text:
Pythagoras in the Forge

This article sheds light on the physical and music-theoretical background of the legend of Pythagoras in the Forge and proves that this legend could have a realistic basis. It is based on a previous publication from 2012, and the appropriate German speaking Wikibook, which was widely translated with www.DeepL.com/Translator (free version).

Preface
The connections between sounds and numbers were not only studied in antiquity. In the Middle Ages, music, together with arithmetic and geometry, belonged to the four liberal arts of the quadrivium. These subjects still offer a rewarding field for music-theoretical considerations and investigations, and this concerns various vocal temperaments still in use today as well as, for example, music-aesthetic aspects or tonal theory. The author hopes that these remarks on the ancient legend can contribute to awakening or consolidating interest in the subject.

The invention of music
(* around 570; † after 510 B.C.) is said to have invented music, according to legend, through his visit to a forge. This does not mean that there had been no music before, but that he is said to have been the first to give music a theoretical basis by assigning the ratios of the natural numbers six, eight, nine and twelve to the pure musical intervals prime, fourth, fifth and octave.

The following table shows the frequency ratios of such four tones with the exemplary frequencies 1200, 1600, 1800 and 2400 hertz:

In pairs, four Pythagorean tones can produce a total of four different lower-frequency combination tones, which result from the difference in the frequencies of the two respective tones under consideration. With respect to each of the four Pythagorean tones, the combination tones each have an integral multiple of one-half, one-third, one-fourth or one-sixth of their frequency. With the exemplary frequencies given in the table above, the four combination tones with the frequencies 200, 400, 600 and 800 hertz thus result. Because of the quite rational ratios, all combination tones also sound in harmonic unison with the four Pythagorean tones.

The following table shows the four Pythagorean tones c', f', g' and c" with the vibration numbers of their tone frequencies, which correspond to the concert pitch A with 440 hertz:

Tradition in antiquity
Unfortunately, no writings by Pythagoras exist (he may not have left any at all), and the oldest sources date from many centuries after his death. recorded Pythagoras' discoveries at least 600 years after his death.

But even these records have not survived, so that we have to resort to the late antique Latin writing De institutione musica ('Introduction to Music) by, which was written only about 1000 years after Pythagoras and presumably also refers to Nicomachus, among others. In any case, in the tenth chapter of De institutione musica it is described "how Pythagoras investigated the relationships of the harmonic sounds."

According to the legend of Pythagoras in the forge, he passed by a workshop "by a divine hint" and noticed the harmony of the individual tones caused by five different hammer blows. Because he suspected that the individual tones were caused by the type and force of the hammer blows, he induced the craftsmen to change the tools. He noticed that the individual tones were not connected with the craftsmen, but with the tools and that the tools, which resonated together, were in certain whole-number weight relationships to each other.

According to the eleventh chapter of De institutione musica, he would have subsequently investigated these relationships when varying the tension weights of strings and finally also with the monochord, and also researched different lengths and thicknesses of the strings.

Tradition in the Middle Ages
Another 500 years later, i.e. 1500 years after Pythagoras' work, the medieval music theorist and Benedictine (* around 992; † 1050) in his Micrologus, also in Latin, again refers to Boethius. In the twentieth chapter, Guido mentions "how music was invented from the sound of hammers".

This tradition of the legend of Pythagoras mentions that he passed a forge where forging was said to have been done with five hammers on an anvil. In the older tradition of Boethius, however, there is no mention of the smiths leading the hammers or of an anvil.

Contradictions
A physical analysis of the facts that have been handed down reveals a number of contradictions.

For this purpose, we consider an idealised hammer head in the form of a cuboid rod with the greatest length $$l$$. Its volume $$V$$ together with its cross-sectional area $$A$$ results in:

 * $$V = l \cdot A$$

The mass $$m$$ is at a density $$\rho$$:

 * $$m = V \cdot \rho = l \cdot A \cdot \rho$$

The weight force $$F$$ of the hammer head can be directly calculated from the mass $$m$$ by the proportional constant of the acceleration due to gravity $$g = 9.8 \frac {\text {m}} {\text {s}^2}$$ can be calculated:

 * $$F = m \cdot g = l \cdot A \cdot \rho \cdot g$$

The natural frequency or pitch $$f$$ of hammerheads made of the same material is usually not inversely proportional to their weight $$F$$, but depends essentially on their exact geometry. The longer the geometric extension in a body, the lower the natural frequency in this direction or of the associated longitudinal vibration mode. The lowest audible frequency is therefore correlated with the greatest length $$l$$ of the hammer head.

The natural frequency $$f$$ of hammerheads, however, is practically not audible at all because it lies in a frequency range that is too high. The speed of sound $$v$$ in steel is about 5000 metres per second, and with a typical forging hammer head length $$l$$ of 10 to 16 centimetres, $$f = \frac{v} {2 \cdot l}$$ results in natural frequencies between 15 and 25 kilohertz, which cannot be perceived in connection with a pitch.

Finally, it should be noted that the tensile weight $$F$$ of a string of length $$l$$ is neither proportional nor inversely proportional to the frequency $$f$$ of the string vibrations or to the pitch. Rather, it is proportional to the square root of the tension weight $$F$$. Furthermore, the pitch is inversely proportional to the length $$l$$ and the thickness $$D$$ of the string:

 * $$f \propto \frac{\sqrt{F}}{l \cdot D}$$

Attempted explanation
These contradictions can be eliminated if the following facts are considered or taken into account:

 * Pythagoras may have witnessed or even accompanied the complicated and elaborate construction of the Tunnel of Eupalinos, over 1000 metres long, on his native island of Samos.
 * During Pythagoras' lifetime, the monumental Heraion of Samos was built of limestone and marble.
 * The Latin word faber does not have to be translated as blacksmith, but can also be translated as craftsman.
 * There were certainly more workshops and craftsmen for stone working than for metal working at that time.
 * The Latin word fabrica means workshop and not smithy.
 * Workshops in which at least four craftsmen forged at the same time with hammers of different sizes were likely to have been rare.
 * With chisels the pitch $$f$$ is in the well audible range.
 * In chisels the pitch $$f$$ of the longitudinal vibrations is inversely proportional to their length $$l$$.
 * For chisels of equal cross-sectional area $$A$$, the pitch $$f$$ is therefore also inversely proportional to their length $$l$$, to their volume $$V$$, to their mass $$m$$ and to their weight $$F$$.
 * The pitch $$f$$ of a vibrating string is inversely proportional to its length $$l$$.
 * The pitch $$f$$ of a vibrating string is inversely proportional to its thickness $$D$$.

The following sound examples illustrate the pitches of five metal rods or chisels of different lengths when mechanically excited along the longitudinal axis with one blow, for example by a hammer. The metal bars all have the same cross-sectional area, and the lengths as well as the natural frequencies and the pitches are in a ratio of 12 to 9 to $$\sqrt {72}$$ to 8 to 6 length units.

The metal bars with the integer length units produce harmonious sounds in all combinations, whereas the metal bar with a non-rational length ratio of $$\sqrt {72}$$ sounds dissonant to all others.

With some corresponding and plausible assumptions, a scenario emerges that could have happened in Pythagoras' time, without any contradictions with physical laws:

If the events of Boethius' tradition, which mentions neither forges nor anvils, took place in a workshop for stonemasons and was inaccurate in the point of naming the tools to the effect that not only the hammers but ensembles of chisels of the same cross-section but of different lengths and hammers were meant, the sounds and pitches would have been clearly audible and caused by hammer blows but attributable to the chisels. Under this assumption, the integer ratios of the pitches would have been identical to those of the lengths or weights of the chisels and completely independent of the craftsmen and the hammers used.

When experimenting with a monochord and constant string tension and texture, Pythagoras would have found exactly the same relationship between string length and pitch for a certain string thickness and exactly the same relationship between string thickness and pitch for a certain string length as between chisel length or chisel weight and pitch. A string twice as long with the same thickness or a string twice as thick with the same length will sound exactly one octave lower than the string with the same thickness or length.

The ratios observed here with the products of the two natural numbers two and three correspond to the consonant intervals octave, fifth, fourth and prime. In relation to any fundamental, the corresponding four Pythagorean tones result in a so-called tetrachord.

Further investigation of these ratios finally yielded the diatonic scale of the seven tones A - B - C - D - E - F - G. This heptatonic scale forms the basis for the ancient Systema Téleion of the Greeks, which developed in the centuries after Pythagoras, as well as for the four main church keys Protus, Deuterus, Tritus and Tetrardus, which developed in the centuries after Boethius.

The ancient investigations with the tension weights of strings may have been carried out, but they are neither sufficient nor necessary for these findings. If the tension of the string is doubled, the result is a frequency increased by a factor of the square root of two (≈ 1.4142), which corresponds to a tritone interval commonly perceived as dissonant. Nevertheless, this irrational number was also known to both the Pythagoreans and, long before, the Babylonians.

Compositions
In harmony theory, the four Pythagorean tones are of great importance, as they form the framework of one of the most frequently used final cadences consisting of tonic, subdominant, dominant and tonic. The folowing example shows the cadence C major - F major - G major - C major with the respective root of the four chords in the bass voice.

