Musical practice and theoretical mathematics: Grocheio in Paris
Carol Williams1 and John N. Crossley2
‘It would make a fascinating study to trace the incursion of arabic numerals into the study
of musica.’ (Christopher Page.3)
Abstract
Grocheio stands at the frontiers between, not just different cultures, but two different sets
of cultures in thirteenth century Paris: scientific and philosophical; Greek and Islamic.
The Greek tradition from Pythagoras onwards, handed down by Boethius and others,
provided the basic theories for both number and music. Although Islamic culture had
absorbed much Greek philosophy, especially bringing Aristotle to a new prominence, it
also added great advances in mathematics. Hindu-Arabic numerals had just begun to be
used, intermittently, in Europe, but they changed the mathematics one could perform.
The Aristotelian approach used empirical observation rather than relying simply on
reflective thought. Parisian practice of music, both liturgical and popular, began to be
examined empirically by Grocheio. The conflicts engendered by these new and
orthogonal views of the world are epitomized in Grocheio's text. We shall explore these
conflicts, showing how Grocheio questioned the accepted theory. Sometimes this was
because it did not fit music as played and sung. At other times it was because it relied on
idealized and accepted metaphors (which had been used for purposes of classification and
clarification) rather than being based on scientific method. We also discuss the influence
that the new notation for numbers may have had on Grocheio and what we can learn of
their use in Grocheio’s time and subsequently.
Introduction
Paris in the thirteenth century found itself on the frontiers between two different sets of
cultures. On the one hand were the revitalized scientific and philosophical ones, and on
the other Greek and Islamic. The Greek tradition from Pythagoras onwards, handed down
by Boethius, provided the basic theories for both mathematics and music. Relatively little
studied at the beginning of the twelfth century, but dominant by the late thirteenth,
Aristotle's writings deeply influenced Grocheio. The Aristotelian approach used
empirical observation rather than relying simply on reflective thought. Parisian practice
of music, both liturgical and popular, began to be examined empirically by Grocheio.
Although Islamic culture had absorbed much Greek philosophy, especially bringing
Aristotle to a new prominence, it also added great advances in mathematics. HinduArabic numerals had just begun to be used, intermittently, in Europe, but they changed
the mathematics one could perform. Unknown until the tenth century, in common use in
the fourteenth, Hindu-Arabic numerals facilitated the arithmetic of music, and perhaps
also its notation. Calculation and empirical observation began to displace Pythagoreanism
and, to a lesser extent, the use of analogy in the writing of Grocheio.
The conflicts engendered by these new and orthogonal views of the world are epitomized
in Grocheio's text. We shall explore these conflicts, showing how Grocheio questioned
accepted theory. Sometimes this was because it did not fit music as played and sung. At
other times it was because it relied on idealized and accepted metaphors (which had been
used for purposes of classification and clarification) rather than being based on scientific
method. We also discuss the influence that the new notation for numbers may have had
on Grocheio and what we can learn of their use in Grocheio’s time and subsequently.
Music and Mathematica in Paris
Thirteenth-century Paris led the European musical world in new methods of composition.
The centres of musical importance in Paris included the Benedictine abbey of St
Germain-des-Prés, the Augustinian abbey of Ste Geneviève, the Abbey of St Denis,4 as
well as the cathedral of Notre Dame. The writings of teachers active in its schools were
highly esteemed during the twelfth and thirteenth centuries; no centre in the medieval
world of learning was as important as Paris. In the West, in general, the study of music
theory, as a branch of Mathematica, is attested by numerous treatises that have portions
devoted to music, as for example, by Robert Grosseteste (c1170--1253).5
Mathematica was studied in Paris but it is not clear that its subject matter was given wide
attention.6 By this time Mathematica broadly encompassed certain natural sciences, for
example optics and mechanics, as well as the subjects traditionally associated with the
Quadrivium, namely Arithmetic, Geometry, Astronomy and Music. So Mathematica was
rather different from our present day ideas of Mathematics, and Music was an integral
part of Mathematica.
