my opinion that the first four pitches-which the Greeks claim were invented by Mercury-existed rather before the Flood: that they were sought after and discovered by Jubal himself, and later arranged and tabulated for teaching purposes. Then, after the Flood, they were handed down by the above-mentioned sons of Noah, in their original state, to the new nation and new peoples. 10In that treatise on music, which Boethius-so often referred totranslated from Greek into Latin,?2 we read that it was the most simple of all types of music at that early stage because it consisted entirely of only four pitches. 11So it was called tetrachordum because it contained four strings. 12The first and fourth strings produced between them the consonance of a diapason, the middle strings together comprised a tone, and produced a diatessaron and a diapente with the outer strings. The following diagram demonstrates all this with the following four letters: 13if A to B is a diatessaron, and B to Cisa tone, then clearly the distance from A to C must be a diapente. 14A gain, if from B to D we have the distance of a diapente, and the interval between A andB is a diatessaron, then the distance between A and D will be an absolutely perfect diapason. 31 This reference to Japhet cannot be traced. 32The treatise would be Nicomachus’s lost Peri mousikes, cited in De inst. mus. 1,20 (205,28-206,7).*

15Hoc tetrachordum omnium primum ac vetustissimum creasse Graeci iactitant ac fecisse Mercurium, sed illud magis traditum renovatis hominibus a Noe succe[sso]ribus, opinor, post diluvium. Diapason intensa Diatessaron intensa | Diapente intensa— Diapente remissau Diatessaron remissa Diapason remissa in figura: (Diatessaron) intensam A

15The Greeks claim that Mercury invented and constructed this tetrachord, which is the first and oldest of them all. However, I think that it was handed down after the Flood by the descendants of Noah to humankind which had been given a new lease of life. The diapason in ascent The diatessaron in ascent A The diapente in ascent B C D The diapene—!The diatessaron in descent in descent The diapason in descent

1Capitulum quintum: 2 Addidisse philosophos suprascriptis chordis et alias undecim per intervalla temporum, nec unquam illos excessisse quindecim numerum. 3Porro philosophi Graeci quatuor illis nervis adhuc undecim alias apposuere chordas per intervalla temporum, quod qualiter aut quomodo si scire cupis, lege Boetium in primo suae musicae libro, ibique reperies non id modo quod quaeris, sed ipsorum quoque nomina philosophorum. 4Sensati namque viri quod omnis vox aut continua sit aut intervallis suspensa non ignorabant, et idcirco se posse plures adhuc tantillo numero superaddere chordas non dubitarunt. SAttamen quindenum numerum transcendere nunquam voluerunt. 6Est enim vox illa continua qua prosas legere solemus et historias verba: loquendo verbis subiungere, quae nimiruma sola natura moderari potest ne sit infinita, dum loqui tantum valeat homo seu legere quantum anhelitus eius duret aut queat respirare. 7Quae vero dicitur intervallis suspensa vox est qua musicos elevare solent homines aut deprimere sonos; quae quidem et ipsa modum non habet si non refrenetur a natura, nam ultra non gravat homo vocem quam valeat sonos alacriter exprimere, nec adeo si sit prudens scandit in altum ut dubitet deficere. 8Quisnam oro fere mortalium, si vocem elevet ultra quintam decimam, fere non deficiat, aut si tantum illam relaxet, confusus non erubescat? NexYr A4vHSr Addidisse philosophos......Capitulum Quintum om H repies pro reperies A valet pro valeat A vero om H deprimere pro exprimere A oro in marg H oro fere dele H

Chapter V: 2The philosophers added eleven other pitches at different times to the ones mentioned above: they did not add any beyond fifteen. 3Further, the Greek philosophers added eleven more pitches, at different times, to the four already in existence. If you wish to know the nature of these, and how they arose, you should read the work of Boethius-the first book of his . treatise on music.33 There you will find not only what you seek, but also the names of the philosophers themselves. 4Perceptive men as they were, they were aware that all pitch is either continuous or interrupted by means of intervals.34 Therefore, they did not doubt that it was possible to add more pitches to the pre-existent small number; however, they were never willing to exceed the total of fifteen. “That line which we call continuous’ is one which we normally use for reading prose passages and for joining words to each other when reciting historical accounts.35 This clearly can be prevented from being infinite only by nature, for a man can speak or read only as long as his breath lasts, or as long as he can catch a second breath. 7That vocal line which is _ referred to as being ‘interrupted by intervals36 is the one which men use to raise or lower musical sounds. This has no limit apart from the fact that it is controlled by nature; for a man does not lower his voice beyond the point at which he can produce sounds effortlessly.37 Nor, if he is sensible, does he sing so high that he is uncertain about reaching it. 8I ask you, what man's voice would not break if he raised it higher than the fifteenth pitch, or if he lowered it to the same extent, would he not blush in confusion? 33De inst. mus. 1,20 (206,7ff). 34Cf, De inst. mus. 1,12 (199,3-4): Omnis vox aut OVVEYN est, quae continua, aut Soo Tn pario, quae dicitur cum intervallo suspensa. Note ambiguity in Latin between ‘voice’ and ‘pitch,’ 35Cf. De inst. mus. 1,12 (199,5-6): Et continua quidem est, qua loquentes vel prosam oratiorem legentes verba percurrimus. 36Cf, De inst. mus. 1,12 (199,4): ... quae dicitur cum intervallo suspensa. 37Cf. De inst. mus. 1,13 (200,1-5).

9Quod si quis se vidisse voces humanas obiecerit ad praefatum numerum superandum aptissimas, respondemus quod paucae sunt ad totius humani generis comparationem, et de his quae raro accidunt non datur regula generalis. 10Egit in hoc ergo per omnia prudenter philosophorum auctoritas, quae nobis ad cantandum docendum ac discendum, et omne quod canitur, si velimus describendum, optime providit, nihilque superfluum, aut inepte per incuriam ordinatum reliquit. praefactum pro praefatum A comperationem A

9Now if anyone disagrees, and claims that he has encountered human voices easily able to go beyond these limits, we reply that they are very few-relative to the size of the human race—and that we cannot form a general principle from these phenomena which occur so rarely. 10So the authority of the philosophers has proceeded wisely here in all respects— an authority which has admirably provided for us a means of teaching and learning the art of singing, and a method of describing, if we wish, everything which is sung. It has passed down to us nothing which is superfluous, or any ideas which are inadequately presented because of carelessness.

