Few excerpts from "The Harmony of the World by Johannes Kepler",
(an English translation of "Harmonices Mundi", published at Linz, 1619)
translated with an Introduction and Notes by E.J. Aiton, A.M. Duncan,
and J.V. Field, published by American Philosophical Society; April 1997,
(Memoirs of the American Philosophical Society, Vol 209).
Note 1: In the end I have included Kepler's comments on
Fludd's work in their entirety.
Note 2: The title pages and contents of the English translation
can be seen at:
http://www.iki.fi/~kartturi/Kepler/www.library.vcu.edu/pdfgif/erpdf/J2001-FALL-568.pdf
including the introduction and the list of chapters of the book V,
plus its first chapter, On the Five Regular Solid Figures.
For the whole book, see for example at Amazon:
http://www.amazon.com/exec/obidos/tg/detail/-/0871692090/qid=1058220594/sr=8-2/ref=sr_8_2/002-6520022-2575238?v=glance&s=books&n=507846
or:
http://www.amazon.com/exec/obidos/ASIN/0871692090/starshopcom-wireless-20/002-6520022-2575238
From the page viii of translators' preface:
When Kepler declared that it was of no account whether
his book would be read by the people of the present or
of the future, he evidently sensed that it might not
be understood by his own contemporaries. He has had to
wait somewhat longer for understanding than the hundred
years (*) he anticipated. Yet today his world harmony is
seen to possess an essential element of truth.
Modern astronomers, of course, do not express the
idea in terms of musical harmonies but attribute the
structure of planetary systems to the operation of
principles of dynamic resonance. In a more general
sense, Kepler's use of formal causes is in line with
the modern physicists' use of symmetry principles
in the investigation of nature.
(* See the page 7/13 (page 391 of the book) of the
above PDF-document at VCU Libraries Electronic Reserves)
The chapters of Book III
Chapter
I. The origin of the consonances from their own proper
causes.
II. On the seven harmonic divisions of the string, and
the same number of forms of Minor consonances.
III. On harmonic means, and the trinity of consonance.
IV. The origin and nomenclature of the conventional or
melodic intervals.
V. The division and nomenclature of the consonances
by their conventional intervals.
VI. On the kinds of melody, hard and soft.
VII. The proportion of all the eight conventional sounds
of one octave.
VIII. The splitting of the semitones and the arrangement
of the smallest intervals in the octave.
IX. On the stave, lines, signs, and letters representing
sounds; on the system, the notes and the musical scale.
X. On the tetrachords and the syllables, ut, re, mi, fa,
sol, la.
XI. On the combination of the major systems.
XII. On the impure consonances which originate
from combination.
XIII. On simple tuneful melody.
XIV. On modes or tones.
XV. Which modes fit which moods.
XVI. On figured melody or melody in harmony.
(What Harmonized or Figured Melody Is.)
+ Appendix: Political Digression on The Three Means.
On page 161, Chapter II of the Book III, the footnote 55
of the translators:
The foregoing propositions established divisions of the
string which produce all the consonances; that is,
unison (1:1), octave (1:2), fourth (3:4), fifth (2:3),
major third (4:5), minor third (5:6), major sixth (3:5),
and minor sixth (5:8). The following propositions will
show that there are no further harmonic divisions of
the string. This is essential to Kepler's theory,
for such further divisions would introduce dissonances.
On page 178: "This therefore, is the origin of the dissonant
melodic intervals, to which we shall give their names a little later on."
Associated footnote 72 by translators:
The reader may find it more convenient to have the names
at this point. They are major tone (8:9), minor tone (9:10),
semitone (15:16), and diesis (24:25).
On page 163, chapter II of the book III, we have:
Corollaries
I. The harmonic divisions of a single string are seven in number,
not more.
II. The expansion of the numbers which are characteristic of divisions
occurs in the following manner. To begin with, the whole is expressed
in the form of a fraction, that is to say with unity above as numerator,
and unity below for denominator. Then each number separately is put as
a numerator, and the sum of the two is put as a denominator in each case.
Hence from any given fraction two branches arise, until from the sum
occurs the number which indicates an unconstructible figure.
__ 1/6 ..... 7
1/5 __
.. 5/6 ..... 11
1/4
/ ..
