The Sleepwalkers (188 page)

But
again
the
facts
did
not
tally
with
the
scheme,
and
had
to
be
explained
away
by
ingenious
reasoning.
The
15-sided
polygon,
for
instance,
is
construable,
but
does
not
produce
a
musical
consonance.
Moreover,
the
number
of
construable
polygons
is
infinite,
but
Kepler
only
needed
seven
harmonic
relations
for
his
scale
(octave,
major
and
minor
sixth,
fifth,
fourth,
major
and
minor
third).
Also,
the
harmonies
had
to
be
arranged
into
a
hierarchy
of
varying
degrees
of
"knowability",
or
perfection.
Kepler
devoted
as
much
labour
to
this
fantastic
enterprise
as
to
the
determination
of
the
orbit
of
Mars.
In
the
end
he
succeeded,
to
his
own
satisfaction,
in
deriving
all
his
seven
harmonies,
by
certain
complicated
rules
of
the
game,
from
his
perfect
polygons.
He
had
traced
back
the
laws
of
music
to
the
Supreme
Geometer's
mind.

In
the
sections
which
follow,
Kepler
applies
his
harmonic
ratios
to
every
subject
under
the
sun:
metaphysics
and
epistemology;
politics,
psychology
and
physiognomics;
architecture
and
poetry,
meteorology
and
astrology.
Then,
in
the
fifth
and
last
book,
he
returns
to
cosmology,
to
complete
his
vertiginous
edifice.
The
universe
he
had
built
in
his
youth
around
the
five
perfect
solids
had
not
quite
tallied
with
the
observed
facts.
He
now
brought
the
two-dimensional
shadow
army
of
polygons
to
the
rescue
of
the
beleaguered
solids.
The
harmonic
ratios
must
somehow
be
dovetailed
in
between
the
solids
to
fill
the
gaps
and
to
account
for
the
irregularities.

But
how
could
this
be
done?
How
could
the
harmonies
be
fitted
into
the
scheme
of
a
universe
full
of
elliptic
orbits
and
nonuniform
motions,
from
which,
in
fact,
all
symmetry
and
harmony
seemed
to
have
departed?
As
usual,
Kepler
takes
the
reader
into
his
confidence,
and
for
his
benefit
recapitulates
the
process
by
which
he
arrived
at
his
solution.
At
first,
he
tried
to
assign
the
harmonic
ratios
to
the
periods
of
revolution
of
the
various
planets.
He
drew
a
blank:

"We
conclude
that
God
the
Creator
did
not
wish
to
introduce
harmonic
proportions
into
the
durations
of
the
planetary
years."
¹³

Next,
he
wondered
whether
the
sites
or
volumes
of
the
various
planets
form
a
harmonic
series.
They
do
not.
Thirdly,
he
tried
to
fit
the
greatest
and
smallest
solar
distances
of
every
planet
into
a
harmonic
scale.
Again
no
good.
In
the
fourth
place,
he
tried
the
ratios
between
the
extreme
velocities
of
each
planet.
Again
no
good.

Next,
the
variations
in
the
time
needed
by
a
planet
to
cover
a
unit
length
of
its
orbit.
Still
no
good.
Lastly,
he
hit
on
the
idea
of
transferring
the
observer's
position
into
the
centre
of
the
world,
and
to
examine
the
variations
in
angular
velocity,
regardless
of
distance,
as
seen
from
the
sun
.
And
lo!
it
worked.

The
results
were
even
more
gratifying
than
he
had
expected.
Saturn,
for
instance,
when
farthest
away
from
the
sun,
in
its
aphelion,
moves
at
the
rate
of
106
seconds
arc
per
day;
when
closest
to
the
sun,
and
its
speed
is
at
maximum,
at
135
seconds
arc
per
day.
The
ratio
between
the
two
extreme
velocities
is
106
to
135,
which
only
differs
by
two
seconds
from
4:5
–
the
major
third.
With
similar,
very
small
deviations
(which
were
all
perfectly
explained
away
at
the
end),
the
ratio
of
Jupiter's
slowest
to
its
fastest
motion
is
a
minor
third,
Mars'
the
quint,
and
so
forth.
So
much
for
each
planet
considered
by
itself.
But
when
he
compared
the
extreme
angular
velocities
of
pairs
of
different
planets,
the
results
were
even
more
marvellous:

"At
the
first
glance
the
Sun
of
Harmony
broke
in
all
its
clarity
through
the
clouds."
¹⁴

The
extreme
values
yield
in
fact
the
intervals
of
the
complete
scale.
But
not
enough:
if
we
start
with
the
outermost
planet,
Saturn,
in
the
aphelion,
the
scale
will
be
in
the
major
key;
if
we
start
with
Saturn
in
the
perihelion,
it
will
be
in
the
minor
key.
Lastly,
if
several
planets
are
simultaneously
at
the
extreme
points
of
their
respective
orbits,
the
result
is
a
motet
where
Saturn
and
Jupiter
represent
the
bass,
Mars
the
tenor,
Earth
and
Venus
the
contralto,
Mercury
the
soprano.
On
some
occasions,
all
six
can
be
heard
together: