CHAPTER I
proportions | scale | harmony
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previous page:
Fig. 4 : Modulor.
The harmonic proportional scale developed by Le Corbusier in order to ‘‘make the beautiful easier’’ is an emblematic
icon of the modernist period. It is nonetheless linked to similar attempts that are twenty-five centuries older by its will to
create works that are commensurable to human beings: Le Corbusier saw the metric system as disconnected from the
human scale, and wanted to find a new alliance between architecture and its inhabitants. One of the main critics that were
adressed to this scale was that his module was based on a six-feet man, because that was the most common size of the
hero in most English novels: besides the anecdoctical - and likely humoristic- aspect of this choice, the module is drawn
from a somewhat idealized view of the Western man - it is based on men as they should be, instead of men as they are.
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I – PROPORTION, SCALE, HARMONY
I  -  1  •    Proportions
At first glance, the word proportion seems simple to define, since its main application field today is
mathematical. However, a brief look at its historical origins demonstrates that its significance has undergone considerable transformations throughout history, which is not surprising, considering the essential
role it played in almost all fields of knowledge (especially those belonging to the quadrivium) up to the
late Renaissance, and because of the cosmological significance it still bears today. The historical and
theoretical studies of proportions have generated a large number of studies and analysis; proportions
actually constitute one of the central concepts in the development of major segments of science.
The first formalization of proportions has been established by Euclid around 300 BC as a particular relation
between three quantities 1 (A is to B what B is to C), which was called analogia. Etymologically speaking,
this Greek word designates a progressive sequence of equivalent ratio (logoi). From then, the meaning
of the two words has evolved towards describing several related concepts. The meaning of analogia
differs from the contemporary definition of analogy, which designates a relation between four quantities
(A is to B what C is to D); and which leads to the figurative meaning to the word by which an analogy
can be established between two pairs of things, in a well-known mechanism of reasoning and demonstration whose application to the relation between music and architecture has been nicely exposed
in a recent musicology thesis, of whom we will talk more in detail in chapter V 2. This last meaning is
close to the implicit definition used by Vitruvius in the first known architectuappendixral treaty in history
(1st century BC), in the famous paragraphs in which he associates the proportions between the parts of the
human body to the proportions between the elements of a temple 3. During the Middle Age, proportio
referred only to a ratio, which is a relation between two numbers; proportio as analogia was referred
1 | The livre V (book V) develops the mathematical theory of proportions for continuous quantities , the livre VII (book VII) for
discrete quantities. See Vitrac, B: Euclide – Elements; II, livres V à IX, Presses Universitaires de France, Paris, 1994.
2 | Duhamel, P. – Polyphonie parisienne et architecture au temps de l’art gothique (1140-1240), Peter Lang, Bern, 2010.
3 | See appendix 1.1 for these paragraphs, which appear in Vitruvius Pollio, Marcus - De Architectura, Book III, chapter 1, par.
III; multiple editions. I referred mainly to the Harvard 1914 translation: Morgan, Morris H.: Vitruvius, the ten books on architecture, with illustrations and original designs, prepared under the direction of Herbert Langford Warren, Harvard University Press,
1914. For a beautiful fac-simile of the latin text, see the 1511 edition by printer Giovanni Tacuino, Venezia, whose full title is: M.
Vitruvius per Iocundum solito castigator factus cum figuris et tabula ut iam legi et intelligi possit. Several English translations are
also available online. See for instance penelope.uchicago.edu/Thayer/E/Roman/Texts/Vitruvius/3*.html.
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to as proportionalitas. From the 13th century, as it has been shown by Rommevaux 4, new theories of
proportions considered the ratio of two terms as their central object; the proportion (proportio) was
associated with its original meaning. At this time however, the ratio was not considered as a number nor
as a quantity, but as a quantitative property of the quantities entering the ratio: the unification by which
ratios entered the realm of numbers did not occur before the 19th century, with the introduction of the
set of real numbers. This is the reason why particular ratios bore specific names – they were denominated, which means that they were given specific names that differed from the words used for numbers 5.