/ 4/5 ............ 9
1/3
\
/ \
/ 3/4 ................... 7
/
1/1 -- 1/2
. \
. \ 2/5 ................... 7
. \ /
. /
. 2/3
. \ 3/8 ............ 11
. \ ..
. 3/5
. ..
. 5/8 ............ 13
.
.
.
.
.
-- 1/2 Same
I found these seven divisions of the string first with hearing as guide,
in other words the same number as there are harmonies not greater
than a single diapason (= octave). After that I dug out the causes
both of the individual divisions and of the number of the total,
not without toil, from the deepest fountains of geometry. Let the
diligent reader read what I wrote about these divisions 22 years
ago in The Secret of the Universe (Mysterium cosmographicum),
Chapter XII (*), and ponder how in that passage I was under a delusion
about the causes of the divisions and the harmonies, mistakenly
striving to deduce their number and the reasons from the number
of the five regular solid bodies; whereas the truth is rather that
both the five solid figures and the musical harmonies and divisions
of the string have a common origin in the regular plane figures.
The footnote 58 of the translators:
The seven harmonic divisions were illustrated in exactly the same way
as here in the Mysterium cosmographicum, Chapter 12.
At that time, as he relates, Kepler attempted to derive the harmonies
from the regular solids but later found the causes of the harmonies
in the constructible polygons. Evidently he had expected to find
inspiration in Ptolemy's Harmonica but when he was eventually
able to read the work, he found that his own theory of the causes
of the harmonies was wholly original.
AK's note: In OEIS-terms we proceed each branch of the
fraction-tree A020651/A086592, until we encounter a denominator
which belongs into A004169: 7,9,11,13,14,18,19,21,...,
"Regular polygons not constructible with ruler and compass".
On page 168, Chapter III of the Book III, the footnote 65
of the translators:
The development of the modern idea of musical chords proceeded
in two stages. The first involved the recognition, above all by Zarlino,
that three simultaneously sounding notes derived by a mathematical
division of the fifth formed a unity. The second stage began with
Johann Lippius, who introduced the idea of chord inversion.
Kepler contributed nothing to this second stage, for he always
regarded the base note as the reference note, a fact which to some
extent restricted his concept of major and minor tonality.
See Michael Dickreiter (1973): Der Musiktheoretiker Johannes Kepler
(Bern and Munich), 154.
In modern terminology, the six chords described by Kepler consist
of the major and minor chords in root position and in first and
second inversion, but he regarded all of them as independent.
On page 193, Chapter VII of the Book III, the footnote 89
of the translators:
Kepler's scales in hard and soft melody (with the modification
introduced in the next chapter for the hard kind) are equivalent
to the Aeolian and Ionian modes introduced in 1547 by Henricus
Glareanus, which have remained as the modern minor and major
scales respectively. Kepler, like Zarlino, however, also classified
the traditional modes as soft or hard, according to whether they
contained minor or major thirds and sixths. Kepler's genera soft
and hard relate more to the melodic concept of the traditional
Church modes than to the harmony-oriented music of the
seventeenth to the nineteenth century. Yet by emphasizing the
fundamental distinction of two types of tonality, he anticipicated
the modern concept more strongly than Zarlino.
See Gioseffo Zarlino (1558), Istitutioni harmoniche (Venice), pp. 182 and 210.
The traditional modes were obtained by taking different notes
as the starting point of the central octave of the two-octave scale.
See Ptolemy, Harmonica, Book II, Chapters 8-11.
On Kepler's concept of tonality, see Dickreiter (1973), 160-170.
On page 230, Chapter XIV of the Book III, the footnote 150
of the translators:
Kepler here follows the erroneous nomenclature of Henricus
Glareanus as a result of which he interchanges the Phrygian
and the Dorian modes. ...
From the Chapter II
"The Number and Kind of the Faculties of the Soul in Accordance with the
Harmonies", of the book IV, at page 309 we find:
However, this perception of the harmonies in the inferior faculties
of the soul is dull and dim, and in a sense material, and under a cloud
of ignorance; for they do not know that they perceive, as when we see
something but do not notice that we are seeing it. Such are those
emotions and terrors celebrated by the Stoics, which are natural and
unintentional, and involuntary. Such also is the natural feeling of
hate or love, especially with a remarkable predisposition, as judging
the goodness of another soul, or its resemblance to one's own,
by the symmetry of the parts of the body and the qualities of voice
and temperament, he is wonderfully inflamed towards it. Therefore,
the crazy youth loves the girl; and he does not know why, nor what
he loves in her most of all, because no courtesan whom he meets can
surpass her, if it is an improper love, nor can any marriageable girl,
if it is a legitimate one. But if a physiognomist comes on the scene,
he finds in both personalities some resemblance of character; and
if the characters are defective, they give occassion for perpetual strife
in the marriage, but if they are good, for perpetual tranquility in
life.