ICapitulum sextum: ‘© 2Harum quindecim vocabula chordarum ac interpretationes earum. 3Chorda Prima: collectis itaque prout tradit Boetius quindecim tantummodo chordis, primam et omnium gravissimam vocavere philosophi Graece proslambanomenos, hoc est 'additam' vel 'appositam' seu 'acquisitam', eo quod quamquam ultima fuerit inventa, caeteris tamen sit necessario praelata. 4Chorda Secunda: Hypate hypaton idem est quae gravissima gravissimarum, quod quippe vocabulum antequam esset proslambanomenos iure sibi competebat, cum tunc esset prima. Nunc vero tenet illud adhuc quamvis illa tono sit altior et secunda. 5Chorda Tertia: Parhypate hypaton iuxta gravissimam interpretatur gravissimarum, quae cum ab illa solo minori semitonio remota sit, suum optime gerit nomen ac vocabulum. 6Chorda Quarta: Lichanos hypaton ‘index’ appellatur non ab re 'gravissimarum', nam cum antiquitus esset in loco tertio distans a secunda chorda tono, mox indici monochordum occurrebat tangenti digito. 1. 2. 4, A Sr H 5r (chordarum) ut (ac) dele A esse pro esset A

IChapter VI: 167 The terms used for these fifteen pitches, with their meanings. 3The first pitch: I have listed the pitches-only fifteen in number-as Boethius has handed them down to us.38 The first and deepest of them all the philosophers called by the Greek word proslambanomenos, which means 'added', ‘placed next to', or 'newly-acquired'.39 The reason for this is that although it was the last to be invented, it is of necessity put in front of the others. 4The second pitch: the Hypate Hypaton. This likewise is the pitch which is the ‘lowest of the lowest’ because this term was properly appropriate for it before the existence of the proslambanomenos. It claimed this term as its right since at that time it was the first pitch. Now, it continues to have this term applied to it, although it is the second pitch and a tone higher than the first. SThe third pitch: the Parhypate Hypaton means ‘next to the lowest of the lowest’: as it is positioned only at a minor semitone's distance from it, it well deserves its title and its terminology. 6The fourth pitch: the Lichanos Hypaton is rightly referred to as ‘the index finger of the lowest pitches’. for although in ancient times it occupied the third position at a tone's distance from the second pitch, it subsequently corresponded with the index finger as it touched the monochord. 38For general description of the fifteen pitches, see De inst. mus. 1,20; for Boethius’s translations of the names of the pitches, see De inst. mus. 4,3. 39Boethius refers to the proslambanomenos as ‘added’ (addita) in De inst. mus. 1,20 (211,22); he further describes it with the term adquisitam in 4,3 (309,21). Boethius does not use the term apposita with respect to the lowest pitch. Martianus Capella De nuptiis likewise uses the term adquisitus (9,931). The naming of notes and their interpretations in the following lines does not follow Boethius or Martianus Capella.

7Chorda Quinta: Hypate meson 'gravissima' recte vocitatur 'mediarum', distans enim a praecedenti vicina tono, sicuti voces finit gravissimas, ita quidem inchoare probatur et medias. 8Chorda Sexta: Parhypate meson interpretatur ‘iuxta gravissimam mediarum, nam ab illa solo scitur distare minori semitonio, quod est initium mediarum et origo. 9Chorda Septima: Lichanos meson Latine sonat ‘index mediarum', quae tono distans a sua praecedenti socia, nomen alterius Lichanos a loci similitudine sortitur ac vocabulum. 10Chorda Octava: Mese, quae dicitur 'Media' jure duplici tale nomen habere debet, a subdita namque sibi vicina tono se tantum elevans, primum finit diapason a principali proslambanomenos vocula, rursusque sedens in loco medio, mater est omnium sequentium vocularum ac domina, 11Chorda Nona: Paramese ‘iuxta Mese' non immerito dicitur, nam si sit ab illa plus minusve tono remota, nusquam de septem diapason speciebus apparet una. 12Chorda Decima: Trite Diezeugmenon optime quidem interpretatur tertia disiunctarum, est etenim a mese locum habens tertium, quamquam ad paramese minus reddat semitonium. 10. quae om A (rursus) que supra lin H om A (sequen)tium supra lin H Chorda X H

7The fifth pitch: the Hypate Meson is rightly called the ‘lowest of the middle pitches’, for, lying at a tone's distance from its aforementioned neighbour, it forms the limit of the lowest pitches, and is accepted as being the start of the middle register. 8The sixth pitch: the Parhypate Meson means 'the pitch which is next to the lowest in the middle register’, for it is known to lie at a distance of only a minor semitone from that pitch, which forms the beginning, and is the basis of the pitches of the middle tetrachord. 9The seventh pitch: the Lichanos Meson; this term, in Latin index mediarum, means ‘the index finger of the middle pitches’. It lies at a distance of a tone from its neighbour mentioned above. It obtains the name and the terminology of the second Lichanos because of its similar position. 10The eighth pitch: the Mese which means ‘the middle pitch’, has to have such a name on two counts. It is only a tone higher than its lower neighbour, and it forms the upper limit of the first diapason, which begins with the principal pitch, the proslambanomenos. Further, it occupies a central position, and is the mother, and ruler of all the following pitches. 11 The ninth pitch: the Paramese is rightly referred to as ‘the pitch lying next to the mese', for if it lay at more or less than a tone's distance from it, one of the seven diapason species would in no sense be realized. 12The tenth pitch: the Trite Diezeugmenon is well translated as ‘the third of the disjunct pitches’, for it occupies the third position from the mese, but is only a minor semitone's distance from the paramese.

13Chorda Undecima: Paranete diezeugmenon ‘iuxta nete disiunctarum' versa de Graeco significat in Latinum, eo quod triten uno tono superans ad neten quoque tonando semel ascendat. 14Chorda Duodecima: Nete diezeugmenon a verbo Graeco neate iuxta Boetium ‘inferior’ appellatur 'disiunctarum', tono namque praescriptam paranete chordam superans, novissimam omnium disiunctarum vocem emittit atque iuniorem. 15Chorda Tertia Decima: Trite hyperboleon iure 'tertia' nominatur 'superiorum' aut 'excellentium', altius namque solo minori semitonio quam nete vicina sua prodiens, etsi certe graviorem, tertium nihilominus a superiori nete locum occupat. 16Chorda Quarta Decima: Paranete hyperboleon ‘iuxta nete' dicitur ‘superiorum' eadem ratione qua paranete diezeugmenon, scilicet pro eo quod ad illam per tonum pergit similiter. 17Chorda Quinta Decima: Nete hyperboleon“inferior superiorum' interpretata probatur eodem argumento quo nete diezeugmenon, videlicet ultima namque superiorum est, sicut illa disiunctarum, soloque tono vicinam sibi subditam superat. 13. 14. 15. (quoque) semel (tonando) dele H Chorda XII H parinete A vocum pro vocem A Chorda XIII H at pro aut A pergat H inferior.......... diezeumenon in marg A

13The eleventh pitch: the Paranete Diezeugmenon means ‘next to the nete in the disjunct pitches’ when translated from Greek into Latin, because it is higher than the trite by a whole tone, and has to climb a tone to reach the nete. 14The twelfth pitch: the Nete Diezeugmenon, derived from the Greek word ‘neate’ according to Boethius, is referred to as the lowest of the disjunct pitches,“ for it lies at a tone's distance above the previous paranete. It produces the newest and most recent sound of all the disjunct pitches. _ The thirteenth pitch: the Trite Hyperboleon is rightly called the third pitch in the highest or outstanding pitches. It appears at only a minor semitone's distance from its neighbour the nete. Even though it occupies a deeper position, nevertheless it is only three pitches down from the higher nete. 16The fourteenth pitch: the Paranete Hyperboleon is so called because its position is adjacent to the nete in the highest pitches, on the same principle as the paranete diezeugmenon, which similarly travels the distance of a whole tone to reach its neighbour. 17The fifteenth pitch: the Nete Hyperboleon is rightly translated as ‘the lowest pitch in the highest pitches’ for the same reason as the nete diezeugmenon is so called—because it is the last of the upper pitches, just as the other is the last of the disjunct ones, and is higher than its neighbour underneath it by only a tone. 40Cf De inst. mus. 1,20 (206,25): + Quasi neate id est inferior. ‘Neate’ is a Doric variant of ‘nete’.

