According to legend, Pythagoras discovered the foundations of musical tuning by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously. According to Nicomachus in his 2nd-century CE Enchiridion harmonices,Weiss, Piero, and Richard Taruskin, eds. Music in the Western World: A History in Documents. 2nd ed. N.p.: Thomson Schirmer, 1984. 3. . Pythagoras noticed that hammer A produced consonance with hammer B when they were struck together, and hammer C produced consonance with hammer A, but hammers B and C produced dissonance with each other. Hammer D produced such perfect consonance with hammer A that they seemed to be "singing" the same note. Pythagoras rushed into the blacksmith shop to discover why, and found that the explanation was in the weight ratios. The hammers weighed 12, 9, 8, and 6 pounds respectively. Hammers A and D were in a ratio of 2:1, which is the ratio of the octave. Hammers B and C weighed 8 and 9 pounds. Their ratios with hammer D were (12:8 = 3:2 = perfect fifth) and (12:9 = 4:3 = perfect fourth). The space between B and C is a ratio of 9:8, which is equal to the musical whole tone, or whole step interval ().
Hammer Weight Interval Chisel In C A 6 C B 8 G C 9 F D 12 C'
The legend is, at least with respect to the hammers, demonstrably false. It is probably a Middle Eastern folk tale.Kenneth Sylvan Guthrie, David R. Fideler (1987). The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings which Relate to Pythagoras and Pythagorean Philosophy, p.24. Red Wheel/Weiser. . These proportions are indeed relevant to string length (e.g. that of a monochord) — using these founding intervals, it is possible to construct the chromatic scale and the basic seven-tone diatonic scale used in modern music, and Pythagoras might well have been influential in the discovery of these proportions (hence, sometimes referred to as Pythagorean tuning) — but the proportions do not have the same relationship to hammer weight and the tones produced by them.Christensen, Thomas, ed. The Cambridge history of Western music theory. Cambridge: Cambridge University Press, 2002. 143. .Burkert, Walter (1972). Lore and Science in Ancient Pythagoreanism, p.375. . Cited in Christensen 2002, p.143. However, hammer-driven chisels with equal cross-section, show an exact proportion between length or weight and Eigenfrequency.Markus Bautsch: Über die pythagoreischen Wurzeln der gregorianischen Modi, English: About the Pythagorean roots of the Gregorian modes, Mater Dolorosa (Berlin-Lankwitz), March 2012, retrieved on 31 August 2017
Earlier sources mention Pythagoras' interest in harmony and ratio. Xenocrates (4th century BCE), while not as far as we know mentioning the blacksmith story, described Pythagoras' interest in general terms: "Pythagoras discovered also that the intervals in music do not come into being apart from number; for they are an interrelation of quantity with quantity. So he set out to investigate under what conditions concordant intervals come about, and discordant ones, and everything well-attuned and ill-tuned." Whatever the details of the discovery of the relationship between music and ratio, it is regarded as historically the first empirically secure mathematical description of a physical fact. As such, it is symbolic of, and perhaps leads to, the Pythagorean conception of mathematics as nature's modus operandi. As Aristotle was later to write, "the Pythagoreans construct the whole universe out of numbers". The Micrologus of Guido of Arezzo repeats the legend in Chapter XX.
Contents of the legend According to the oldest recorded versionNikomachos von Gerasa, Handbook of Harmonics 6, translated by Anja Heilmann: Boethius' Music Theory and the Quadrivium, Göttingen 2007, pp. 345–347, cited literally in Iamblichos von Chalkis, On Pythagorean Life 115–121, translated by Michael von Albrecht: Iamblichus. Pythagoras: Legend – Doctrine – Way of Life, Darmstadt 2002, pp. 109–113. of the legend, Pythagoras, who lived in the 6th century BC, sought a tool to measure acoustic perceptions, similar to how geometric quantities are measured with a compass or weights with a scale. As he passed by a forge where four (according to a later version, five) craftsmen were working with hammers, he noticed that each strike produced tones of different pitch, which resulted in harmonies when paired. He was able to distinguish Octave, fifth, and fourth. Only one pair, which formed the interval between fourth and fifth (a major second), he perceived as dissonant. Excitedly, he ran into the forge to conduct experiments. There, he discovered that the difference in pitch was not dependent on the shape of the hammer, the position of the struck iron, or the force of the blow. Rather, he could associate the pitches with the weights of the hammers, which he measured precisely. He then returned home to continue the experiments.