Editors' footnote 28, on the preceding page tells:
Horoscopes, which take their name from the ascendant (horoscopus),
the degree of the zodiac rising above the horizon at the moment of birth,
involve much else besides the influence of the aspects that was
accepted by Kepler. Of judicial astrology in general, he remarked that
it was like the stupid daughter who, thanks to her incantations,
clothes and feeds her mother (astronomy), who is as wise as she is poor.
See Tertius interveniens, thesis VII (KGW 4, p. 161). Indeed Kepler
owed his own position to his employers' interest in astrology;
the writing of horoscopoes and prognostications of events was part
of his official duties. Among the Kepler manuscripts in Leningrad
are some 800 horoscope diagrams in which he has written in dates and
the corresponding positions of the planets, though only a few are
accompanied by interpretations. See List (1971), p. 129. Some of
his forecasts of events were in fact remarkably successful and
served to enhance his reputation. In his Prognosticum for 1618
(KOF, vol. 1, pp. 481-483), for example, he predicted great
upheavals in the world in May; and on 23 May of that year,
the thirty years' war started when two Prague councillors were
thrown from the windows of the Town Hall.
Kepler's most famous horoscope was the one that he wrote for
his patron Albrecht von Wallenstein in 1608 (KOF, vol. 1, pp. 386-391).
Later, in 1625, interpreting the horoscope at Wallenstein's request,
Kepler predicted that in March 1634, there would be a terrible
confusion in the country that would affect him. In fact Wallenstein
was assasinated on 25 February 1634. The augmented horoscope may be
found in KOF, vol. 8, pp. 348-358. If Kepler had lived to see
this prediction come to pass, he would have judged it to be merely
a coincidence. When he appended to this horoscope the remark,
"I have only devised horoscopes when I was sure that my work was
intended for somebody who understands philosophy and is not affected
by contradictory superstition" (KOF, vol. 8, p. 348), it seems
clear that he would have preferred Wallenstein not to take it
seriously, though he was not in a position to question his powerful
patron's faith in astrology.
Kepler's commentary on R. Fludd, Utriusque cosmi maioris scilicet et
minoris, metaphysica, physica atque technica historia (Openheim, 1617-1618).
(AK's note: A few illustrations from this book can be viewed at
Beinecke Rare Book and Manuscript Library at:
http://highway49.library.yale.edu/photonegatives/
by entering "Fludd" as a keyword to the search form).
This work is an attempt to describe the philosophy of the harmonious
design of the cosmos and the corresponding harmonies of man.
Besides drawing on the Cabala and the Hermetic texts, Fludd
also assimilates the ideas of the Rosicrucian philosophy, of
which he was one of the chief exponents. The Rosicrucian manifestos,
known briefly as the Fama and Confessio, published in 1614 and 1615
respectively, were followed in 1616 by a strange alchemical romance
with the title Chymische Hochzeit Christiani Rosencreutz, which
has been attributed to Johann Valentin Andreae. For Fludd, harmony
meant Pythagorean numerological relationships, running through
the three worlds of the empyrean, the heavens and the elements,
and binding together the macrocosm and the microcosm. But he
held alchemy to be the true science, penetrating into the hidden
depths of nature. Fludd attacked Kepler in a pamphlet of 1621,
claiming that Kepler had treated only the outer movements of nature,
while he himself had penetrated the hidden depths. Kepler
replied at length in his Apologia pro opere Harmonices Mundi (1622)
(KGW 6, pp. 381-457), rejecting Fludd's Pythagorean numerology
and poking fun at his claim to possess deeper perception. When
Fludd renewed his attack, Kepler just ignored it. On Fludd, see
Frances A. Yates (1972): The Rosicrucian Enlightenment (London), pp. 70-90.