These four Pythagorean notes are, for example, a central motif of the Impromptus (opus number 5) by (* 1810; † 1856) for piano of 1833 on a theme by.

Gregorianian Chant

The first transcriptions of the melodies of Gregorian chant are made with adiastematic neumes, in which the direction of the pitch for the following tone could be recorded upwards, to the same pitch or downwards, as well as the approximate duration of the tones. It was not until Guido of Arezzo introduced line notation with diastematic neumes in the 11th century that the intervals of diatonic melodies could also be precisely notated. In the various traditions from the Middle Ages, there are slightly different melodic progressions for the liturgical Latin texts. C and F clefs were already used, but a tuning pitch was not yet available, so that the absolute pitches are not fixed despite the naming of the seven tones on the diatonic scale.

Ad te levavi
The Gregorian antiphon Ad te levavi is sung as the Introit on the first Sunday in Advent. The melody in the VIIIth tone (tetrardus plagalis) with the finalis (final tone) G begins on the text "Ad te levavi animam meam". The Latin text from the Nova Vulgata with the first three verses of the 25th Psalm and the corresponding Hebrew letters Aleph, Beth and Ghimel reads as follows:

 * Psalm 25 (24),1-3A
 * 1 Aleph. Ad te, Domine, levavi animam meam,
 * 2A Beth. Deus meus, in te confido; non erubescam.
 * 2B Neque exsultent super me inimici mei,
 * 3A Ghimel. etenim universi, qui sustinent te, non confundentur.

The text of the first verse appears again in Psalm 143 (Nova Vulgata):

 * Psalm 143 (142),8
 * Auditam fac mihi mane misericordiam tuam, quia in te speravi.
 * Notam fac mihi viam, in qua ambulem, quia ad te levavi animam meam.

The melody restored after the Graduale Novum consists of twenty notes in the first verse, fourteen of which belong to the Pythagorean tetrachord c - f - g - c' and the remaining six can be considered as ornaments or passing notes. The melodic section ends on the note F, the repercussa (the sustaining note or tenor) is the C.

The following table gives the frequency of the Pythagorean tones for the four sections of Psalm 25 of the Introit according to the version of the Graduale Novum:

In all four sections, the four Pythagorean tones predominate, even clearly in the first two sections. This coincidence is quite striking, and it seems as if the anonymous medieval composer wanted to point us to the Pythagorean origin of music theory and the systems of ancient and Gregorian modes with this first piece of Gregorian repertoire in the Christian church year.

Factus est repente
Another example from the Gregorian repertoire is the Communion of Pentecost Sunday Factus est repente de caelo sonus (Made is suddenly from heaven a sounding) in the VIIth tone (Tetrardus authenticus) with the tenor D and the finalis G. The text describes the Pentecost event in which the Holy Spirit descended on the Christian community with tongues of fire and brought about the speaking in tongues. Except for the first three notes of the strongly accented group neume (porrectus flexus) with the two top notes f' and the passing note e' a semitone below, the elemental melody of the first verse consists only of the Pythagorean notes g – c' – d.

High Baroque

There are two works from the end of the 17th century with explicit reference to the legend, both composed by pupils of the Italian composer (* 1632; † 1687).

The sounds caused by hammer blows were introduced in 1690 by the French-German organist and composer (* 1653; † 1704) with the organ composition Nova Cyclopeias Harmonica in C major and set to tones in 3/4 time. This composition is framed by an aria consisting of two parts of 16 bars each. Otherwise, it comprises eight variations of 21 bars each on the theme Ad Malleorum Ictus Allusio (To allude to the blows of the hammers) and ends with the chant Summo Deo Gloria. The individual pieces are all built on the fundamental notes c-f-g-c', which are variedly played around and harmonised. In the last four bars of the Aria and in the last seven bars of the Variations, at least one of these Pythagorean tones can be heard on every beat:

In 1692, the German violinist, composer and court conductor (* 1646; † 1712) published the seven orchestral suites dedicated to the Elector of Bavaria Maximilian II Emanuel:

 * Pythagorische Schmids=Fuencklein
 * Bestehend in unterschiedlichen Arien / Sonatinen / Ouverturen / Allemanden / Couranten / Gavotten / Sarabanden / Giquen / Menueten / &c.
 * Mit 4.Instrumenten und beygefügten General-Baß, Bey Tafel=Musicken / Comœdien / Serenaden / und zu anderen fröhlichen Zusammenkunfften zu gebrauchen.

The main keys of the seven suites for solo violin are F major, D major, G major, D minor, F major, D major and B flat major.

Late Baroque
The German composer (* 1685; † 1750) created a magnificent work in 1741 with the Goldberg Variations, which, like the Nova Cyclopeoas Harmonica by Georg Muffat written over fifty years earlier, consists of a two-part aria with variations. The two Ariae reveal a number of similarities.

As a young man, Johann Sebastian Bach had already composed an equally outstanding work for the organ, namely the Passacaglia and Fugue in C minor (Bach-Werke-Verzeichnis 582). The eight-bar main theme of the Passacaglia consists of fifteen notes, ten of which correspond to the Pythagorean tones.

Concluding remarks

The legend of Pythagoras in the forge may be based on an actual incident. Regardless of the question of which of the regularities described here were actually investigated and found in antiquity, inaccuracies have obviously occurred in the medieval and modern traditions.

Furthermore, unhistorical additions have been made to the legend that has been handed down, but these need not be considered further for the interpretation of Boethius' tradition. Nevertheless, inaccuracies in the traditions and the additions and changes that were not in line with actual practice have certainly contributed to the fact that even the oldest reports on Pythagoras' investigations have been relegated to the realm of legends by many authors - but perhaps quite unjustly according to the above explanations.

The German theoretical physicist  (* 1901; † 1976) wrote in his 1937 essay Thoughts of Ancient Natural Philosophy in Modern Physics:

The abstractness of the modern concept of the atom and of the mathematical forms which serve today's atomistics as an image for the multiplicity of phenomena already leads over to the second basic idea which the exact natural science of our time has taken over from antiquity: the thought of the sense-giving power of mathematical structures.

The harmonies of the Pythagoreans, which Kepler still believed to find in the orbits of the stars, have been sought by natural science since Newton in the mathematical structure of the dynamic law, in the equation formulating this law.

The successes of this view of nature, which has in part led to a real mastery of the forces of nature and thus decisively intervened in the development of mankind, have proved the belief of the Pythagoreans right to an unforeseeable degree.

This turn means a consistent implementation of the programme of the Pythagoreans insofar as the infinite multiplicity of natural events finds its faithful mathematical image in the infinite number of solutions of an equation, such as Newton's differential equation of mechanics.

Dedication
The main author thanks his teacher (*1929). He introduced him to medieval church music with his profound knowledge of the Middle Ages and Gregorian chant.

Summary of the project

 * Target audience: Musicians, historians, natural scientists
 * Learning objectives: Integer-rational relationships based on an ancient legend.
 * Book sponsorship/contact person: User:Bautsch
 * Are co-authors currently desired? Yes, very much so. Corrections of obvious errors directly in the text; content please via discussion.
 * Guidelines for co-authors: Wikimedia-like.

title: Pythagoras in the Forge


text: Research Methods in Information Science/Printable version

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text: Finnish/Printable version

title: Finnish/Printable version


text:
The cave painting in Magura Cave

This paper shows parallels between the paintings in the Magura Cave in Bulgaria and written creation myths. Furthermore, some depictions are examined in archaeoastronomical terms, and it is shown which elements may be connected to an early calendar. It is based on a previous publication in the appropriate German speaking Wikibook, which was widely translated with www.DeepL.com/Translator (free version).

Hypotheses
The investigations described pursue two main hypotheses:
 * 1) The older seven first scenes of the Cave painting in Magura Cave are related to a creation myth.
 * 2) The seven sequentially depicted scenes show a coherent creation story (from right to left):
 * 3) The Bringer of Light.
 * 4) The Celestial Bull in the present-day constellations of Taurus, Aries and Cetus.
 * 5) Seven figures corresponding to the seven wandering stars.
 * 6) The separation of heaven and earth with the creation of living beings.
 * 7) The special place of human beings in creation.
 * 8) The creation of water.
 * 9) The completion of creation.

Irrespective of these unprovable hypotheses, this paper will show that the above-mentioned scenes may have served thousands of years ago as stations and memory aids for the narration of a creation myth as well as for its illustration. The depictions in the cave paintings of Magura Cave are therefore indications that Neolithic users of the cave knew and passed on a creation myth that shows numerous similarities to other creation myths that have been handed down in writing.

Introduction
Creation myths and calendars exist in many ancient cultures. They often answer some questions about the origins of human beings and their environment and can justify religious traditions, such as the celebration of festivals and the associated keeping of calendars. Myths were passed on orally for thousands of years, and before they could be written down, after writing was invented, they may also have been pictorially recorded. This may have served as a pictorial supplement to the spoken or sung narratives (today the term multimedia would be used), but may also have been used as an aid to memory so that important details were not inadvertently omitted or reproduced in the wrong order.

