§41.5. An Ektypic Example, Taken from Astronomy, for the Theorem of the Melodic (Value-Form: Horizontal Step)
In my Hörende Mensch,
p. 191 ff., I gave a harmonic analysis, from specific viewpoints, of
the distances between the planets, which when reduced by octaves
produce a closed scale of major-minor character. There one can read
about the results, arising from the enharmonics of two steps of this
scale, significant for the speculative knowledge of a “disruption
factor” connected a priori to the birth of the solar system.
As mentioned often in this book, the modern numerical formulation of
precise scientific laws and relationships is free of prejudice in
itself, but in the majority of cases it obscures the morphological
background of these laws more than it illuminates them. Especially
typical of this are numbers in modern astronomy.
The voluminous work of the astronomer Wilhelm Kaiser attempts to regain
a morphological, formal grasp of the astronomical cosmos, as was
mentioned in §24a.2. In this work, especially in the second book of his
Geometrische Vorstellungen in der Astronomie
(Selbstverlag des Verfassers, Subingen, Canton Solothurn, Switzerland,
1933), we harmonists find a wealth of previously unknown proportional
relationships that merit deeper harmonic analysis.
One of the most significant discoveries made by W. Kaiser appears to me
to be what he called the “inversion spheres of the planets” (ibid., 143
ff.). If we set the average distance of Earth from the sun as 1
[astronomical] unit, then Jupiter's distance from the sun at aphelion
(the point in its orbit at which it is farthest from the sun) is 5.45 =
60/11
times that of Earth. W. Kaiser now produces the following table (ibid.,
162) using Earth's average distance from the sun as a unit 1:
Upper limiting sphere with the radius 5.45 = J = 60/11 of Jupiter's domain of movement
Asteroids |
P = 3 |
6/11 |
11/6 |
109/11 |
= T Saturn |
Mars |
M = 180/121 |
3/11 |
11/3 |
20 |
= U Uranus |
Earth |
E = 1 |
11/60 |
60/11 |
119/4 |
= N Neptune |
Venus |
V = 3/4 |
11/80 |
80/11 |
119/3 |
= W Pluto |
Mercury |
K = 50/131 |
5/66 |
66/5 |
72 |
= X Gaea |
Radial ratios and radii of the planets' orbits (from W. Kaiser)
Kaiser writes: “In the middle columns are the inverse ratios, which, multiplied by the value J = 60/11, always yield the radius of the corresponding planetary orbit. For Saturn, for the sake of the summary, the fraction 11/6 is written instead of the more precise ratio 109/60; through this one finds that all these ratios have something to do with the number 11, including the bottommost 66/5,
because 6 × 11 = 66. This summary also contains 11 spheres, 11 special
domains in the ether, so that the number eleven is in fact a realistic
symbol for the conditions existing here.”
As one can see, the reciprocal symmetry of this table hints towards the
inclusion of a hitherto undiscovered trans-Neptunian planet (Gaea); its
further justification, as well as the inner meaning of this peculiar
mirror-image quality of the inner and outer planets, can be studied
further in W. Kaiser's work.
Such pronounced reciprocities are always a sure sign for us harmonists
that yet more harmonic relationships must be present-considering that
the “P” system itself is reciprocal through its < 1 and > 1
sectors. If one now examines the average distances of the planets from
the sun (where the Earth-sun distance = 1) by means of octave
potentiation and reduction-a typical harmonic operation that Kepler
used often in his Harmonice Mundi-one obtains the following values:
Planets |
av. dist. from sun |
Octave operations
(reduction to the octave 1.000-2.000) |
nearby tone-values |
tem-pered |
Mercury ☿ |
0.39 |
0.39 → 0.78 → 1.56 |
1.58 as |
128/8 |
log 678 |
as
gis |
Venus ♀ |
0.72 |
0.72 → 1.44 |
1.44 ges |
36/25 |
log 526 |
ges
fis |
Earth ♁ |
1 |
1 |
1 c |
1/1 |
log 000 |
c |
Mars ♂ |
1.52 |
1.52 |
1.52 xg |
32/21 |
log 608 |
g |
Asteroids |
3.04 |
3.04 → 1.52 |
1.52 xg |
32/21 |
log 608 |
g |
Jupiter ♃ |
5.2 |
5.2 → 2.6 → 1.3 |
1.3 øf |
13/10 |
log 378 |
f |
Saturn ♄ |
9.54 |
9.54 → 4.77 → 2.385 → 1.192 |
1.2 es |
6/5 |
log 263 |
es |
Uranus ♅ |
19.19 |
19.19 → 9.585 → 4.7975 → 2.39875 → 1.19937 |
1.2 es |
6/5 |
log 263 |
es |
Neptune ♆ |
30.11 |
30.11 → 15.055 → 7.5275 → 3.76375 → 1.881875 |
1.875 h |
15/8 |
log 907 |
h |
Pluto ♇ |
39.6 |
39.6 → 19.8 → 9.9 → 4.95 → 2.475 → 1.2375 |
1.25 e |
5/4 |
log 322 |
e |
Gaea (Kaiser) hypothetical |
72 |
72 → 36 → 18 → 9 → 4.5 → 2.25 → 1.125 |
1.125 d |
9/8 |
log 170 |
d |

Figure 428
As one can see and hear, the distances of the planets, under this
harmonic analysis, approximately and with the addition of the
hypothetical “Gaea,” form a definite “harmonic” minor scale c d es a g as h c, with the variants e and ges. The reciprocity of W. Kaiser's “inversions,” 6/11 11/6 etc., is expressed in this scale in the equal interval ratios upwards and downwards from the “middle” f (Jupiter).