From the Renaissance, the meaning of proportions oscillated between a single ratio of two numbers and
an equivalence between two ratios. It may be legitimate to suppose that this was due to the progressive
clarification that occurred at that time between the notions of numbers and quantities. If two distance
intervals are in the proportion of 2 to 3, it means that the quantities defined by the measure of the lengths
(which are quantities) are in the same ratio as the pure numbers 2 and 3. Thus the ratio 2/3 came to symbolize and represent any proportion between the equivalent quantities (in Euclid’s and Vitruvius’ sense) 6,
and to identify them in an abstract form, in the same way as the triangle 3:4:5 symbolizes and represents
all triangles whose sides are obtained by multiplying each of these numbers by the same constant.
Physically speaking, in modern science, the main characteristic of any proportion is represented by a
pure number: it corresponds to the ratio of two measured quantities from which all non-numerical elements are eliminated. As an example, a physical dimension such as speed results from the division of a
distance by a time, and its unit (meter/second) is physically considered as a dimension. Even if speed is
inversely proportional to time and proportional to space, it is not generally considered as a proportion
as such. This is not the case for other physical quantities resulting from the division of two quantities of
identical nature: for instance, the Helmholtz number in acoustics has no physical dimension. It results
from the division of a length by another (the division of a length by a wavelength). When divided, the two
dimensional units simplify to one and disappear from the result. The same thing can be said from one of
the major constant of quantum physics, namely the fine-structure constant. Dimensionless constants are
of particular importance in physics since they keep the same value in every measurement system. This
leads to a paradox, since any measurement needs a dimension; the international system of units (SI) solved this situation by declaring that the unity, and not one, was the actual unit for dimensionless constants.
4 | Rommevaux, S. – De la proportion au rapport; in Proportions, Acts of the LIe International Colloquium on Humanist
Studies; Brepols Publishers,Turnhout, 2011; p. 27.
5 | For instance, the ratio 3 / 2 was called ‘‘sesquialter’’. incomparable) and the 4 / 3 ratio was called ‘‘epitrite’’.
6 | Pierre Cayé note that it would be useless to look for any other meaning than numerical in Vitruvius’ definition of ‘‘analogia’’. See Cayé, P. - Ars sine scientia, Architecture et mathématiques palladiennes; in Musique et Philosophie; edited by
Dufour, H., Fauquet, M., & Hurard, F. ; Klincksieck, Paris,1992.
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The fact that the unity is considered as a dimensional unit unveils the proportional nature of any measurement. Even dimensional units corresponding to the most common dimensions (length, time, mass)
are originally defined as proportions. Measuring one of these quantities is equivalent to determining its
proportional relation with respect to a particular measurement unit, which is an object or a phenomenon
from the physical world defined as a standard. The kilogram is defined as the mass of the standard kilogram in platinum-iridium, a metallic cylinder located on the third basement of the Weight and Measures
Pavilion in Sèvres, near Paris. The second is currently defined as the duration of 9 192 631 770 periods
of the radiation emitted during the transition between the two hyperfine levels of the ground state of a
specific atom (the isotope 133 of caesium). The meter is defined from the second: it is equal to the distance travelled by light in one 299 792 458th of a second, in a complete vacuum. Note that the definitions
of the second and the meter involve only integer numbers: this seemingly insignificant characteristic is
another illustration of the proportional nature of any measurement.
Fig. 5 : The standard kilogram under its three glass bells. Any mass measurement corresponds to the defi-
nition of the proportional relation between the measured object and the standard kilogram.
Measuring a dimension such as length is thus an implicit two-steps process. The first step consists in
finding the proportion between the distance separating two points and the distance travelled by light
in the prescribed time lapse; the second consists in associating this proportion with the corresponding
ratio between two numbers: length A is to length B what number C is to number D. Measuring the
mass of an object amounts to repeat this sequence of operations, starting from the ratio between
the quantity of matter that the object contains and the quantity of matter contained in the standard
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kilogram (see fig. 5). The measure of a time lapse starts by dividing the duration of an event by the
duration of about nine billion periods of the aforementioned radiation. By abstracting ratios, and by
associating them with specific quantities or numbers, they acquire an a-temporal existence that becomes independent of any event or measure. Any measurement corresponds to the act of mapping the
ratio of quantities determined at the moment of the measurement – a ratio that did not exist prior to the
operation – with one of the ratios that permanently live in the realm of numbers. The relation between
ratio and proportion can thus be written as such: a ratio is an analogy in the mathematical sense of the
term, in which one of the sides of the equivalence is a ratio of pure numbers. At the cosmological level,
this means that all measurements of the Universe are made from proportions.