There are many examples of how characters evolved from iconographic images. In the ancient European Vinča culture, in whose area the Magura cave was also located, prehistoric signs were already used in the Neolithic (Danube script).

For the northern hemisphere of the earth, the following can be stated: The direction of writing from right to left corresponds to the long-term course of the sun and the moon along the ecliptic in relation to the fixed star sky. This course can be most easily detected in the case of the Moon, since within a sidereal month (27,322 days) it moves once completely through the fixed star sky. The ancient Indians and the Bedouins long ago divided this self-contained and eternally repeating path into 27 and 28 lunar houses respectively. The direction of writing from left to right corresponds to the daily course of all the celestial bodies in relation to the horizon from rising in the east to culmination on the meridian in the south to setting in the west.

There are numerous indications that our ancestors already referred to the events to be observed in the sky, which they could still perceive completely without light pollution. In this respect, it seems indispensable to consider archaeoastronomical aspects in the interpretation of the ancient representations. There is no evidence that people in ancient times were less intelligent than in modern times, but they had accumulated less knowledge than that which we have at our disposal today. The eternal course of the stars, the planets, the moon and the sun can be described with increasing precision if observed for a sufficiently long time, but cannot be explained without further ado. Consequently, mythical stories were invented for the then inexplicable processes and connections in the sky, especially in the night sky, which were subsequently spun out and spread more and more. Many religions refer to these processes, which were described as divine, among other things to determine calendars and the dates of certain feast days.

In the Magura Cave there are a total of over one thousand pictorial representations from various prehistoric periods. Dating seems to be problematic, the old cave paintings are said to be created at least 2600 BC, but the oldest ones are said to be considerably earlier. This wikibook deals mainly with the seven older depictions, which seem to tell a picture story starting from the entrance of the cave, and also takes up some of the depictions deeper in the cave.

Description

The Magura Cave is located about two kilometres northwest of the Bulgarian town of Rabisha in the very northwest of the country. The highest elevation of the mountain is 461 metres above sea level and about 170 metres above the water level of Lake Rabisha, which lies directly to the north. This lake has an extension of just under two and a half kilometres in a west-east direction and a good one and a half kilometres in a north-south direction. To the east, a plain gently slopes down to the Danube for more than 20 kilometres. The Danube forms the national border with Romania here. The Black Sea lies a good 400 kilometres away in this direction. To the south are the western foothills of the Balkan Mountains with their two mountain ranges, "Veneza" and "Vedernik". The border with Serbia runs in a westerly direction 15 kilometres away. From Magura Mountain you can see northwards to the Transylvanian Alps (Southern Carpathians).

The Magura Cave is a karst cave formed about 15 million years ago by the dissolution of limestone, with an air temperature of 12 degrees Celsius. It has a total length of two and a half kilometres, is now equipped with electric lighting and can, however, only be visited in parts and as part of a guided tour all year round.

The cave is located in the eastern tip of the region that was shaped by the Vinča culture around 7000 years ago, where ceramics were worked. Prehistoric signs, the Vinča symbols, which apparently served religious purposes, are also known from this period. The cave paintings seem to date from several periods, going up to the Bronze Age. The artists of the older cave paintings worked with bat guano, made from excrement, about 5500 years ago. There are a total of six halls of different sizes in the main gallery and three more galleries off to the side. At the entrance to the cave, the largest hall of the complex is the Triumph Hall (also known as the Arch Hall), which is over 100 metres long, up to 58 metres wide and up to 28 metres high.

The Magura Cave was already used by people in the Palaeolithic Age. An Indo-European group of peoples called the Thracians inhabited it for several centuries in the Bronze Age. The Triumph Hall was inhabited by numerous people. The cave was also used later, as ceramic artefacts from the Iron Age were found.

In 1984, the cave was inscribed on the UNESCO World Heritage Tentative List. Since 2009, there has been an exhibition of the cave paintings in the entrance area. Similar paintings and others depicting the sun and crescent moons can be found in the Bailovo Cave, near Sofia..

Creation myth

The advanced civilisations of the Stone Age lived not only south but also west of the Black Sea. The permafrost boundary was always north of this region during the last cold period, and the land bridge between the Sea of Marmara and the Black Sea could be walked on. The Black Sea was fed with fresh water via numerous rivers. The entire coastal region was therefore well supplied with drinking water and marine life and thus represented a well-suited habitat for the inhabitants of that time. It cannot be ruled out that the people of all the bordering regions already exchanged trade goods, stories and information as well as knowledge by land at that time, and that this left common traces in the artefacts, traditions and myths of the different peoples. It would therefore not be surprising if the development of societies in all directions from the shores of the Black Sea had revealed numerous similarities and parallelisms. Due to the increasing warming of the earth and the glaciers melting in the process, the water levels of the oceans and inland seas gradually rose by more than one hundred metres, and on the shallow coastal regions of the Black Sea, the water displaced the shore several hundred kilometres inland. The land bridge between the Sea of Marmara and the Black Sea broke up sometime between 7500 and 9500 years ago, and the Mediterranean and the Black Sea equalised their sea levels. From then on, people had to use boats to travel between the western and southern coastal regions of the Black Sea, which could no longer be reached by land.

The first seven stations of cave paintings in Magura Cave apparently date from an earlier period of paintings in Magura Cave. They reveal many elements of a creation myth, such as a Bringer of Light, the seven, heaven and earth, living beings, man and woman, or water. In further depictions, a calendar and elements of a Flood are evident.

All in all, the paintings in the Magura cave not only reveal astonishing references to Genesis (Bereshit of the Jewish Torah and First Book of Moses of the Christian Old Testament), but also to the Sumerian Atraḫasis Epic and Gilgamesh Epic, which speak of a Celestial Bull and also of a Flood. The Flood narrative is considered a supplement to the creation narrative. The Sumerian primeval ocean or cosmic water was called Abzu ("primeval anian"). The Babylonian creation myth Enuma Elish tells on seven clay tablets, for example, of the attributes of the seven winds and of a storm flood as well as of the creation of the constellations by the Sumerian deity Marduk, first mentioned in the third millennium BC. Very similar motifs also appear in ancient Chinese folk tales passed down orally, such as the tales of The Incarnation of the Five Ancients and of the creator goddess of the human race Nü Wa.

In the Magura Cave there are almost a thousand other drawings in addition to the cut-outs presented here. It is therefore conceivable that these narrate a longer "Magura epic" that clearly goes beyond the accounts of the creation of the world and the Flood.

The following seven sections deal with the first seven and apparently oldest stations in the Magura cave. The corresponding cave paintings are interpreted herein as an account of the creation of the world.

Here is a brief simple retelling of the creation myth described in more detail below:

 * → Retelling of the Creation Myth from the Magura Cave

First station
Separation of light and darkness

 * It rises morning after morning, perceptible to everyone, in the east.

Second station
Creation of the Celestial Vault

 * Attentive and persevering observers see them eternally passing through this gateway into the light.

Bulls seem to have had a special meaning for people since time immemorial and were painted as early as around 40000 years ago in El Castillo Cave in Cantabria, Spain, and in the limestone cave of Lubang Jeriji Saléh on Borneo in Indonesia. There are numerous other examples: The aurochs in the Chauvet Heights on the Ardèche in southern France is said to be about 31000 years old. Members of the Magdalenian culture painted the bull paintings in the cave of Lascaux in the French Dordogne about 16000 years ago, as well as the bison in the cave of Niaux in the French Pyrenees and the steppe bison in the cave of Altamira in northern Spain (Cantabria) about 13000 years ago. In Switzerland, several so-called "moon horns" with rows of holes dating from the late Bronze Age, very reminiscent of the shape of a bull's head, have been found at various archaeological excavation sites. There are also life-size representations of bulls with long horns at least 6000 years old in the dolmens at Locmariaquer in the department of Morbihan as well as on the offshore island of Gavrinis in the Gulf of Morbihan in southern Brittany in France.

The second sign from Magura bears a strong resemblance to the Celestial Bull. Towards the top two long horns could be depicted, the body is open towards the bottom and shows in the middle something like a footprint of something entering this bull. The depiction has the same orientation as the asterism Celestial Bull in the western evening sky, through which all seven wandering stars visible to the naked eye (the Sun, the Moon and the five planets Mercury, Venus, Mars, Jupiter and Saturn) emerge from a star-poor region at night. Thus they enter from the particularly dark sky segment of the Funnel of Thuraya through the Golden Gate of the Ecliptic into the light of the brighter objects along the ecliptic. In doing so, these seven beings are virtually born out of the Celestial Bull and subsequently make their orbits in the light of the stars along the ecliptic.

This asterism includes parts of today's constellations of Taurus, Aries and Cetus, respectively parts of the corresponding Bedouin constellations Lamb (Arabic: al-hamal, at the star Hamal (α Arietis)) with the "belly" of the lamb (Arabic: al-buṭayn, at the star Nair al Butain or Bharani (41 Arietis)) and the "many little ones" in the "fat tail" of the Lamb (Arabic: al-thurayya, the open star cluster of the Pleiades respectively the Seven Sisters (Messier 45)) as well as the Following (Arabic: al-dabarān, the Red Giant Aldebaran (α Tauri)) and the "amputated hand" (Arabic: al-kaf al-jadhma, the star Kaffaljidhma and Kaffaljidh respectively (γ Ceti)) with the star Menkar (α Ceti) at the Funnel of the Thuraya. The Pleiades are the most conspicuous celestial object in this asterism, they are located on the back of the Celestial Bull and could be regarded as the original source of the seven wandering stars.

