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).

The copper engraving Duynkirchen by Eberhard Kieser from the Thesaurus philopoliticus by Daniel Meisner (* 1585; † 1625) published in 1626.
On the left Pythagoras with an angle measure and on the right three blacksmiths with hammers at an anvil with five staves, on which the lettering "Guido" is shown.

The Latin and German texts of the images have the same content and read as follows:

Triplicibus percussa sonat varie ictibus incus.
Musica Pythagoras struit hinc fundamina princ(eps).

Der Amboß von drey Hämmern klingt, darauß dreyerley thon entspringt.
Pythagoras hie die Music findt, das hett kein Eselskopff gekönt.

In English:
The anvil rings with three hammers, and from it springs three sounds.
Pythagoras finds the music here, that no donkey's head can do.

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

Pythagoras of Samos (* 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 four integers 6, 8, 9 and 12 and their relations 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:

Intervall	Prime	Quarte	Quinte	Oktave
$f$	$f_{6}$	$f_{8}$	$f_{9}$	$f_{12}$
$f$	1200 Hz	1600 Hz	1800 Hz	2400 Hz
${\frac {f}{f_{6}}}$	${\frac {1}{1}}$	${\frac {4}{3}}$	${\frac {3}{2}}$	${\frac {2}{1}}$
${\frac {f}{f_{8}}}$	${\frac {3}{4}}$	${\frac {1}{1}}$	${\frac {9}{8}}$	${\frac {3}{2}}$
${\frac {f}{f_{9}}}$	${\frac {2}{3}}$	${\frac {8}{9}}$	${\frac {1}{1}}$	${\frac {4}{3}}$
${\frac {f}{f_{12}}}$	${\frac {1}{2}}$	${\frac {3}{4}}$	${\frac {2}{3}}$	${\frac {1}{1}}$

Example with the four Pythagorean tones c', f', g' and c" in notation with treble clef.

• Sound examples with the four Pythagorean tones c', f', g' and c" in pure Pythagorean tuning and all 15 combinations
• Variant A.
• Variant B.

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:

Tone name	Tone frequency $f$ in hertz	Frequency ratio to first Tone c
c'	261,6	${\frac {1}{1}}$
f'	349,2	${\frac {4}{3}}$
g'	392,0	${\frac {3}{2}}$
c"	523,3	${\frac {2}{1}}$

The Typus arithmeticae from the Margarita Philosophica of 1503 by the philosopher Gregor Reisch (* 1467; † 1525) with Boethius (left) and Pythagoras of Samos (right)

Guido of Arezzo (* c. 992; † 1050) instructs Bishop Theobald of Strasbourg († 1082) on the monochord. Vienna, Austrian National Library, 12th century.

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. Nicomachus of Gerasa 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 Boethius, 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 Guido of Arezzo (* 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.

Rod with length $l$ and cross-sectional area $A$.

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.

Vibrating string with length l and tension force F.

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.

Four Pythagorean chisels with the in harmonic length and mass and natural frequency ratios 12:9:8:6.

• Pitches of five metal bars of different lengths
• Basic tone with 12 length units.
• Fourth with 9 length units.
• Fifth with 8 length units.
• Octave with 6 length units.
• Tritone (exactly half an octave) with ${\sqrt {72}}\approx$ 8.485 units of length. The corresponding chisel is not shown in the picture above and has a length between the two chisels with 8 and 9 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.

Two parallel monochords on a common resonance box in Deutsches Museum in Munich.

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.

The four Pythagorean notes c - f - g - c' are depicted 'blue.

Hearing example of the final cadence in C-Major.

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

Gregorianian Chant

Illumination "Pythagoras in the Forge" to the work "De musica cum tonario" by Johannes Cotto (around 1100) from a collected manuscript of the Cistercian Abbey of Aldersbach. The third chapter describes the invention of music by Pythagoras. Among other things, the work also contains extensive instructions for the composition of monophonic Gregorian chant and the organum.

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 beginning of the antiphon Ad te levavi from the first Sunday in Advent after the Graduale Novum notated in square notation with a C clef. The four Pythagorean tones c - f - g - c' are represented in blue.

Audio sample of the beginning of the Introit Ad te levavi from the first Sunday of Advent after the Graduale Novum.

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:

Verse	Final tone	Number c	Number f	Number g	Number c'	Sum of Pythagorean	Number others	Sum all	Part of Pythagorean
1	f	1	3	8	2	14	6	20	70,0%
2A	g	0	4	8	10	22	10	32	68,8%
2B	f	0	2	4	11	17	12	29	58,6%
3A	g	0	2	13	3	18	17	35	51,4%

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.

The beginning of the antiphon Factus est repente from Pentecost Sunday according to the Graduale Romanum notated in square notation with a C clef. The Pythagorean tones g - c' - d' are represented in blue.

Audio sample of the beginning of the Communion Factus est repente from Pentecost Sunday after the Graduale Romanum.

High Baroque

Frontispiece of the first volume of the "Musurgia Universalis, sive Ars Magna Consoni et Dissoni" published in Rome by the jesuit Athanasius Kircher (* 1602; † 1680) in 1650. In the lower left corner there ist Pythagoras. Next to him on the right there are three smithies, who hit an anvil with their hammers.

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 Jean-Baptiste Lully (* 1632; † 1687).

The sounds caused by hammer blows were introduced in 1690 by the French-German organist and composer Georg Muffat (* 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:

Score of the last seven bars of the first variation Ad Malleorum Ictus Allusio of the composition Nova Cyclopeias Harmonica of Georg Muffat. The Pythagorean tones C, F and G are marked blue.

Audio sample of the last seven bars of the first variation Ad Malleorum Ictus Allusio of the composition Nova Cyclopeias Harmonica by Georg Muffat.

Title page of the seven orchestral suites "Pythagorische Schmids=Fuencklein" by Rupert Ignaz Mayr from 1692 with an illustration by the German painter Johann Andreas Wolff (* 1652; † 1716).

Canon with blue Markings at the four Pythagorean tones g'-c"-d"-g" on the title page of the seven orchestral suites "Pythagorische Schmids=Fuencklein" by Rupert Ignaz Mayr from 1692..

In 1692, the German violinist, composer and court conductor Rupert Ignaz Mayr (* 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.

Canon in G major on the title page of the seven orchestral suites "Pythagorische Schmids=Fuencklein" by Rupert Ignaz Mayr from 1692.

Four-part canon in G major on the title page of the seven orchestral suites "Pythagorische Schmids=Fuencklein" by Rupert Ignaz Mayr from 1692.

Late Baroque

The German composer Johann Sebastian Bach (* 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.

The theme of the Passacaglia in C minor (BWV 582) by Johann Sebastian Bach with the blue Pythagorean tones depicted..

Theme of the Passacaglia in C minor (BWV 582) by Johann Sebastian Bach.

Concluding remarks

Pythagoras in a forge with four stonemasons

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 Werner Heisenberg (* 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 Lorenz Weinrich (*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.