Any investigation involving proportions must deal with the essential distinction between the three
notions of number, quantity and measurement. Duration, which is a quantity, is measured in seconds,
which yields a time: time is the measure of a duration. In the same way, mass is the measure of a quantity
of matter, and length is the measure of a distance. Proportion as such cannot result from the division
between two quantities, but only from the division of the measure of these quantities, which yields
the ratio itself, expressed as a number: the dimensional units simplify and disappear, leaving a pure, a
– dimensional number – or, equivalently, a number whose dimensional unit is one. Through this absence
of dimensionality, a set of proportions can determine the whole set of dimensions for a building of any
size: the definition of a basic, unitary dimensional module automatically gives all the related measurements for the building. It will also yield dimensions for a building of a different size; or for an object that
is not a building; or for a temporal event or phenomenon that is not an object; or for any element of the
physical world whose measurements or dimensions can be mapped on a set of numbers, which is equivalent to say that it possesses a geometry. To use a contemporary analogy, a set of proportions amounts
to a hidden genetic code, or more precisely, a hidden morphogenetic code. The code alone does not
contain all the instructions to generate the object: it is useless by itself. It contains the numbers that
will determine the relative quantities of each ingredient needed to create the object; the nature of the
object, which becomes the equivalent of a phenotype, will depend on the nature of these ingredients,
and on the universe in which the morphogenetic code is expressed.
As revealed by the overwhelming presence of proportions in all spheres of life in classical antiquity,
this means that a proportion extracted from a given object or phenomenon, such as the proportional
division of a string to produce consonant sounds, can be transposed to any other object or phenomenon. The different aspects of the transposition of musical scales to architectural proportions, along
with their projection on the human body, are only possible because proportions are not rooted in the
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physical world: they are meaningless numbers, just like binary numbers resulting from the scanning of
objects or events by a computer system. In that sense, the transposition of proportional sequences
between objects pertaining to different realms is not dissimilar to today’s experiments in which the
formal, meaningless nature of digital information is exploited to transpose data from a particular
realm, such as weather observations, to another one, such as acoustics or music. Such hybridizations,
which seem associated with the most sophisticated contemporary technologies, find their origin in the
oldest mythical ventures.
The precise meaning and significance of numbers and proportions is twofold. First, they carry the significance of their own status as elements of the realm of numbers. Second, they have a significance coming
from their association with some object, event or phenomenon, from which they are extracted in some way.
This induced meaning is what enables sequences of numbers or proportions to symbolize or represent
some elements of the physical reality, to the extent that a particular chain of numbers uniquely associated with a given element can become the exclusive signature of that element and denote it with
the same precision as a substantive. Proportional ratios can be used to construct chains of numbers
connected by a relation that can be deduced from the chain alone, without reference to the physical
world. They possess their own logic and their own structure, which was a major factor in the symbolical
and mythical load attributed to particular sequences in the ancient world. Among such sequences,
those that are associated with musical scales have been the object of a lot of attention during centuries.
I  -  2  •    Scales
The meaning of the word scale needs some disentanglement. Not considering its homonyms that are
out of our field of investigation, it can be associated with three notions. All three are fairly known, but I will
reformulate them here in order to focus on those of their aspects that are relevant for the present work.
The first one describes an ascending or descending series of values that can be represented spatially by
a ladder-like scheme. The values do not need to be equally spaced, but there must be some constant
relation between neighbouring values. Proportional sequences are easily represented by this method.
The second pertains to the realm of music, and corresponds to a sequence of notes defining a mode,
from which a melody can be composed; more precisely, in the physical sense of the term, musical scales
are not defined by the notes themselves, that is, by the frequencies, but by the sequence of intervals
between these notes or frequencies. Though specific to the English language (it is not the case in
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French, German and Spanish, to name a few), the homonymy between these two definitions of scale is
likely due to the fact that a musical scale can be defined by dividing a vibrating string into intervals of
equal length, and by listening to the sounds thus produced. The frequency is inversely proportional to
the length of the interval played, and can be represented spatially by a scale-like diagram similar to the
division of the string.