Third station
Creation of the celestial bodies

 * The seven being in perpetual motion in the heavens.

 * We are the we-are-there.

This figure is followed on the third wall of the cave passage by the depiction of a mythical-looking group with a seven figure, which could stand for the seven wandering stars coming out of the Celestial Bull. They can therefore be interpreted as the children of the Celestial Bull born of the Celestial Bull (for the terms "sibunstirri" and "septemtriones" in this context compare: German Wikibook on the Pleiades), namely the "sun", the "moon" and the five planets "Mercury", "Venus", "Mars", "Jupiter" and "Saturn":

Perhaps also the Pleiades, which is part of the Celestial Bull, is meant by these seven beings. These Pleiades were equated with the nymphs of the same name in ancient Greece.

In the cave paintings, a great many of such air-spirit-like figures with arms raised to the sky appear altogether.

Some figures in the cave paintings clearly resemble today's constellation Orion with its three belt stars. The brightest seven stars of this constellation, which was already named in antiquity, are:

After the creation of the firmament, the Old Testament describes in verses 16 to 19 the creation of the Sun (ruler of the day), moon (ruler of the night) and stars are mentioned. The five planets visible to the naked eye and also walking in relation to the fixed star sky, Mercury, Venus, Mars, Jupiter and Saturn, were obviously counted among the stars:

16 Gott machte die beiden großen Lichter, das große zur Herrschaft über den Tag, das kleine zur Herrschaft über die Nacht, und die Sterne. 17 Gott setzte sie an das Himmelsgewölbe, damit sie über die Erde leuchten, 18 über Tag und Nacht herrschen und das Licht von der Finsternis scheiden. Gott sah, dass es gut war. 19 Es wurde Abend und es wurde Morgen: vierter Tag.

Fourth station
Separation of heaven and earth / Creation of living beings

 * Here the eternal heavenly, there the transient earthly.

The fourth station is horizontally divided into two parts and in the upper part (Heaven) it takes up the figurative representations of the seven heavenly figures with raised hands. The lower part shows people as well as land animals and birds (Earth), but no fish. In the first larger composition of drawn figures, therefore, the upper half shows several celestial figures, whereas below is an earthly scene with people and animals (living creatures). The two parts are separated on the rock surface by a long horizontal edge.

With this depiction, the motif "In the beginning God created heaven and earth" is taken up from the very first verse of the creation story in the Old Testament. On the fifth and sixth day, it describes what happened on earth afterwards as follows:

20 Dann sprach Gott: Das Wasser wimmle von Schwärmen lebendiger Wesen und Vögel sollen über der Erde am Himmelsgewölbe fliegen. 21 Und Gott erschuf die großen Wassertiere und alle Lebewesen, die sich fortbewegen nach ihrer Art, von denen das Wasser wimmelt, und alle gefiederten Vögel nach ihrer Art. Gott sah, dass es gut war. 22 Gott segnete sie und sprach: Seid fruchtbar und mehrt euch! Füllt das Wasser im Meer und die Vögel sollen sich auf Erden vermehren. 23 Es wurde Abend und es wurde Morgen: fünfter Tag. 24 Dann sprach Gott: Die Erde bringe Lebewesen aller Art hervor, von Vieh, von Kriechtieren und von Wildtieren der Erde nach ihrer Art. Und so geschah es. 25 Gott machte die Wildtiere der Erde nach ihrer Art, das Vieh nach seiner Art und alle Kriechtiere auf dem Erdboden nach ihrer Art. Gott sah, dass es gut war.

In the Ten Commandments of the Jewish Tanakh the following is commanded:

4 Du sollst dir kein Kultbild machen und keine Gestalt von irgendetwas am Himmel droben, auf der Erde unten oder im Wasser unter der Erde.

In the Psalms there are numerous references to heaven and earth, such as the following:

Psalm 8 "Die Herrlichkeit des Schöpfers – die Würde des Menschen": 2 HERR, unser Herr, / wie gewaltig ist dein Name auf der ganzen Erde, der du deine Hoheit gebreitet hast über den Himmel.

Psalm 50 "Der rechte Gottesdienst": 1 Ein Psalm Asafs. Gott, ja Gott, der HERR, hat gesprochen, / er rief die Erde vom Aufgang der Sonne bis zu ihrem Untergang. 2 Vom Zion her, der Krone der Schönheit, ging Gott auf in strahlendem Glanz. 3 Unser Gott möge kommen und nicht schweigen; Feuer frisst vor ihm her; um ihn stürmt es gewaltig. 4 Dem Himmel droben und der Erde ruft er zu, um sein Volk zu richten: 5 Versammelt mir all meine Frommen, die den Bund mit mir schließen beim Opfer! 6 Da taten die Himmel seine Gerechtigkeit kund; weil Gott selbst der Richter ist.

Psalm 57 "Geborgenheit im Schutze Gottes": 6 Erhebe dich über den Himmel, Gott! Deine Herrlichkeit sei über der ganzen Erde! 12 Erhebe dich über den Himmel, Gott! Deine Herrlichkeit sei über der ganzen Erde!

Psalm 85 "Bitte um Frieden und Gerechtigkeit": 12 Treue sprosst aus der Erde hervor; Gerechtigkeit blickt vom Himmel hernieder.

Psalm 89 "Verheißung an David und Klage über die Verwerfung seines Hauses": 12 Dein ist der Himmel, dein auch die Erde; den Erdkreis und was ihn erfüllt hast du gegründet.

Psalm 103 "Loblied auf den barmherzigen und gerechten Gott": 11 Denn so hoch der Himmel über der Erde ist, so mächtig ist seine Huld über denen, die ihn fürchten. 19 Der HERR hat seinen Thron errichtet im Himmel, seine königliche Macht beherrscht das All.

Psalm 106 "Sündenbekenntnis Israels angesichts seiner Geschichte": 20 Denn herabgeschaut hat der HERR aus heiliger Höhe, vom Himmel hat er auf die Erde geblickt.

Psalm 108 "Gott, Geborgenheit und Schutz seines Volkes": 6 Erhebe dich über den Himmel, Gott! Deine Herrlichkeit sei über der ganzen Erde!

Psalm 113 "Loblied auf Gottes Hoheit und Liebe zu den Geringen": 5 Wer ist wie der HERR, unser Gott, der wohnt in der Höhe, 6 der hinabschaut in die Tiefe, auf Himmel und Erde?

Psalm 121 "Der Hüter Israels": 2 Meine Hilfe kommt vom HERRN, der Himmel und Erde erschaffen hat.

Psalm 124 "Israels Dank für die Befreiung": 8 Unsere Hilfe ist im Namen des HERRN, der Himmel und Erde gemacht hat.

Psalm 134 "Nächtliches Loblied im Tempel": 3 Es segne dich der HERR vom Zion her, er, der Himmel und Erde erschaffen hat.

Psalm 146 "Preislied auf Gott, den Helfer der Armen": 6 Er ist es, der Himmel und Erde erschafft, / das Meer und alles, was in ihm ist. Er hält die Treue auf ewig.

Psalm 148 "Lobpreis auf den Herrn, den König des Kosmos": 13 Loben sollen sie den Namen des HERRN,/ denn sein Name allein ist erhaben, seine Hoheit strahlt über Erde und Himmel.

In Christianity, the dichotomy between heaven and earth is thematised in the Lord's prayer from the sixth chapter of the Gospel of Matthew:

9 So sollt ihr beten: Unser Vater im Himmel, geheiligt werde dein Name, 10 dein Reich komme, dein Wille geschehe, wie im Himmel, so auf der Erde.

Fifth station
Creation of man

 * The perpetual condition for the creation of new human life.

The fifth station shows two figures that look like a woman and a man and at the same time bear resemblance to the heavenly beings from stations three and four.

The depiction is reminiscent of a couple dancing with their arms stretched upwards (in the fifth position). According to the natural philosopher Paracelsus, who was active in the first half of the 15th century, the sylphs or sylphids, which are often depicted in exactly this way, are mythical spirits of the air and are regarded as an example of the spiritualisation of matter.