1Capitulum septimum: 2Has quindecim chordas hic more Graeco per solum genus diatonicum divisas. 3His igitur expeditis, ac quindecim illis phthongorum vocum sive sonorum vocabulis satis ad propositum interpretatis, scire non erit absonum Graecis antiquis, valde curiosis, unam harum quindecim vocum nequaquam suffecisse distinctionem, immo tribus illas permutando tonos ac semitonia distinxere modis, primam utpote modulandi formam genus diatonicum, secundam autem enarmonicum, tertiam vero chromaticum appellantes. 4Veruntamen de solo diatonico, quod primum est verum ac perfectum, tractare dispono, quamquam de caeteris etiam duobus in totum silere nolim, captato tempore siquidem et loco congruo. 5Mater enim Ecclesia de tribus his generibus solum diatonicum ad omne quod canere velis aptissimum elegit, aliis reprobatis duobus, Deum utique laudare volens ad libitum, idque totum quod curiosum est magisve difficile quam consonuma se reiciens. 6Has igitur Graeci quindecim chordas in quatuor primo divisere tetrachordis, primum appellantes tetrachordum hypaton, id est 'gravissimarum', secundum autem tetrachordum meson, hoc est ‘mediarum’, tertium tetrachordum diezeugmenon, id est 'disiunctarum', quartum autem tetrachordum hyperboleon quod sonat 'superiorum'. 1, 2. 3. 4, 6. A 5r H 6r hic supra lin H (Gracco) divisas (per solum) dele H add A divisas in marg Hom A Hae pro His A vocalibus pro vocabulis A siquidem tempore A in pro et A tetrachordum in marg H est om À

173 Chapter VII: 2These fifteen pitches at this point are classified in the Greek fashion only by the diatonic genus. 3Having settled these matters then, and having explained, sufficiently well for our purpose, the fifteen terms used for the phthongi pitches or sounds, it will not be out of place for us to be aware that, as far as the ancient Greeks were concerned*-and they were a very intellectually curious people-one version of the succession of fifteen pitches was not enough. They therefore distinguished them in three ways by changing the order of tones and semitones. The first type of singing they called the diatonic, the second the enharmonic, and the third the chromatic. 4However, I am inclined to deal only with the diatonic type, which is the first truly perfect type, though I would not wish to remain wholly silent about the other two, if there is time, and in the appropriate place. SMother Church chose only the diatonic type of the three as being the most suitable for what you feel inclined to sing-and rejected the other two. She wished nothing else but to praise God at will while rejecting all that is laboured, or is too difficult to be pleasing to the ear. 6These fifteen pitches then the Greeks divided up into four tetrachords 4! They called the first Tetrachordon Hypaton -that is, the one made up of the lowest pitches. The second was called Tetrachordon Meson, containing pitches in the middle register. The third they called Tetrachordon Diezeugmenon -the Tetrachord of the Disjunction. The fourth was the Tetrachordon Hyperboleon containing the highest sounds. De inst. mus. 1,20.

7Contemplantes namque philosophi solis tribus diatessaron differentiis inesse totam harmoniae virtutem-quicquid enim ultra fit replicatur et unum est-omnem istius modi per illam divisere vocum ordinem, duo simul tetrachorda connectentes, quibus tres exprimi probantur illius varietates. 8Nil aliud enim est hic tetrachordum quam perfecta consonantia diatessaron. 9Est autem eius prima species ab hypate hypaton in hypate meson secundum Graecos, quae currit per minus semitonium et tonum et tonum, et hoc, ut dixi, per solum genus diatonicum. !0Secunda vero pergit a parhypate hypaton ad parhypate meson, per tonum et tonum ac minus semitonium, a lichanos autem hypaton in lichanos meson tertia, currens utique per tonum ac minus semitonium atque tonum. 11Cernis ergo quod sumant hic a semitonio minori tetrachorda semper exordium, quodque primum sit ab hypate hypaton in hypate meson, cui connectitur, ut patet in hac figura, statim secundum. (Figura in pagina 176) 7. 8. omnem om A Nihil pro Nil A hic supra lin H (quod) que om A (hypa)te (hypaton) om A

7The philosophers believed that the entire virtue inherent in the tonal structure lay solely in three species of diatessaron-for whatever lies beyond that range is a duplication and a reiteration-and they divided up every such order of pitches by this scheme, connecting together two tetrachords, by means of which three varieties of structure are produced. 8Here, the tetrachord is nothing but the perfect consonance of a diatessaron. "The first species of diatessaron extends, according to the Greeks, from the hypate hypaton to the hypate meson.42 It progresses as follows: minor semitone, tone, tone, and this, as I have said, applies only in the diatonic genus. !0The second species extends from the parhypate hypaton to the parhypate meson, using the following order: tone, tone, minor semitone. The third extends from the lichanos hypaton to the lichanos meson in this order: tone, minor semitone, tone. 11You see therefore that here the tetrachords always begin with the interval of a semitone, and that the first tetrachord extends from the hypate hypaton to the hypate meson, to which is connected conjunctly the second tetrachord, as the following diagram makes clear. (Diagram on page 177) 42The ordering of species follows Boethius’s second ordering of species, that more consistent with traditional Greek theory: see De inst. mus. 4,14 (339,12-15), and Bower/Boethius p. 151,n. 76. The order “according to the Greeks” may be found, for example, in Cleonides Eisagoge 9 (JanS. p. 195-98), and Ptolemy Harmonica 2,3,50.