As it is well known, the construction of musical scales has been the object of a number of attempts and
variations in history. What is generally less known is that about all these variations are born from the
attempt to reconcile two incommensurable series, namely the powers of 2 and the powers of 3. The
natural resonance of a vibrating string produces a fundamental tone, whose frequency is doubled by
dividing the length of the string by two; it also produces several overtones, each of them corresponding
to the division of the strings in segments of equal length. The easiest to hear results from the division of
the string by the next integer, which is three: it naturally yields a tone whose frequency is three times the
fundamental one. The note thus emitted is situated within the upper octave; through an operation that
is induced by the striking similarity between a note and its own double, it is brought one octave back
to be located between the fundamental and its first octave, where it becomes the fifth, thus creating
the first two elements of the scale. The frequency of this fifth corresponds to 3/2 of the frequency of the
fundamental. The next divisions of the scale are produced by finding all successive fifths and by dividing
their frequencies by multiples of two so that they all come back within the original scale, up to the point
where a certain number of fifths will exactly correspond to a certain number of octaves.
The essential problem to which is confronted anyone trying to create a musical scale by this method is
that there is no way of finding, starting from a given note, another note that is simultaneously produced
by a succession of octaves and by a succession of fifths, and this for a simple mathematical reason: at the
frequency level, each octave is produced by powers of two, and each fifth by powers of three (more precisely by powers of 3/2, but that does not change the problem); and no power of two is also a power of three.
In other words, no combination of octaves can lead to an integer combination of fifths. This problem
lies at the centre of the question of consonant sounds and of musical harmony. It has been extensively
explored since antiquity, and can easily be formalized today. Its contemporary formulation would be to
find two integers M and N such as:
F * (3/2) M = F * 2N
M represents the number of fifths and N the number of octaves required to produce the same frequency
from a fundamental frequency F.
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By placing back all notes within the same octave, through division by the adequate power of 2, and by
simplifying by on both sides, we obtain the following equation:
(3/2) M = 2
Which is the simplest mathematical way to write the fundamental equation of musical harmony.
By solving it for M, we obtain the irrational number 1,70951129… - a number that cannot be written as
the ratio of two integers. Through a mathematical method called approximation by continuous fractions 7, it can however be approximate by fractions that get progressively closer to its exact value,
without ever reaching it.
This method yields the following values:
1/1 2/15/3 12/7 41/24 53/31 306/179 665/389 15601/9126 …
By using the number of fifths specified by the numerator, and the number of octaves specified by the
denominator, the fractions lead to frequencies that are progressively closer. The differences between
the ratio of integers and the irrational number M are successively:
71% 29% 4,3% 0,48% 0,12% 0,017% 0,0014% 0,00003% 0,0000005% …
The first two values are too large to be useful. Each of the following can be used to define a particular
musical scale – and it has actually been the case in musical theory. We can note however that the higher
are the terms involved in the ratio, the smaller is the difference; but the full reconciliation between the
two scales only occurs at the infinite.
The third meaning of the word scale is associated with an architectural concept that is simultaneously
considered as one of the most important and one of the most difficult to define. It has been explored in a
lot of theoretical texts, from Vitruvius’ architectural treaty to the most contemporary works. No consensus
seems to emerge, apart from an a-contrario definition: the scale of a building is not related to its dimensions,
and has little to do with its proportions. Philippe Boudon describes the “imposing multiplicity of phenomena involved in the notion of scale in architecture” 8. Saying that a building is at human scale can mean
that a human being can size it through his own dimensions: it should not look like if it were made for giants
7 | A fairly accessible paper on the method of continuous fractions, including a historical recapitulation, can be consulted
online at URL http://www-math.mit.edu/phase2/UJM/vol1/COLLIN~1.PDF (accessed 3/03/2014). Original reference: Collins,
Darren E – Continous Fraction; in MIT Undergrad Journal of Mathematics, vol. 1, June 1999; pp. 11-20.
8 | Boudon, P. - ‘‘Échelle’’ en architecture et au-delà; Annales de la recherche urbaine, no 82, March 1999, Paris; pp. 5-13.
47
or dwarfs; it should not bring about a feeling of awe or oppression, nor should its spaces look ridiculously
small or cumbersome. A number of rational discourses in architecture have tried to use the size and
proportions of the human body as a basis for establishing a scale of proportions from which all the dimensions of the building are derived. The results of such processes were diversely successful.