The creation story in the Old Testament describes the creation of man on the sixth day:

26 Dann sprach Gott: Lasst uns Menschen machen als unser Bild, uns ähnlich! Sie sollen walten über die Fische des Meeres, über die Vögel des Himmels, über das Vieh, über die ganze Erde und über alle Kriechtiere, die auf der Erde kriechen. 27 Gott erschuf den Menschen als sein Bild, als Bild Gottes erschuf er ihn. Männlich und weiblich erschuf er sie. 28 Gott segnete sie und Gott sprach zu ihnen: Seid fruchtbar und mehrt euch, füllt die Erde und unterwerft sie und waltet über die Fische des Meeres, über die Vögel des Himmels und über alle Tiere, die auf der Erde kriechen! 29 Dann sprach Gott: Siehe, ich gebe euch alles Gewächs, das Samen bildet auf der ganzen Erde, und alle Bäume, die Früchte tragen mit Samen darin. Euch sollen sie zur Nahrung dienen. 30 Allen Tieren der Erde, allen Vögeln des Himmels und allem, was auf der Erde kriecht, das Lebensatem in sich hat, gebe ich alles grüne Gewächs zur Nahrung. Und so geschah es. 31 Gott sah alles an, was er gemacht hatte: Und siehe, es war sehr gut. Es wurde Abend und es wurde Morgen: der sechste Tag.

Sixth station
Creation of the water

 * In the water is heavenly power, through the water is earthly life.

The horizontally aligned zigzag lines in the sixth part are reminiscent of water waves. The primeval flood consisted of water. Water is an essential basis of life for plants, animals and humans. Creation myths often mention that the primordial water is divided into the water in the sky and the water in the rivers, lakes and seas. The zigzag lines of the Magura cave paintings appear in both the upper and lower parts of the depictions.

The creation story in the Old Testament describes the primeval flood and the water even before the creation of man:

6 Dann sprach Gott: Es werde ein Gewölbe mitten im Wasser und scheide Wasser von Wasser. 7 Gott machte das Gewölbe und schied das Wasser unterhalb des Gewölbes vom Wasser oberhalb des Gewölbes. Und so geschah es. 8 Und Gott nannte das Gewölbe Himmel. Es wurde Abend und es wurde Morgen: zweiter Tag. 9 Dann sprach Gott: Es sammle sich das Wasser unterhalb des Himmels an einem Ort und das Trockene werde sichtbar. Und so geschah es. 10 Und Gott nannte das Trockene Land und die Ansammlung des Wassers nannte er Meer. Gott sah, dass es gut war. 11 Dann sprach Gott: Die Erde lasse junges Grün sprießen, Gewächs, das Samen bildet, Fruchtbäume, die nach ihrer Art Früchte tragen mit Samen darin auf der Erde. Und so geschah es. 12 Die Erde brachte junges Grün hervor, Gewächs, das Samen nach seiner Art bildet, und Bäume, die Früchte tragen mit Samen darin nach ihrer Art. Gott sah, dass es gut war. 13 Es wurde Abend und es wurde Morgen: dritter Tag.

In this context, two points stand out that represent a divergence between the sequence of the cave paintings and the biblical creation story:

 * This block with the second and third days of creation appears only after the sixth day of creation, otherwise the sequence is identical.
 * Of all the verses in the first chapter of the creation story in the Old Testament, only verses 11 to 13, which deal with plants dependent on light and water, are not reflected in the pictorial representations of the first seven stations of the Magura cave.

Great masses of water later put people to the test during the Flood, which is apparently thematised in a later station (see below).

Seventh Station

Reflection on the number seven

 * The seven stands for a mystical perfection.

The seventh station is somewhat hidden behind a wall and shows seven distinguishable symbols with circles, arcs and crosses.

In this context, it should also be pointed out that these symbols from the Magura Cave have very clear similarities to some of the letters of the oldest known alphabet in the 4000 year old Proto-Sinaitic script, from which the Phoenician alphabet emerged and thus, even later, the Aramaic, Hebrew and Greek alphabets:

According to the creation story in the Old Testament, on the seventh day the work of creation was completed. This can be read at the beginning of the second chapter:

1 So wurden Himmel und Erde und ihr ganzes Heer vollendet. 2 Am siebten Tag vollendete Gott das Werk, das er gemacht hatte, und er ruhte am siebten Tag, nachdem er sein ganzes Werk gemacht hatte. 3 Und Gott segnete den siebten Tag und heiligte ihn; denn an ihm ruhte Gott, nachdem er das ganze Werk erschaffen hatte.

The number seven is a special number because it is the only number of the numbers from two to ten that is not divisible with all other numbers. On the seventh day of creation, the world order is established and the mythical creation is completed. The seven days of a week correspond to a quarter of the moon. The story of creation begins in German, Latin (Vulgate) and Hebrew (Bereshit) with these seven words or 28 letters:

The verse reads in Hebrew (from right to left):

בְּרֵאשִׁית בָּרָא אֱלֹהִים אֵת הַשָּׁמַיִם וְאֵת הָאָרֶץ

Written in Latin letters, this reads approximately as follows:

bereschit bara elohim et haschamaijim weet haarez

In Latin:

The Latin Vulgate contains the following seven words here:

In principio creavit Deus caelum et terram.

In the Einheitsübersetzung into German, the first sentence of the Bible also consists of seven words:

Im Anfang erschuf Gott Himmel und Erde.

In English:

In the beginning God created the heavens and the earth.

There are seven wandering stars visible to the naked eye eternally moving along the ecliptic through the fixed star sky. Other astronomical objects related to the number seven are the seven brightest stars of the asterisms Orion, Big Dipper, Little Dipper, Head of the Bull or Pleiades respectively the seven brightest objects of the fixed star sky near the ecliptic that all lie within an arch of 180 degrees behind the dark Funnel of Thuraya:

 * Pleiades (Taurus)
 * Hyades (Taurus)
 * Aldebaran (α Tauri)
 * Pollux (α Geminorum)
 * Regulus (α Leonis)
 * Spica (α Virginis)
 * Antares (α Scorpio)

The seven symbols of the cave drawing at the seventh station are reminiscent of the seven celestial figures of the third station as well as partly of astronomical symbols still in use today.

In the celestial segment of these four signs of the zodiac, measured from the star Deneb Algedi (δ Capricorni) in the constellation Capricornus, the "Tail of the Billy Goat", there are no stars close to the ecliptic with a magnitude of 3.5m or brighter. Only in the Golden Gate of the ecliptic in the constellation Taurus do the Pleiades, the Hyades as well as Aldebaran exceed this brightness, and that considerably. This means that all the fixed stars lying near the ecliptic in this quadrant are surpassed in brightness by several hundred other stars of the night sky, as well as very clearly by the seven wandering stars.

After the Moon leaves the constellation of Capricornus, it passes through the three dark and inconspicuous constellations of Aquarius, Pisces and Aries along its path on the ecliptic line for seven days, month after month, before entering the light of the Golden Gate of the ecliptic in the constellation of Taurus.

Flood
The zigzag lines visible at the sixth station are taken up several times in the further course of the cave paintings, where they could possibly be related to the depiction of the Flood. Immediately before the calendar-like depictions, which can be seen in connection with a reordering of the world, there is, for example, a strong horizontally aligned zigzag line in the upper part of the drawing, which seems to wash over all the trees, fields and creatures like a torrent. Below this flood wave, there is also a square matrix of forty points - the Flood is said to have lasted forty days according to the written tradition in the Old Testament.

In the sixth chapter of Genesis, a grave measure is announced because of the depraved and violent behaviour of mankind:

11 Die Erde aber war vor Gott verdorben, die Erde war voller Gewalttat. 12 Gott sah sich die Erde an und siehe, sie war verdorben; denn alle Wesen aus Fleisch auf der Erde lebten verdorben. 13 Da sprach Gott zu Noach: Ich sehe, das Ende aller Wesen aus Fleisch ist gekommen; denn durch sie ist die Erde voller Gewalttat. Siehe, ich will sie zugleich mit der Erde verderben.

In the seventh chapter we are then told how the Flood caused for this took place:

4 Denn noch sieben Tage dauert es, dann lasse ich es vierzig Tage und vierzig Nächte lang auf die Erde regnen und tilge vom Erdboden alle Wesen, die ich gemacht habe. ... 10 Als die sieben Tage vorbei waren, kam das Wasser der Flut über die Erde. 11 Im sechshundertsten Lebensjahr Noachs, am siebzehnten Tag des zweiten Monats, an diesem Tag brachen alle Quellen der gewaltigen Urflut auf und die Schleusen des Himmels öffneten sich. 12 Der Regen ergoss sich vierzig Tage und vierzig Nächte lang auf die Erde.

The open star cluster of the Pleiades had a special significance in many cultures due to its unusual and impressive sight. It was also used as a calendar star, according to whose rising and setting agricultural and seafaring activities were aligned, as is documented, for example, by the Greek poets around 700 BC or  (* zirka 310 vor Christus; † 245 vor Christus). Hesiod also mentions in his text that the Pleiades are regularly not visible for forty days and nights in spring, as they are then outshone by the light of the passing sun. The star-loving people of Babylon are said to have burnt forty reeds in the morning sky out of joy at the return of the Pleiades. Also the term quarantine (from the French "quarantaine (de jours)" = "forty days") is also connected with the Pleiades or with the duration of the Flood.

At the end of the eighth chapter of Genesis, in the account of the end of the Flood, there are several references to the central and elementary role that a calendar will henceforth have for the surviving people:

Niemals, so lange die Erde besteht, werden Aussaat und Ernte, Kälte und Hitze, Sommer und Winter, Tag und Nacht aufhören.

