Pythagoras in the forge
Pythagoras in the forge is an ancient legend that describes how Pythagoras discovered in a forge that simultaneous hammer blows produced melodious tones when the weights of the hammers were in certain integer ratios. This observation led him to experiments on the vibrating string of a monochord , which became the basis for the music-theoretical description of intervals . With the knowledge gained in this way, Pythagoras founded music theory. The legend had the consequence that Pythagoras in theRoman imperial times and in the Middle Ages was generally referred to as the inventor of "music", by which the music theory was meant.
The legend is only attested in Greek during the Roman Empire, older sources are not known and may have been lost. Over the centuries the narrative has been modified. It was only in the 17th century that it could be shown that the description of the legend could not apply, because the pitch of the hammer hardly depends on the weight of the hammer and the vibrations of the hammer itself are practically inaudible. However, the legend is still treated like a credible report in more recent publications.
Regardless of the question of how and by whom the discovery of musical numerical relationships attributed to Pythagoras actually occurred, the formulation of these numerical relationships is the first traditional mathematical description of a physical fact, for the correctness of which experimental observations were cited as evidence.
Content of the legend
According to the oldest surviving version of the legend, Pythagoras, who lived in the 6th century B.C. I was looking for a tool with which acoustic perceptions can be measured, such as geometric quantities with a compass or weights with scales. As he passed a blacksmith's shop, where four (according to a later version, five) craftsmen were at work with hammers, he noticed that the individual beats produced tones of different pitches that resulted in pairs of harmonies. He could distinguish between octaves , fifths and fourths . He found only one pair, which resulted in the interval between fourth and fifth (major second ), to be dissonant . Then he happily ran into the smithy to make experiments. He found that the difference in pitch does not depend on the shape of the hammer, the location of the iron being struck, or the force of the blow. Rather, he was able to assign the pitches to the weights of the hammers, which he measured precisely. He then returned home to continue the experiments there.
On a peg that was attached to the wall at an angle across the corner, he hung up four strings of the same length, strength and twist, which he weighed down by tying different weights. Then he struck the strings in pairs, sounding the same harmonies as in the forge. The most heavily loaded string with twelve weight units resulted in an octave with the least loaded string, on which six weight units hung. It was shown that the octave is based on the ratio 12: 6, i.e. 2: 1. The tightest string resulted in a fifth with the second loosest (eight weight units) and a fourth with the second tightest (nine weight units). It follows that the fifth is based on the ratio 12: 8, i.e. 3: 2, the fourth on the ratio 12: 9, i.e. 4: 3. The ratio of the second tightest string to the loosest string was again 9: 6, i.e. 3: 2, a fifth, for that of the second loosest to the loosest with 8: 6, i.e. 4: 3, a fourth. For the dissonant interval between fifth and fourth it was found that it is based on the ratio 9: 8, which agreed with the weight measurements already carried out in the forge. The octave turned out to be the product of the fifth and fourth:
Then Pythagoras extended the experiment to various instruments, experimented with vessels, flutes, triangles , the monochord, etc .; he always found the same numerical proportions. Finally, he introduced the names for the relative pitches that have been used since then.
With the invention of the monochord to investigate and demonstrate the harmony of pairs of strings with different integer length ratios, Pythagoras is said to have introduced a convenient means of showing the mathematical basis of music theory that he had discovered. The monochord - ancient Greek κανών kanōn , called regula in Latin - is a resonance box over which a string is stretched. There is a scale on the box. The device is equipped with a movable bridge, which can be shifted to divide the oscillating length of the string; the graduation can be precisely determined using the graduation. This enables the intervals to be measured. Despite the name “monochord”, which means “single string”, there were also multi-string monochords with which the intervals could be made to sound simultaneously. However, it is unclear when the monochord was invented. Walter Burkert dates this achievement only after the time of Aristotle , who apparently did not yet know the device; accordingly it was only introduced long after Pythagoras' death. Leonid Zhmud, on the other hand, thinks that Pythagoras probably carried out his experiment, which led to the discovery of the numerical relationships, with the monochord.