... Along with it I should also bring to a close the appendix
promised at the entry to Book V, if the affinity of the subjects
did not incite me also to give a satisfaction to those who have been
in contention with me, such as Mr. Robert Fludd, the Oxford physician.
He filled his book on the Microcosm and the Macrocosm, published
a year ago, with reflections on harmony, so that I should not omit
to mention him in my book but should briefly show the reader the
subjects on which there is agreement between him and myself,
and those on the other hand on which we differ.
That author, then, has promised two volumes, one of which,
written on the Macrocosm, has already seen the light, while
the second, on the Microcosm, is still awaited. The former volume
embraced two treatises, and also brought them out at different times.
For I saw the first treatise, on the threefold cosmos, after the
autumn Frankfurt Fair of the year 1617, and the second on the Arts,
which he calls the apes of cosmic nature, at the Easter Fair of
the following year, 1618. In the second treatise, then, he has
placed music among the arts, embracing up to a point the subject
matter of my Book III. However, in the earlier treatise, which is
contained in seven books, he has allotted the third book to cosmic
music, taking the same title as I put at the head of my entire work.
However, he has taken on the subject matter of my Book IV and Book V.
Therefore, we shall start with his artificial music. He tells us of
that in seven books, in the first of which he reviews the authorities,
the nomenclature, and the force exerted on the minds of men.
On the authorities, or the history of the discovery, I have said
nothing, or little, inasmuch as my intention is to reveal the causes
of things which are natural. The necessary nomenclature I have embraced
in my definitions throughout; the superfluous I have omitted.
I deal with the force of music in minds in Chapter XV of Book III
and throughout Book IV. In the second book the author attacks
the actual subject matter, which he speaks of as intervals and times.
Again, I have said absolutely nothing about times, or the length or
brevity of sounds; for they are arbitrary, and do not need inquiry
into causes. He calls certain intervals simple which for me are the
smallest melodic dissonances, the major and minor tone, the semitone,
and the diesis. He considers the others, which I call consonances,
as compounds of these. But I have expressly refuted this opinion
of the ancients, that the consonances are composed of smaller intervals,
which are, so to speak, prior by nature, in Chapter IV of my Book III,
showing that the smaller intervals on the contrary arise from the
consonant intervals which are larger than themselves. In his third book
he expounds the musical system or scale, which is a main part of my
Book III from Chapter IV to Chapter IX. The author's remaining four
books are on the practical side, which I do not even touch.
For in Book IV he gives advice on the measure of the beat, and on
its various modes, and on the value of the notes in them, on
which I have a very few points in Chapter XV of Book III and in
Chapter III of Book IV. In V he has advice on the composition of
figured melody, an art which I do not profess. In VI he also
digresses to various musical instruments, to which I had not even
given thought. Last, in VII he reveals a new instrument himself.
In these last four chapters he differs from me in the way in
which a practitioner does from a theorist. For where he writes on
instruments, I enquire into the causes of things, or of consonances;
and where he gives instruction on composing a tune for several voices,
I provide mathematical derivations of many features which occur
naturally both in choral and in figured melody. Consequently, there
are also many pictures in his work; in mine, mathematical diagrams
organized with letters. Notice also that he takes great delight
in topics which are hidden in the darkness of riddles, whereas I strive
to bring topics which are wrapped in obscurity out into the light of
understanding. The former is familiar to chemists, Hermeticists, and
Paracelcians; the latter is considered their own by mathematicians.
Furthermore, indeed, in Books II and III, where he is dealing with
the same subject matter as myself, this is the difference between us:
what he takes over from ancients, I draw out from the nature of things
and establish from the very foundations. He applies what he has got
in a confused (on account of the varying opinions of his sources)
and uncorrected form: I proceed in the natural order, so that everything
is set right according to the laws of nature, and confusion is avoided;
so much so that I do not even relate what has been established to
the opinions of the ancients, except where no confusions follows.
Thus at the point where I expressly refute the ancients' treatment of
the causes of consonance, he follows the ancients, without the hazard
of hesitation: he does not even give a thought to the truer causes.
In a word, in the discipline of harmony, one plays the part of a vocal
and instrumental musician, the other of a philosopher and mathematician.
