Sonorous images
From the Book of Harmonics by Hans Kaiser

The general concept of this chapter is that there is the possibility to find, obtain and build sonorous images, meaning that they can be perceived both by the eyes and by the ears at the same time.

We are not talking, mind you, about connotations similar to the musical notes or any other written form of the word. Of course we could use completely different signs from those normally used (c, d, e, f, …).

The geometric nature of Harmonics wouldn’t be affected by this.

This is the core of the sonorous figures, where sonorous and numerical proportion finds direct expression.

The consequence is that any harmonics diagram is a sonorous image. Sonorous images involve three senses at the same time: touch (numbers and measures), sight and hearing (sounds) and they join in a mental act. As its application, the Author proposes the study of the sonorous image of the human image, showing at first its external harmonious proportion and then its sonorous content. This research goes back to old times. The Author reminds us of the human figures drawn by the Egyptians, the famous Canon of Polykleitos, the rules of Italian Renaissance. He refers to (fig. 338) the study by Wineken on the proportions of the male and female human body. As you can see in the image, man’s height is divided into twelfths – viz. in fifths; woman’s height is divided in thirds (*).

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(*) The interval is the distance between two sounds. Any interval is expressed by a number that establishes the relation existing between two sounds.

For example: C-D, interval of a second; C-A, interval of a fifth. The interval is calculated according to the succession of our notes, counting the initial note and the final note and implying the intermediate notes.

For example: C-G= interval of a fifth, C (D, E, F) G
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In this case the division in twelfths means the formation of an interval of a fifth calculated according to the development of harmonic sounds where, taking as fundamental note C, the twelfth harmonic is a G, viz. a sound with an interval of a fifth.

The division in fifth means the formation of an interval of a third, again calculated according to the development of harmonic sounds. The harmonics according to the note C are the following:

Therefore the relation between the two is:

This is a relevant result, being the relation between a major third (4/5 ex; woman) and a minor (5/6 ex; man) (*), which confirms the importance of the number five with regard to sex.

If we take the C as a basis, we have the following values:

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(*) The interval of major third is always calculated according to the succession of harmonics sounds. Between C 3 and E 3, viz. between the forth and fifth harmonics (45), an interval of major third, namely an interval made by two tones, is created. The interval of minor third (made by a tone and half) is between the fifth and sixth harmonics, viz. between E 3 and G 3.
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Therefore the relation would be: ces (C) cis (*)
It looks like the relation man/woman, in the sense of harmonics, originates two chromatic notes (**), symmetric to the fundamental (C) almost as if the two bodies were both generated by a unique androgen unmanifested reality.
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*) The word ‘ces’ corresponds to C flat (Cb), whilst the word ‘cis’ stands for C sharp (C#).
(**) By chromatic notes we mean two notes at a distance of a semitone that keep the same name. For example: C-C#, or C-Cb.
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The Author carries on by examining the figure of the female body (fig. 339) through the method of equitonals, and he concludes that its main points and parts are determined by lengths that, together, make the common diatonic musical scale:
c d e f g a h c (*)

Still with regards to the human figure, he considers the Canon of Villard by Honnecourts, further harmonics divisions, whose construction is reported below (**):
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(*) this is the transcription in Italian of the musical notes:

(**) Kaiser dedicated a special leaflet to this canon used by the Builders of Cathedrals of the middle Ages. You can see an extract mentioned in this Commentary, page 144.
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Be A B or CD the segment to divide. Let’s draw a rectangle with proportions 1:2.
Now let’s draw the diagonals AC, DB and the angle DSC; we already have Villard’s scheme.

It offers us two values of parts:

1) At the intersection of the diagonals, half (1/2 of the segment) AB;
2) At the intersection of the diagonals with the oblique SC, SD the division by three of the segment AB or more precisely the points 1/3 e 2/3.

The rest follows automatically.

Let’s draw from the point of intersection of the diagonals the parallel to the line AB and we obtain the points E and F. The lines SE and SF give, with their intersection with the diagonals, the division by 4 (1/4 e 3/4). Then we draw from 1/3 and 2/3 the parallel GH, the lines SG and SH; this gives us, at the points of intersection with the diagonals, the division by 5 (1/5 e 4/5). In theory, we could continue the division endlessly and obtain all the rational division of the segment AB. This method of distribution as it appears in fig. 341 has the advantage of allowing not only vertical division, but also horizontal ones, not examined by the mentioned Wyneken. We refer to the thorough exam of the figure to observe the harmonious disposition of both the external image of man and his main organs (heart, head, etc.).