In Music, at Notre Dame in Paris, not only organum but also monophonic and polyphonic
conductus attained their richest forms; the creation of the motet from the discant sections
of organum became a landmark in musical history. Notre Dame also provided the first
system of musical notation that clearly specified rhythmic values as well as pitches. It
also saw the development of major new musical genres. Moreover, the most significant
musical theorists from about 1240 to 1350 all worked at, or had contact with, Parisian
institutions.7
The interaction between music and mathematics by the late thirteenth century is
particularly evident in the well-known treatises of Johannes de Muris, a musician,
mathematician, astronomer and teacher at the Sorbonne. Muris's treatises found a wide
dissemination throughout Western Europe and his mathematical work, Musica
speculativa secundum Boetium,8 was an important musical text transmitted by nearly 50
manuscripts through the fourteenth and fifteenth centuries.9 His universal fame is an
indication of the great interest in music of all kinds at the University of Paris in the
Middle Ages. It points to the high position that the city held and sustained in the study of
music through its cathedral and university.
Grocheio’s treatise attests to his mathematical knowledge. It is contained in a fourteenth
century manuscript in the British Library (GB-Lbl Harl. 281) and sits in the midst of a
collection of treatises on music of the same era including ones by Guido de Arezzo,
Pseudo-Odo and Guy de St Denis. It is very distinctive, however, in that it does not give
examples requiring musical notation as do some of the others, but it has far, far more
numbers in it than any other. The way that Grocheio embraces numbers indicates that he
was very comfortable with them. By contrast the other treatises rarely employ numbers
written in numerals (as opposed to words). We are certain that Grocheio lived and
worked in Paris, presumably teaching somewhere within the University of Paris. It is
quite possible that he taught Mathematica informally as Beaujouan describes.10
Grocheio and Aristotle
Paris played a central role as the major centre for the study of Aristotelian texts in the
Latin West. While some writings were initially recovered by way of Spain and Arabic
philosophical culture, and thus coloured by Muslim Neo-Platonism, it was not until after
the Latin conquest of Constantinople in the second half of the thirteenth century that most
of the original Greek texts of Aristotle and other scientific writers were made available to
western scholars in more or less unadulterated form.
As Mews and McKinnon11 have shown, Aristotle's thought permeates Grocheio's work.
The latter used his empirical observation of music, as performed in Paris, on his way to
try to find out an adequate theory, and a true classification, of music. Grocheio’s method
is strikingly different from, but obviously related to, the arguments from analogy that can
be said to have driven intellectual enquiry from the time of Pythagoras. He does not
abandon the old views but modifies them, especially in the light of experience.
In the past Pythagorean number theory, according to Aristotle, had developed from the
discovery that the ratios of the harmony or modes could be expressed numerically. This
had led to the claim that ‘the whole heaven is a harmonia and a number’ and to the notion
of the harmony of the spheres. It produced the belief that 24, the number of notes on the
aulos, ‘equals that of the whole choir of heaven’. All this, according to Aristotle,
proceeds out of the mistaken fundamental idea that real things are numbers.12
Boethius (480-524) treated both mathematics and music. The foundation of music as
espoused by him, particularly in his De Institutione Musicae, is profoundly Pythagorean,
since it is based on the whole-number ratios seen to produce harmonious intervals of
sound. Boethius reports the tale of how Pythagoras related the properties of a solid object
to the pitch of sound that it produced when caused to vibrate. Though other early
scholars also included the same tale, it was Boethius who ‘became the principal fount and
methodological model of music theory in the Middle Ages’.13 Nonetheless, Grocheio
does not accept the view of these theorists. Here he is clearly following Aristotle, for he
says on folio 40r:14
[2.6] All these [theorists] have taken the foundation for their position in this, that
proportion, as they say, is found firstly and in itself in numbers and is attributed
through numbers to other things. But this foundation is not undoubted among the
disciples of Aristotle. For they would say perhaps that proportion is first among
prime qualities and natural forms if an utterance is assigned in order to signify
this. But it does not belong to this work to consider who of these may be speaking
the truth, but where the first principles of the sciences are considered.15
[2.6] reflects the Pythagorean number theory that espoused the idea that everything was
made of number. On the other hand Euclid and Aristotle differentiated between numbers
and ratios.16 Grocheio follows this view from the second sentence quoted.
