C13 The Development of the Western tonal System

The organ had a predecessor in the ancient water organ, which is said to go back to Ctesibius of Alexandria,[i] a mathematician, engineer and inventor who lived in the third century BCE. Charlemagne was given a water organ as a gift, but there is a gap of several hundred years in our knowledge between this time and the early organs of the Middle Ages, and we only have a rough idea of what the keyboards of those instruments were like. Some depictions show sliders that are more comparable to today's register pulls and do not yet show any chromatic sequence. Others have keys the width of a hand that had to be operated with the hands or elbows, giving rise to the German expression "beating the organ"[ii]. And yet others demonstrate the dexterity of the player, which requires a smooth keyboard.[iii] The following depiction of a portative also shows that the earlier key sequence was still a uniform one, because two rows of organ pipes correspond to two rows of keys. And this division has been deliberately avoided.

The term keyboard, which contains the Latin clavis, may have ancient roots. At least, we hear from Sebastian Virdung about the clavichord (lat. clavis "key", corda "string") in 1511: «Wer aber darnach der sey gewesen / der das erfunden oder erdacht hab, mocht ich nye erfahren / wer auch das instrument nach den selben Schlüsseln / als clavichordium hab getauffet oder genennet / waiss ich nit»[iv]

The title of his book Musica getutscht und ausgezogen indicates that he is reproducing a Germanized (getutscht) version in excerpts (ausgezogen). The underlying, probably Latin source text has remained unknown until now.

Since drop bolt and locking spring locks were already known in ancient times, the mental connection with the key can have very far-reaching roots, especially since appearance and movement could have a much more vivid relationship to one another than is the case today. With locking spring locks in particular, it is sufficient to push the key in to gain access. A turning movement is not necessary.

In light of the content presented, it seems sensible to examine the keyboard as we know it today and as it was drawn by Arnaut de Zwolle, including the possibility of a numerical symbolic implementation. In analogy to the typewriter, whose keys naturally refer to the alphabet and the character set, we must first consider the available sound material, because it is obvious that the meaning and purpose - especially since we are dealing with a tool - are much more important than the form of a keyboard as such. A keyboard can only ever be the reflection of an underlying ordering system.

When Pythagoras was already studying the vibration of individual strings, he was confronted with the fact that they always vibrate at the same time in whole-number multiples of the fundamental frequency. So they don't just produce a single tone, but provide a rich spectrum of sounds, which should become the starting point for all subsequent considerations.

The reason for their vibration behavior lies in the limiting situation of the string ends. All vibration forms of the string have exactly the same two resting points - called vibration nodes - at the end of the string in common, because they are simply predetermined. The overall picture of their wave-like movements therefore always has a symmetrical character, whether they are even or odd. Vibration forms other than integer ones are not possible.

The following table shows the sound spectrum of a single vibrating string up to the 16th partial tone, where we use the variable x for the fundamental frequency. 

Partial tone spectrum of a string

If we hear the sound of two strings simultaneously, we are dealing with the same spectrum twice. There is complete agreement, i.e. consonance (Latin con "together" and sonare "to sound"). The situation is that equally high notes sound clear and pure, while slight deviations cause volume fluctuations that we immediately perceive as unpleasant disturbances. And the speed of these volume fluctuations increases as the distance increases. In acoustics, this is called beats.

It is easy to imagine the chaos that ensues when two strings that do not quite match each other with all their partial tones come into contact with each other. A multitude of vibrations are created, which also overlap each other. This simply overwhelms the human ear's perception, as if several people were talking to one person at the same time. It is not for nothing that we speak of detuning in a deliberate ambiguity.

If we shift the spectra further, we suddenly notice that partial similarities arise that sound pleasant, not in the sense of a subjective sense of beauty, but due to the noticeable fact that the disturbing beats suddenly disappear. This is always the case when the shift itself is integer. The following table shows the comparison of the sound spectra of two sounding strings up to the 16th partial tone, which are tuned to a factor of 3 apart. 

Two partial tone spectra in comparison

The resulting similarities (highlighted in red) are quite high, but of course not as high as with complete similarity. However far you want to extend these comparisons into the highly complex, the basis always remains the integer proportionality of the partial tones to one another, whereby a certain order is introduced into the system by the fact that things repeat themselves when doubled. If you only look at what happens between x and 2x, you gain access to the entire tonal system. For this reason, the fundamental frequencies in relation to one another take on central importance, because read as fractions, they are numerical values between 1 and 2.

