PEDAGOGICAL

Part Two: Kepler and the Principle of Inversion

by Fred Haight

        There is a wonderful paradox in Kepler's method. He
approaches everything knowing what he is looking for, and finds
it; but at the same time, makes lawful, yet unexpected
discoveries. He knew, that in discovering that the principle of
non-constantly changing curvature, was the actual
"characteristic" of the solar system, he had negated not only the
Ptolemaic, but Copernican universes. What new "characteristic",
or "curvature", would he discover in music?
        After examining the intervals formed by the ratios of the
aphelial, and perihelial motions of individual planets, Kepler
then examines the extreme intervals of different planets with one
another. He finds that the intervals between the planets form two
scales, both running from G to g. It is notable that Kepler does
not try to find all the modes accepted at that time, but only
two. (5) These are usually translated as major, and minor. -
although Kepler's terms, hard and soft, are better (6). His
emphasis on these two types can be found in Chapter Six of Book
Three, and as early as Chapter Twelve of Mysterium
Cosmographicum.
        This author has long been intrigued by the revolutionary
change from the modal system, to that of 24 keys These are two
lattices that are completely different. The problem is, that
since they are formal "fixed systems", you can't find the
revolution by comparing the two lattices, since both are
relatively untrue! You need to find the DISCOVERIES of physical
principle, in order to find the revolution.
        As far as I know, no-one else besides Kepler examines only
two scales that are minor and major scales of the SAME OCTAVE.
(The "lattice" of the 24-key system, considers relative major and
minor scales that share the same key signature to be the most
closely related, such as C major, and A minor, where inversion is
not as much a question).
        This change, from the "modal" to the "tonal" universe is
usually seen as taking place gradually, over almost two centuries
(7), but discoveries are made by MINDS, and the principle of
inversion, along with the related Lydian principle is, I believe,
the key to this change.
        Kepler's soft scale, corresponds to what we would today call
the G "natural" minor scale. However, his hard scale differs from
the G major scale, in that it features an F natural rather than
F# (7). Here is the most important point in this entire
writing.This might seem like a deficiency, until you realize that
these two scales are INVERSIONS OF ONE ANOTHER! Try it! The hard
scale is the old Mixolydian mode, but is it not also C major,
from G to G, which is the inversion of G minor? (G A Bb C D Eb F
G inverts onto G F E D C B A G). This is unique to Kepler! I
certainly don't know everything being discussed in Kepler's time,
but I can think of no "theorist" who does this. They all tend to
see scales as fixed, a priori existences, rather than arising
from a form of CHANGE, such as inversion.
        Inversion is important for all of Kepler's work, (8) and he
places much importance on these two types of scale.

        From Chapter Five of Book Five:

        "Accordingly, the musical scale or system of one octave with
all its pitches, by means of which natural song is transposed in
music, has been expressed in the heavens by a twofold way, and in
two, as it were, modes of song."

        In chapter nine, when discussing that none of these
harmonies can occur by chance:

        "[L]east of all that very subtle business of the distinction
of the celestial consonances into two modes, hard and soft,
should be the outcome of chance, without the special attention of
the Artisan:
        "Accordingly you won't wonder any more that a very excellent
order of scales has been set up by men, since you see that they
are doing nothing but aping God the Creator and to act out, as it
were, a certain drama of the ordination of the celestial
movements.
        "But there still remains another way whereby we may
understand the two-fold musical scale in the heavens, where one
and the same system but a twofold tuning is embraced, one at the
aphelial movement of Venus, the other at the perihelial."

        Kepler associates these two scales, with his discovery of
the cause of the intervals in his second great archetype.
        In chapter nine, propositions 24-35, he locates a major
(hard) sixth between the aphelia of the Earth and Venus, and a
minor (soft) sixth between their perihelia. The two movements
differ then by a diesis (the smallest kind of half-tone). He says
that the two planets that SHIFT the genus of harmony SHOULD
differ by this interval. Thus the half-tone is the characteristic
interval of inversion. Although both scales use perihelial, and
aphelial motions, he says that "THE HARD SCALE IS DESIGNATED
PROPERLY ONLY IN THE APHELIAL MOVEMNETS, THE SOFT, ONLY IN THE
PERIHELIAL" (Ch 9-prop 33). He also says that "Saturn and the
Earth embrace the hard scale more closely, Jupiter and Venus, the
soft" (prop 35) (9)
        So, the question is, is Kepler really the discoverer of
Bach's principle of inversion, and, for that matter, the 24-key
system? First, keep in mind that we speak of the PRINCIPLE of
inversion, which means that it is more than just the mechanical
act of inversion, which is much more ancient than Kepler, or
Bach.
        Remember the example from "Grammar and Science":

        "Please: feed the cat"
        "To what?"

        "The meaning of either of the two statements is located in
the Bach-like mental act of inversion."