Hippasus of Metapontus , an early Pythagorean (late 6th and early 5th centuries BC), conducted quantitative research on musical intervals. In contrast to the alleged experiments of Pythagoras, the experiment with freely swinging circular plates of different thicknesses attributed to Hippasus is physically correct. Whether Archytas of Taranto , an important Pythagorean of the 5th / 4th centuries Century BC Chr., Has carried out relevant experiments, is unclear. Presumably he was more theoretician than a practitioner in music, but he referred to the acoustic observations of his predecessors. The musical examples he gives in support of his acoustic theory concern wind instruments; He does not mention attempts with stringed instruments or individual strings. Archytas started from the wrong hypothesis that the pitch depends on the speed of propagation of the sound and the force of the impact on the sound body; in reality the speed of sound in a given medium is constant and the force only affects the volume.
Interpretation of the legend
Walter Burkert is of the opinion that the legend, despite the physical impossibility, should not be regarded as an arbitrary invention, but rather has a meaning that can be found in Greek mythology . The Idaean dactyls , the mythical inventors of blacksmithing, according to a myth, were also the inventors of music. A very old tradition, in which the mythical blacksmiths were portrayed as connoisseurs of the mystery of magical music, evidently already assumed a connection between the art of blacksmithing and music. Burkert sees the legend of Pythagoras in the smithy as a late transformation and rationalization of the ancient myth of Dactyl: In the Pythagoras legend, the smiths no longer appear as owners of old magical knowledge, but they become, without wanting to, become - albeit ignorant - "Teachers" of Pythagoras.
In the early Middle Ages called Isidore of Seville biblical blacksmith Tubal as the inventor of music; later authors followed him in this. This tradition again shows the idea of a relationship between blacksmithing and music, which also occurs in non-European myths and legends. Tubal was half-brother of Jubal , who was regarded as the forefather of all musicians. Both were sons of Lamech and thus grandsons of Cain . In some Christian traditions of the Middle Ages, Jubal, watching his brother Tubal, was equated with Pythagoras.
Another explanation is proposed by Jørgen Raasted, followed by Leonid Zhmud. Raasted's hypothesis states that the starting point for the creation of the legend was an account of the experiments of Hippasus. Hippasus used vessels called sphaírai . This word was mistaken for sphýrai (hammers) through a scribal mistake , and instead of Hippasus' name that of Pythagoras was used as the author of the experiments. From this the legend of the blacksmith emerged.
Basis of music theory
The whole numbers 6, 8, 9 and 12 correspond with the lowest note (number 12) to the pure intervals fourth (number 9), fifth (number 8) and octave (number 6) upwards:
|Relation to the
largest number 12
Such pure intervals are perceived as beating-free by the human ear , since the volume of the tones does not vary. In musical notation , these four Pythagorean tones can be expressed, for example, with the tone sequence c ′ - f ′ - g ′ - c ″:
If this tone sequence is not viewed from the lowest but from the highest tone (number 6), there are also a fourth (number 8), a fifth (number 9) and an octave (number 12) - in this case, however, downwards:
|Relation to the
smallest number 6
The fifth and the octave also appear in relation to the fundamental tone in series of natural tones , but not the fourth or their octavings . This fourth tone does not occur in valveless brass instruments and in the flageolet tones of stringed instruments .
Significance for the later further development of the sound systems
The further investigation of intervals consisting of octaves, fifths and fourths and their multiples finally led from diatonic scales with seven different tones ( heptatonic ) in Pythagorean tuning to a chromatic scale with twelve tones. Thereby disturbed the Pythagorean fifths Wolf : Instead of pure fifths As - It and Des - As rang the around the Pythagorean comma detuned Quinten Gis - It and Cis - As.
With the advent of polyphony in the second half of the 15th century, in addition to the octave and fifth, the perfect third became decisive for the major and minor triads. Although this tuning could not be achieved on a twelve-step keyboard, it could be achieved well in the mid-tone tuning . Their disadvantage was that not all keys of the circle of fifths were playable. To remedy this deficiency, tempered tunings were introduced, but with the disadvantage that the pure third sounded rougher in some keys. Nowadays, most 12-key instruments are tuned equally , so that the octaves sound perfectly pure, the fifths almost pure and the thirds sound rough.