Sun Calendar

In the rear part of the cave there is a complex representation of a sun with various attributes, in which numerous references to a solar calendar could be established.

Lunar calendar
But there are also several indications of a lunar calendar. In the following, some further findings are presented:

In particular, there is a depiction of the old light at the Moon's morning last with a reclining crescent moon followed by fifteen dash-shaped marks to the left and a sixteenth circular figure with a downward bulge that could symbolise the Full Moon with a glittering path below. From the old light of the moon to the new moon and waxing moon to the full moon, there are sixteen days.

On the Bronze Age Nebra Sky Disk there are three different representations of a basic circular structure, which according to the Austrian prehistorian Paul Gleirscher can be related to the three lunar phases of the old light, the first quarter and the full moon, as they could also be depicted in the sequence of the Magura cave painting:

Also in the ancient Egyptian lunar calendar, which was already in use in the Neolithic period, the month did not begin with the invisible new moon, but already with the just visible old light of the moon's morning last.

In the section of the Magura cave further back with the depiction of the solar calendar, fields with 27 or 28 points can be seen, which indicate that people already counted and knew the length of the sidereal month at that time. After this time, the moon is again in the same lunar house (Arabic: manazil al-qamar), respectively at the same ecliptical longitude or at the same star of the fixed star sky.

Furthermore, fields with 19 and with 25 points appear in the cave paintings. As with the Calendar Stone of Mnajdra, these could indicate the Meton Cycle as well as the number of seven-day weeks within synodic periods or the completed lunar quarters between the equinoxes or solstices.

Closing words
If our ancestors can be assumed to have the same creativity as we have today, it is not surprising that lunar and solar calendars as well as creation myths were created on the basis of the knowledge that was initially passed on exclusively orally. The artistically gifted contemporaries were able to create and leave behind corresponding pictorial representations with the tools available at the time. By documenting astronomical observations and events, for example, in the form of tally sheets, tables, lists or even drawings, more and more knowledge could be gathered over the millennia, which could be evaluated with the help of human cognition and used to predict future events.

It is certainly very difficult to interpret these early documents today, and even more difficult to prove proven correlations, but nevertheless it cannot be denied that there are striking and remarkable parallels in various traditions and forms of representation.

It would be desirable if prehistoric and archaeological research could also take into account to a greater extent the undoubtedly existing and non-trivial astronomical knowledge of those who were active at the time. This requires a thorough understanding of the astronomical facts and contexts, which presupposes a corresponding education.

Summary of the project

 * Target audience: Interested and curious people
 * Learning objectives: Archaeoastronomical origins with reference to creation myths.
 * Book sponsorship/contact person: User:Bautsch
 * Are co-authors currently desired? '''Yes, very much so. Corrections of obvious errors directly in the text; content please via discussion.
 * Guidelines for co-authors: Wikimedia-like.

title: The cave painting in Magura Cave


text:
The cave painting in Magura Cave/ Retelling

Assuming that a coherent creation story is told in the first seven scenes depicted in the Magura Cave, a retelling is found in this chapter.

The Bringer of Light

1,1

In the beginning our world was in darkness, and there was only an immeasurable primeval flood without living creatures.

1,2

Then the great Bringer of Light appeared above the horizon and hovered over the primeval flood.

1,3

From then on there was light by day and darkness by night.

The Celestial Bull

2,1

In the night sky the great Celestial Bull appeared, who had first already given birth to the sun as ruler of the day.

2,2

From him were now also born countless other lights in the firmament.

2,3

They came out of him into the darkness to shine also in the night.

The Seven Wandering Figures

3,1

After the sun, the Celestial Bull also gave birth to the moon as ruler of the night.

3,2

Then he created five more great lights as companions of the sun and moon.

3,3

All seven great lights will forever pass by the many little lights as rulers of the ages.

The Heaven and the Earth

4,1

The heaven is eternally populated by the lights.

4,2

Under heaven, in its image, the earth was created.

4,3

As the lights inhabit the heavens, so the living creatures inhabit the earth.

The Humans

5,1

In the midst of the many living creatures on earth, human beings were also created.

5,2

Humans were made as woman and man.

5,3

They should multiply and respect the lights of heaven.

The Water

6,1

The waters of the primeval flood entered the sea and were thereby separated from the land of the earth.

6,2

From then on, the water could also flow in heaven and on the land.

6,3

This water quenches the thirst of living creatures and makes plants and food grow.

The Completion

7,1

Thus the heavens with their lights and the earth with its living creatures were completed.

7,2

The seven walking lights in heaven will watch over the earth from the beginning to the end.

7,3

They serve as visible signs of the perpetual cycle of days, weeks, months and years.


text:
Computation of an acoustic Transmissionline

This Wikibook explains, how the characteristics of an acoustic transmissionline can be computed. It is based on the appropriate German speaking Wikibook, which was widely translated with www.DeepL.com/Translator (free version).

Introduction
Transmissionline (transmission line) is the name given in loudspeaker acoustics to a channel, usually folded several times, through which the sound emitted to the rear by a loudspeaker passes. At the end of this channel is an opening through which the sound can exit. Under suitable conditions, this sound can be used to extend and improve the amplitude frequency response of an electrodynamic loudspeaker cabinet at the low-frequency end.

Based on idealised theoretical approaches to the radiation behaviour of loudspeakers and using the theory of quasi-homogeneous absorbers, this book deals with how the low-frequency range of a transmission-line loudspeaker cabinet can be dimensioned and acoustically damped. Because of the simple mathematical formalism, functions of complex numbers are used for this.

Loudspeaker radiation pattern

The amplitude frequency response $$A_0$$ of an electrodynamic loudspeaker with the total quality factor $$Q$$ and the natural frequency $$f_0$$ is calculated theoretically over the sound frequency $$f$$ as follows

 * $$A_0 (f) = \frac {\frac {f} {f_0}} {\sqrt {\frac {1} {Q^2} + \left( \frac {f} {f_0} - \frac {f_0} {f} \right)^2}}$$.

In the resonance case $$f = f_0$$ this equation simplifies to.

 * $$A_0 (f_0) = Q$$

With increasing frequency, however, the frequency response drops again, contrary to the statement of the equation above, because the loudspeaker membrane can no longer resonate due to its mechanical inertia. Further deviations occur in practice because natural frequencies other than $$f_0$$ also make small contributions to the frequency response. These natural frequencies are generally caused by unwanted vibration modes of the loudspeaker membrane.

Free speaker

If a loudspeaker is operated at the frequency $$f$$ without a baffle or enclosure, then at the point $$x = 0$$ (i.e. at the loudspeaker surface) over the time $$t$$ the sound wave emitted to the front interferes with the following

 * $$A_v = A(f,t) = A_0(f) \cdot e^{-i \omega t}$$

and the sound wave radiated to the rear

 * $$A_h = -A(f,t) = -A_0(f) \cdot e^{-i \omega t}$$

with the $$e$$, the  $$i = \sqrt {-1}$$ and the  $$\omega = 2 \pi f$$ destructive at all times:

 * $$ A_{tot} = A_v + A_h = 0$$.

This means that the total amplitude frequency response $$ A_{tot}$$ becomes zero as the two sound waves cancel each other out. This is also called.

Loudspeaker with baffle

The problems with the free loudspeaker can be circumvented if the loudspeaker is installed in a baffle that is as wide as possible in relation to the largest sound wavelength to be radiated. However, such a baffle usually has to have very large dimensions. At a minimum frequency of, for example, 30 hertz, the baffle would have to extend laterally by several metres.

 * $$ A_{tot} = A_v = A(f,t) = A_0(f)\cdot e^{-i \omega t}$$

where $$A_v$$ represents the forward radiated sound wave, which in this case is identical to the total radiated $$ A_{tot}$$.

Sound labyrinth

An alternative would be to install the loudspeaker in a so-called sound labyrinth. The sound energy of the sound wave emitted to the rear is destroyed here by multiple reflection and multiple scattering in combination with absorption. However, such a sound labyrinth must also be very large so that the sound can be absorbed to a large extent. As with the baffle, the disadvantage is that the sound energy radiated backwards by the loudspeaker is not used.

Closed loudspeaker cabinet

The ideal behaviour of a sound labyrinth is not achieved with closed cabinet loudspeakers. In addition to the sound wave $$A_v$$ radiated to the front, the sound waves $$A_h$$ radiated to the rear, which are reflected by the inner walls of the enclosure, can also escape through the speaker opening. As a result of the reflections, standing waves also occur, which have a negative effect on the amplitude and phase frequency response of the loudspeaker cabinet. Although this can be limited by filling the inside of the speaker cabinet with a sound-absorbing medium and by not aligning the cabinet walls parallel, this effect can never be completely avoided. Furthermore, especially at low frequencies, sound pressure can be so high that feedback occurs on the speaker cone, which, like the reflected sound waves, has a negative effect as a non-linear effect.

The total radiated sound wave $$ A_{tot}$$ results from the superposition of all sound waves:

 * $$ A_{tot} = A_v + B(A_h) + C(A_h)$$,

where the function $$B(A_h)$$ represents the sound waves damped and reflected in the loudspeaker cabinet and $$C(A_h)$$ the non-linear components due to feedback of the sound.