120mne quidem tetrachordum est sola diatessaron, sed si duo coniunxeris, mox tres diversas efficis; quarta namque replicatur ut hic recte comprobatur, nec illud quod est simile diversum dicitur. His duobus tetrachordis surgit istud heptachordum Secunda species | ciatessaron Tertia species siatessron | a prima simils haec Prima species —— diatessaron | Tetrachordum Tetrachordum meson hypaton 3 à o E 3 3 Eg Ee 3 238 e $ E F <eiRr£dgrfdif ¢ El 3 ae E ZL

12Every tetrachord is made up of one species only of diatessaron, but if you join two tetrachords together, you quickly produce three species. The fourth is a repetition, as is rightly pointed out here, and what is identical is not referred to as different. From these two tetrachords arise this heptachord The second species of diatessaron The third species of diatessaron The whole of this is The first species identical to the of diatessaron first (species) | | E The tetrachord of the middle E a E 5 $ È 5 v 2 o 8 © The tetrachord of the bottom = 2 oO = g E S E 5 3 E 2 © 8 [= E vw 2% VU. 4 5 = E R E I M Zeek PERLE REE RS 6 OCR 2 6 CV 6 2

ICapitulum octavum: 2 Alia duo tetrachorda primis duobus simillima; cur sint ab illis per unum tonum disiuncta. 3Hic natura rerum satis exercuit solertiam philosophorum: aut certe coacti sunt in duas partes tonum secare, vel humanas aures trium discordia tonorum sibi succedentium vehementer offendere. 4In illo enim quem supra depinximus heptachordo diapason haberi non potest, eo quod ibi sint septem tantummodo chordae sive voces, ipsa vero consonantiarum consonantia minus quam octo chordas habere non debet. SGraeci namque proslambanomenos nunquam in divisione vocum annumeraverunt, quoniam divisis iam longo tempore chordis ab hypate hypaton, quae tunc erat prima sicut hic ponitur, ad complendum bisdiapason, et illa, prout in eius interpretatione tactum est, illi superadditur. 6Cum ergo proslambanomenos, quae prima nunc est omnium, ad eam quae prima prius erat tonum semper habeat, et ad eam chordam quae media dicitur, optimum diapason efficiat, quod et illa chorda quae mese sequitur, ab ipsa tono distet integro necesse est. 7Alioquin una de septem diapason speciebus tota perit, et si servaveris eam, trium tonorum, ut dixi, discordiam pessimam incurris. 8Cum tribus namque tonis contiguis humanae naturae concordia non est. ?Qua de causa, philosophi totis diebus altercari maluere cum tritono quam unam de septem diapason auferre de numero. PD» A 6r H 6v simillia pro simillima A diximus pro depinximus A habere pro haberi H

IChapter VIII : 2Two other tetrachords, which had the same structure as the first two: why they were separated from them by the interval of a tone. 3At this stage, the facts of nature were enough to tax the ingenuity of the philosophers: there is no doubt that they were compelled either to divide the tone into two parts, or violently offend human ears with the dissonant interval produced by the succession of three tones.*3 4For a diapason cannot be contained within the heptachord which we have described above, because the latter possesses only seven pitches or pitches. Also, the most perfect consonance needs to contain no fewer than eight pitches. SThe Greeks never included the proslambanomenos in their classification of sounds until, as is mentioned in Boethius' account, it was added on to the hypate hypaton to complete the bisdiapson.44 For some time, the pitches themselves had been classified starting with the hypate hypaton—which at that time was the first pitch— which is how it is placed here, in the preceding diagram. 6Since therefore the proslambanomenos, which is by this time the first pitch, always lies at a tone's distance from the pitch which used to be the first, and since it produces a perfect diapason with that pitch which is called the mese, it is necessary that the pitch following the mese lies at a tone's distance from it. 7In the absence of the proslambanomenos, one of the seven diapason species is entirely destroyed; and if you retain it, you produce the dissonance of the interval produced, as have said, by three successive tones, the worst possible one. 8For human nature has no agreement with three whole tones adjacent to each other. 9For this reason, the philosophers preferred to argue with the tritone all the time rather than be deprived of one of the seven diapason species. 43For Johannes’ suppositions on the Greek view of the tritone, see below, Pars prima 1.9.14. 44See De inst. mus. 1,20 (211,21ff). I assume here eius refers to Boethius’s account.

10Hinc est quod sequentem paramese chordam ab ipsa mese per tonum integrum elongarunt, tertiumque tetrachordum a primis duobus seiunctum et cum quarto sequenti ligatum diezeugmenon, hoc est disiunctum, nominant, sicque durum a parhypate meson in paramese vilipendentes tritonum, optimam ab hypate hypaton in eandem paramese diapason conservant. 11Tonus ergo soluit hic tetrachordum tertium a chorda mese tritonum generando, quod ligare poterat cum caeteris minus semitonium dulcisonam diapason enervando, pergens videlicet a paramese versus nete diezeugmenon, ut alia tetrachorda per semitonium minus, tonum et tonum. !2In qua nete chorda siquidem et quartum tetrachordum huic tertio connectitur, non aliter quam et illa duo prima simul in hypate meson connexa sunt, tenditque similiter ad nete hyperboleon chordam ultimam per minus semitonium, tonum ac tonum. 13Nullamque prorsus inter haec quatuor tetrachorda video distantiam, cum et ista duo tres diatessaron demonstrent species sicut et illa, nisi quod ibi tantummodo graves, hic autem acutae voces resonent. 14Quod totum esse verum hic depicta probabit figura. (figura in pagina 182) 10. 12. 13. 14. tertium namque A est supra lin A huic supra lin H in om À (prorsus) est dele H om A video om A apparebit pro probabit A

10]: was for this reason that they separated the following pitch-the paramesefrom the mese itself by the distance of a whole tone. To the third tetrachord they applied the term diezeugmenon which means ‘disjunct’, for it was separated from the first two tetrachords, and joined to the fourth which followed it. Thus, they preserved the perfect diapason extending from the hypate hypaton to the same paramese, paying little regard to the harsh tritone between the parhypate meson and the paramese. 11 The distance of a tone therefore separates here the third tetrachord from the mese, thus creating the interval of a tritone; were it to be joined by a minor semitone to the rest, this would result in the weakening of the sweet-sounding diapason. This tetrachord then extends from the paramese to the nete diezeugmenon- like the other tetrachords, in that it contains the progression minor semitone, tone, tone. 12At this nete, the fourth tetrachord is joined to the third, in the same way as the first two tetrachords are joined at the hypate meson; it extends to the nete hyperboleon-the last pitch-in a similar way, using the progression minor semitone, tone, tone. 13] myself see no difference at all between these four tetrachords, since both pairs produce the three species of diatessaron; however, there is the fact that one pair produces only deep pitches, and the other high. 14The diagram which I have drawn here will prove all this to be true. (Diagram on p. 183)

Haec duo ligata simul sunt caeteris simillima, nisi quod voces acutas occupant et illa graves; in quo quidem ostenditur quod quicquid voces resonant, totum quippe dirigitur triplici diatessaron. His duobus tetrachordis surgit istud heptachordum —Secunda species diatessaron — Tertia species Diatessaron ——- Tota orina similis haec | | Tetrachordum hyperboleon | Prima species diatessaron] | Tetrachordum diezeugmenon hNypertboelon hPyaprearnbeotleon Thyrpeirbtoleon Smeitnouism dNiezetugmenon dPiaerzaeungemtenon Tdirezieutgmenon Smeimntounisum Paramese Tonus Tonus. Tonus Tonus