Grocheio also displays a wide range of theoretical mathematical knowledge, both
Boethian and Aristotelian. He has a good grasp of technical matters, such as perfect
numbers, that is to say, numbers such as 6 and 28, where all the divisors add up to the
original number. Thus 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 = 14.
In his considerations on music he strives to make a classification of all music into musica
simplex, musica composita and a third kind17 musica ecclesiastic[a]. In making his
classification, empirical observation often supersedes the various arguments by analogy.
In particular, in his discussion on the number of consonances, he takes an empirical
approach (see especially [2.4], [2.5], [2.6], [2.9], folio 40r). He considers the views of
others and then questions why, if the consonances are based analogically on the number
of kinds of proportion, there are not the same number of consonances as there are kinds
of proportion, namely five. In order to follow his argument we need to consider various
kinds of proportion.
A superparticular proportion is a proportion of the form (x + 1):x, for example, 3:2 or
5:4 or 9:8. Nowadays we would often express this as a fraction: thus 3/2 or 5/4 or 9/8. A
superpartient proportion is a proportion where the first number is greater than the
second, for example 12:8 or 7:5. Sometimes we have sequences of proportions. Thus if
we consider two cubes (or cubic numbers), for example 8 = 2 times 2 times 2 and 27 = 3
times 3 times 3, then between them are the numbers 12 and 18 which lie in equal
proportions, that is to say, the proportions between adjacent numbers in the list 8, 12, 18,
27 are always 2:3. There are always two such numbers between any two cubes. If the
cubes are a3 and b3, then these numbers can be written as a2b and ab2 and the ratio will
always be a:b.
Grocheio’s argument is in essence as follows:18
Some say that consonances are infinite in number, others that there are three,
arguing by numbers following Pythagoras and Nicomachus the arithmetician.
Plato said in the Timaeus that between two cubes two proportions are always
found. Boethius followed this in his De Musica.
All said proportion is found in number and number is found in everything.
However, if proportion comes from numbers they do not show the cause or the
number of consonances.
Although he eventually concludes in [3.1], folio 40r, that there are indeed only three
consonances, he is still dissatisfied (in [2.10], folio 40r) that, from his empirical
observations, it ‘seems difficult to define the reason for the number of consonances.’
[2.10] Further, if a consonance be natural, it can be recognised by its end. That
which is natural is best shown from its end, as Aristotle says in the second book
of the Physics: ‘For initially the end moves the efficient force and finally
completes the work.’ If, indeed, we are dealing with music, knowledge of it is
sufficient through form. Therefore, because of these things, and of many others, it
seems difficult to define the reason for the number of consonances.19
Sometimes, however, he is still moved by analogy. This is unsurprising as Grocheio is
cognisant of his predecessors, and for them analogy was a favourite way of understanding
the world. Thus in [4.8] folio 40v, (which we quote in full later) he focuses on ‘7 gifts of
the spirit and 7 planets in heaven and 7 days in a week’. In [3.3], in concluding his
discussion on the number of consonances, he invokes the Trinity. Here he manages to
combine both analogy and experience saying, ‘And perhaps just as He is in the glorious
Trinity, so in a certain way He teaches through this experience’. He then continues with a
delightful analogy with the Trinity.