For our purposes, we will consider that scale acts as a midterm between proportion and harmony: it is
a representation, physical or through numbers, of a particular sequence of quantities related to each
other by the same proportion or ratio. As for the Le Corbusier’s Modulor, but also as it is the case for
about any artistic or architectural venture from classical antiquity to the end of Renaissance, the use of
specific sequences to build, compose or create something is seen as the path towards harmony.
I  -  3  •    Harmony and harmonics
The word harmony and its derivative harmonics are more complex, and deserve a brief etymological investigation. It is etymologically known that the first literal meaning of harmony is mechanical: it
corresponds to the junction of two wheel gears, which most likely generated a figurative meaning close
to perfect adjustment. This meaning has subsisted today in anatomy, to describe a flat articulation
between two bones in which the bones are linked together by an almost invisible, slightly indented articulation, like those that can be found in the upper jaw 9. Considering the cosmological connection that
has existed between the notion of harmony and the human body at several historical periods, this persistence of the word to designate a part of the human anatomy is worth noticing. From there, the word
itself followed several geographic paths that led it through Latin areas (harmonia, in the Roman empire)
towards Italy (armonia), Spain and Provence (armonia), Portugal (harmonia) and France (harmonie),
from which came the English word.
The meaning of the word has simultaneously spread to encompass more than two elements, to designate the correct arrangement or adjustment of several parts. It has given the antique Greek derivative
harmoste, which is all the most interesting since it designated a governor that was imposed by Sparta
to the defeated countries at the time of the Spartan hegemony (4th century BC): the harmoste was the
one who united and organized the conquered societies in what could be called a kind of pax Lacedaemonia, and was also supposed to act as a conciliator 10. Linguists have established a lineage between
harmony, harmoste and the Latin root armus for shoulder, from which emerged arma, weapons
11
and
9 | This use can be found for instance in Winslow, J.-B. – Exposition anatomique de la structure du corps humain; D’Houry
et Vincent, Paris, 1776; pp. 39-40.
10 | Daremberg, C. and Saglio, E. - Harmostai; in Dictionnaire des antiquités grecques et romaines, tome 3, vol. 1, 1999, p. 10.
11 | ‘‘Weapons’’ translates as ‘‘armes’’ in French.
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artus, member 12; which in turn leads to the Indo- European root ar, to prepare or to obtain. This root
has given the words arpana, to orient, to arrange, to order and to ornament, in a group of meanings
that is coherent with the Greek association between beauty and order; and the verb arpayani, which
adds to the previous meanings the sense of raising something or someone, and which led to the past
participle arpila, elevated 13. Arpila is considered to be at the origin of the name of the Alps – and of
mount Olympus, in an etymological loop that brings us back to Greece, which is all the most fascinating
since, as we know, mount Olympus (see fig. 6) is the cosmological centre of all the Greek pantheon, in a
cosmos that is sovereignly ruled by the very notion of harmony.
Fig. 6 : Mount Olympus
The word ‘‘Olympus’’ is etymologically connected to the word «Harmony», through an IndoEuropean root designating the notion of being raised, or to rise (Frokor | 2007)
If we add that the same Sanskrit root is at the origin of the word arpisa, which means heart, we see
that the roots of harmony dive and ramify into a series of meanings that could likely explain part of the
importance and power of the concept 14.
12 | Littré, E. – Dictionnaire de la langue française, tome 2, 1863, Paris; p. 1984.
13 | Burnouf, Émile and Leupol, L. – Dictionnaire classique sanscrit-français: où sont coordonnés, revisés et complétés les
travaux de Wilson, Bopp, Westergard, Johnson, etc., contenant le dêvanagâri, sa transcription européenne, l’interprétation,
les racines et de nombreux rapprochements philologiques, Maisonneuve, Paris, 1866; p. 49.
14 | It would be tempting to associate the same root to the prefix of the verb «to arrange», since harmony is closely related
to the notion of arrangement; but the syllab ‘‘ar’’ in this word is only a suffix, which does not support the hypothesis of a direct
descent.