15These two conjunct tetrachords are totally identical to the others, apart from the fact that these occupy a high register, and the others a low one. It is this fact which demonstrates that whatever the pitches sound, the whole is controlled by the three species of diatessaron. From these two tetrachords arise this heptachord The second species of diatessaron The third —— species of diatessaron | The whole of this The first species of diatessaron is identical to the first (species) Tetrachord of the highest The disjunct tetrachord 4 5 e [e] 2 e YQ = o © o 38 & D = 3 E © Y [a = € 6 0 ESS = 4. 3 vo © © © E o © cs s se o [= 8 Y 3 E 5 D I 8 E 5 © È So 2 © E = Kai 2 2 [=] o 3 & E 2Q © 598 gu eo 8 € 32 $ o E 5 7ZPREFZZEAGEERA

1Capitulum nonum: 2Cur quintum sit inventum tetrachordum et cum chorda mese ligatum. 3Nunc autem ad id quod de tribus tonis contiguis supra motum est convertere stilum nos oportet. 4Circa quod primo quaerendum est ad quid philosophi Graeci quintum illis quatuor addere voluerunt tetrachordum, et cum satis allegata mese chorda totaliter connectere, cum praesertim illa quatuor sufficiant ad canendum tetrachorda, sic, ut vides, in optimo genere diatonico distincta, tamque decenter per diapason ac diapason simul adunata. >Videtur praeterea superfluum ac inane totum istud tetrachordum, quoniam etsi nomina mutata sint chordarum, est tonus identidem et tonus inter triten paraneten netenque synemmenon, sicut inter triten paraneten ac neten diezeugmenon erat. 6Sed si rem diligenter consideremus, non parvam inter has chordas differentiam esse videmus. 7Providi namque philosophi, trium illorum tonorum discordia, quae cadit a parhypate meson in paramesen, concitati, rursus et aliam inter mesen et paramesen constituere chordam triten synemmenon, hoc est, tertiam coniunctarum illam ea de causa qua et triten hyperboleon vocitantes. 8Quae procul dubio tonum ab ipsa mese in paramesen secat et dividit, sed non aequaliter, dum ad mesen minus reddit semitonium, et ad parhypate meson per consequens non iam tres tonos successivos, immo veram diatessaron generat. „rom AGvH7v inventum sit A quatuor om A propterea pro praeterea A actom A mesem et paramesem A

IChapter IX: 2Why the fifth tetrachord was invented, and why it was joined to the mese. 3]t is necessary now to devote some of my text to what we mentioned above concerning the three adjacent tones. 4With regard to this, we must first ask ourselves why the Greek philosophers wished to add a fifth tetrachord to the other four, and join it conjunctly to the mese, a pitch I have mentioned enough already, since these four tetrachords are more than adequate for singing purposes, set out thus, as you can see, in the excellent diatonic genus, and combined together so aptly between one diapason and another. Further, this tetrachord seems to be totally useless and superfluous, because even if the names of the pitches are changed, there is still a progression of two tones between the trite, the paranete and the nete synemmenon, likewise between the trite, the paranete and the nete diezeugmenon. But if we consider the matter carefully, we see a considerable difference between these pitches. 7The philosophers, provident as they were, were disturbed by the dissonance produced by the three tones which occurs between the parhypate meson and the paramese. They placed another extra pitch between the mese and the paramesethat is, the trite synemmenon-the ‘third of the conjunct pitches’. They named it on the same basis as the term adopted for the trite hyperboleon. 8Clearly, this cuts and divides into two parts the whole tone which lies between the mese and the paramese-but not into equal halves. The pitch lies at a minor semitone's distance from the mese; consequently, the distance to the parhypate meson is not now three successive tones, but rather a true diatessaron is produced.

9A paramese vero distat apothome, quod est maius semitonium, idque totum non cernitur solum oculo, sed et manu de facili tangitur in monochordo. 10Quotiens ergo necesse est hoc uti quinto tetrachordo, relictis tetrachordi diezeugmenon chordis, totiens naturalem deserere paramesen, et hanc arte factam triten synemmenon recipere nos oportet; !!tuncque tonum inter ipsam et paraneten synemmenon quae prius trite diezeugmenon erat proferre, rursusque' tonum ab ipsa paranete synemmenon ad neten synemmenon quae chorda paranete diezeugmenon fuerat, ut ad tempus tetrachordo disiunctarum seposito quod naturale totum est atque facillimum, quintum istud difficile recipiatur quod currat etiam per minus semitonium ac tonum et tonum. 12Difficile dico quidem et non naturale quoniam, ut vides, arte quadam hic tonus, quod pauci capiunt, dividitur, minorique semitonio cum chorda mese ligato, maius quod et apothome cum sequenti disiunctarum minori semitonio iungitur. 13Hanc itaque Graeci maluere cum pessimo tritono iugem habere colluctationem quam unam, ut dixi, non posse describere diapason speciem, nec ab eis ob aliud tetrachordum istud synemmenon est cum chorda mese copulatum, nisi quo tanta trium tonorum duritia mutaretur in diatessaron perfectam. 14Quid amplius? Tolle, si potes, tritonum, et nil valet istud tetrachordum. 15Quo viso, melius est ipsum ab aliis naturalibus atque semper necessariis tetrachordis, ut hic depingendo seiungere, quam ab eo quod per se clarum est intricare. 10. naturalem in marg H paramesem A recipere..... paranete synemmenon in marg A 11. ad nete synemmenon om À chorda scripsi chordam HA 12. quod pauci capiunt in marg H et? om A 14. potesses A

The pitch lies at a distance of an apothome from the paramese, that is, a major semitone. This can not only be wholly seen by the eye, but easily produced by fingering the monochord. 10Whenever we find it necessary to use this fifth tetrachord, then we have to abandon the pitches in the disjunct tetrachord, and by the same token, dispense with the natural paramese and replace it with the trite synemmenon, which is created by artifice. Then it was necessary to produce the distance of a tone between this pitch and the paranete synemmenon-previously the trite diezeugmenon-likewise a tone between the latter and the nete synemmenon-previously the paranete diezeugmenon. Consequently, having set aside temporarily a totally natural and easy concept, that is, the disjunct tetrachord, the difficult fifth tetrachord was accepted, which was likewise made up of the progression semitone, tone, tone. 12I emphasize that this is a difficult and unnatural feature, for as you see, by means of certain skilful procedures, the whole tone is at this point dividend into two segments, a point which few appreciate; the minor semitone joins itself to the mese, and the major-that is the apothome-is joined to the minor semitone with the next of the _ disjunct pitches. 13 And so the Greeks preferred to be involved in this continuous battle with the dreadful tritone rather than not be able, as I have said, to realize one of the diapason species. There was no other reason to join this other tetrachord, that is the synemmenon,with the mese than to change the considerable dissonance produced by three tones in succession into a perfect diatessaron.45 14What more is there to say? Take away the tritone, if you can, and this fifth tetrachord has no validity. 15Now that I have clarified this point, it is better to separate this from the other natural, and totally necessary tetrachords, by describing it as I have done here, rather than, because of it, complicate a matter which, in itself, is quite self-evident. 45Conceming this extended argument concerning the synemmenon tetrachord and the tritone, see Introduction, pp. 42-44.