[3.3] For there is one first harmony, like a mother, which is called the
diapason by the ancients, and another, like a daughter, contained in it,
called the diapente, and a third proceeding from them which is named the
diatessaron and these three, sounded at the same time, give the most
perfect consonance. And perhaps certain Pythagoreans influenced by a
natural inclination sensed this, not having dared, however, to express it in
such words but speaking of it in numbers through metaphor.20
Grocheio and algorism
The numerals which we familiarly use today, namely the Hindu-Arabic numerals, made
their way only slowly from India to the West. Coming from Asia, they were discussed in
a lost work by al-Khwarizmi,21 who gave his name to the process in the Latin West. He
came from Khwarezm, which is now Khiva, Uzbekistan. Then the numerals came
through the Maghreb into Europe. The transmission through Europe seems to have
begun in Spain and Italy. It gave rise to a major change in thinking. As Rashed says,
‘L’histoire de l’arithmétique arabe dans la civilisation latine, n’est ... pas moins que
l’histoire de l’interaction entre deux civilisations.’22
Roman numerals had been used for a long time and, although the abacus was in common
use, the answers to calculations were written down in Roman numerals. Because of this
the numerals do not reflect the workings of the abacus. When we turn to the HinduArabic numerals and their use we find two novelties. On one hand there are the actual
symbols used, the ‘figures’, 9.8.7.6.5.4.3.2.1 and 0,23 and on the other there is the system
of (decimal-)place notation, where moving to the left means multiplying by 10. The
actual figures had been used by Gerbert around 1000 A.D. in his abacus, but he did not
use the place notation. What was strange to practitioners in the thirteenth century about
the use of the numerals was the way that one figure could mean different things
depending on how far to the left it occurred. Thus ‘3’ might mean ‘three’ or ‘thirty’ or
‘three thousand’, while the Roman ‘L’ always and only meant ‘fifty’. Hindu-Arabic
numerals appear therefore to have been regarded as different from letters and seem,
originally, to have been treated in the same way as diagrams, rather than as letters from
the alphabet. (See also below and n.30.)
The two novelties also permitted a very different kind of calculation. Previously, numbers
were written in Roman numerals but calculations were carried out on an abacus which
was, in principle, like those we see in oriental shops today, but in practice was a board
with movable stones, or even just dust in which one drew. With the advent of HinduArabic numerals together with the place notation system, it was now possible to write
down the whole of a calculation, on paper, as a permanent record of how the result was
achieved. By contrast, the working on an abacus disappeared as the calculation
progressed. However, this new technique of calculating, known as algorism, only reached
England in the twelfth century. Acceptance of this quite revolutionary system was slow
and spasmodic but was greatly facilitated by the popular accounts of Sacrobosco and
Villedieu.24
There is some hint as to how novel the Hindu-Arabic figures were in the earliest
European version we have of (what is believed to be) work of al-Khwarizmi.25 The scribe
in the thirteenth century manuscript in Cambridge (Cambridge University Library, Ii.vi.5,
folios 104r-111v) has left far larger gaps than necessary between the two dots standardly
enclosing a numeral. Indeed, in some cases in this Cambridge manuscript, the numeral is
omitted altogether,26 though in the recently discovered New York manuscript (New York,
Hispanic Society of America, H C 397/726, 17r-24v) all the Hindu-Arabic numerals are
present.27 (This manuscript is probably also thirteenth-century.)
Besides the Harley manuscript of Grocheio in the British Library mentioned above there
is another at Darmstadt (D-DS 2663).28 The two manuscripts are very similar, but they
differ in their use of Hindu-Arabic and Roman numerals. This is perhaps symptomatic of
the less than total acceptance of such numerals, which were only just beginning to be
used in the thirteenth and fourteenth centuries.
What seems important for Grocheio, and for us, is that the Hindu-Arabic numerals are
‘figures’. That is to say, they are akin to diagrams rather than extra letters added to the
Roman alphabet.29 He carries this idea of using novel signs to denote, in his case, musical
notes, into his discussion of musical notation for the longa, in [18.3] on folio 46r.
[18.3] And they used to find certain general signs and indeterminate figures
in order to represent sound, by which they were not able to represent cantus
or sound adequately. And so others added a definition. For they placed one
square figure, having a straight line descending or ascending from the righthand part, which they called a long, and they distinguished it in a twofold
[way]: a perfect and an imperfect long.
The word figura appears only to have come into use in this sense of a symbol for a
numeral with the use of algorism, thereby replacing30 the Greek word schema (whence
our English word ‘scheme’) around Grocheio's time.
The use of numerals in the manuscripts
The two manuscripts of Grocheio’s work differ in their use of numerals. The Darmstadt
one is consistent in using Roman numerals, the Harley sometimes uses Roman,
sometimes Hindu-Arabic.31 Thus we find:
Harley, [1.3], folio 39v: Invenit unum in dupla proportione ad alterum sicut sunt
.12. ad .vi.