        Focus on the words, "mental act". Don't look for letters
turned upside down, or words running backwards and forwards at
the same time. Praxiteles' sculpture, is like polyphony, not
because of resemblances in the art forms themselves, but because
both create paradoxes that must be resolved, in the mind.
        The principle of inversion, also addresses the question of
discoveries of PHYSICAL principle, that negate the Ptolemaic
systems, whether astronomical, or musical; as both Kepler's and
Bach's discoveries do. So, Kepler's discovery of the role of
inversion in the solar system, points towards exactly such a
principle, as the basis of a necessary revolution in music.
        This will have the musical priesthood howling. Such people
always wish to believe that music is a hermetically sealed tuna
sandwich, growing into whatever it does, as a purely internal
matter. But, when a scientific discovery about the universe is
made, art must change itself, according to that discovery. Thus,
Kepler, and Leonardo, revolutionize music as much, or more, than
any composer. We can only imagine what future works of art will
come out of Lyndon LaRouche's discoveries!

       - Kepler's Little Joke on the Lydian Interval? -

        In Book Three, there is little or no mention of the Lydian
principle. In Book Five, there are hints of it. The most
well-known reference is the question of the destroyed planet, but
there are others.
        Chapter Six of Book Five, ends in an unusual way. Kepler
finds, between the aphelial, and perihelial motions of the Earth,
a half- tone, to which he assigns the tones G Ab. The last of his
marginalia reads thus:

        "The Earth sings MI, FA, MI so that you may infer even from
the syllables that in this our domicile MIsery and FAmine obtain"

        In Guido's hexachord system, Mi-Fa always refers to a
half-tone within a single hexachord, but across hexachords, it
refers to the "devil's interval." If you remember a previous
pedagogical entitled "Bach's Little Joke on the Lydian interval",
the pedantic verse prohibiting the Lydian interval was:

        "Fa contra mi- mi contra fa
        est Diabolus in Musica"

        Bach's answer was:

        "Fa-mi mi-fa est tota Musica"

        This proscription was very well-known. Kepler refers to
Mi-Fa as a half-tone in Book Three, but he could not have been
unaware of the "Lydian" implication of Mi-Fa, as his pun on the
"dissonance" of misery and famine, attests. (10)
        Perhaps making fun of this devilish ban was common to great
Humanist thinkers.

Notes

        (5) In Chapter Fourteen of Book Three, Kepler makes his own
attempt to determine the modes. At first, he contemplates twelve
modes, based on twelve tones, but ends up with variants of the
old church modes. But, in Book Five, Kepler is only concerned
with two modes.
        (6) Hard and soft, which are used in German today, seem to
correspond more to the inversion intervals of arithmetic-harmonic
mean (which Kepler discusses in the Third Book), than do major
and minor. In the hexachord system, C is the natural hexachord, G
(the arithmetic mean of the C octave) hard, and F (the harmonic
mean) soft- yet; they are not referred to as major and minor.
Kepler refers to major and minor thirds as hard and soft. The
major third is the arithmetic mean of the fifth, and the minor
third, the harmonic. Take the fifths above the three first tones
of the natural, hard, and soft hexachords, C-G, G-D, and F-C.
Divide them at the arithmetic (hard) mean, or major third:

        G     D     C

        E     B     A

        C     G     F

        And you will have all the tones of the C major scale. Then
divide the same fifths at the harmonic (soft) mean, or minor
third:

        G     D     C

        Eb    Bb    Ab

        C     G     F

        And you have all the tones of the "natural" minor scale.
Here you have both scales derived as inversions of the division
of the octave and the fifth.
        (7) It is usually seen as being being finalized with Bach's
1722 Well-tempered Clavier, which is passed off as "demonstrating
that one could play in all 24 keys on the same instrument, if it
is well-tempered." A more interesting question is: did anyone
ever compose in keys such as Gb major, or G# minor before?
        (8) In chapter five, through an adjustment in the orbit of
Mars, he does add the F# to the hard scale. The same scale is a
kind of transposed Lydian scale, because he adds a C# from the
aphelial motion of Mercury.

G A B C C# D E F# G

        (9) (insert footnote on Kepler's discoveries of
non-linearity of inversion from "Riemann for anti-dummies")
        (10) Kepler finds, in the aphelial motion of Saturn to the
perihelial, the ratio of 4:5, a hard third, but in Jupiter 5:6, a
soft third. The two compound into a fifth, 3:2. (Ch 9-prop 11). G
B D is an ascending major third and minor third, compounding to a
fifth. D Bb G is a descending major third, and minor third.
Kepler finds Saturn at G B, and Jupiter at G Bb.
        (11) Although stressing the role of consonance, Kepler
throughout, underlines the role of dissonance. In Chapter nine,
prop twelve, he cites the cube, and octahedron as being defined
by the octave and double octave, but sees the sphere between them
as 1: the square root of two, the ratio of the Lydian interval.
Thus he sees this ratio as being between the solids, which is of
interest, since the Lydian interval often lies between the keys.
In the early part of Book Five, he says that some solids are wed
to each other through rational proportions, and others divine
proportions (Golden Section), such as the wedding of the
dodecahedron, and the icosahedron, which are between the Earth,
and it's neighbours. When considering merely the perihelial, and
aphelial proportions, he finds the Lydian interval between Earth
and each of its neighbours. He rejects that determination ,
though, for the ratios of the motions. In reading these hints at
the Lydian principle, I keep thinking of Kepler's own footnotes
to the Mysterium Cosmographicum, written twenty-two years later,
and how his own thinking had developed. What might he have
written twenty-five years after Harmonice Mundi?

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