The four Pythagorean tones in music
In music, play the four harmonic Pythagorean tones in the pentatonic scale , especially on the first, fourth, fifth and eighth tone of the diatonic scales (especially in major and minor ) and in the composition of cadences as fundamentals of tonic , subdominant and dominant outstanding Role. This tone sequence often occurs in final cadences with the corresponding chords :
The four Pythagorean tones appear in many compositions. The first tones of the early medieval antiphons Ad te levavi and Factus est repente consist essentially of the four Pythagorean tones, apart from a few ornaments or pointed tones.
Another example is the beginning of the Passacaglia in C minor by Johann Sebastian Bach . The theme consists of fifteen tones, of which a total of ten tones and in particular the last four tones were drawn from the tone sequence.
Absolute pitch of hammers
The natural frequency of steel hammers, which can be moved by human hands, is mostly in the ultrasonic range and therefore inaudible. Pythagoras may not have perceived these tones, especially if the hammers had an octave difference in pitch.
Pitch depending on hammer weight
The vibration frequency of a longitudinally freely vibrating solid is usually not proportional to its weight or volume , but is proportional to its length, which changes only with the cube root of the volume with a similar geometry .
For the Pythagorean hammers with a similar geometry, the following ratios apply (information in arbitrary units of measurement):
|Ratio to the
|Hammer head length /
|Ratio to the
Pitch as a function of the string tension
The assumption that the vibration frequency of a string is proportional to the tension force does not apply, rather the vibration frequency is proportional to the square root of the tension force. In order to double the oscillation frequency, four times the tensile force has to be exerted and a weight that is four times as heavy has to be hung on a string.
Integer frequency ratios
|Integer ratio of
The fact that a tone with the basic frequency is in consonance with a second tone with an integer multiple (with and ) of this basic frequency results directly from the fact that the maxima and minima of the tone oscillations are synchronized in time , but can also be as follows be explained:
This difference in turn has an integer ratio to the basic frequency :
For all integer multiples of the basic frequency for the second tone, there are also integer multiples for the beat frequency (see adjacent table), so that all tones sound consonant.
Rational frequency relationships
There is also a consonance for two tones whose frequencies are in a rational relationship to . The frequency of the second tone results from:
The result for the beat frequency of the two simultaneously sounding tones is:
Under this condition, the basic frequency is always an integral multiple of the beat frequency (see adjacent table). Therefore, there is no dissonance either .
Longitudinal vibrations and natural frequency of solids
To estimate a metal block, consider a homogeneous cuboid with a maximum length and made of a material with the speed of sound . For the vibration mode, this has the lowest natural frequency of along its longest side ( longitudinal vibration ) with antinodes at both ends and a vibration node in the middle
The pitch is therefore independent of the mass and the cross-sectional area of the cuboid, the cross-sectional area may even vary. Furthermore, the force and the speed when striking the body do not play a role. At least this fact coincides with the observation attributed to Pythagoras that the perceived pitch did not depend on the hands (and thus the strengths) of the craftsman.
Bodies with more complex geometry , such as bells , beakers or bowls , which may even be filled with liquids, have natural frequencies, the physical description of which is considerably more complex, since here not only the shape, but also the wall thickness or even the location of the impact must be taken into account. Here, under certain circumstances, transverse vibrations are also stimulated and audible.
A very large sledgehammer (the speed of sound in steel is approximately = 5000 meters per second) with a hammer head length = 0.2 meters has a natural frequency of 12.5 kilohertz. With a square cross-sectional area of 0.1 meters by 0.1 meters, with a density of 7.86 grams per cubic centimeter, it would have an unusually large mass of almost 16 kilograms. Frequencies above around 15 kilohertz can no longer be heard by many people (see listening area ); therefore the natural frequency of even such a large hammer is barely audible. Hammers with shorter heads have even higher natural frequencies, which are therefore in no way audible.