But these proportions do not just express a calculated frequency ratio. You can read the precise position of the first matching partial tones from them. 3:1 therefore means - to stay with the example given - that the 3rd partial tone of the lower-tuned string and the 1st partial tone of the higher-tuned string are at the same height, see table. This is because we are dealing with a kind of parallel shift, because the relationship of the fundamental tones to each other affects all partial tone relationships equally. In our case it is 3:1. Up to this point we are actually dealing with an observation of natural conditions and not with human intervention.

The first scales arose from the need to sort the tonal material according to size, with the five-step scale (pentatonic) still predominating in antiquity. In the early Middle Ages, this was supplemented by a further tone (hexatonic). The term solmization goes back to the procedure of singing the tones on different syllables, for which the John Hymn by Paulus Diaconus (722-799) provided the syllables for the then six-step scale. This system is said to have been introduced by Guido of Arezzo (born around 992). The seventh note of the scale, SI, which again exceeds the tonal range chosen by Guido, was later formed from the initial letters of Sancte Ioannes. It assumes a seven-step scale. (Heptatonic)

Ut queant laxis

Resonare fibris

Mira gestorum

Famuli tuorum

Solve polluti

Labii reatum

Sancte Ioannes.[v]

The seven-step structure is based on the alphabetical naming: A-B-C-D-E-F-G. The B inserted between B and C in the 12th century has to do with the splitting of the note B into a higher variant (b durum) and a deeper variant (b molle), which were distinguished in writing by a square (b quadratum) and a round (b rotundum) form of the letter b. The b quadratum gave rise to today's natural sign and the sharp, while the b rotundum became the accidental b. Because of the visual similarity of the B quadratum to the letter h and the resulting use of the h typeface for the b quadratum, the designation H for the 7th step of the scale was used in Germany in the 16th century. In the Anglo-Saxon language area, our B is still called B today, while there our B (as the lowering of the B by a semitone) is called "B flat".[vi] The following illustration shows a portative with a very reduced number of inserted semitones. 

St. Caecilie with portative, Master of the Bartholomew Altar, detail, Wallraf-Richartz-Museum, Cologne

Prim 1:1, Octave: 2:1, Quint 3:2, Quart 4:3, gr. Sext 5:3, gr. Terz 5:4, kl. Terz 6:5, kl. Septim 7:4, grosse Sekund: 9:8, gr. Septim 15:8, kl. Sekund 16:15, Tritonus 45:32.

It is a mix of natural harmonic intervals, such as the pure fifth (3:2), which can be tuned directly from the fundamental note, with constructions that result from the addition or subtraction of intervals. The effort to further pursue the relationships of pure intervals and their relationships to one another goes back to Pythagoras, who mathematically compared 7 octaves and twelve fifths, as well as to Didymus, who compared two whole tones and the major third.[vii] Behind the structure of scales are harmonic systems that become more complex with each addition of additional tones, taking into account increasingly smaller tonal distances. In order to clarify the question of numerical symbolic reference, however, these do not need to be presented specifically in this study.

Popular piety has a certain way of looking at things, which is expressed by drawing analogies. This can be easily illustrated by a completely different example, namely the passion flower. It got its name because the three styles of the pistil were seen as the three nails of the crucifixion, the five stamens as the five wounds, the staminodes as the crown of thorns of Christ, the shoots as the scourges, and the 10 petals as the 12 disciples, excluding Judas and Peter - who are excluded as a result of the betrayal and the threefold denial.

Passion flower, passiflora incarnata

Its name is Passiflora incarnata, which means: the incarnate. This puts it in a long line of plants that have been associated with the Bible or the church in a similar way, such as angels trumpet, crown of thorns, bishops weed, Joseph’s coat, Jerusalem thorn, St John’s Wert, to name just a few.

But let us return to the order of the tonal system. The term harmony, as shown, is related solely to the degree of the partial tone agreement shown. In this respect, the interval series listed - starting with the prime and ending with the tritone - describes a gradual decrease in the proportions of agreement, because the larger the numerical values, the further the partial tones of equal pitch are from their respective fundamental tones and more and more other dissonant partial tones get in between. This explains the term Diabolus in Musica[viii] for the tritone, because it is in some way in opposition to the consonance.

The development described up to the division of the octave into twelve went hand in hand with the desire to use pure intervals, because only because of the freedom from beats could the pitch intervals be precisely defined - so the musical need of attentive listeners was naturally to drive differentiation further. This is evident from numerous proposals for keyboards that have more than 12 keys per octave. In technical terminology, these are referred to as broken keyboards. In 1548, Gioseffo Zarlino had Domenico de Pesaro build him a harpsichord with 19 keys per octave, the Clavemusicum omnitonum by Vito di Trasuntino had 31 keys per octave, the Archicembalo by Nicola Vicentino had 36 keys per octave on two manuals. And for the same reason, Michael Praetorius writes:

«Darumb ist sehr gut und hochnötig, dass in den Orgeln und Clavecymbeln, welche zu Concerten in der Music gebraucht werden, das schwarze Semitonium Dis, und womöglich, auch das Gis duplieret würde.»[ix] 

It is not going too far to say that classical harmony theory with regard to keyboard instruments came to a temporary end around Arnaut's time because considerations other than purely musical ones played a dominant role. Suddenly people were no longer looking for the "acoustic clean room", the paradisiacal embodiment of consonance, and Arnaut was already showing the keyboard as we know it today. Up to this point, every key had its own distinctive characteristics, which can still be implemented today by instruments such as the violin. But at piano concerts, two very different tonal systems collide.[x]

We will find out why this is the case if we follow the process further. The consequence of not making further subdivisions was that people had to be interested in the distribution of the beats that had to be accepted, which they had previously tried to avoid wherever possible. The problem of musical temperament, which scholars have grappled with for centuries, is precisely due to the fact that not a single tuning that uses pure intervals works mathematically in the 12-part tonal system. The purity of the octave always remained untouched. Only equal temperament was able to overcome this problem. The factor 12√2, which mathematically underlies the equal semitone steps, was not a known quantity, however, to Pythagoras or to medieval scholars, and was not an acoustically desirable goal because not a single interval with the exception of the octave sounds pure in it. For a long time, people stuck to the beat-free intervals when tuning - remained dissatisfied and got into many disputes - and this was clearly due to the impressively clear acoustic experience of pure intervals.

Since ancient times, the relationship between world order, divine plan and the recognition of basic structures of reality (ontology) has run parallel to the pure observation of facts. Pythagorean numerology already linked the movement of the stars with the harmonious relationships of the vibrating string.[xi] The metaphysics of the Trinity, inspired by Augustine, in particular, took on a particular form in the High and Late Middle Ages, particularly in the work of Bonaventure, Thomas Aquinas and Nicholas of Cusa.[xii] Against this background, the concrete relationship between triad and Trinity appears unexpectedly late and was probably first encountered by Cyriacus Snegassius (Cyriak Schnegass 1546-1597).

Purely tuned sounds are to be understood as special, regardless of subjective experience. The partial tone relationships alone make this clear. Equal temperament has nothing comparable to offer. Its advantages lie elsewhere.[xiii]

If we look at the current key division from C to C, and because the underlying construction principle of a keyboard can be easily illustrated in this section, we count 13 keys. There are 12 semitones between them.

The 8 diatonic keys (from the Greek Διατονικός, going through the tones) are white and their names are: C, D, E, F, G, A; B, C. The 5 chromatic keys in between - the name (ancient Greek χρῶμα, color) reveals that they are only gradual "changes of color" of the higher or lower starting tone - are now colored black. Their names depend on whether they are viewed as a raising of the lower tone, in which case they are C sharp, D sharp, F sharp, G sharp, A sharp, or as a lowering of the higher starting tone, in which case the same keys are D flat, E flat, G flat, A flat and B flat. In natural-harmonic thinking, however, these keys would have to be split in two - Praetorius called it duplicated - because C sharp and D flat are not identical in terms of the tonal system.

The graphical structure of the octave is the result of the fact that there are semitones between B and C and between E and F. This results in two divisions of 3 and 4 keys in the front area. Including the chromaticism, there is a twelve-part division. Graphically, the section between HC and EF in the back area is divided into 5 parts and that between EF and HC into 7 parts. In addition to the diatonic keys, there are also the chromatic keys, which are arranged separately and appear as shortened keys between the whole tone intervals. This brings order to the system and creates a visual structure that makes it much easier to find a specific note. It is worth noting that Arnaut's keyboard still starts at B - because the division of pitches in his time was made from B-B and not from C-C,[xiv] which is easy to see when we look at his entire keyboard labeling. However, this does not change the construction principle as such. It had already been noticed that he did not yet have a precise knowledge of the practical division process of a keyboard. He draws straight lines between the semitones B and C and thus divides the notes into octaves. He did not know that he would have had to do this between E and F as well and that he would have had to divide the keys back and forth independently in each section. The notches in the paper make it clear that he divided the notes by moving the compass along a drawn line. This is why the size of the deviations between the back and front divisions increases towards the treble. There is no doubt that this approach was wrong even then, because such stumbling blocks are not deliberately sawed into the keys. The extreme is the line between e'' and f'' (red circle).