From the octave operation applied to this analysis, as with all octave
operations, one might begin to think that maybe any seven random
numbers, packed into an octave, would produce seven tones and thus a
scale. The former idea, the 7 tones, is definitely the case, but the
latter, that this would produce a scale, is absolutely wrong, as anyone
can easily find from experimenting; the starting ratios must be chosen
so that they produce an approximate scale or an image similar to a
scale. Admittedly, we must not forget that we have used the average
distances as a measure. The distance of a planet from the sun varies
during its elliptical orbit between the limits of its perihelion and
aphelion. These are often significant, depending on the “eccentricity”
of orbit; e.g. for Mercury:
Potentiated by octaves
↓ |
Perihelion |
Average |
Aphelion |
|
0.31 |
0.39 |
0.47 |
|
0.62 |
0.78 |
0.94 |
|
1.24 |
1.56 |
1.88 |
|
e |
gis-a |
h |
|
└─────────────────────┘ |
|
fifths |
The characteristic (tempered) value is a full fifth; in the above scale
there must therefore be the “tolerance” of a fifth around the tone as.
Regardless of this, it can be no coincidence for those versed in
morphology that the average distances of the planets yield a complete
scale when the distance between sun and Earth is set equal to 1,
approximate enough if not pure-tonal. The reader can now easily perform
the monochord testing of the exact average planetary distances, reduced
by octaves, or better still the perihelions and aphelions reduced by
octaves, whose values can be found in any collection of mathematical
formulae. In testing this he will convert the decimal values into
monochord lengths, according to the formula:
Decimal fraction
(octave 1.00 - 2.00) : 2 = x : 1200 (Monochord)
x = (Decimal fraction · 1200)
2
For x we
obtain the string segment in millimeters on a 1200-mm monochord, for
example: for the average octave-reduced distance of Mercury, 1.56:
x = |
1.56 × 1200
2 |
= 936 (position on the 1200-mm monochord) |
As one can see from Table 481, this position 936 is near to the position 937.5, which represents the tone 25/32 fes.
We have obtained the above scale from the harmonic analysis of the
average octave-reduced planetary distance, assuming W. Kaiser's “Gaea”
exists; this last trans-Neptunian planet has not yet been discovered,
but due to the surprising inversion symmetry, its existence is very
likely-if it can be discovered at all with our current optical methods.
The distance from Earth to the sun has been used as the unit for this,
as is usual in astronomy textbooks.
Another planetary distance scale is given in my Hörende Mensch,
p. 191 ff. The average distances are analyzed there, and their
logarithms on base 10 are compared with our partial-tone logarithms on
base 2, while the distance from Mercury to the sun is used as a
measuring unit (= 1). The familiar planets sufficed there, without the
hypothesis of an undiscovered one. The juxtaposition of these two
logarithmic systems, and the emergent scale, yielded a series of
interesting results; the problem of Lucifer especially seemed to be
surprisingly clarified, at least in cosmic terms, and to become
understandable in mythological and psychic terms. This will be
discussed in §53 and §54.