A large steel anvil with a length of 0.5 meters has a natural frequency of only 5 kilohertz and is therefore easily audible.
There are a variety of compositions in which the composer prescribes the use of anvils as a musical instrument . The two operas from the musical drama Der Ring des Nibelungen by Richard Wagner are particularly well-known :
- Das Rheingold , scene 3, 18 anvils in F in three octaves
- Siegfried , Act 1, Siegfried's Schmiedelied Nothung! Emergency! Envious sword!
Materials with a lower speed of sound than steel, such as granite or brass , generate even lower frequencies with congruent geometry . In any case, there is no mention of anvils in the early traditions and of the audible sounds of the anvils in the later versions of the legend, but the sounds are always attributed to the hammers.
It is possible to compare metal rods , such as chisels from stonemasons or riving knives for breaking stones , in order to arrive at an observation similar to that attributed to Pythagoras, namely that the vibration frequency of tools is proportional to their weight . If the metal rods, neglecting the tapered cutting edges, all have the same uniform cross-sectional area A , but different lengths l , their weight is proportional to the length and thus also to the vibration frequency, provided that the metal rods are excited to longitudinal vibrations by blows along the longitudinal axis (sound examples see in the box on the right).
Strings can be attached to a bridge on two sides . Exactly the other way around than in a solid body with longitudinal vibrations, the two webs create the boundary conditions for two vibration nodes ; therefore the antinode is in the middle.
The natural frequency and thus the pitch of strings with their length are not proportional to the tension , but to the square root of the tension. In addition, the frequency increases and does not decrease with a higher pulling weight and thus higher clamping force:
Nonetheless, the oscillation frequency at constant tension is strictly inversely proportional to the length of the string, which can be directly demonstrated with the monochord - allegedly invented by Pythagoras .
The earliest mention of Pythagoras' discovery of the mathematical basis of musical intervals is found in the Platonist Xenocrates (4th century BC); since it is only a quote from a lost work of this thinker, it is unclear whether he knew the legend of the blacksmith. In the 4th century BC In BC, criticism of the Pythagorean number theory of intervals has also been expressed - albeit without reference to the Pythagorean legend; the philosopher and music theorist Aristoxenus thought it was wrong.
The oldest surviving version of the legend is presented - centuries after the time of Pythagoras - by the New Pythagorean Nicomachus of Gerasa , who recorded the story in his Harmonikon encheiridion ("Handbook of Harmony") in the 1st or 2nd century AD . For his representation of the numerical relationships in harmony, he refers to the philosopher Philolaos , a Pythagorean of the 5th century BC.
The famous mathematician and music theorist Ptolemy (2nd century) knew the weight method handed down by legend and rejected it; However, he had not recognized the falsity of the weight experiments, but only criticized their inaccuracies in comparison with the precise measurements on the monochord. He probably got his knowledge of the legendary tradition not from Nicomachus, but from an older, now lost source.
The imperial music theorist Gaudentios , who is difficult to classify chronologically, described the legend in a version that is slightly shorter than that of Nicomachus in his harmonics ḗ eisagōgḗ (“Introduction to Harmony”). The Neoplatonist Iamblichus , who was active as a philosophy teacher in the late 3rd and early 4th centuries, wrote a biography of Pythagoras entitled About the Pythagorean Life , in which he reproduced the legend of the blacksmith in the version of Nicomachus.
In the first half of the 5th century, the writer Macrobius went into detail in his commentary on Cicero's Somnium Scipionis on the legend of the blacksmith, which he described in a similar way to Nicomachus.