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From these etymological considerations, we will propose the hypothesis that harmony is not only an
emerging phenomenon, but that it has the status of an operator. We will present arguments supporting
this hypothesis in the next chapters. Harmony operates at two levels. Looking upstream, it is a static
concept that describes the correct and proper arrangement of all the elements of a whole. To generate
harmony implies to know the position of each element within the whole, as well as the rules that control
their arrangement and determine the best and proper way to do it. There is one precise moment in
time – or a precise location in space – from which harmony begins to appear. For a musician, it may
be the moment where the hands play a chord, after each finger of each hand has been precisely and
carefully positioned. For the architect, it may be the moment where the last stone is put into place, when
the interplay between all elements begins to deploy in space. For the wanderer, it may be the moment
when a particular point of view inside a building suddenly reveals the perfection of its underlying order.
Looking downstream, harmony operates as an ordinator, in the sense of a demiurge, and as a reminder
of the order of the world: it describes and prescribes the ways things ought to be and events ought to
unroll. The strange etymological association between harmony and weapons has the curious effect of
imbuing harmony with coercive overtones. As we will see and as opposed to its common interpretation,
historically speaking, harmony is not a peaceful concept.
Harmony is the result of a precise preparation process during which all elements are installed and arranged
together to obtain a particular result, the nature of which depending strongly on the historical context in
which it is put to work. In the literature I covered, a coherent explanation on the intentions beyond the
desire for harmony is much harder to find than the prescriptions for its emergence, and the descriptions
of its effects. What should be done and obtained is often the object of long elaborations; the reasons
why this result is looked for are often briefly summed up, as if the reasons were so obvious that they did
not deserve more than a few lines. But the omnipresence of the concept across history and geography,
as well as its demiurgic role, demonstrates that its stakes are both essential and primordial. For a number of civilizations, through harmony, nature is trying to communicate something about its most subtle
fabric, and to teach societies how to live and act in relation to itself: it incorporates and establishes the
laws of the universe, which in turn define the laws that allow people to live together. Its armed wing,
which makes human societies comply with its prescriptions, is the closely related concept of harmonics.
The first occurrence of the word harmonic, which derives from the older word armonica, can be traced
back to the 16th century (around 1560) 15, where it was used as an adjective. The same word can be found
in the 14th century in the writings of beacon and polymath Oresme 16. In ancient Greece, harmonics
15 | T. Hoad (ed.) - «Harmonic» in The Concise Oxford Dictionary of English Etymology, Oxford University Press; Oxford, 2003.
16 | Interestingly enough, Oresme has demonstrated an important property of a mathematical series today known as «harmonic», namely that it was divergent. See Oresme, N. - Tractatus de commensurabilitate; quoted in Pesic, P - Music and the
Making of modern Science; MIT Press, Cambridge, 2014; p. 33.
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refers to the science of harmony, and is conventionally dated from around -330: the oldest known musical
treaty, written by Aristoxenus of Tarentum, consists of three volumes bearing the title Elements of
Harmonics 17. As soon as the 4th century BC, Aristoxenus’ musical theory rejects the use of abstract
mathematical ratios, and considers that only the ear should evaluate the precision of the musical scale:
in the first historical schism between mathematics and harmony, Aristoxenus postulates that the relation
between the musical instrument and harmony was the same as the relation between the body and the
soul, in a reasoning that corresponds to our contemporary definition of analogy (harmony is to the
instrument what the soul is to the human body); the soul was seen as resulting from the harmonious
dialogue between the four elements constituting the body.
The use of ‘‘harmonics’’ as a noun to describe what ancient Greeks called «consonant sounds» is attested
in English from 1777
18
: it describes a series of sounds generated from a single tone by multiplying
or dividing its frequency by a given number. The word harmonics usually describes the sounds resulting from divisions or multiplications by an integer number; but particular kinds of harmonics, called
a-periodic or non periodic harmonics, can be produced by using non-integer or irrational numbers.
Instruments of the bell family produce such sounds, whose potential for composition has been explored
in 20th century musical works after Risset and Mathews’ demonstration that musical sounds were essentially made of non-periodic signals 19.The Modulor subdivisions can be seen as spatial harmonics, since
they present the same kind of relation to each other than the subdivisions of the monochord, though
the proportional factor is not the same.
Harmonic as an adjective appears in several physical and mathematical expressions. The locution harmonic functions appeared during the 19th century: it corresponds to a specific class of mathematical
functions with important properties. They are defined as solutions of the Laplace equation, which has
been developed by French mathematician Pierre-Simon de Laplace (‘‘the French Newton’’) to solve
propagation problems such as the diffusion of heat in solid bodies. They will be discussed in chapter
VII; for the moment, we will only mention that harmonic functions must enforce two criteria: they must
be continuous and differentiable twice.