Darmstadt, [1.3], folio 56v: Invenit unum in dupla proportione ad alterum sicut
sunt .12. ad .6.
Further on in the same paragraph Harley has a gap or erasure and Darmstadt incorrectly
repeats ‘.12. ad .6.’, the same as the previous proportion. In fact it should be ‘.12. ad .8.’,
the proportion for the diapente.32
Whether the uncertainties surrounding the use of the Hindu-Arabic numerals have
anything to do with the extra large spaces in the Cambridge manuscript, mentioned
above, remains to be investigated.
Now the use of Hindu-Arabic numbers in Grocheio’s treatise as it is expressed in the
Harley manuscript is unpredictable. In the discussion concerning the number of
consonances a range of practices appears. Three is spelt out as ‘tres’ in reference to the
three consonances and to the three perfections in sounds [folio 40r] and again in the
wonderful analogy, mentioned above, of the mother as diapason, the daughter as diapente
and a third proceeding from them – the diatessaron, where the ‘three’ referring to
consonances is spelt out, not in figures, but as ‘tres’ [folio 40r].
The Harley manuscript sometimes uses the Latin words for numbers, and occasionally
Roman numerals. Thus in [4.8] folio 40v, we find four ‘7’s and once the Latin word for
‘seven’:
Others, however, reduce them all to 7. ... These people, however, take
the source of their saying from the sayings of the poets... saying that
there are 7 gifts of the spirit and 7 planets in heaven and 7 days in a
week, by the multiple repetition of which the whole year is measured.
And similarly they say there are seven concordances in sound.33
Again in [4.9], folio 40v- 41r we find the number of principles written as Roman ‘vii’:
For the seven stars, with their forces, have sufficed for the diversity of
generation and decay of the whole universe. And therefore, it was
reasonable to posit the 7[ written ‘vii’] principles in human art... 34
However, the Harley scribe appears to have pronounced the Hindu-Arabic numerals in
Latin for he uses Hindu-Arabic numerals in combinations to be read in Latin, e.g.
‘.4.us’(in [20.5], folio 47r) for ‘quartus’ and ‘.3.bus’ for ‘tribus’ in [4.11].
Grocheio certainly makes use of Hindu-Arabic numerals for simplicity when he writes in
[4.13], folio 41r, of a proportion as ‘.256. ad .243.’
There are several possible explanations for the differences in the recording of the
numerals in the two manuscripts. For example, perhaps the scribe (or scribes) did not
understand Hindu-Arabic numerals and just copied them like an artist rather than a writer
without trying to understand them. It might also have been that the only characters (in the
sense we use the word today) that he thought of were Roman letters (which of course
includes Roman numerals) and he treated the unfamiliar Hindu-Arabic numerals more
like diagrams (see footnote 25 above). A third possibility, which looks more plausible for
the Harley manuscript, is that the manuscript was dictated, and that the scribe was
completely at ease, writing down either the Latin word for the number or the Roman
numeral or the Hindu-Arabic numeral, depending on which way he envisaged it. What is
clear, however, is that Hindu-Arabic numerals were not the standard, even when these
manuscripts were written.
The situation with proportions is more complicated and it must be remembered that
Euclid and Aristotle distinguished these from (pure) numbers. In practical terms it was
easy to create a proportion of, say, 9:8 on a monochord, thereby producing a tone. But if
one wanted to produce a wider interval, it was quite practical to do this in stages by using
simple superparticular (see above) or other proportions, and then repeating them. In terms
of the arithmetic, however, Hindu-Arabic numerals greatly facilitate the description. Thus
on folio 41r we find:35
[4.14] Further, a ditone is a concordance containing 2 tones, which compared to a
preceding sound is seen to be proportional as 81 is to 64.
This comes immediately from constructing the (superparticular) proportion: 8 units plus
one unit to 8 units twice, successively yielding a proportion 9:8 times 9:8 = 81:64. Surely
no-one would wish to construct the latter proportion directly.