The strongest aftereffect among the ancient music theorists who took up the story was achieved by Boethius with his textbook De institutione musica (“Introduction to Music”), written in the early 6th century , in which he described the knowledge of Pythagoras first in the smithy and then to Home portrays. It is unclear whether he proceeded from the representation of Nicomachus or from another source. In contrast to the entire older tradition, he reports five hammers instead of assuming four hammers as the earlier authors did. He claims that Pythagoras discarded the fifth hammer because it produced a dissonance with all of the other hammers. According to Boethius' account (as with Macrobius), Pythagoras checked his first assumption that the sound difference was due to different strength in the arms of the men by having the blacksmith swap hammers, which led to a refutation. About the experiments in the house of Pythagoras, Boethius writes that the philosopher first hung weights of the same weight as those of the hammers in the forge and then experimented with pipes and beakers, all experiments leading to the same results as the first with the hammers . Using the legend, Boethius addresses the scientific and epistemological question of the reliability of sensory perception. What is essential is the fact that Pythagoras was initially stimulated by sensory perception to pose his questions and to form hypotheses and that he achieved irrefutable certainty through empirical testing of hypotheses. The path of knowledge led from sensory perception to the first hypothesis, which turned out to be erroneous, then to the formation of a correct opinion and finally to its verification. Boethius recognizes the necessity and the value of sensory perception and the formation of opinions on the way to insight, although as a Platonist he principally distrusts sensory perception because of its susceptibility to error. Real knowledge only arises for him when the law has been grasped, with which the researcher emancipates himself from his initial dependence on unreliable sensory perception. The researcher's judgment must not be based merely as a sensory judgment on empirical experience, but rather it must only be made when, through deliberation, he has found a rule that enables him to position himself beyond the realm of possible sensory illusion.
In the 6th century the scholar Cassiodorus wrote in his Institutiones that Gaudentios had traced the beginnings of "music" back to Pythagoras in his report on the legend of the blacksmith. What was meant was music theory, as with Iamblichus, who, with reference to the blacksmith's story and the experiments described there, had called Pythagoras the inventor of "music".
In the early Middle Ages , Isidore of Seville briefly mentioned the legend of the blacksmith in his etymologies , which became an authoritative reference work for the medieval educated, adopting Cassiodor's formulation and also calling Pythagoras the inventor of music. Since Cassiodorus and Isidore were first-rate authorities in the Middle Ages, the idea spread that Pythagoras had discovered the basic law of music and was therefore its founder. Despite such blanket formulations, medieval music theorists assumed that music had existed before Pythagoras and that the "invention of music" meant the discovery of its principles.
In the 9th century, the musicologist Aurelian von Réomé reported the legend in his Musica disciplina ("Music Theory "). Aurelian's presentation followed in the 10th century Regino von Prüm in his work De harmonica institutione ("Introduction to the theory of harmony"). Both set great store by stating that Pythagoras had, through a divine providence, been given the opportunity to make his discovery in the forge. Even in antiquity, Nicomachus and Iamblichus had spoken of a daimonical disposition, and Boethius had made divine counsel out of it.
In the 11th century, the legend material was processed in the Carmina Cantabrigiensia .
In the first half of the 11th century, Guido von Arezzo , the most famous music theorist of the Middle Ages, told the legend of the blacksmith in the last chapter of his Micrologus , starting from the version of Boethius, whom he named. In the introduction, Guido remarked: Also, a person would probably never have researched anything specific about this art (music) if the divine goodness had not finally brought about the following event at her suggestion. The fact that the hammers 12, 9, 8 and 6 weighed units and thus produced the melodious sound, he attributed to God's providence. He also mentioned that Pythagoras invented the monochord based on his discovery, but did not elaborate on its properties.
The work De musica by Johannes Cotto (also known as John Cotton or Johannes Afflighemensis) was illustrated with the blacksmith scene around 1250 by an anonymous illuminator from the Cistercian Abbey of Aldersbach .
The medieval music theorists who told the legend of the blacksmith based on the version of Boethius also included Juan Gil de Zámora (Johannes Aegidius von Zamora), who worked in the late 13th and early 14th centuries, and Johannes de Muris and Simon Tunstede in the 14th century , in the 15th century on the threshold of modern times Adam von Fulda .
As an opponent of the Pythagorean view, according to which the consonances are based on certain numerical relationships, Johannes de Grocheio emerged in the 13th century , who started from an Aristotelian point of view. Although he expressly stated that Pythagoras had discovered the principles of music, and related the legend of the blacksmith with reference to Boethius, whom he considered trustworthy, he rejected the Pythagorean consonance theory, which he wanted to reduce to a mere metaphorical speech.