The stages of the construction are the following (fig. 343, 344, 346, 349 and 350).

Index 1. Original, undifferentiated cell. Index 2. Both octaves (*) C' and C appear; they are the axis of vertical symmetry and the two tonal higher directions. Nothing new in the tonal sense. Index 3. Two new sounds appear. G and F, the dominant (**). Index 4. No new sounds; other octaves.

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(*) By octave we mean the interval between two notes with the same name, implying the other seven internal notes. For example C (D, E, F, G, A, B) C; 1 (2 - 3 - 4 - 5 - 6 - 7) 8

(**) The notes G and F are considered, in our musical system, as dominant notes in the C major scale. Dominant means in general a note with a particular ‘tension’ in a scale and in a musical composition.
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In a figurative sense the two lower directions appear; the Author calls them static because they contain only identical tonal values, C' and C. Therefore they are opposite to the two mentioned above, which have a precise dynamic character. Index 5. The major and minor thirds originate and the possibility of major and minor chords. Index 6. This index as well (2 x 3), as all the even indexes, doesn’t produce any tonal news. In the figure is outlined the whole of points and values obtained so far, to highlight the closed nucleus of the senary. It is important to observe the melody made by perimetric sounds followed in order: it is the melody of the border. We must say that every index is characterized by its own melody. Index 7 has been excluded, because of the law of the Generator.

Index 8. A certain spatial structure appears for the first time. The points are not closed tight, one against the other any more. In the form we can see open spaces.

Nothing new in a tonal sense (except for the melody of the border!)

Index 9. The sounds D and B originate. The intervals of a second, so far absent, appear. The consequence is the possibility to trace circles and ellipses of musical scales, such that the sounds of their border are like a concluded musical scale.

Furthermore, this index originates a contest of counterpoint (intended in its genuine sense); at this stage a specific differentiation enters the biological form.

a) Axis of vertical symmetry.
It divides the form in two parts: greater than 1 and less than 1, right and left. This explains the presence of two clearly distinct polarities in the human form.

There is another polarity, though, present in every part of it: the major and minor tone. The latter doesn’t depend at all on the simple numerical expressions.

b) Two separate pairs of directional axis. The superior one is dynamic, the inferior is static. The correspondence with the higher and lower limbs of man, hands and feet, is obvious.

Note that the exam of the values and not of the numbers allows the understanding of the different function of these pairs.

c) Sonorous circles on the left and the right and in the higher part (3 + 3). They seem to represent the biological parallel of the appearance of double sensorial organs (eyes, ears).

d) The central nucleus: true solar plexus from which all directions and impulses irradiate. It is the true starting center of the harmonic and morphologic development.

e) Counterpoint. Every sound present in a part of the form has its exact counterpoint in the symmetric one. Furthermore, because of the combination of two Diagrams chosen there is also congruence. In a biological sense, this reality justifies and ensures the transformation of the organic form; this is nothing but the material and visible expression of melody and counterpoint.

f) Two central ellipses, high and low. We’ve already talked about this with the Index 10.

It is justified to assume that they indicate the cerebral sphere and the sexual function. Their polar collaboration in the whole of the form is obvious; their true harmonic meaning, though, is visible only by observing their sonorous content.
Where the first enharmonic divisions appear in the sonorous development, we can find both; around a center of identical value (C) two different values of B and D appear.

The enharmonic structure that they have seems to indicate the deep need for a dialectic motion of the two spheres of action, in and between themselves.

The Author warns that when we want to study in such a way the growth of an organic form or when we want to verify its harmonic impulses, we can’t expect exact results. These sonorous figures show and reveal the inner tendencies of development. It is not about finding through them the formal differences between hand and foot, but about discovering their psychic reciprocity. Another consideration is that melodics occur (becomes in a certain period) whilst a chord is, whilst it coexists spatially. Furthermore, in the formal expression melodics can be assimilated by graphic or a drawing; the chord, on the opposite, is similar to the pictorial or the coloristic whole. As an example of ektypic application, the Author proposes the Canon by Villard by Honnecourt, examined and discussed earlier. He points out that if we develop it in the senary, we keep constant the basis and vary the height in the measure of 1/2, 1/1, 2/1, we obtain three figures (427) that can describe the essential outline of three completely different architectonic styles.

Egyptian, Romanic and Gothic. In the first the earth prevails, in the second we have a balance, whilst in the third the celestial element prevails. Three octaves, three styles, three civilizations.