To recapitulate: all these harmonic analyses of planetary distances
that lead to scales or similar images begin with space (distances) and
lead both to a spatial coexistence (intervals) and a temporal
succession (scales). The reader who has “time” should perform further
tests with other distances as units, e.g. Sun-Jupiter = 1, and play the
planet-tones with their characteristic intervals (perihelion and
aphelion) on the monochord, noting the locations on separate sheets of
paper and attempting to hear the various possible scales.
§41.6. Kepler
In this book, it is impossible to thoroughly discuss Kepler's Harmonice Mundi-the
work that was most important to him, and a treasure in many ways for
modern harmonics. After the completion of this textbook, I published a
more thorough digest of the content of Harmonice Mundi in the Schweizer Rundschau (Oct.-Nov. 1946, pp. 545-553).
What could be said in summary I attempted to say in Hörende Mensch,
p. 171 ff. The plea there for a new collected edition of the works of
this genius has been answered in the excellent translations of Prodromus, Marswerk, and Harmonice Mundi,
by Max Caspar (see the Bibliography for this chapter). These new
editions, revisions, and translations give us the duty as harmonists to
view Kepler's works as he would have wanted them to be seen: not as dry
formulaic articles on astronomy, but as the brilliant unveiling of a
speculative thinker and scholar, believing in the harmony of the
universe, to whom the formulae were not ends in themselves, but merely
means and tools for the proof of his great harmonic formal ideas.
Here I will remark only that Kepler, in his Harmonice Mundi,
attempted to find harmonies in the planetary distances. The idea that
guided him in this was to coordinate the planetary orbits with the
circumferences of the so-called “five regular solids,” to “box” them
together so that they corresponded to the approximate distances. Kepler
tried to realize this in his first work, Prodromus (later reissued by him with corrective notes as Mysterium Cosmographicum), and he approached it anew under a harmonic aspect in Harmonice Mundi,
but no longer found the precision he was seeking, although the
morphological value of the five-solid scheme as the background of the
distances of the planetary orbits from the sun still seemed correct to
him. He then set about analyzing the distances of the aphelions and
perihelions of the individual planets harmonically amongst themselves.
Regarding the extreme distances, i.e. the greatest and smallest
distances from the sun (since the planetary orbits are ellipses), for
the individual planets, Kepler found harmonic intervals only for Mars
and Mercury; “however, if one compares the extreme distances of the
various planets to each other, the first light of harmonics dawns” (Harmonice Mundi, Book V, Ch. 4). There, in close approximation, are the ratios
2/1 5/3 4/1 3/1 27/20 12/5 and 243/160.
But this did not satisfy him either. He then sought harmonies not in
the distances, but in the velocities, in the daily heliocentric
movements. Here, in the extremes of the individual planets (see
Caspar's translation, p. 301, as well as my Hörende Mensch,
p. 177), he found harmonic ratios everywhere, with such close
approximation that it could not be coincidental. A further
investigation of these ratios, using our familiar operation of interval
potentiation, then led him to discover his famous Third Law, which he
used not as an end in itself, but as foundation for the proof of the
existence of his heavenly harmonics. We know this from the fact that
this law was given as the eighth of thirteen
fundamental theorems (Book V, Ch. 3), which are “necessary in the
observation of heavenly harmonies.” Furthermore, regarding this law, he
writes (Mysterium Cosmographicum, tr. by Caspar, 2nd ed., Ch. 21, note 8, p. 137): “An explanation for the eccentricities has been found, not from the ideas just presented, but from the harmonies.”
For the reader of this book, no further proof is necessary that for
Kepler, the main issue was not the (modern) formula of his Third Law: f12 : f22 = r13 : r23
(“the squares of the orbital periods have the same ratio as the cubes
of the major axes”) but the fact that this formula should be the exact
proof for the harmonies existing de facto
in the planetary movements and distances-harmonies that achieve their
simplest sensory and psychical realization in music, which for Kepler
is therefore an image of the heavens, i.e. of the planetary system. Why
else would he devote pages in the conclusion of Book V to hundreds of
these harmonic ratios, as the crowning of his work, with minute
precision and love? And why else would he give, in the same chapter,
all the laws of music, chords, melodies, major and minor, the scales,
and finally the entire chord of creation in notes?
And all this, not before the harmonic derivation of his law, but after it, as proof of the great harmonic form-idea of the Harmonice Mundi, the “harmony of the world.”