In mathematics, the harmonic proportion corresponds to Euclid’s proportio, which describes a relation
between three terms A, B and C such that A/B = B/C; in geometry, the harmonic division corresponds
to a relation between four segments connecting four collinear points, such that CA/CB = AD/DB. We
17 | Some researchers consider this title as likely erroneous. See Macran, H.S. - The Harmonics of Aristoxenus; Georg Olms,
Verlag, Darmstatd, 1902; p. 88.
18 | Harper, D. - Online Etymology Dictionary; www.etymonline.com (accessed 07/09/2014).
19 | Risset, J.-C. & Mathews, M.V. - Analysis of Musical Instrument Tones; in Physics Today; vol. 22, no 2, 1969; p. 23.
51
will not elaborate on the adjective harmonious, which is a subjective quality of mostly empirical definition.
It describes the quality of an object, phenomenon or environment where seemingly different elements
work in collaboration to generate a positive feeling for the observer. Though it is applied to a wide variety
of situations, it is deprived of the complexity of significations that characterizes the word ‘‘harmony’’.
I  -  4  •    The cosmological status of proportions
Proportions were everywhere present in the prescientific cosmos. As it is widely known, they were the
unifying concept underlying the notions of truth and beauty; as opposed to a common extension of the
term, which assimilates it to a ratio between two quantities, the original meaning of proportio implies
at least three elements. In the Timaeus, Plato uses the word analogia to describe a particular mathematical proportion requiring at least three quantities; he presents it as the most beautiful of all relations;
no beauty or order could exist without it:
But two things cannot be rightly put together without a third; there must be some bond of union
between them. And the fairest bond is that which makes the most complete fusion of itself and the
things which it combines; and proportion is best adapted to effect such a union 20.
Proportion in the general sense was actually described by the word summetria, or symmetry, a
word whose meaning is much wider than the elementary mirror operation for which it stands today.
As Viollet-le-Duc puts it:
We must do justice to the Greek, who invented the word symmetry, in that they never give it such
a flat meaning.(…). To stick to symmetry as inaugurated during the XVIth century, and mainly during
the XVIIth century, in Italy and in France, is an intellectual infirmity 21.
The ancient meaning of symmetry is actually very close to its etymology, which corresponds “to measure
with”, or “in common measure”. Vitruvius writes it clearly:
Symmetry also is that agreeable harmony between the several parts of a building, which is the result
of a just and regular agreement of them with each other; the height to the width, this to the length,
and each of these to the whole 22.
The simultaneous use of agreeable harmony and just and regular agreement in this sentence seems to
reduce this definition to a tautology. But the original Latin text does not actually use these words, though
they appear in several French and English translations. In Viollet-le-Duc dictionary, the agreeable harmony
becomes the word accord, by which a musical link is established: this word has the two meanings of
agreement and musical chord, that is, the simultaneous emission of consonant sounds. This is not a
20 | This passage comes from a section identified as 31cd in all versions of the Timaeus, of which several versions can be
found on line. For this particular translation: http://www.ellopos.net/elpenor/physis/plato-timaeus/genesis.asp?pg=5 (last
accessed 6/08/2014).
21 | Viollet-Le-Duc, E.E. - Dictionnaire raisonné de l’Architecture française du XIe au XVIe siècle; vol. 8, Gründ, 1869; p. 511.
22 | Vitruvius, book I, chap. 2, par. 3.
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unique example: the vocabulary of architecture is full of words whose meaning can be directly transposed to music. For instance, Vitruvius describes his proportional system by saying that all elements of
the whole building must be dimensioned from multiples of a unit element that he calls the modulus or
tone: tone is directly related to the field of music.