16Istud quintum tetrachordum totum erit superfluum, si non discors ac nimius nobis occurrat tritonus quem delet diatessaron. Prima species diatessaron Tetrachordum synemmenon id est, coniunctarum Nesyntemenon Parsaynemtenon Trsiyntemenon Smeitnouism Tonus Tonus Mese

16The fifth tetrachord will be totally superfluous, if the tritone does not strike us as discordant, and excessively so. This diatessaron does away with this tritone. The first species of diatessaron Tetrachord 'synemmenon' that is, of the conjunct pitches Nseyntemenon Pasyrnaenmetenon Trsyinetmenon sMeimntonre Tone Tone Mese

ICapitulum decimum: 2Ad sonitum malleorum Tubal-Cain; Jubal concepisse totam in numeris musicam consistere. 3Quoniam de diviso tono mentionem fecimus ac de monochordo, putavi non esse vanum parumper ad Jubal redire, quoque ritu subiectam invenit esse numeris musicam breviter aperire, quatenus in hoc saltem me vidisse Boetii musicam evidenter appareat ac eius arithmeticam non nescisse, et priusquam dispersa superius illa quinque simul aggregentur tetrachorda, viso quod nil a me loquar aut novi quippiam de proprio cerebro cudam, lector lectioni fidem adhibeat, et si forsan dubitaverit, ad fontem relicto rivo properet. 4Jubal igitur ille dum iam multis diebus, uti credendum est, a natura cantasset, aliosque suos coaevos ad id ipsum provocasset, non ei suffecit sonos auditu tantum discernere, quin potius coepit paulisper in dies meditando causas inquirere. SQui cum apud se talia crebro cogitaret, ac omnino cur sic soni permixti consonent aut dissonent investigare vellet, audit una dierum super incude fratris sui Tubal-Cain, qui faber erat, resonare tonum, diatessaron, diapente, simulque diapason, et ait: "Mutate quaeso malleos ac iterum percutite, non enim parvum aut in vestris bracchiis aut in ipsis malleis latere sentio naturae secretum". 1. A7rH8v 3, evidenter in marg H (quinque) tetrachorda dele H 4, 5. (viso) quoque add A letor A forsan in marg H coaevos suos A sufficit A tantum auditu A cepit A aut pro audit A simul cum pro simulque A

IChapterX: 2Regarding the sound of the hammers of Tubal-Cain; Jubal discovers that music consists entirely of numbers. 3Since I have mentioned the division of the whole tone, and also the monochord, I thought it not a waste of time to return to the subject of Jubal for a while, and to show briefly how he discovered music to be subservient to number. To this extent at least it should be clear that I have consulted the De Musica of Boethius, and that I am familiar with his De Arithmetica.. Before the five tetrachords-treated separately above-are brought together as a single topic, let the reader-seeing thatI say nothing on my own account, or forge any new idea in my own mind-accept on faith what he reads: and, if he entertains any doubts, let him return directly to the source and abandon the rivulet.46 4Jubal#? therefore, we must assume, had been singing naturally for many days, and encouraging his contemporaries to do the same. At this point, it was not enough for him to distinguish the pitches by ear alone; rather he began, as the days went by, to tum gradually to enquiring into the reasons for these phenomena. His thoughts turned to such matters frequently, and he was generally interested in investigating why combinations of sounds were either consonant or dissonant. One day, he heard the sound of the tone, the diatessaron, the diapente and the diapason at the same time upon the anvil of his brother Tubal-Cain, who was a blacksmith. He said to his brother: "Please change the hammers around, and strike again, for I feel that a wonderful secret lies either in your arms, or in the hammers themselves”. 46This passage conceming faith on the part of the reader may represent a literary topos modelled on similar passages in De inst. mus., 1,19 (205, 19ff) and 1 33 (222-23). 47The narrative of this chapter rather closely follows De inst. mus. 1,10, but name of Jubal is substituted for that of Pythagoras-as noted by Johannes himself in sentence 10 below. The same Christian tradition of substituting Jubal or Tubal Cain for Pythagoras can be traced back to Egidius of Zamora (fl. ca. 1260-80), see GS2, p. 372a. Burtius (Florum libellus, p. 76) makes Pythagoras the hero of the legend.

6Dubitavit siquidem vir sensus habens ob longam aetatem exercitatos ne causa tantae novitatis quam aure captabat inesset ferientium viribus, sed nec sic aliud ibi concipiens quam quod ante senserat, malleos concitus ponderavit. 7Quo facto, singula singulis comparans pondera malleis, quosdam ab invicem duplo distare deprehendit numero, quosdam autem epitrito sive sesquitertio, quosdam nec non sesquialtero, sed et quosdam simul sesquioctavam reddere vidit proportionem. 8Exempli gratia: si quis libras duodecim ad novem comparet, mox proportionem sesquitertiam habet, sed si novem ad octo sesquioctavam, et si rursus duodecim ad octo sesquialteram, quod si duodecim ad sex consideret, invenit duplam. 9Haec autem subiecta demonstrat figura quae solus musicus capere solet, nam qui se musicum reputat ignorans arithmeticam, haud secus quam si se rhetorem praedicet nesciens grammaticam. . 10Tradunt Graeci Pythagoram invenisse figuram, sed magis puto consonum opinari dictum Jubal suum fratrem Tubal-Cain frequentasse fabricantem, qui ferro pater extitit ac aere malleantium. 6. 7. inesse A deprehendit in marg H 8, 9, (consideret) et add solis musicis À

6Jubal was a man whose senses were sharpened by the long passage of time, and he doubted that the reason for this new experience which was reaching his ears lay in the strength of the beaters themselves. However, though he did not imagine that there was any factor other than what he had previously observed, in a state of excitement, he weighed the hammers. 7Having done this, he compared the weight of each hammer, and discovered that some were twice the weight of others. Some produced the ratio 4:3, some the ratio 3:2, and others 9:8. 8For example, if one balances twelve pounds against nine, one easily arrives at the ratio 4:3. Again, nine pounds in relation to eight produces the ratio 9:8. Twelve pounds set in relation to eight produces the ratio 3:2, and twelve pounds in relation to six the ratio 2:1.* 9The following diagram explains ' these laws, which only the musician can grasp. For the person who considers himself to be a musician, while he at the same time is ignorant of arithmetic, can . be compared to the person who claims to be an orator, but has no knowledge of grammar. 10It is traditional amongst the Greeks to claim that Pythagoras invented this diagram, but I think it is a more convincing idea to believe that the said Jubal visited his brother Tubal Cain the blacksmith, who was reputed to be the father of those who forge with iron and with bronze.