The question of whether Kepler's “harmonies” are still correct today
can be answered absolutely in the affirmative. A comparison of Kepler's
aphelions and perihelions in regard to the distances from the sun with
the current values shows, with the exception of Saturn's aphelion, only
very minor differences; the same is the case for the daily heliocentric
movements, since these fluctuate around average values, despite the
“perturbations” that have since become known. Uranus and Neptune also
show Kepler's harmonic ratios, and looking through the abovementioned
work of W. Kaiser, one discovers so many further typical harmonic
ratios that one will find Kepler's “Harmonice”
proven once again, under aspects of which Kepler himself knew nothing.
And as for the “failure” of his polyhedron theory, i.e. the
theoretically possible existence of only 6 planets through the
existence of only 5 regular solids, this is only connected with the
actual harmonic ratios insofar as he derived musical intervals from the polyhedron ratios in his Harmonice Mundi,
and there found the theoretical anchor for the harmonies suspected to
exist in the planetary system. These are present above all in the
planetary movements; and Kepler would have been the first to abandon
the polyhedron theory, and would have concentrated completely on
harmonics, whose laws he would have proven for the other planets with
equal delight and enthusiasm, as he did for the planets known to exist
during his time. Kepler's harmonics are still just as valid today as
they ever were-for those with ears to hear them. Indeed, we can
construct them much further and more broadly with current harmonic
resources, especially in terms of the symbolic-spiritual side; the “P”
system was unknown to Kepler.
With this, however, Kepler's significance for modern harmonics is in no way exhausted. In his Harmonice Mundi,
especially in Book IV, there is a series of fundamental and
exceptionally important observations for the foundation of harmonic
starting positions (if I may thus express myself); in any case, the
entire work is saturated with a wealth of principial harmonic
observations, examples, ektypic references, etc. We must fittingly and
justly view Kepler's Harmonice Mundi and A. von Thimus's Harmonikale Symbolik
as the two great harmonic cornerstones of modern times, the exhaustive
study of which should be the first and foremost duty of every
harmonist. In Kepler's other works there is also a great deal of
important material for harmonics, for example his commentary on the
harmonics of Claudius Ptolemy, many letters to and from Kepler, and so
forth.
§41.7. Bibliography
Besides the passages cited in the text in my Hörende Mensch, Grundriß, and articles in the Schweizer Rundschau: the works of astronomer W. Kaiser, and those of the gestalt theorist Christian von Ehrenfels. Regarding Kepler:
the great new critical collected edition (ed. M. Caspar et al., O.
Beck-Verlag Munich, starting 1938), of which volumes 1, 2, 3, 4, and 6
have been released as of now (1944); and M. Caspar's translation of the
work on Mars (Neue Astronomie, ibid. 1929), Mysterium Cosmographicum (Das Weltgeheimnis, Munich, Oldenburg 1936), and Harmonice Mundi (Munich, Beck 1929). Moreover, Caspar's Johannes Kepler in seinen Briefen (Munich, Oldenbourg 1930) and the Biographia Kepleriana (Munich, Beck 1936). Other writings on Kepler and others: Apelt, Reformation der Sternkunde (Jena 1852); L. Günther, Die Mechanik des Weltalls (Leipzig 1909); and the standard translation (with introduction and commentary) of Harmonice Mundi by W. Harburger (Johannes Keplers Kosmische Harmonie,
Dombücherei, Inselverlag, Leipzig 1925), in which the “table of
geometric-musical-astrological-astronomical correspondences” is
noteworthy.
E. Britt's booklet, Tonleitern und Sternskalen,
tr. by Felix Weingartner (Leipzig 1927) is mentioned here only as a
curiosity, and as an example of how one should not engage in such
investigations. In harmonics we are used to somewhat more thoroughness
and exactitude; above all one should never use the tempered system as a
basis here, only pure-tonal values and ratios. Viktor Goldschmidt, in
his Über Harmonie im Weltraum
(Ostwalds Annalen der Naturphilosophie V, p. 51 ff.), investigates the
planetary distances, among other things, by means of his “law of
complication.”