It would certainly have been in keeping with Arnaut's logic of numerical symbolic content if the construction of the keyboard had been included in the rules of his other drawings. He would then not have had to consider it as something separate. However, for technical reasons, the underlying principle of order is to be found less in the graphic construction of the keyboard than in the structure of the entire tonal system - because, as we have seen, the octave is not treated as a whole in the keyboard division as a tonal space between C and C that needs to be divided up. Due to the need to cut the keyboard straight through at the location of the diatonic semitones, the impression is created that the higher C no longer belongs there, but is already part of the next section. This means that the last semitone step is deducted. However, this is purely a technical fact. It is therefore more appropriate to represent the tonal space of an octave in a circular form, as shown in the following illustration:

Representation of the octave in circular form

Circle of fifths in analogy to the last Judgment 

Diletski, Nikolay, Grammatika, musikijkogo pinija, 1677

Much more common, however, is the representation of the so-called circle of fifths, in which the tones are shown in the tuning order. The increase in complexity can already be seen in the large number of accidentals in the sense of necessary "color changes" that increase towards F sharp/G flat - i.e. downwards - especially when you realize that the two tones are not the same and neither of them divides the octave in the middle - which would have to happen in order to close the circle. Instead, a spiral is created and the octave is broken down into countless small parts. The Riemann Music Encyclopedia lists 262 tones per octave.[xv] Using cent calculations, they are broken down into 1200 parts.[xvi] Against this background, the numerous variations in tuning theory are almost like the search for the Holy Grail.[xvii]

In the language of numerical symbolism, the octave forms the greatest possible consonance and encompasses the entire musical cosmos like a circular arc. As the name suggests, it contains eight elements that correspond to the eight bliss. With the addition of the five chromatic keys, the five wounds can be associated.

The total number (8+5) of the 13 keys now present corresponds to the number of people at the Last Supper. In a circular representation, analogous to the face of a clock, the beginning and end, alpha and omega coincide, leaving 12. The six whole tone steps included represent the six gifts of the Holy Spirit. The divisibility of 12 by 4 alone makes one think of the 4 evangelists. The divisibility by 3 of the Trinity. The 1 is present in all tones and is the epitome of consonance and inner peace. Dissonance, on the other hand, is the work of the devil. In the circular representation, consonance (prime/octave) is at the top, opposite it is dissonance, evil. It thus corresponds to the wheel of fortune, which carries the lucky one up and lets the unlucky one, in the language of the image, "get under the wheel" and expose them to unfortunate circumstances. But the wheel does not just symbolize happiness, it also stands for fate and the course of the world. The following illustration shows a depiction from the Codex Buranus, the medieval song collection from 1230 that was found in Benediktbeuern and whose name became widely known through Carl Orff's Carmina Burana.

The Wheel of Fortune: Book Illumination in the Codes Buranus, Bavarian State Library Munich

The handwritten text is added to the depiction of the wheel: left - regnabo ('I will rule'), above - regno ('I rule'), right regnavi ('I have ruled'), below sum sine regno (I am without rule). The wheel itself has eight spokes. In front of it appears Fortuna, who, like the god, determines the course of the world as the "unmoved mover".[xviii] This in turn shows parallels to numerous depictions of the Last Judgement.

The following illustration shows the Last Judgement by Hans Memling, which is about distinguishing good from evil. God the Father appears as ruler of the world (Pantocrator), here on a rainbow that closes in a full circle. God the Father has the globe directly at his feet. The Archangel Michael weighs the good and bad deeds of those who have just risen from the graves and decides whether they will be ascended or descended. The inside of the left altar wing shows the gates of paradise in the style of a cathedral portal. Right on the steps leading up to the portal, Peter, with the key in his hand, greets the blessed, who receive their new garments from the hands of the angels before entering.

Hans Memling, World Court, Gdansk National Museum

The right wing, on the other hand, shows the fall of the damned, the fire of hell and a number of demonic beings. The trumpeting angels within the band of clouds correspond to the mention in Matthew: "He will send out his angels with the sound of a trumpet, and they will gather his elect from the four winds."

Christ at the Center of the Cosmos, Fresco in the Basilica of Mtskheta, Georgia

The above illustration[xix] shows the zodiac[xx] with the 12 well-known symbols in the outer ring, with the sun at the top and the moon at the bottom. Both have a human face, with the profile of the moon looking up at the sun. This creates a contrast between day and night, or light and darkness. There are 14 symbols in total. In the middle ring, Christ is surrounded by 12 apostles or their symbols, making a total of 13. In the center, God the Father sits enthroned within a square standing on its tip, surrounded by the four evangelists. This illustration is informative, although the numbers alone are less consistent, but the conceptual connection is clearly reflected. The entire cosmos is designed according to a divine plan. A world view is revealed here that recognizes divine ordering principles in all elements of creation - including the cosmos - which corresponds to Augustinian tradition. The cosmic dimension in particular is made clear in Johannes Kepler's books on world harmony:

"I feel myself seized by an inexpressible rapture at the divine spectacle of heavenly harmony. […] For we see here how God, like a human architect, has approached the foundations of the world according to order and rule."[xxi]

Keppler's analogy between creator and builder can already be found in the Bible (e.g. Heb, 11:10). In particular, in the letter of the Apostle Paul to the Ephesians (2:20) it says:

"So then you too are no longer strangers and aliens, but fellow citizens with the saints, and members of the household of God, built on the foundation of the apostles and prophets, and of which Jesus Christ is the cornerstone, in whom the whole building, fitted together, rises into a holy temple of the Lord, in whom you also are being built into a dwelling place of God by the Spirit."