We believe we have now begun the task quoted from Page at the beginning:
It would make a fascinating study to trace the incursion of arabic numerals into
the study of musica.36
Conclusion
We have observed the conflicting streams at whose confluence Grocheio stands. We have
seen the incursion of the Aristotelian approach, opposing the earlier dominance of
Boethius and, under Aristotle's influence, Grocheio empirically observing music in Paris
in order to understand and to classify it. However, the traditions of the past are not
completely overturned. They are modified in the light of experience and observation, not
just because of pure theory and speculation.
The influence of Hindu-Arabic numerals and algorism is more subtle. We do not know to
what extent using the new signs (figurae), as opposed to the old letters, opened
Grocheio’s mind, for we do not know exactly what was in the original manuscript.
Nevertheless the text demonstrates a significant knowledge of mathematics and a facility
with numbers, which is quite dramatic. In 1977, Evans37 wrote an excellent analysis of
the change from the abacus to algorism. Page quotes from it:38 ‘it may be no coincidence
that the numerical bases of musical notation did the same [i.e. took a new turn at this
time].’ Given Grocheio’s mathematical prowess it would seem a natural extension to
create new symbols in music, now that new symbols had been introduced in mathematics.
We have still much to learn from Grocheio, not only about music but also how a holistic
approach, which he had, can lead to deep insights. Fortunately Grocheio was a great
teacher also: his text is a pedagogical lesson in itself. It manages to combine the
conflicting demands of Pythagorean and Aristotelian philosophical approaches, and
Grocheio seems to have been remarkably well-versed in mathematics as well as music.
1
School of Historical Studies, Monash University, Clayton, Victoria, Australia 3800. Email:
Carol.Williams@arts.monash.edu.au
2
School of Computer Science and Software Engineering, Monash University, Clayton, Victoria, Australia
3800. Email: John.Crossley@infotech.monash.edu.au
3
Christopher Page, Discarding Images: Reflections on Music and Culture in Medieval France, (Oxford
[England] and New York: Clarendon Press; Oxford University Press, 1993)., 131.
4
Gordon Anderson, ‘Paris I’ in New Grove Dictionary of Music and Musicians, ed. Stanley Sadie,
Macmillan, London, 1980.
5
Grosseteste’s writings on music are contained in De artibus liberalibus, and in the treatise on phonetics
De generacione sonorum. See Samuel Harrison Thomson, The Writings of Robert Grosseteste, Bishop of
Lincoln, 1235-1253 (New York: Kraus Reprint, 1971). It was his view that music regulates motion in time
and space, both celestial and earthly and that it serves natural philosophy, having the power to restore to its
true state any deficiency of harmony or proportion. See Mary Berry, Groseteste, Robert (Grove Music
Online) (ed. L. Macey, [accessed 31 August 2004]; available from http://www.grovemusic.com. Also
Nancy Van Deusen, ‘Thirteenth-Century Motion Theories and Their Musical Applications: Robert
Grosseteste and Anonymous IV’ in The Intellectual Climate of the Early University: Essays in Honor of
Otto Grundler, ed. Nancy Van Deusen, Studies in Medieval Culture (Kalamazoo: Medieval Institute
Publications, 1997). However Grosseteste does not appear to have written on mathematics (Charles
Burnett, Algorismi vel helcep decentior est diligentia: the Arithmetic of Aadelard of Bath and his Circle,
Mathematische Probleme im Mittelalter, der lateinische und arabische Sprachbereich, ed. Menso Folkerts,
Wiesbaden, 1996, 245, n.79), though he did know how to write Hindu-Arabic numerals (Burnett, Why we
read Arabic numerals backwards, Ancient & Medieval Traditions in the Exact Sciences: Essays in Memory
of Wilbur Knorr, ed. Patrick Suppes, Julius M. Moravscik and Henry Mendell, CSLI Publications,
Stanford, California, USA, 2000, 201).
6
Guy Beaujouan, ‘L’enseignement de l’arithmétique élémentaire à l’Université de Paris aux XIIIe et XIVe
siècles’ [1954], reprinted in Par raison de nombres (Aldershot, 1991) XI, 98,; see also Chartularium
Universitatis Parisiensis, ed. Heinrich Denifle and Emile Chatelain, 4 vols (Paris, 1891-99), 1:78, no. 20.