1Capitulum undecimum: 2Quibus proportionibus numerorum Jubal adaptari voluit consonantias vocum atque sonorum. 3At Jubal, his cognitis, non nesciens voces acutas e pluribus ac velocioribus quam graves fieri motibus, omnemque pluralitatem ad paucitatem non aliter haberi quam si numerus comparetur ad numerum, in epitrito numero iudicat esse diatessaron consonantiam, eo quod illam inter duos eiusdem numeri malleos aure concepisset, motusque suos se sic invicem habere non ambigit tam graves quam acutos. *Certus quoque per arithmeticam ex epitrito numero et sesquioctavo gigni sesquialterum, in illo tonum, et in hoc diapente constituit, iuxta quod in malleis talium proportionum resonare praesenserat. SEt quidem bene, scimus enim ex diatessaron et tono diapente fieri, et ex sesquitertia cum sesquioctava sesquialtera generari, propter quod necesse est ut tonus sesquioctavam et diapente sesquialteram occupet. Porro duplam inveniens proportionem ubi dulcis diapason resonabat in malleis, totam in duplo numero naturam eius esse censuit, cum praesertim diapente cum diatessaron aut e converso componere diapason aspiceret, quemadmodum epitritus numerus et sesquialter duplum generat, quod totum sequens figura monstrabit. PD= A7v H9r voluit adaptari A proportionem A praesenserat (praesenserat) dele H

IChapter XI: With which numerical ratios Jubal wished to relate the consonances produced by sounds and pitches. 3Having made these discoveries, Jubal became aware that high pitches are produced by more and faster vibrations than are deep pitches; also that every greater quantity has the same relation to a smaller quantity as one number has in relation to another.48 He decided that a diatessaron was produced by the numerical ratio 4:3, because he had heard this interval produced by hammers of the same numerical ratios; he was sure that the low and the high vibrations had the same interrelation. 4He established too, using arithmetic, that the ratios 4:3 plus 9:8 produce the ratio 3:2. He established that the 9:8 ratio produced a whole tone, and the 3:2 ratio a diapente, according to what he had previously realised to be happening with hammers of the same relative sizes. 5And indeed he did well; for we know that a diapente is made from the combination of a diatessaron and a whole tone, and that the 3:2 ratio is produced from 9:8 plus 4:3. Consequently, it must be that the 9:8 ratio produces a whole tone, and that the 3:2 ratio produces the diapente. After this, he discovered the 2:1 ratio-that is when the sweet diapason sounded on the hammers-and he decided that its entire nature depended upon this duple ratio, especially since he saw that the combination of diatessaron and diapente-or vice versa-constitute a diapason, just as the 4:3 ratio added to the 3:2 ratio produces the 2:1 ratio The following diagram will make clear all of this. 48Cf, De inst. mus. 1,3 (190,21-30): ...acutae voces spissioribus et velocioribus motibus incitantur ... Ex pluribus enim motibus acumen quam gravitas constat. In quibus autem pluralitas differentiam facit, ca necesse est in quadam numcrositate consistere. Omnis vero paucitas ad pluralitatem ita sese habet, ut numerus ad numerum comparatus.

THic est tonus, haec diatessaron, haec diapente et haec diapason quae profecto veris suis partibus aequis ac integris totam monochordi metiuntur chordam, neque sunt aliae phthongorum vocum aut sonorum aggregationes quae tantam arrogare sibi praesumant omnino gloriam. 8De quo quippe monochordo post haec disputare cogito, monstrare volens oculo tonum ac diatessaron cum reliquis perfectis consonantiis nonnisi per praefatas proportiones in chorda posse creari vel resonare, videlicet per duplam, per sesquialteram, per sesquitertiam, ac per sesquioctavam proportionem, partes quoque toni principales esse maius atque minus semitonium palpare velle disponens, nec ullatenus in aequa tonum ipsum dividi. 9Nullum enim est aliud penes musicos ita verum approbans instrumentum, et hoc propter continuum ibi varie per praescriptas proportiones ac iustissime compartitum. !0Omnia siquidem, ut ait sapiens, in mensura, numero et pondere consistunt. on aequis in marg H . haec om A praefactas A in chorda in marg H (iustissime) pa dele H

7This diagram shows the tone, the diatessaron, the diapente, and the diapason. All of these, with their true, equal and perfect parts, certainly measure out the total length of the string on the monochord. There are no other combinations of sounds, phthongi or pitches which can in fact presume to claim such status. 81 intend to examine after this the nature of the monochord, as I wish to demonstrate to the naked eye that the tone and the diatessaron, together with the other perfect consonances, cannot be produced and sounded on the string without using the numerical ratios mentioned above-2:1, 3:2, 4:3 and 9:8. Iam disposed to wish to examine the fact that the principal parts of the tone are the major and minor semitones, and that in no way can the tone be divided into two equal parts. There is no other instrument which proves the truth in such a way as this one, as far as musicians are concerned: and this is because the unbroken whole is there divided variously and absolutely correctly according to the ratios mentioned above. 10AII these principles-so the wise man tells us-are dependent on measurement, on number, and on weight.4? 49Cf. Sap. 11,21: sed omnia in mensura et numero et pondere disposuisti.*

11 Hic Jubal prior cecinit ac primus artem reperit, omne sumens iudicium, ut hic patet in numeris; quam demum in marmoribus sculpsit ac in lateribus ne pereat diluvio vel solvatur incendio. Proportio dupla Proportio sesquialtera -Proportio sesquitertia XII Proportio Proportio sesquioctava | sesquitertia VI Diatessaron Tonus | Diatessaron Diapente Diapason

11This Jubal was the first to sing, and indeed the first to discover the art by undertaking every investigation, as is demonstrated here in the numerals. He eventually sculptured this in marble and in brick lest it perish in a flood or be destroyed by fire. The duple ratio — — The sesquialteral ratio —The sesquitertial-—The sesquioctaval ratio The sesquitertial ratio XII ratio IX Diatessaron VII Tone Diapente Diapason Diatessaron

ICapitulum duodecimum: Numeros eiusdem esse naturae voces et sonos. 3Nemini quidem grave videri debet, si numeros eiusdem esse naturae dicam et sonos, cum etsi lingua sileat aut scribere cesset calamus, non aliud ipsa natura clamet. 4Nam oro, quid est numerare nisi de decem unitatibus crescendo quasdam summas varias aut decrescendo congregare, totumque si sit opus, in primam unitatem et omnium matrem resolvere. SDecem etenim varii tantummodo numeri sunt, et quicquid ultra numeretur non est novum, sed unum et idem totiens quotiens volueris replicatum. 6Quidve, quaeso, canere nisi de solis sex sonis tam varias quas supra vidisti vocum resonantias concreare, seu idem resumendo semper nunc voces intendere, nunc versus suam originem reflectere? 7Sex namque dumtaxat varii soni tres diatessaron species exprimunt, ut supra visum est, nihilque novum ultra canitur, sed replicatum est. 8Nullam ergo prorsus inter sonos ac numeros distantiam video, nisi quod vox sit continua dinumeratio, cantus autem vox cum intervallo suspensa. Quid plura? Motus est ab unitate numerus in unitatem, et motus est a sono in sono cantus identidem. !0Quemadmodum enim ab unitate movetur in aliam unitatem binarius, sic et tonus ab uno sono transit in alium ac semitonium minus. eeMAW A 8r H 10r . clamat A voces nunc A ultra in marg H