Let us now pass on to another passage of the author's, in which
he introduces music into the cosmos. Here the difference between us
is of immense size. First, what he endeavors to teach us as harmonies
are mere symbolism. Of them I say what I said of Ptolemy's symbolism,
that they are poetic or rhetorical rather than philosophical or
mathematical. This is the spirit of the whole of this work, as is
evident even from the title of Macrocosm and Microcosm. For in the
second volume he will undoubtedly endeavor to demonstrate this noble
thesis, that the ideas of the whole great cosmos, and of all its parts,
are found in man. This same spirit is also that of the first volume, as he
divides the whole cosmos into three regions; and there in accordance with
the most celebrated axiom of Hermes he makes the higher things similar
or analogous to the lower. However, for this analogy to succeed
in all cases, the points of comparison on either side often have
to be dragged in by the short hairs. My opinion of analogies is clear
from the digression at the foot of my Book III: in other words, although
the analogy of proportions in geometry is something formal, in respect
of the actual quantitative matter, which is indefinite and undetermined,
yet in respect of harmonic proportions it can be considered rather as
a material property of the harmonic proportions. For since the harmonic
proportions define a certain quantity, analogies on the other hand are
apt to extend themselves to infinity, and thus counterfeit the material
property of infinity.
However, I also deal with some points similar to what he says on
the Microcosm in my work, such as what I say in Book IV. I make the Earth
a living creature: but that is for quite different reasons. For I do
not contend that there is a pure analogy between the Earth and living
things, and neither do I mean that the archetype of a living thing has
been taken from the Earth itself; but the proposition which I mean to
demonstrate is simply that those works which are seen on the Earth's globe
cannot come about from the motions of the elements, or from the properties
of matter on their own, but bear witness of the presence of a soul.
In that case for my arguments to be understood it was necessary to
adduce the various functions of the soul in the body of a living creature.
Let us now come closer to the foundations on which Rovert Fludd erects
his cosmic music. First, he takes possession of the whole cosmos and all
its three parts, empyrean, celestial, and elementary: I, of the celestial
alone, and not the whole of it, but only of the motions of the planets,
so to speak, against the zodiac. He trusts the ancients, who believed
that the force of the harmonies comes from abstract numbers, and considers
it sufficient if he demonstrates that there is consonance between any of
the parts, in whatever way he expresses them in numbers, not caring what
sort of units are combined together in that number: I teach that harmonies
should never be sought when the things between which the harmonies are
cannot be measured by the same quantitative measure, in such a way
that with respect to quantity the proportion between them is the same
as there is with respect to length between two strings at the same tuning.
Consequently, he divides the whole cosmos into three equal parts by means
of a radius, taking it as sufficiently well known that they are far from
equal, but for the sole reason that the first unit is the elementary world,
the second the aethereal world, and the third the empyrean. And in fact
the units cannot be depicted otherwise than by equality of lines.
But I, unless astronomy bears witness that the units share the same
quantitative measure, do not in any way employ them as units for
numbering the harmonic proportions. He, however, standing on his
principles, and erecting a pyramid on the great circle of the Earth
as base, sets its vertex at the very apex of the empyrean heaven;
and dividing its height into three equal sections (just as if he had
in absolute truth had equal units), he counts how many parts belong
to the empyrean, how many to the celestial, how many to the elementary.
For the top of the elementary region on this showing is twice as far
away from the top of the celestial region as from the top of the
empyrean; and in the division of the pyramid, with respect indeed
to the axis, three equal sections belong to the three regions, but
with respect to the triangle on the axis, one unit belongs to
the empyrean, three units to the celestial, and five to the elementary.
Finally, with respect to volume or fatness of shape, one unit belongs to
the empyrean, seven units to the celestial, and nineteen to the elementary.
Now what shall I say about the other, contrary pyramid of light,
of which he makes the worshipful Trinity itself the base, at the topmost
apex of the empyrean heaven, and places the vertex on the very Earth?
Since he mingles these two pyramids with each other, and elicits
musical proportions from the mixture, he is attempting something
entirely different from the intention of my work. For he compares
light (which bestows form and spirit) and matter, two things which
are completely different from each other, and to which quantities
do not in any way belong in the same respect; but I admit, as terms
in forming harmonic proportion in the universe, only those things
which admit quantities in the same respect, for instance the motion
of Mars and the motion of Jupiter, both diurnal. The difference between
us consists equally of this fact also, that he ascribes to the elementary
region four degrees of obscurity and darkness, because, he says,
everything has four quarters, certainly no less than three thirds
or five fifths; and in fact all four belong to the Earth, three to water
(and therefore, it is in fact transparent), two to air, and one to fire.