We encounter comparable representations of God the Father and the cosmos in a wide variety of forms, particularly in the design of church domes as imaginary vaults of heaven, whereby the number of windows was of course also sometimes taken into account.[xxii] In general, the whole-number proportions of harmony theory have not remained without influence on architectural theory. The Roman architect Vitruvius already mentioned the importance of knowledge of the theory of sound and the mathematical relationships of tones in connection with the training of an architect.[xxiii] However, it should be noted that musical harmony, as described, results from the decomposition of the octave, taking into account the partial tone relationships and beats, while in architecture it develops from a freely assembled construction with the proviso that all parts enter into a relationship with the whole. For example, in Andrea Palladio (1508-1580) all details of the column orders and their members are proportional details that relate to the lower column diameter as a module. Against this background, the attempt to transfer the proportion of the golden ratio to the division of the octave, as Thorvald Kornerup did, appears to be an interdisciplinary overreach that simply ignores the fact that in music, listening is the main focus and one cannot avoid proportioning the beat speeds, i.e. putting them in relation to one another.[xxiv] Hans Kayser, who reproduced the division of the golden ratio on the monochord, even considers it to be a mistake as an aesthetic criterion based on the listening experience. It is an irrational number, "a purely intellectual quantity"[xxv] with no inner connection to people. Even if, figuratively speaking, they are different currencies, they are still means of payment. Because the golden ratio is also based on the desire to establish a relationship - and that is precisely the concern of the analogy-recognizing, i.e. compassionate person. The fact that the determined number is not equally applicable in music does not change its ability to put three segments in wonderful relation to one another. It divides a segment AB+BC in such a way that the smaller AB is in the same relation to the larger BC as the larger is to the entire segment AC. And it is precisely this kind of tuning and relating that is meant by harmony.

It is therefore quite logical to consider musical intervals (Latin intervallum, space, distance)[xxvi] as distances and divisions of distances, which can be done independently of the given partial tone spectrum and the interference that occurs. But this shows that the theory of proportion and harmony can in no way be considered compatible, even if on an abstract numerical level it appears as if, for example, the interval ratio of the fifth of 3:2 and Palladio's calculation of the height of a room in the ratio 3:2[xxvii] - we remember that this proportion was also used in Arnaut's Trinity symbolism - are also in terms of content one and the same universally valid law or the same proportion of beauty applied in an interdisciplinary manner. To differentiate, one should probably say: artistically: yes, scientifically: no. Because appearances are deceptive. Although these are mathematical procedures, the consideration of the various calculations is based on an analogy. And not everything that has a similar appearance - in this case the calculation result - can be traced back to the same cause. It is certainly a magical thought that a building was built according to musical proportions, as if architecture had been able to borrow something from the beauty of its immaterial sister, which in turn, just as symbolically, is connected to the kingdom of God. This is what is so magical about suggestive communication: things are known to one another, understand one another and intentionally relate to one another out of pure empathy. It also says:

"If two of you on earth agree about anything they ask, it will be done for them by my Father in heaven. For where two or three are gathered together in my name, there am I in the midst of them."[xxviii]

It is not without reason that Martin Schleske points out that the word "agreement" used in the translation in the Septuagint is based on the verb symphonein "sounding together".[xxix] Here we are dealing with a successful multi-media implementation of what we are used to describing as paradisiacal. Palladio himself puts it this way:

"He [God] is the supreme and absolutely good being and the embodiment of supreme perfection. It is therefore very appropriate that everything that is dedicated to him should be carried out with all the perfection of which we are capable."[xxx]

The compassionate person has an affinity for analogies, appreciates a system of order in which one can find one's way around and sees oneself as part of a wonderfully interwoven whole. Only in this way can one understand why even the capitals of the columns take their name from the reference to people (Latin capitellum, little head, caput, head).

Reason, on the other hand, has no sensorium for beauty - otherwise there would actually be universally valid criteria of beauty that would be recognized worldwide, like the multiplication table that can be reliably applied everywhere. Causality identifies the different, which comes in analogies, as alien to nature and immediately declares itself not responsible for this area. Therefore, art history has the task of working bipolarly, i.e. applying two different criteria of assessment. These connected differences, as Robert Brentano[xxxi] put it, form the backbone of the studies.