He hung four equally long, equally strong, and equally twisted strings in succession on a peg attached diagonally to the corner of the walls, weighting them differently by attaching different weights at the bottom. Then he struck the strings in pairs, and the same harmonies resonated as in the forge. The string with the heaviest load of twelve units, when paired with the least burdened string carrying six units, produced an octave. Thus, it was evident that the octave was based on the ratio 12:6, or 2:1. The most tense string yielded a fifth with the second loosest string (eight units), and a fourth with the second tightest string (nine units). From this, it followed that the fifth was based on the ratio 12:8, or 3:2, and the fourth on the ratio 12:9, or 4:3. Again, the ratio of the second tightest string to the loosest, with 9:6, or 3:2, yielded a fifth, and the ratio of the second loosest to the loosest, with 8:6, or 4:3, yielded a fourth. For the dissonant interval between fifth and fourth, it was revealed that it was based on the ratio 9:8, which coincided with the weight measurements carried out in the forge. The octave proved to be the product of the fifth and fourth:
Pythagoras then extended the experiment to various instruments, experimented with vessels, flutes, triangles, the Monochord, etc., always finding the same numerical ratios. Finally, he introduced the commonly used terminology for relative pitch.
Further traditions With the invention of the monochord to investigate and demonstrate the harmonies of pairs of strings with different integer length ratios, Pythagoras is said to have introduced a convenient means of illustrating the mathematical foundation of music theory that he discovered. The monochord, called κανών (kanōn) in Greek and regula in Latin, is a resonating box with a string stretched over it. A measurement scale is attached to the box. The device is equipped with a movable bridge, which allows the vibrating length of the string to be divided; the division can be precisely determined using the measurement scale. This enables measurement of intervals. Despite the name "monochord", which means "one-stringed", there were also multi-stringed monochords that could produce simultaneous intervals. However, it is unclear when the monochord was invented. Walter Burkert dates this achievement to a time after the era of Aristotle, who apparently did not know the device; thus, it was introduced long after Pythagoras' death.Walter Burkert: Weisheit und Wissenschaft, Nuremberg 1962, p. 353 and note 28. On the other hand, Leonid Zhmud suggests that Pythagoras probably conducted his experiment, which led to the discovery of numerical ratios, using the monochord.Leonid Zhmud: Wissenschaft, Philosophie und Religion im frühen Pythagoreismus, Berlin 1997, pp. 193–196; see also Barbara Münxelhaus: Pythagoras musicus, Bonn–Bad Godesberg 1976, pp. 28 f.
Hippasus of Metapontum, an early Pythagorean (late 6th and early 5th centuries BCE), conducted quantitative investigations into musical intervals. The experiment attributed to Hippasus, involving freely oscillating circular plates of varying thicknesses, is physically correct, unlike the alleged experiments of Pythagoras. It is unclear whether Archytas of Tarentum, an important Pythagorean of the 5th/4th centuries BCE, conducted relevant experiments. He was probably more of a theoretician than a practitioner in music, but he referred to the acoustic observations of his predecessors. The musical examples he cites in support of his acoustic theory involve wind instruments; he does not mention experiments with stringed instruments or individual strings. Archytas proceeded from the mistaken hypothesis that pitch depends on the speed of sound propagation and the force of impact on the sound-producing body; in reality, the speed of sound is constant in a given medium, and the force only affects the volume.Walter Burkert: Weisheit und Wissenschaft, Nuremberg 1962, pp. 362–364; Leonid Zhmud: Wissenschaft, Philosophie und Religion im frühen Pythagoreismus, Berlin 1997, pp. 196–199. Skepticism regarding Archytas' acoustic experiments is expressed by Carl A. Huffman: Archytas of Tarentum, Cambridge 2005, pp. 129–148; see also pp. 473–475. He points out that Archytas mainly relies on information from his predecessors and everyday experience.
Interpretation of the legend Walter Burkert is of the opinion that despite its physical impossibility, the legend should not be regarded as an arbitrary invention, but rather as having a meaning that can be found in Greek mythology. The Idaean Dactyls, the mythical inventors of blacksmithing, were also, according to myth, the inventors of music. Thus, there already existed a very ancient tradition associating blacksmithing with music, in which the mythical blacksmiths were depicted as possessors of the secret of magical music. Burkert sees the legend of Pythagoras in the blacksmiths as a late transformation and rationalization of the ancient Dactyl myth: In the legend of Pythagoras, the blacksmiths no longer appear as possessors of ancient magical knowledge, but rather, without intending to, they become - albeit unknowing - "teachers" of Pythagoras.Walter Burkert: Weisheit und Wissenschaft, Nuremberg 1962, p. 355.
In the Early Middle Ages, Isidore of Seville referred to the biblical blacksmith Tubal as the inventor of music; later authors followed him in this. This tradition once again shows the idea of a relationship between blacksmithing and music, which also appears in non-European myths and legends.Walter Burkert: Weisheit und Wissenschaft, Nuremberg 1962, p. 355; Barbara Münxelhaus: Pythagoras musicus, Bonn–Bad Godesberg 1976, pp. 38 f., 46. Tubal was the half-brother of Jubal, who was considered the ancestor of all musicians. Both were sons of Lamech and thus grandsons of Cain. In some Christian traditions of the Middle Ages, Jubal, who observed his brother Tubal, was equated with Pythagoras.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).