Partially ventilated loudspeaker cabinet

The high sound pressure is avoided in the so-called partially ventilated loudspeaker cabinets by a small opening that serves to equalise the pressure. If such loudspeaker cabinets are well damped and do not have parallel cabinet walls so that no standing waves can build up, they represent a quite good compromise, but the sound energy radiated to the rear is not used for sound reproduction here either.

Bass reflex loudspeaker cabinet

In bass-reflex loudspeaker cabinets, the natural frequency of the loudspeaker cabinet is deliberately used to amplify the sound radiation of the loudspeaker cabinet mostly in the low frequency range. The loudspeaker cabinet has an additional, usually forward-facing tube opening and thus acts as a Helmholtz resonator with the amplitude function $$B_v(f,t)$$. However, it is problematic to achieve a uniform frequency response in this way because there is usually too much amplification in a frequency band that is too narrow, and as a result the two sound waves radiated forwards from the loudspeaker $$A_v$$ and from the additional opening $$B_v$$ can be very poorly matched to each other. Even with bass-reflex loudspeaker cabinets, as with closed loudspeaker cabinets, standing waves can naturally occur in addition.

The total radiated sound wave $$ A_{tot}$$ thus essentially results from the superposition of two sound waves:

 * $$ A_{tot} = A_v + B_v$$,

where the function $$A_v$$ represents the sound wave radiated forward and $$B_v$$ represents the sound waves radiated out of the bass-reflex port.

Transmissionline without damping

The sound wave $$A_h$$ emitted backwards from the loudspeaker can be conducted through a sound channel with the length $$X_0$$, at the end of which there is an opening. This ideally resonance-free sound channel was named transmission line by Arthur R. Bailey, a Briton who taught at the University of Bradford. The sound wave $$A_{h,TL}$$ emerging from this opening can interfere with the sound wave $$A_v$$ radiated forward by the loudspeaker. In the case of the undamped sound channel, the amplitude of the interfered sound wave results from the sum of the two components:

 * $$A_v = A_0(f)$$ as in the case of the free loudspeaker

and

 * $$A_{h,TL} = -A_0(f)\cdot e^{-i \omega \cdot {X_0 \over c_0}}$$.

Here $$c_0$$ is the speed of sound in a gaseous sound medium (in air at 20° C and 60% relative humidity is $$c_0 = 344 {\frac {m} {s}}$$).

At the loudspeaker front, $$A_{tot,TL}$$ without damping thus results in:

 * $$A_{tot,TL} = A_v + A_{h,TL} = A_0(f)\cdot \left(1 - e^{-i \omega \cdot {\frac {X_0} {c_0}}}\right)$$

Qualitative description
In the 1960s, Bailey came up with the idea of allowing the sound energy radiated backwards by a woofer to "die" in a sufficiently long, damped sound channel that is open at the end, which he described in two articles in the trade magazine Wireless World. Damping is an integral part of Bailey's design.

The damping, the speed of sound and thus also the wavelength of the sound waves are frequency-dependent (dispersion) in such a low-pass filter. The lower the frequency or the longer the wavelength of the sound waves, the less they are damped. The higher frequency components are converted into thermal energy by the acoustic damping, and the very low frequency sound waves still emerging from the transmission line at the end can improve the amplitude frequency response at low frequencies if there is suitable interference with the direct, forward emitted sound waves of the woofer used.

The first commercial loudspeaker cabinets built according to this principle were first brought to market by the Cambridgeshire-based company Radford, which cooperated with Bailey, and later by IMF Electronics and in Germany by Lautsprecher Teufel, among others.

The sound wave emitted from the loudspeaker to the rear is increasingly attenuated in the sound channel, which is damped with long-fibre sheep's wool, for example, but very low frequencies can leave the sound channel only weakly attenuated and be used for sound reproduction. According to Bailey, the length of the duct should be at least about a quarter of the wavelength of the lower cut-off frequency, but it can also be longer. It can be bent or folded in various ways to fit in a conventional enclosure, but reflections (for example, with curved or bevelled surfaces) should be avoided.

Since this "non-resonant loudspeaker cabinet" (Bailey) destroys a large part of the sound energy radiated to the rear, the efficiency of transmission line loudspeakers is relatively poor, but thanks to powerful (transistor) amplifiers, this is usually not an issue. When correctly constructed, transmission line speakers are characterised by very good, sound-neutral (low) bass reproduction and very good impulse behaviour. However, due to size, weight and the relatively high manufacturing costs, they could not establish themselves on the market in the long run.

Quantitative calculation
In the following, the facts described by Bailey essentially only qualitatively are described quantitatively by means of corresponding acoustic formulae.

If the transmission line is damped with a porous, fibrous absorber, the sound wave amplitude $$A_{h,TL,D}$$ emerging from the transmission line can be approximated according to Fridolin Peter Mechel with the theory of quasihomogeneous absorbers:

 * $$A_{h,TL,D} = -A_0 (f)\cdot e^{-i \omega \cdot {X_0 \over c_0} \cdot \sqrt{1 - i \cdot {r \over {\rho_0 \cdot \omega}}}}$$

Where $$\rho_0$$ is the density of the gaseous sound medium (for dry air at 20° C is $$\rho_0 = 1.2 {\text{kg} \over \text{m}^3}$$) and $$r$$ is the specific flow resistance of the porous absorber material in Newton seconds per biquadrate metre. The greater the specific flow resistance, the greater the sound attenuation. It depends mainly on the material, the fibre fineness and the density $$\rho$$, i.e. the packing density, of the absorber and the degree of filling in the transmission line. Furthermore, it depends on the uniform distribution of the absorber, but also on the direction of the fibres and the sound frequency, so that only approximate guide values for some porous, fibrous absorbers are given in the following table:

For the entire transmission line loudspeaker cabinet, the following frequency-dependent amplitude results at the front of the loudspeaker cabinet with damping:

 * $$A_{tot,TL,D} = A_v + A_{h,TL,D} = A_0(f)\cdot \left(1 - e^{-i \omega \cdot {X_0 \over c_0} \cdot \sqrt{1 - i \cdot {r \over {\rho_0 \cdot \omega}}}}\right)$$

The amplitude curve results from the magnitude of the complex function $$A_{tot,TL,D}$$. This must be adjusted according to the parameters total quality of the woofer $$Q$$, length of the transmission line $$X_0$$ and specific flow resistance $$r$$ in such a way that the amplitude curve is as even as possible at the low frequencies. The two figures opposite are intended to illustrate the influence of the two parameters $$X_0$$ and $$r$$ for a given $$Q$$.

The parameters determined to be suitable can be implemented in the construction of a transmission line loudspeaker cabinet. However, the correct damping in particular must be checked experimentally and corrected if necessary. Furthermore, when driving the woofer and midrange driver, the amplitude resulting from the above equation must be

 * $$|A_{tot,TL,D}|$$

and phase difference

 * $$arg(A_{tot,TL,D})$$

at the crossover frequency between the system of transmission line and woofer on one side and midrange driver on the other side are balanced by means of the crossover.

Empirical values for the design of a transmission line loudspeaker

The cross-section of the transmission line directly behind the loudspeaker should be at least as large as the loudspeaker cone area or even slightly larger. Along the transmission line its cross-section can be reduced by a maximum of 25 percent.

The woofer should have a relatively high overall quality factor $$\left(Q \approx 1\right)$$.

The optimum value of the specific flow resistance $$r$$ must be checked and adjusted experimentally in the finished loudspeaker cabinet (for example with a frequency generator or with a noise generator and frequency analyser), as it is difficult to estimate for an absorber built into the transmission line.

The crossover is used to distribute the electrical signal to the woofers, midrange and, if necessary, the tweeters. To avoid undesirable acoustic effects, the lateral distances between midrange and woofer as well as between tweeter and midrange should be kept as small as possible. In addition, all speakers should be flush with the cabinet and lie in one plane.

The exit port of a transmission line can be considered constructively like a subwoofer.

Summary of the project

 * Target audience: This Wikibook is aimed at acoustically interested readers with mathematical training.
 * Learning objectives: This Wikibook describes a complete way to calculate a damped acoustic transmission line with complex-valued functions. Building on basic considerations of sound generation and radiation of loudspeakers, the functioning of such a transmission line is worked out through step-by-step extensions and additions.
 * Book sponsorship/contact person: User:Bautsch
 * Are co-authors currently desired? '''Yes, very much so. Corrections of obvious errors directly in the text; content please via discussion.
 * Guidelines for co-authors: Wikimedia-like.


text:
User:Saltrabook/Sustainable Development Goals - Journal Club

Researchers and academic staff in universities are key players in promoting SDGs. Contribution of scientists in different disciplines enlightens policy-makers and politicians on certain goals that need to be addressed in the future as well as on the extent of success or failure in achieving a certain goal. The SDGs are based on the presence of serious global problems that need to be addressed and solved through research and innovation. Researchers are in a position to provide genuine solutions and inventions to various national and international problems facing the pathway to achieving the 2030 agenda [10]. Local and international authorities need to adopt research results and recommendations made by academic staff and put them into a practical agenda to strengthen partnership with all stakeholders and to accelerate the practical steps toward achieving SDGs. Will the goal to strengthen the STD 17 goals mainly are most often good health and well-being for all. What we can do is to ask who will and can implement this wishes it has been expressed in the in the manuscript. What should be done by the shareholders going through one by one, legal, ship gup owners, seafarers, unions, schools, government. What is the need for further research. goal is to bring this knowledge to all shareholders and to ask especially students all companies to discuss what can be done.