Furthermore, it is worth noting – again rationally – that the naturally harmonious octave division is an asymmetrical division, which in turn would not have been tolerable in architecture until deconstructivism, because the human blueprint played a very important and exemplary role, especially in the Renaissance.

Likewise, the movement of the planets is subject to its own laws, which cannot simply be derived and transferred from the vibration behavior of a string. If you look closely, nobody does this, but the assumption that both can be traced back to one cause, the Creator, assumes a commonality in terms of their origin, which can be reconstructed by looking at parallels.

This view comes from a time when scientific disciplines were not yet separated from one another and were subject to the primacy of the church. "Your will be done, on earth as it is in heaven," says the Lord's Prayer. Basically, everything that can be counted can be related to one another on an abstract level using numbers.

Otherwise, we encounter the division into twelve, the duodecimal system, in the number of months of the year, in the hours of the day - whereby we find the 13-1 symbolism again in the saying: "Now it's striking 13", in the 12 signs of the zodiac, in the units of measurement of feet and inches,[xxxii] in the longitude and latitude, in the angle measurement and in counting in dozens.

Bloodletting Man, Strasbourg Calendar, State Graphic Collections, Munich

Christianity was able to adapt the much older mysticism of numbers and reinterpret it in its own way.[xxxiii] And in medicine, the so-called "bloodletting men" or "bloodletting tables" were an integral part of almanacs and calendars. In addition to a general description of the movements of the celestial bodies, seasons and agricultural activities in the rhythm of the moon and sun, these also contained a calculation of the months and days that were considered particularly favorable or unfavorable for bloodletting and also showed the places where the bloodletting should be carried out. Even recipes for varnishes or essences were compiled according to astrological symbolism of proportion.[xxxiv] The astronomical tables therefore served doctors as a valuable guide for the professional practice of their art. Scholars like Arnaut were particularly trained in the overview of these things and were critically examined for the conformity of their thinking with the Bible - not only within the framework of the Inquisition. If they were religious, they naturally also constantly questioned themselves in this regard. The struggle of natural science for its independence runs through history as a long process of tough internal and external conflicts and was to flare up again when Charles Darwin presented his work: On the Origin of Species.

Regardless of this, there is still an extensive ontological literature in which individual scientific observations are used in the sense of ordering principles and do not necessarily have to take on the character of proofs of God. One representative of this genre is Hans Kayser (1891-1964), when he wrote: "First and foremost, harmony is a holistic theory. That is, it attempts to understand the world and humanity from a holistic point of view and listening. The scientific means for this in harmony is the primal phenomenon of the number of tones."[xxxv] Rudolf Haase (1920-2013), founder of the Hans Kayser Institute for basic harmonic research at the Vienna University of Music, continued the studies in his spirit. And the work of Joachim Ernst Berendt, which he presented with Nada Brama, The World is Sound, to name just a few representatives of this genre, should be seen in the same context.

It is noteworthy that the investigations of Robert Hooke (1625-1703) into the elastic behaviour of solid bodies had completely different consequences than the investigations carried out by Pythagoras on the vibrating string. This difference is undoubtedly related to the acoustic effect on the physical, psychological and mental well-being of humans. From a scientific point of view, humans are only a measuring instrument that can be used to a limited extent because their complex perception influences the result. Their affinity for holistic observation is given a freedom in ontological studies that science does not allow itself. Therefore, such treatises - even if they contain respectable academic achievements - have the character of speculation from their perspective, because they are satisfied with sufficient plausibility where evidence is required. And plausibility is already created by bringing together numerous different observations with analogy instead of causal reference. The strictly scientific point of view is expressed in the words of Sigmund Freud:

"Psychoanalysis, I believe, is incapable of creating its own particular worldview. It does not need it, it is a piece of science and can therefore join the scientific worldview. But this hardly deserves the grandiose name, because it does not look at everything, it is too incomplete, and makes no claim to completeness or system formation. Scientific thinking is still very young among humanity, and has not yet been able to overcome too many of the major problems. A worldview based on science has, in addition to the emphasis on the real external world, essentially negative traits, such as modesty towards the truth, the rejection of illusions. Anyone who is dissatisfied with the current state of things, who demands more for his immediate appeasement, can get it wherever he can find it. We will not blame him, we cannot help him, but we cannot think differently because of him either."[xxxvi]

Freud's criticism of ontology is directed less against the content, which he does not even name in the quoted conclusion to his lecture, than against the method of trying to describe something as a whole when one only knows individual parts. For science itself, a world view is utopian in that it would require the end of all questions - i.e. omniscience. But then it would not be a world view in the sense of a projection, but would have to be described as a description of the facts of the world.

