Image 10
Emergence of Three Types of Architecture
from: Lehrbuch der Harmonik by Hans Kayser (1891-1964)
Format 1 : 2 (Octave)
This
image, like the right side of the preceding one, comes from a “harmonic
division canon”: If one takes any rectangle and draws diagonals in it,
then divides the rectangle in two through the intersection point of the
diagonals and draws more diagonals in the newly created rectangles,
proceeding this way at one's discretion, the intersection points always
produce harmonic intervals. This division canon was discovered by Hans
Kayser in medieval architecture.
Regarding
Image 10, Kayser writes: “This harmonic analysis not only gives us an
idea of how the ancient architects, who were doubtless familiar with
this 'rational segment division,' may have proceeded in their designs.
It also gives us something far more important: an inner
characterization for these three styles in relation to each other. The
lengthening of the monochord-the increasing of the octave space-can be
observed psychically as an enlargement of the psychical configuration
space. Whereas in the Egyptian aspect, the tone-lines still adhere to
the 'earthly' and allow only the pyramid as a prototypical form,
Romanesque architecture introduces the 'tower' and thereby allows for
basilical symmetry. In the Gothic aspect, this tower is of commanding
importance and pulls all other forms upwards along with it, reaching
the maximal expansion, the greatest possible upsurge of the
architectonic measure as a symbol for the relationship of the earthly
and the human to the divine.
“Naturally, for these three types of building, one cannot measure
'after the fact' whether the pyramid's angles are exactly in tune,
whether the ratio of church and roof in the Gothic aspect is exactly
'right,' and so forth. As for all 'sound-images,' it is not immediately
essential to make special individual measurement analyses, but instead
to produce the evolution of the model-here of three architectural styles-from a unified idea.”
Peter Neubäcker

Image 11
The Proportions of the Temple of Athena at Paestum
from: Architektur und Harmonie by Paul v. Naredi-Rainer
Format 8 : 15 (Major Seventh)
Whereas
the previous image relates to ideal observations on archetypes of
fundamental architectonic principles, this and the next image examine
existing buildings concretely in terms of the harmonic measures at
their basis. The book Architektur und Harmonie by Paul v. Naredi-Rainer contains many such investigations. Regarding this image he writes:
“The
flourishing of the Pythagorean school, approximately contemporary with
the construction of the Temple of Athena, makes it seem possible that
Pythagorean speculations on numbers were reflected in the Temple's
architectonic expression. The numbers of the Temple's axial
measurements, 40 × 96, can be derived from the numbers of the Tetraktys
(1 2 3 4), which the Pythagoreans held sacred: division by 4, the
measure of half the span, yields 10 as the sum and 24 as the product of
the Tetraktys numbers (40 : 4 = 10, 10 = 1 + 2 + 3 + 4; 96 : 4 = 24, 24
= 1 × 2 × 3 × 4).
“This already points to the central meaning that is attributed to the
reciprocal correspondence of numbers and tones in Pythagorean thought.
Hans Kayser, who revived Pythagorean harmonics in our century,
performed a harmonic analysis of the Temple of Paestum, translating the
rational number ratios into musical intervals. Inspired by the romantic
idea of viewing architecture as 'frozen music,' Kayser perceives
numbers and proportions not as 'purely intellectual quantities,' but as
'tones, intervals, and melodic types, “nomoi,”
which are identical to the forms in our psyche and express them in many
direct ways as mere numbers and measurements.' Amidst all the
fascination emerging from this harmonic approach, the question must be
asked of whether the ancient architects were as familiar with the
philosophical and mathematical theories of their time as, for example,
the humanistically educated architects of the Renaissance-Kayser, in
any case, is convinced that 'where the architect knew nothing of
harmonic proportions, he still applied them, simply because instinct
and feeling led him to arrange his plans and ideas according to these
proportions.'”

Image 12
Musical Intervals in the Façade of the Palazzo Rucellai
from: Architektur und Harmonie by Paul v. Naredi-Rainer
Format 9 : 10 (Whole-Tone)
Regarding
this image, Naredi-Rainer writes: “The architectural aesthetics of the
Renaissance are saturated with the Pythagorean-Platonic idea that the
harmony of the cosmos is built upon musical number ratios; its first
and most significant verbalization was in Alberti's tract on
architecture, De re aedificatoria libri decem, which was written in the mid 15th
century but first printed in Florence in 1485. The fundamental
aesthetic principle, the 'concinnitas,' is manifested according to
Alberti in certain numbers and proportions, which appear most clearly
in music. Thus one should obtain the entire law of the relationship
from the musicians, who know these numbers best.
“... The three-story façade of the Palazzo Rucellai, subdivided into
pilasters and cornices, is now 7 axes across, but was originally
conceived upon 5 axes. The measuring unit, the Florentine braccio equal
to 58.3 cm, is visible in the width of the finely carved pilasters,
which together with the fascia organize the wall surface into a
grid-patterned arrangement and frame the 'outer panels' thus formed.
However, this grid is in no way uniform, but is subtly varied through
the differences in story height and axis width: the middle axis
accentuated by the doorway is wider than the other axes by the ratio of
a whole-tone (5 2/5 : 4 4/5 = 9 : 8). Subtracting this difference in width of 3/5 braccia from the total width (30 3/5
braccia) yields the round measure of 30 braccia, whose ratio to the
total height is that of a minor third (36 : 30 = 6 : 5). The
complementary interval to the minor third, completing the proportion of
the façade panels, is the major sixth. It determines the proportion of
the central panel on the 1st story above the ground floor, engraved with a coat of arms, the Piano Nobile (9 : 5 2/5 = 5 : 3). The outer panels of the remaining axes have the proportions of the major seventh on the 1st upper story, the minor seventh in the 2nd story, while the outer panels of the middle axes on the 2nd
story represent the ratio of a major sixth, which is constructed in a
quasi-mirror image relationship from the ratio of the height of the
doorway to its width (including the doorframe). The high rectangles of
the window apertures repeat the form of the outer panels; they yield
the ratios of the fourth and major third.”