IChapter XII: That pitches and sounds have the same natural ratios as numbers. 3No-one should be seriously concerned because of my claim that numbers and pitches share natural properties: even if my tongue were silenced and my pen ceased to write, nature herself would make the same claim. 4For, pray, what is counting, if not the production of different amounts by addition or subtraction— out of the ten basic units-and, if one's efforts are to be complete, to resolve them back to the initial unity, the mother of all?50 SThere are only ten different numbers, and any reckoning beyond these is not new, but one and the same thing repeated as often as you wish. 6What is singing, I ask, other than to create all the different vocal sounds which you saw above, out of just six sounds, or, using them over and over again, raising the pitch of the voice, and at other times reverting to the original pitch? 7For there are but six distinguishable sounds which produce the three types of diatessaron, as I previously made clear: nothing new is ever sung beyond these, but is a mere repetition. 8I am aware of no great difference between sound and number, unless perhaps we can say that spoken sound possesses the quality of continuous delivery, and that a melodic line is made up of such a sound divided up by intervals.5! 9What need of more? Number moves from one whole to another, and melody likewise moves from one sound to another. !0In the same way as a two-fold number progresses from one unit to another, likewise there exists a tone or minor semitone between one sound and another. SOCf. De inst. arith. 1,14 (30, 26-28): Dicitur autem primus et compositus, quod nullus eum alter numerus metiatur practer solam, quae cunctis mater est, unitatem. 51See above, n. 34.

11Et sicut ex tribus motibus unitatis in unitatem surgit ternarius, et ex quatuor quaternarius, sicque de reliquis, ita quidem ex tribus sonorum motibus constat ditonus ac semiditonus, et ex quatuor oritur diatessaron et tritonus caeterique musici motus iuxta suam uniuscuiusque propriam qualitatem et quantitatem. 12Concrescunt numeri, conscendunt et soni. 13Decrescunt numeri, descendunt et soni. 14Infiniti sunt numeri, infiniti sunt et soni. !9Prodeunt ab unitate numeri, prodeunt ab unisono soni. !6Reducuntur ad unitatem omnes numeri, resolvuntur in unisonum omnes soni. 17Numerat homo quantum habet ad numerandum; canit homo quantum ab imo scandit in altum. !8Ergo cantor non dicas "quid habent numeri cum cantibus?," quoniam adeo soni subiecti probantur esse numeris ut nullum omnino verum de motibus musicis proferri valeat iudicium, si non fuerit per descriptas ibi proportiones tam in multiplici quam superparticulari diligenter discussum. 19Nam ut vides in ipsa figura, quatuor motus musicos integros simplices ac primos quatuor illis proportionibus subscriptos, ac sicut sesquitertia cum sesquioctava producit sesquialteram, et ipsa sesquialtera cum sesquitertia, vel e converso duplam proportionem, ita diatessaron cum tono diapente perfectum, ac idem diapente cum diatessaron aut e converso diapason perfectam, non aliter progredi potes eosdem componendo motus usque in infinitum. 20Si namque duplam cum sesquialtera componas, ut hic: II-III-VI, triplam componis, in qua diapason diapente perfecta resonet, sed non simplex consonantia. 11, (numeri) a dele H consendunt A uniuscuiusque scripsi unusquisque HA 18. soni om À iuditium À (aut) e (converso) om A

ll The number three arises out of three progressions from unity to unity; the number four from four progressions. Thus it is with the rest of the numbers. Likewise, the ditone and the semiditone are established by the progression of three pitches. From the four-fold progression arise the diatessaron and the tritone. Other musical progressions come into being, each according to its own characteristics and its range. 12Numbers become greater, sounds become higher. 13Numbers decrease, sounds become lower. 14Numbers are infinite, sounds are also infinite. 15Numbers are born of the single unity, sounds are born of the one initial sound. !6All numbers finally resolve back to this initial unity, all sounds back to the one sound. !7The range of man's counting depends on how much he needs to count; man's singing range depends upon how high he reaches, starting with the lowest pitch. 18Therefore, dear singer, you are not to say “What do numbers have in common with melodies?", since sounds are proved to be so subservient to numbers that no value judgement can be brought to bear on musical progressions unless there has been a careful analysis according to the numerical ratios I have described above-both of the multiple and superparticular species. !9For as you see in the diagram, the four prime, simple, complete musical progressions are written under the four ratios. The 4:3 ratio added to the 9:8 ratio produces the 3:2 ratio. The latter, added to the 4:3 ratio-or vice-versa-produces the 2:1 ratio.°2 Likewise the diatessaron with a whole tone added to it produces a perfect diapente, and the same diapente with the diatessaron - or vice versa - produces a perfect diapason. In just the same way, you can progress by compounding the same movements to infinity. 20]f you couple the 2:1 ratio with the 3:2 ratio in this manner, 2:4:6, you then produce the ratio 3:1 as a result of which a perfect diapason and diapente is sounded. This is not however a simple consonance. 52Concerning addition of consonances to create others, cf. De inst. mus. 2,26 (258, 19-27).

21Si vero triplam et sesquitertiam copules, non dubium quin extremi numeri surgant in quadruplam ut hic: I-III-III proportionem, quae sicuti dupla simplicem diapason, ita compositam bisdiapason consonantiam generat. 22Quid ultra dicam? 23Nulla fit utique de sonis ac vocibus concordia sive discordia canendo quae non cadat etiam in his proportionibus simul aggregatis numerando. 24Quod ut lector facilius capere valeat ubicumque tractandum est de numeris, hoc expleto primo libro, secundum a quinque generibus inaequalitatis inchoare disponimus, ubi tota natura numerorum ac inaequalium proportionum habetur. 25EXPLICIT LIBER PRIMUS. 21. ut hic 1 111 1111 in marg H (ut hic) apparet add et dele A 24. quae om A letor A in aequalitatis A proportionem pro proportionum A

211f you couple together the 3:1 and 4:3 ratios, clearly the outside numbers produce the 4:1 ratio in this manner, 1:3:4, This produces the compound consonance in the same way as the 2:1 ratio produces the simple diapason. 22What more need I say? 23Assuredly, no consonance or dissonance exists in singing which does not also fall within these ratios compounded by the arithmetical process. 2450 that the reader may grasp with greater ease the subject matter whenever we must deal with number, now that the first book is complete, I am disposed to begin the second with the five types of inequality, when I will consider the whole nature of numbers and unequal ratios. 25HERE ENDS THE FIRST BOOK.