Early modern age
Franchino Gaffurio published his work Theoricum opus musice discipline ("Theoretical Music Theory") in Naples in 1480 , which appeared in a revised version under the title Theorica musice ("Music Theory") in 1492 . In it he presented a version of the legend of the blacksmith that surpassed all earlier representations in terms of detail. He started from the version of Boethius and added a sixth hammer in order to accommodate as many notes in the octave as possible in the narrative. In four pictorial representations he presented musical instruments or sound generators with six harmonic tones each and indicated the numbers 4, 6, 8, 9, 12 and 16 assigned to the tones in the lettering. To the four traditional ratios of the legend (6, 8, 9 and 12) he added the 4 and the 16, which represent a tone a fifth lower and another tone a fourth higher. The entire tone sequence now extends not just over one, but two octaves. These numbers correspond, for example, to the tones f - c '- f' - g '- c "- f":
The painter Erhard Sanßdorffer was commissioned in 1546 to design a fresco in Büdingen Castle in Hesse , which is well preserved and which, based on the forge of Pythagoras, represents the history of music like a compendium.
Also Gioseffo Zarlino told the legend in his book Le Istitutioni harmoniche ( "The principles of harmony"), which he published in 1558; like Gaffurio, he used the representation of Boethius as a basis.
The music theorist Vincenzo Galilei , the father of Galileo Galilei , published his pamphlet Discorso intorno all'opere di messer Gioseffo Zarlino ("Treatise on the works of Mr. Gioseffo Zarlino") in 1589 , which was directed against the views of his teacher Zarlino. In it he pointed out that the information given in the legend about the loading of strings with weights is incorrect.
1626 appeared in the Thesaurus philopoliticus of Daniel Meisner an engraving of Eberhard Kieser entitled Duynkirchen on which only three blacksmiths can be seen on an anvil. The caption in Latin and German reads:
- Triplicibus percussa sonat varie ictibus incus.
- Musica Pythagoras struit hinc fundamina princ (eps).
- The anvil of three hammers sounds from which dreyerley thon rises.
- Pythagoras finds music here that no donkey's head could have done.
A few years later, the matter was definitely resolved after Galileo Galilei and Marin Mersenne discovered the laws governing the vibrations of strings. In 1636 Mersenne published his Harmonie universelle , in which he explained the physical error in the legend: The oscillation frequency is not proportional to the elasticity, but to its square root .
In the 19th century, Hegel assumed the physical correctness of the alleged measurements communicated in the Pythagoras legend in his lectures on the history of philosophy.
Werner Heisenberg emphasized in an essay published for the first time in 1937 that the Pythagorean "discovery of the mathematical conditionality of harmony" is based on "the thought of the power of mathematical structures that give meaning", a "basic idea that the exact natural science of our time has adopted from antiquity"; the discovery attributed to Pythagoras is "one of the strongest impulses of human science at all".
Even more recently, representations have been published in which the legend is reproduced uncritically without reference to its physical and historical falsehood, such as in the non-fiction book The Fifth Hammer. Pythagoras and the disharmony of the world by Daniel Heller-Roazen .