Again elsewhere he subdivides every region, belonging either to an
element or to a heaven, into three spaces, top, middle, and bottom,
which the agreement of the senses does not follow in every case.
You see that his units are again arbitrary. However, he proceeds on
that account to establish a diatessaron between Earth and water,
and to relate its three intervals, a tone, a tone and a semitone,
to the three spaces, top, middle, and bottom, since the former have
definite quantities arising from their causes, the latter not even
boundaries from nature, but measures which are plainly indeterminate
drawn from these very general principles; and so on. But I have set
out units which are natural, that is to say the two extreme motions
of each planet (whether diurnal or hourly makes no difference),
expressed by their nature in their definite quantities, in which
to seek harmonies. He seeks harmonic proportions in degrees of
darkness and light, without respect to any motion: I seek harmonies
only in motions. He plucks out a few trivial consonances, and elicits
them from the mixture of his pyramids, from which he conjures up
the cosmos privately depicted in his mind, or deems them to be
represented by it. I have demonstrated that the whole body of
harmonic combinations, with all its parts, is found in the planets'
own extreme motions, according to measures which are certain
and derived from astronomy. Thus for him his conception of the cosmos,
for me the cosmos itself, or the real motions of the planets in it,
are the basis of the cosmic harmony.
From this short discussion I think it is established that,
although knowledge of the harmonic proportions is absolutely
necessary for understanding the crowded secrets of the deepest
philosophy, of which Robert tells, yet even if he has thoroughly
learnt the whole of my work, he will still be considerably further
from those most intricate secrets; and the proportions have departed
from the totally accurate certainty of mathematical derivations.
And now let this also be the end of the Appendix.
WORKS BY KEPLER IN TRANSLATION
Mysterium cosmographicum (Tübingen, 1596, 1621)
Max Caspar (1923): Das Weltgeheimnis. German translation with
commentary (Augsburg; 2nd edition Munich and Berlin, 1936).
A.M. Duncan (1981): Mysterium cosmographicum. The Secret of the universe.
English translation by A.M. Duncan, with introduction and notes
by E.J. Aiton. Preface by I. Bernard Cohen (New York).
Alain Segonds (1984): Le Secret du monde. French translation
with commentary (Paris).
Astronomia nova (Heidelberg, 1609)
Max Caspar (1929): Neue Astronomie. German translation with
commentary (Munich and Berlin).
Jean Peyroux (1979a): Astronomie nouvelle. French translation
with commentary (Paris).
W.H. Donahue (1992): Johannes Kepler. New Astronomy, English
translation with introduction, Cambridge.
Harmonices mundi libri V (Linz, 1619)
Max Caspar (1929): Weltharmonik. German translation with
commentary (Munich and Berlin; reprinted Darmstadt, 1967).
J.V. Field (1979a): "Kepler's Star Polyhedra."
[A translation with commentary of Harmonice mundi, Book II.]
Vistas in astronomy, 23: 109-141.
Jean Peyroux (1979b): L'Harmonie du monde. French translation
with commentary (Paris).
E.J. Aiton, A.M. Duncan, J.V. Field (1997):
"The Harmony of the World by Johannes Kepler".
English translation with an Introduction and Notes.
See at Amazon:
http://www.amazon.com/exec/obidos/tg/detail/-/0871692090/qid=1058220594/sr=8-2/ref=sr_8_2/002-6520022-2575238?v=glance&s=books&n=507846
De nive sexangula (Prague, 1611)
C. Hardie (1966): The six-cornered snowflake. English translation (Oxford).
R. Halleux (1975): L'Etrenne ou la neige sexangulaire. French translation.
Preface by René Taton (Paris).
(AK's note: On the english translation, see Keith Tognetti's page
about "Fibonacci, his rabbits, his numbers and Kepler."
at http://www.austms.org.au/Modules/Fib/
)
Somnium .. seu opus posthumum de astronomia lunari (Sagan and Frankfurt 1634)
E. Rosen (1967): Somnium. English translation with commentary
(Madison and London).