Each of the Journal Club members bring short abstract and a reference.

And try to give an answer to these things. It must not be more than 100 to 200 words in all. The Journal Club document stay in Wiki and link to the news send to CMSS Newsletter, fagskolestidsskrifter og universitet, skoler i det maritime, rederier, ask them to make "Club" to meet 15 minutes and discuss one article
 * 1) What is the need for further research to observe the implementation.
 * 2) What is the need for further research.

Contribution to UNs 17 Sustainable Development Goals [edit | edit source]
Goal 3: Good health and well-being for all workers

Goal 4: Quality Education

Goal 5: Gender Equity

Goal 8: Decent Work and Economic Growth

Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)

Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)

Goal 14: Life underwater observations on compliance with good waste management

Goal 17: Partnerships to achieve the Goals

Task for the month: Find an article published within 3 months clearly support one or more the SDGs goals in maritime health (globally) write a short (100 words) abstract IMRAD structure. - Evaluation: Is the content strong scientifically, y/n What is the message in less than 20 words Which of th SDG goals are supported and how? Impact for seafarers health What should be done by the stakeholders, 1 by 1 Researcher, need more studies, follow up of implementaiton, further studies of etiology etc. Link to original abstract and fulltext possible.


text:
Cookbook:Palmyra Palm Fruit

The palmyra palm fruit, also known as the toddy palm fruit or simply as palm fruit, is the fruit of Borassus flabellifer.

title: Cookbook:Palmyra Palm Fruit


text: Category:Book:Roblox WF Wars

title: Category:Book:Roblox WF Wars


text: Roblox WF Wars/Printable version

title: Roblox WF Wars/Printable version


text:
The Legend of Zelda: The Wind Waker/Species/Greudo

The Greudo people are not seen or even widely spoken of in , with the sole example seen in the game being the main antagonist Ganondorf.


text:
The Legend of Zelda: The Wind Waker/Species/Zora

Backstory
The Zora are an ancient water dwelling species in The Legend of Zelda: The Wind Waker.

Current State
Despite being aquatic, they are not seen in the flooded world the game takes place in, save for a ghost of a Zora sage. However their airborne dependents, the Rito, play a prominent role in the game world.


text:
The Legend of Zelda: The Wind Waker/Species/Kokiri

Backstory
The Kokiri were a people who resided in forests prior to the events of The Legend of Zelda: The Wind Waker. Their appearance in The Legend of Zelda: The Wind Waker as small humanoids who are associated with the forest and clad in green clothing is not too different from their depiction in The Legend of Zelda: Ocarina of Time.

Current State
By the time the events of The Legend of Zelda: The Wind Waker not a single Kokiri is seen alive in the game. Only the ghost of a Kokiri sage is ever seen by Link.

However the Kokiri do play an important role in the game, through their descendants in Korok, who have established one of the largest settlements seen in the game.


text:
The Legend of Zelda: The Wind Waker/Species/Rito

The Rito are a bird like species in the Wind Waker. They are descended from the Zora.

The Rito primarily live on Dragon Roost Island. Many Rito serve as postal workers, and due to their ocupation they can be found in many locations across the game map.


text:
The Legend of Zelda: The Wind Waker/Species/Hylian

The ancient script of the Hylian people is seen as an ancient language in The Wind Waker, though it is not commonly understood by most contemporary characters.


text: Template:Hindi/Exercise

title: Template:Hindi/Exercise


text:
Skaneateles Conservation Area/Invasive species/Invasive animal list

 * List of invasive or potentially invasive plants that could be found at the SCA
 * List of invasive or potentially invasive animals that could be found at the SCA
 * List of invasive or potentially invasive microorganisms that could be found at the SCA

Subclass Oligochaeta (earthworms and allies)

 * }

Hemiptera (bugs, hoppers, aphids)

 * }

Hymenoptera

 * }

Coleoptera (beetles)

 * }

Diptera (flies)

 * }

Ixodida (ticks)

 * }

Decapoda (decapods)

 * }

Rhabditida

 * }

Limnomedusae

 * }

Clupeiformes (herrings)

 * }

Cypriniformes (carp)
Uses Fish Name Etymology Database © Christopher Scharpf and Kenneth J. Lazara
 * }


text:
Category:Pecan pie recipes

Recipes for pecan pie.

title: Category:Pecan pie recipes


text:
Cookbook talk:Vegetarian "Patte"

Proposed merge
The recipe here is nearly identical to Cookbook:Vegan Pate, and if there are no objections I will merge this into that page. -- Nostriker (discuss • contribs) 15:47, 23 January 2022 (UTC)

title: Cookbook talk:Vegetarian "Patte"


text:
Talk:Super Mario 64 DS

Removal of course articles
Looks like course articles such as Super Mario 64 DS/Tall Tall Mountain and Super Mario 64 DS/Bob-omb Battlefield were removed weeks after they were restored last June, with the deletion comment being "phasing out". Why were they removed? Looking at the pages on Strategy Wiki, they seem to be of good value to the book. And the original Super Mario 64 also has course articles. Anerisys (discuss • contribs) 19:15, 23 January 2022 (UTC)

 * Most people won't see this talk page, but I'll put in an undeletion request. Mbrickn (discuss • contribs) 05:18, 24 January 2022 (UTC)
 * I've made an undeletion request at Requests for deletion. Mbrickn (discuss • contribs) 05:23, 24 January 2022 (UTC)
 * Hello. I copied this book earlier. Both pages just had one paragraph. The whole text of the former page was "Tall Tall Mountain can be accessed from a painting on the second floor of Princess Peach's Castle. In the level, there are numerous mushroom platforms to jump from.", and that of the latter was "Bob-omb Battlefield is course 1 of Super Mario 64 DS. To access it, enter the castle, enter the door on the right with no number on the star, and jump inside the painting. No stars are required to access it.", for anyone curious. While too short for individual pages, the courses could be merged into Super Mario 64 DS/Courses. Elominius (discuss • contribs) 03:20, 26 March 2022 (UTC)

title: Talk:Super Mario 64 DS


text:
Super Mario 64/Characters

Throughout the game, Mario will encounter various characters which may damage his health to various severities, but can also have utilities.

Goomba
Gives Mario one unit of damage when bumped, and one yellow coin when stomped.

In Tiny-Huge-Island, they appear as tiny and huge variants respectively, where the tiny causes no damage and disappears without coin when bumped, and one yellow coin when stomped. The huge causes two units of damage when bumped, and one blue coin when stomped. The sound when stumping noticably differs from ordinary goombas.

Bob-omb
Causes no damage when touched, but two units of damage when exploding. Can be picked up and thrown.

Pink Bob-omb
They do not explode. Pressing "B" near them causes them to unlock a cannon. An additional one in Bob-omb battlefield tells a story about rivalry with the normal (grey) Bob-ombs.

Koopa
Causes no damage when bumped, gives a blue coin (worth five coins) and a Koopa shell when stomped. The shell can be used to surf on water and lava, and allows riding up slippery slopes uphill.

A stompable Koopa is found in Bob-omb Battlefield. In some courses, such as Jolly-Roger Bay and Shifting Sand Land, Koopa shells are located in breakable boxes.

Owl
Elevates Marios altitude and provides a birds-eye view of the course. Found in Tall Tall Mountain, Whomp's Fortress, and Snowman's Land.

Fly-guy
Spits fire, which causes three units of damage. Bumping causes two units of damage. Stomping causes Mario to spin, which allows him to glide for long distances through the air. Found in Tall Tall Mountain and Snowman's Land.

Thwomp
Quader brick which tries to throws itself on Mario. Causes three units of damage if it flattens Mario. Needs to be stomped using ground pound. Found in Thromp's Fortress and Bowser in the Sky.

Chuckya
Grabs and throws Mario. Dangerous because located near tall slopes and high platforms such as in Rainbow Ride, Wet-Dry World, Bowser in the Sky, and Tall-Tall Mountain. Can be destroyed by being picked up and thrown by Mario.

Toad
Located throughout the castle. Engaging by pressing "A" makes them tell a story and/or praise Mario. Three of them provide a star, one located in the basement near the entry to "Hazy Maze Cave", one in the upper hallway, and another one in the dark_cyan-walled star room near the top of the castle.

title: Super Mario 64/Characters


text:
Cookbook talk:Basic Chai

Merge and Move
Hello! I propose that the contents of the introduction here be merged into the Cookbook:Chai Tea disambiguation page, while the recipe on this page is kept and renamed to something like Cookbook:Basic Chai. I am happy to make this move if there are no objections. -- Nostriker (discuss • contribs) 22:34, 23 January 2022 (UTC)

title: Cookbook talk:Basic Chai