- Gottfried Friedlein (Ed.): Anicii Manlii Torquati Severini Boetii de institutione arithmetica libri duo, de institutione musica libri quinque. Minerva, Frankfurt am Main 1966 (reprint of the Leipzig 1867 edition, online ; German translation online )
- Michael Hermesdorff (translator): Micrologus Guidonis de disciplina artis musicae, ie Guido's short treatise on the rules of musical art . Trier 1876 ( online )
- Ilde Illuminati, Fabio Bellissima (ed.): Franchino Gaffurio: Theorica musice. Edizioni del Galluzzo, Firenze 2005, ISBN 88-8450-161-X , pp. 66–71 (Latin text and Italian translation)
- Walter Burkert: Wisdom and Science. Studies on Pythagoras, Philolaos and Plato (= Erlangen contributions to linguistics and art studies . Volume 10). Hans Carl, Nuremberg 1962
- Anja Heilmann: Boethius' Music Theory and the Quadrivium. An introduction to the Neoplatonic background of "De institutione musica" . Vandenhoeck & Ruprecht, Göttingen 2007, ISBN 978-3-525-25268-0 , pp. 203–222 ( limited preview in Google book search)
- Werner Keil (Hrsg.): Basic texts music aesthetics and music theory. Wilhelm Fink, Paderborn 2007, ISBN 978-3-8252-8359-9 , pp. 342–346 ( limited preview in Google book search)
- Barbara Münxelhaus: Pythagoras musicus. On the reception of Pythagorean music theory as a quadrivial science in the Latin Middle Ages (= Orpheus series of publications on fundamental questions in music . Volume 19). Publishing house for systematic musicology, Bonn - Bad Godesberg 1976
- Jørgen Raasted: A neglected version of the anecdote about Pythagoras's hammer experiment. In: Cahiers de l'Institut du Moyen-Âge grec et latin . Volume 31a, 1979, pp. 1-9
- Leonid Zhmud: Science, Philosophy and Religion in Early Pythagoreanism. Akademie Verlag, Berlin 1997, ISBN 3-05-003090-9
- Leonid Zhmud: Science, Philosophy and Religion in Early Pythagoreism , Berlin 1997, pp. 193–196; see Károly Simonyi: Kulturgeschichte der Physik. 3. Edition. Frankfurt am Main 2001, p. 62.
- Nikomachos von Gerasa, Handbuch der Harmonielehre 6, translated by Anja Heilmann: Boethius' Musiktheorie und das Quadrivium , Göttingen 2007, pp. 345–347, quoted verbatim in Iamblichos von Chalkis, About the Pythagorean Life 115–121, translated by Michael von Albrecht: Jambly. Pythagoras: Legend - Doctrine - Lifestyle , Darmstadt 2002, pp. 109–113.
- Walter Burkert: Wisdom and Science , Nuremberg 1962, p. 353 and note 28.
- Leonid Zhmud: Science, Philosophy and Religion in Early Pythagoreism , Berlin 1997, pp. 193–196; cf. Barbara Münxelhaus: Pythagoras musicus , Bonn - Bad Godesberg 1976, p. 28 f.
- Walter Burkert: Wisdom and Science , Nuremberg 1962, pp. 362–364; Leonid Zhmud: Science, Philosophy and Religion in Early Pythagoreanism , Berlin 1997, pp. 196–199. Carl A. Huffman expresses skepticism regarding the acoustic experiments of Archytas: Archytas of Tarentum , Cambridge 2005, pp. 129–148; see pp. 473-475. He points out that Archytas mainly relies on information from his predecessors and on everyday experience.
- Walter Burkert: Wisdom and Science , Nuremberg 1962, p. 355.
- Walter Burkert: Wisdom and Science , Nuremberg 1962, p. 355; Barbara Münxelhaus: Pythagoras musicus , Bonn - Bad Godesberg 1976, p. 38 f., 46.
- James W. McKinnon: Jubal vel Pythagoras, quis sit inventor musicae? In: The Musical Quarterly , Vol. 64, No. 1, 1978, pp. 1-28; Paul E. Beichner: The Medieval Representative of Music, Jubal or Tubalcain? (= Texts and Studies in the History of Mediaeval Education 2), Notre Dame (Indiana) 1954; Francis Olivier Zimmermann: La forge et l'harmonie. De Pythagore à Tubalcain et Jubal. In: Zimmermann: Orphée: arts vivants, arts de parole et mélodie ( online ).
- Jørgen Raasted: A neglected version of the anecdote about Pythagoras's hammer experiment. In: Cahiers de l'Institut du Moyen-Âge grec et latin 31a, 1979, pp. 1–9, here: 6 f .; Leonid Zhmud: Science, Philosophy and Religion in Early Pythagoreanism , Berlin 1997, p. 192.
- See Factus est repente , word-melody relationships in Gregorian chant; The first chant of the church year .