Looking back at the development of the keyboard, it can be seen that the scale was already associated with biblical content at the time of heptatonic music. And the verses of the John Hymn by Paulus Diaconus read like a prayer. The division into six directly corresponded to the six gifts of the Holy Spirit, and even today the octave consists of six whole tones. Therefore, we should not look for a single inventor as an outstanding personality who put the tonal system together in numerical symbolism; rather, it is a development spanning many hundreds of years, which was already based in antiquity on the idea that the ordering principle of the gods can be recognized in creation. Today's keyboard contains the elements of its historical development. This can be clearly seen in the inclusion of the B between B and C and in the almost Babylonian confusion of languages ​​used to describe the respective tones. Even today, it is difficult for a piano student to understand why one and the same key should be called something different (e.g. D flat or E flat) depending on where it is being played from, and why there are augmented and diminished intervals in an equal-tempered system. This in some ways reflects the longing for how things should actually be. Against this background, the existence of only one key for D flat and E flat is disillusioning.

The nature of the partial tone spectrum motivates anyone who loves order to search for perfection. In this respect, physical conditions, the experience of beauty and spirituality are closely interwoven and the Western tonal system is the result of ontological considerations. It is important to note that this is not a pure presentation of natural conditions, but an idealized projection. The circle of the circle of fifths is only closed with the help of mathematical construction. If you close it, it sometimes becomes quite inharmonious. In technical terminology, however, one speaks of enharmonics, as if the drastic departure from pure intervals had to be glossed over. (The Greek prefix en, ἐν means: in, with, with). Today's keyboard was created at a time when religious number symbolism was of fundamental importance throughout Europe. Even if their design cannot be documented by a single source, it is hardly conceivable that, despite all efforts, the usability of the material at hand to support the religious intention was overlooked and that the agreement with the numerical values ​​observed by Arnaut must be regarded as a coincidence. Around 200 years before Gregory XIII, who decreed the calendar reform to precisely determine Easter in his bull of February 24, 1582, a similar decision must have been made with regard to the tonal system, because since then the keyboards have had a uniform appearance, with a few exceptions.

If the only aim had been to divide the octave into equal steps, we would expect a keyboard that reflected this regularity, such as the keyboard of the telegraph by George Phelps (1820-1888) with its alphabetical key labels. Whole tones and semitones, if one can even use these terms for this device, alternate with one another. Instead, it was decided to use the C major scale with its specific semitone steps as the basis for the field of music and to characterize all other keys as deviations from it. 

Telegraph by George Phelps, Museum of Arts and Crafts, Paris

This idea would be understandable even without reference to numerical symbols. Here it is sufficient to note that numerous variants were available and that, above all, musical needs would have given reason to divide the octave into more parts than just 12. It is remarkable that the primacy of pure intervals was suddenly abandoned instead of consistently pursuing the path that had been chosen. Instead, the intervals were now exposed to impurity and confronted with dissonance, the acoustic evil - from which one would otherwise have wanted to be redeemed. Here the analogy to the incarnation and the expulsion from paradise is unmistakable. It is therefore not surprising that music has a moving effect, because people can see in it the reflection of their own fate: their existence exposed to adverse circumstances, with only the hope of redemption. The religious perspective has given the Western tonal system its character, which encompasses the greatest possible emotional range between heaven and hell, or healing and painful. In between, all the necessary tones can be mixed, and all emotional states can be evoked.

The world has a rich and astonishing musical heritage to thank for this fact, including the numerous tuning variations up to equal temperament – which clearly shows the inherent problems – the work of Johann Sebastian Bach, who dealt in detail with the given harmonic situation in his Well-Tempered Clavier, as well as countless sacred and secular compositions that benefit from the acoustic tension between beat-free clarity and traumatizing dissonance and use its colorful palette for their compositions – be it in jazz or blues. No other tonal system was created like this. With the keyboard, the user enters a sacred sound sphere, comparable to a church building. Basically, both belong together in a way that was vividly expressed in the form of the medieval portative. From a Christian perspective, there could be no secular music at all, because it recognizes a part of creation in simply everything. And music itself does not primarily serve to create beautiful sounds: the practice of music is prayer and worship. The establishment of the tonal system, in the sense of consciously stopping at a selected point in its development based on numerical-symbolic congruence, is an epochal cultural-historical process simply because of its dimension.

© Aurelius Belz 2013

[i] Vitruvius mentions him in his VII book as the inventor of a water pressure mechanism immediately before the description of the water organ.

[ii] The keys of the carillon are still struck with the fist today. Batel, 1986, 15. A detailed description is given in Williams, 1980, 14ff.