7
Hieronymus de Moravia (Tractatus de musica); Johannes de Garlandia (De mensurabili musica);
Lambertus (Tractatus de musica); Franco of Cologne (Ars cantus mensurabilis); St Emmeram Anonymus;
Anonymus 4; Jacobus of Liège (Speculum musice).
8
Christoph Falkenroth, ‘Die Musica Speculativa des Johannes de Muris: Kommentar zur Uberlieferung
und kritische Edition’, Beihefte zum Archiv für Musikwissenschaft; Bd. 34 (Stuttgart: Steiner, 1992). Also
Susan Fast, ed., Musica ‘Speculativa’, Wissenschaftliche Studien = Musicological Studies; Bd. 61 (Ottawa:
Institute of Mediaeval Music, 1994).
9
Lawrence Gushee, ‘Jehan des Murs’ in New Grove Dictionary of Music and Musicians, ed. Stanley Sadie,
Macmillan, London, 1980.
10
Beaujouan, ‘L’enseignement’, XI, 97.
11
Constant Mews and Leigh McKinnon, ‘Aristotle, Music and Analysis of the Liberal Arts in the Ars
Musicae of Johannes de Grocheio’, this volume, ??-??
12
Aristotle, Metaphysics, 985b32—986a2, 1090a20—23, 1093a28—b4; On the Heavens, 290b21—3.
13
Claude Palisca, ‘Theory, theorists’ in New Grove Dictionary of Music and Musicians, ed. Stanley Sadie,
Macmillan, London, 1980.
14
Our quotations are from a new translation of Grocheio’s work currently being prepared by a team at
Monash University comprising (in alphabetical order) John Crossley, Catherine Jeffreys, Leigh McKinnon,
Constant Mews, and Carol Williams. Numerical references in this paper are to our numbering in this
translation.
15
[2.6] Omnes autem isti fundamentum sue positionis accipiunt in hoc quod proportio ut dicunt primo et
per se in numeris invenitur et per numeros est aliis attributa. Sed istud fundamentum apud discipulos
aristoteles non est certum. Dicerent enim forte proportionem primo esse inter primas qualitates et formas
naturales si vox sit imposita ad hoc signandum. Quis autem istorum verum dicat non est huius negocii
pertractare sed ubi prima principia scienciarum pertractantur.
16
Proportion was regarded as more fundamental than pure number, which is essentially the same as saying
that geometry has precendence over arithmetic.
17
‘Sed tercium genus est quod ex istis duobus efficitur et ad quod ista duo tamquam ad melius ordinantur.
Quod ecclesiasticum dicitur.’
18
[2.4] Quidam autem vulgaliter loquentes dixerunt esse consonancias infinitas. Sed sue positionis nullam
assignaverunt rationem.
19
[2.10] ‘Adhuc autem si consonancia sit naturalis, ex fine cognosci habet. Naturalis enim potius ex fine
demonstrat. ut ait aristoteles. secundo physicorum. finis enim primo movet efficientem et ultimo complet
opus. Si vero musica. eius cognitio sufficiens est per formam. Propter hec itaque et propter talia plura
difficile videtur assignare propter quid de numero consonanciarum.’
20
[3.3] Et forte sicut est in trinitate gloriosa. Ita quodam modo in hac experiencia docet. Est enim una
prima armonia quasi mater, que dyapason ab antiquis dicta est. Et alia quasi filia in ista contenta dyapente
dicta. Et tercia ab eis procedens que dyatessaron appellatur. Et iste tres simul ordinate consonanciam
perfectissimam reddunt. Et forte hoc senserunt quidam pytagorici naturali inclinatione ducti non ausi tamen
sub talibus verbis exprimere sed in numeris sub methaphora loquebantur.
21
See J.N. Crossley and A.S. Henry, Thus spake al-Khwarizmi, Historia Mathematica 17, 1990, 103-131
and Menso Folkerts: Die aelteste lateinische Schrift ueber das indische Rechnen nach al-Hwarizmi. Edition,
Uebersetzung und Kommentar. Muenchen 1997 (Bayerische Akademie der Wissenschaften, Philosophischhistorische Klasse, Abhandlungen, neue Folge, Heft 113).