- Ludwig Bergmann, Clemens Schaefer: Textbook of Experimental Physics. Volume 1, 9th edition. Berlin 1974, Chapter 83: Sound transmitter , section longitudinal vibrations .
- Markus Bautsch: About the Pythagorean roots of the Gregorian modes ( online ).
- Ludwig Bergmann, Clemens Schaefer: Textbook of Experimental Physics. Volume 1, 9th edition. Berlin 1974, chapter 83: sound transmitter , section string .
- Xenokrates fragment 9 H. See also Walter Burkert: Weisheit und Wissenschaft , Nürnberg 1962, p. 57 and Leonid Zhmud: Wissenschaft, Philosophie und Religion im early Pythagoreismus , Berlin 1997, p. 193.
- This passage from Nicomachus' work has been edited, translated into English and commented on by Carl A. Huffman: Philolaus of Croton , Cambridge 1993, pp. 145–165.
- Barbara Münxelhaus: Pythagoras Musicus , Bonn - Bad Godesberg 1976 S. 52nd
- Walter Burkert: Wisdom and Science , Nuremberg 1962, p. 355.
- Macrobius, Commentarii in somnium Scipionis 2,1,8-13.
- Boethius, De institutione musica 1,10–11, translated by Anja Heilmann: Boethius' Musiktheorie und das Quadrivium , Göttingen 2007, pp. 342–345.
- For the scientific and epistemological background, see Anja Heilmann: Boethius' Musiktheorie und das Quadrivium , Göttingen 2007, pp. 205–218.
- Cassiodorus, Institutiones 2,5,1; see Iamblichos, De vita Pythagorica 121.
- Isidore, Etymologiae 3,16,1.
- Barbara Münxelhaus: Pythagoras musicus , Bonn - Bad Godesberg 1976, pp. 15-17.
- Hans Martin Klinkenberg : The decay of the quadrivium in the early Middle Ages. In: Josef Koch (ed.): Artes liberales. From ancient education to the science of the Middle Ages , Leiden 1976, pp. 1–32, here: 24 f.
- Carmina Cantabrigiensia , song 45 and Pythagoras sequence (first half of the 11th century); see Walther Kranz: Pythagoras in the Carmina Cantabrigiensia ( online ; PDF; 2.3 MB).
- Guido von Arezzo, Micrologus 20 ( German translation online ).
- Aldersbacher collective manuscript, Clm 2599, fol. 96 v. ,, Bavarian State Library, Munich
- Barbara Münxelhaus: Pythagoras Musicus , Bonn - Bad Godesberg 1976, pp 16, 76; Frank Hentschel: Sensuality and Reason in Medieval Music Theory , Stuttgart 2000, pp. 148–150 ( online ).
- Walter Salmen : Musical life in the 16th century , Leipzig 1976; black and white photography online .
- Werner Keil (Ed.): Basistexte Musikästhetik und Musiktheorie , Paderborn 2007, p. 56 (translation of Zarlino's text).
- Vincenzo Galilei: Discorso intorno all'opere di messer Gioseffo Zarlino. Florence 1589 ( online ; PDF; 347 kB).
- Daniel Meisner: Thesaurus philopoliticus , Frankfurt 1626, p. 327.
- Georg Muffat: Nova Cyclopeias Harmonica (1690).
- Rupert Ignaz Mayr: Pythagorean Schmids-Fuencklein (1692).
- Werner Keil (ed.): Basic texts music aesthetics and music theory , Paderborn 2007, p. 343.
- Werner Heisenberg: Thoughts of ancient natural philosophy in modern physics . In: Werner Heisenberg: Changes in the fundamentals of natural science , 8th edition, Stuttgart 1949, pp. 47–53, here: 50.
- Arnold Keyserling: History of the styles of thinking. 3. Logical thinking ( online ); Karl Sumereder: Music and Mathematics ( online ); Arnold Keyserling: The New Name of God. The world formula and its analogies in reality , Vienna 2002, p. 71 ( limited preview in the Google book search).
- Daniel Heller-Roazen: The fifth hammer. Pythagoras and the Disharmony of the World , Frankfurt am Main 2014, pp. 14–22 ( online ).