[iii] Such a depiction from the Belvoir Castle Psalter from the 13th century, see Apel, 1967, 12 Fig.4

[iv] "But who it was that invented or thought it up, I could never find out, and who named the instrument according to the same keys as a clavichord, I do not know.”, Virdung, 1511, Eii, f

[v] Literally: "So that the disciples with loose vocal cords may be able to make the wonders of your deeds resound, release the guilt of the stained lip, Saint John" - an allusion to Zacharias, who, according to the report in the Gospel of Luke (Luke 1:22), became mute and whose tongue was released again at the birth of his son John (Luke 1:64f). For the same reason, John the Baptist was the patron of church music before he was replaced by Saint Cecilia.

[vi] Riemann, Music Encyclopedia, 1967, 357. However, this representation clashes with the fact that Arnaut already knew the B inserted between B and C. We also encounter it in a Regensburg manuscript from the 11th century. Bavarian State Library Clm, 14965a, fol.30. The inclusion of the B between B and C can therefore hardly be related to the use of printing types. Ottaviano dei Petrucci (1466-1539) did not patent the printing of music using movable metal types until 1498. The author considers it obvious that the necessary addition was simply made with the letter that follows the G and was therefore "next in line". If one imagines labeled keys, the old names remain in both cases.

[vii] The Pythagorean comma can be calculated using frequency ratios: The unit cent divides the octave into 1200 cents.

[viii] Andreas Werckmeister sometimes used this term for the minor second, 1707, 75f

[ix] Praetorius, 1619, 61

[x] At this point we would like to express our sincere thanks to Burkard Olbrich, who made his guide to the practical understanding of instrument tonality in the orchestra available to the author before publication.

[xi] Riedweg, 2002, 108

[xii] Schadel, 1995, 94

[xiii] The equal temperament takes away the characteristics of the keys in favor of concordance in the overall structure. The beats now serve the interval characteristics and do not have to be stored in an unused place, as was the case with the so-called wolf fifth.

[xiv] This was first introduced by Gioseffo Zarlino (1517-1590)

[xv] Riemann, 1929, Sachteil, Lemma: Tonbestimmung, 1856ff

[xvi] The term was proposed in 1875 by Alexander John Ellis in the appendix to his translation of Hermann von Helmholtz's "Theory of Tone Sensations" as a unit for comparing the size of intervals.

[xvii] At least the following should be mentioned here: the Pythagorean tuning, the meantone tuning and those of Arnolt Schlick (before 1460-after 1521), Lodovico Fogliano (1470-1538/42), Salomon de Caus (1576-1626), Andreas Werckmeister (1645-1706), Jean-Philippe Rameau (1638-1764), Gottfried Silbermann (1683-1753), Johann Georg Neidhart (ca.1685-1739), Francesco Antonio Valotti (1697-1780), Georg Andreas Sorge (1703-1778), Friedrich Marpurg (1718-1795), Johann Philipp Kirnberger (1721-1783) and Thomas Young (1773-1829).

[xviii] Diese Bezeichnung geht auf Aristoteles zurück, Metaphysik XII

[xix] Der Verfasser dankt Prof. Zaza Skhirtladse von der Universität Tiflis für die Zurverfügungstellung des Bildes. 

[xx] The history of the zodiac and its pre-Christian roots cannot be discussed in detail within the scope of this study.

[xxi] Johannes Kepler, Harmonices mundi Libri V, 1618

[xxii] In St. Sophia Cathedral in Kiev, 12 windows illuminate the image of God the Father with the four evangelists.

[xxiii] Vitruvius, liber primus, 1980, 29

[xxiv] Kornerup, 1930

[xxv] Kayser, 1968, 30

[xxvi] While the synonymously used terms spatium and diastema emphasize the distance between the tones, coniunctio refers to the connection as such and mode to the way in which they sound together.

[xxvii] Palladio takes as an example a room with a given length of 9 feet and a width of 4 feet and then looks for a room height that is in the same ratio to both numbers. He multiplies both values, takes the square root and determines the number 6 for the room height. 9:6 and 6:4 are each in a ratio of 3:2 to each other. Palladio, 1570, reprint and translation 1983, 87.

[xxviii] Matthew 18:19

[xxix] Schleske, 2012, 313

[xxx] ibid, 269

[xxxi] quoted in Lowe, 2003 2

[xxxii] Inch is the 12th part of a foot and is itself divided into 12 sections.

[xxxiii] Clifford Pickover has devoted an insightful cross-cultural study to mathematics and the divine, Pickover 1999

[xxxiv] Fontana, 1992, 12

[xxxv] Kaiser, 1968, 32

[xxxvi] Freud, 1990, 197