Symmetry, or to measure with, involves the mythical concept of commensurability. It is in this very concept
that the fundamental impulse for harmony and proportions should be sought. By referring to older Greek
canons, Vitruvius prescribes that a building should define all of its dimensions from a scale derived from
human proportions. The use of a human standard for dimensioning a whole temple, no matter how large,
means that the building enters the realm of objects whose size can be grasped by using the human
body as a measurement unit or standard, both symbolically and geometrically: it remains within the
limits of human understanding. As we will see later, since the temple is a scaled representation of the
cosmos, the same proportional scale allows human beings, by successive steps, to grasp the size of the
whole Universe. Thanks to it, for more than two thousand years, the Universe was finite and commensurable to man. Since any proportional scale can be extended towards the infinite, proportions not only
make it possible to comprehend any dimension, as huge as it may be, but also open a path towards
the transcendence associated with the ideas of infinity and eternity. The status of proportions has also
been associated with their occurrence in about all spheres of daily life. It has been mentioned that their
symbolical power might have been increased by Archimedes’ concept of lever: the proportions between
the arms of the lever is inversely proportional to the strength needed to raise a given weight. By this
observation, Archimedes was able to say that he could raise the Earth, provided he could find a fulcrum.
The implicit idea that emerges here is that very small causes can trigger very large effects: proportions
increase the potential power of mankind almost to the infinite, an idea that is symbolically present
in every proportional scale.
The Pythagorean monochord with its vibrating strings, as well as the flute with its air column, provided
evidences of the correlation that could be established between a material object, a repeatable experiment (the division of the string in equal intervals, or the adjustment of the length of the air column)
and a physical phenomenon (the increase of frequency, inversely proportional to the decrease of the
intervals size). Since the division of a string is a geometric operation occurring in space, the monochord
illustrates the proportional relation between space and frequency, that is, the relation between a spatial and a musical scales. Since there was no reason at the time to suppose the existence of a smallest
possible string element, the string could be divided in an ever increasing number of intervals, producing
higher and higher frequencies in a never-ending sequence: the monochord became the physical device
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that opened a possible way towards the infinite while maintaining the commensurability of the process.
Considering the role of proportions in ancient Greek architecture, through its proportional scales,
music provided a way to describe the cosmos, a path to access the supernatural spheres, as well as the
instructions for creating, through architecture, new elements of the physical reality: for the antique
world, it was both the legitimator and the legislator paradigm. The proportional scheme was seen as a
way to save and maintain the order of the world, a role that it also played, perhaps more radically, in the
neo-Platonician theories of the Renaissance:
Neo-Platonism, which is both the achieved synthesis of ten centuries of antique philosophy and the
matrix of the Western civilization up to today, actually rests on two piers: it is certainly a poiesis, an
instrument for production; (...) but it is also a thought of the phroura, of the watch of reality, of its
protection, of its stability, of its duration, of what Plato called the sôteria, the salvation 23.
As mentioned above, among the natural phenomena that are likely to produce sound harmonics,
vibrating strings and sounding cavities hold an important place. The resonance of a simple, cylindrical
column of air actually produces a purest signal than the vibration of a string; its physical representation
is very close to a pure sinusoid. The question may thus arise to know why antique cosmologies have
used the monochord rather than some kind of flute to generate proportional scales. It actually appears
that flutes have been used in other civilizations at other times in history, and up to the 18th century, for
the very same cosmological purposes as the monochord. The Altar of Heavens in Beijing Imperial City
is a remarkable example of a transposition of sounds to distances. It belongs to a world were several
physical dimensions, such as weigh and volume, were measured through sound phenomena.
24
A full
description of this building appears in appendix 3.12.
Vitruvius treaty is the first reliable source for architectural theories of the ancient world. Since then, but
most likely since much older times, the connection between music and architecture followed a similar
structure, in which the cosmic order was in a way or another understandable through music, transmitted
through it and transposable into architecture, so as to imbue the terrestial world with qualities attributed to the celestial spheres. This connection kept intriguing and inspiring researchers and artists of
all fields for centuries. Despite all advances in knowledge, and despite all rational or scientific attempts
at invalidating it, it remains an active topic of research and creation even today. The following chapters
constitute an attempt to find the origins of this never ending fascination.
23 | Cayé, P. - La question de la proportion: pour un humanisme du quadrivium; in Proportions: Science - Musique - Peinture
& Architecture; ed. by Rommevaux, S., Vendrix, P. & Zara, V., Brepols, no 201, p. 74 (tr. NR).
24 | Zhang, Y. - Altar and Studio: Musical Design in 18th-century Chinese Architecture; in Resonance: Essays on the Intersection of Music and architecture; edited by Muecke, M.W. & Zach, M.S.; Culicidae Architectural Press, Ames, USA, 2007; pp.
197-202. See appendix 3.12.
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