§45.2. Ektypics
One of the most important applications and results of the interval
power series is “enharmonics.” With its theorem foundations, it offers
a precise representation, and we will discuss it by itself in §48. Here
we will only give an example for the I-powers, namely using Kepler's
Third Law, whose harmonics were discussed in §41.6. The most important
interval power, the octave operation, is discussed in §41.5 (analysis
of the planetary distances). Kepler performed the same operation in his
Harmonice Mundi, e.g. Book V, Ch. 3, in which the following table appears:
Orbital periods in Earth days:
octave
reduction
(halving)
↓
(for Saturn and Jupiter) |
Saturn |
Jupiter |
Mars |
Earth |
Venus |
Mercury |
octave
potentiation
(doubling)
↓
(for Venus and Mercury) |
|
10759 12/60 |
|
|
|
|
|
|
|
5379 36/60 |
4332 37/60 |
|
|
|
87 58/60 |
|
|
2689 48/60 |
2166 19/60 |
|
|
224 42/60 |
175 56/60 |
|
|
1344 54/60 |
1083 10/60 |
686 59/60 |
365 15/60 |
449 24/60 |
351 52/60 |
|
|
672 27/60 |
541 35/60 |
|
|
|
|
|
Figure 443
An explanation of this table and its significance in Harmonice Mundi would
take too long here. It represents only a stage along the path Kepler
was on: the path to the proof of precise harmonies in the law of
orbits. I cite it here only to show that Kepler consciously used
interval powers, in this case octave operations, to find certain
numeric proportions in the orbital periods of the planets.
If one is aware of this, and has pursued the many other typical harmonic analyses in Harmonice Mundi,
then one will see the mathematical result of this work by Kepler,
namely his famous Third Law, through entirely different eyes. This law
states that the squares of the orbital periods are proportional to the
cubes of the major axes. This means, in Kepler's harmonic mode of
thought, that the temporal primal intervals of the orbital periods must
be raised to the second intervallic power, and the spatial primal
intervals of the major axes must be raised to the third intervallic
power. This means a power ratio of 2 : 3, i.e. the fifth,
the most important interval after the octave! And if any doubt remains
about the harmonic basis of these operations, Kepler dispels it
himself. In Book V, Ch. 3, Point 11, he clarifies his Third Law as
follows:
“Suppose
the orbital periods of two planets are 27 and 8; their median daily
movements have the ratio 8 : 27; the radii of their orbits thus have
the ratio 9 : 4. The cube root of 27 is 3; that of 8 is 2. The squares
of these cube roots, 3 and 2, are 9 and 4. Now suppose that the
apparent movement of one planet at aphelion is 2, the other at
perihelion 33 1/3.
The median proportions between the median movements 8 and 27 and the
apparent ones are 4 and 30. If the mean 4 yields 9 for the median
distances of the planets, then the median movement 8 yields the
aphelion distance 18, corresponding to the apparent movement 2. And if
the other mean 30 also yields 4 for the median distance between the two
planets, then its median movement 27 yields its perihelion distance 3 3/5. I claim, then, that the aphelion distance of the one planet has a ratio of 18 : 3 3/5
to the perihelion distance of the other planet. Thus, the extreme
distances and the medians, i.e. the eccentricities, are automatically
yielded when the harmonies between the extreme movements of the two planets are arranged and have their orbital periods prescribed.”
Here, and in many other passages, the harmonic origin of Kepler's Third
Law is clearly shown. The very meritorious translator of Harmonice Mundi,
Max Caspar, writes in his notes (p. 386) on Book V, Ch. 3, Point 8:
“Kepler, in the previous section, says not a word about the physical
considerations that led him to the discovery of his Third Law, as he
does with the discovery of his first two laws.” This is completely
correct, but not important with regard to the actual theoretical
backgrounds from which the Third Law was finally obtained. These
backgrounds were harmonic in nature, not physical. If one wishes to
dismiss all these harmonic thought-processes as “aesthetic”-if one is
not indeed babbling from the outset about simple “proof”-then one is
completely overlooking the exact, numerical side of the harmonic
theorems and approaches. These exempt all harmonic thought processes
from the merely aesthetic, and connect them to all objective research
methods; because indeed, all harmonic ratios can be measured and
counted. In harmonics, then, the “aesthetic”-if that is what we wish to
call the forms apperceptible via tone-perception in our psyche-is thus
objectified by means of the tone-numbers.
On the other hand, however, we can evaluate and perceive these numbers
psychically. This is also no longer “aesthetic,” instead something much
more authoritative: harmonics. It was exactly this reciprocal
foundation, this intensification of the objective through the
subjective and vice versa, which so impressed Kepler in his Harmonice, and which casts an equally impressive spell over anyone who studies harmonics.