22
Roshdi Rashed, Preface to André Allard, Muhammad Ibn Al-Khwarizmi. Le calcul indien (algorismus).
Versions latines du XIIe siècle, Leuven, Belgium, 1992, unnumbered page: ‘The history of Arabic
arithmetic in Latin civilisation is nothing less than the history of the interaction between two civilisations.’
23
This was the standard order and followed the Arabic way of writing. It was only later that people began
to write them in the present day order: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
24
There were many copies of of Alexandre de Villedieu, Carmen de algorismo, and John of Holywood
(Iohannes de Sacrobosco), De arte numerandi, circulating in the thirteenth century and indeed they were
rendered into English very early. See Robert Steele, The earliest arithmetics in English, Early English Text
Society, Extra series 118, London: H. Milford, 1922.
25
J.N. Crossley and A.S. Henry, Thus spake. Recently a complete version of this manuscript has been
found by Folkerts, Die aelteste.
26
It is not clear whether the figures were inserted into the Cambridge manuscript at another time or by
another person. It is also possible that the same process was employed in the Harley manuscript since the
space occupied by Hindu-Arabic numerals sometimes appears rather larger than necessary.
27
See Folkerts, Die aelteste.
28
Ernst Rohloff, ed. Die Quellenhandschriften zum Musiktraktat des Johannes de Grocheo. In Faksimile
hg. nebst Übertragung des Textes und Übersetzung in Deutsche, dazu Bericht, Literaturschau, Tab. U.
Indices (Leipzig: Deutscher Verl. F. Musik, 1972). This followed his earlier work: Ernst Rohloff, Studien
zum Musiktraktat des Johannes de Grocheo (Leipzig: 1930) and Ernst Rohloff, Der Musiktraktat des
Johannes de Grocheo nach den Quellen Neu Herausgegeben mit Übersetzung, Deutsche und
Revisionsbericht. Series of Media Latinitas Musica(Leipzig Gebrüder Reinecke, 1943)
29
Burnett has independently noted this in his unpublished manuscript, The semantics of Indian numerals in
Arabic, Greek and Latin (September 2004), writing ‘The retention of the visual order of the Indian
numerals, I think, adds to the evidence for their being conceived as different from written text, ... Numerals
were hors de texte, and like pictures, or geometrical diagrams, they kept the same directionality as they
passed from one language context to another.’
30
See The Oxford Dictionary of English Etymology, ed. C.T. Onions with the assistance of G.W.S.
Friedrichsen and R. W. Burchfield, Oxford 1966, 355.
31
In the other treatises in the Harley manuscript the same variation occurs, but there are very few
occurrences of numerals in any of the other treatises.
32
Page, in Discarding Images, 172, appears to err in his description of Grocheio's use of Roman versus
Hindu-Arabic numerals. Moreover, in Rohloff’s transcription we find: ‘Invenit unum in dupla proportione
ad alterum sicut sunt .xii. ad .vi’
33
Alii autem omnes ad .7. reducunt. ... Isti autem ex dictis poetarum originem sui dicti capiunt. et cum hoc
rationes probabiles adducunt. dicentes esse .7. dona spiritus. et in celo .7. planetas et in septimana .7. dies
quibus multociens resumptio totus annus mensuratur. Et similiter in sonis esse concordantias septem
dicunt.
34
Ad diversitatem autem generationum et corruptionum totius universi .vii. stelle cum earum virtutibus
suffecerunt. Et ideo rationabile fuit ponere in arte <41 r> humana .vii. principia ...
35
[4.14] Dytonus autem est concordantia continens .2. tonos que sono precedenti comparata sic
proportionari videtur sicut .81. ad .64
36
Page, Discarding Images, 131.
37
Gillian R. Evans, 'From Abacus to Algorism: Theory and Practice in Medieval Arithmetic’, British
Journal for the History of Science, 10 (1977), 114-131.
38
Page, Discarding Images, 137, n.70.