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Pythagorean Hammer

The concept of Pythagorean hammers is a part of the lore surrounding the early mathematical and philosophical ideas attributed to Pythagoras and his followers, the Pythagoreans. This idea relates to the discovery of harmonic ratios and the relationship between mathematics and music.

Pythagorean Hammers and Harmonic Ratios:

1. Discovery of Harmonic Ratios:

   • According to ancient anecdotes, Pythagoras discovered the mathematical relationships between musical notes and the lengths of strings or the weights of hammers. One story suggests that he noticed different tones produced by blacksmiths' hammers and investigated the cause.
   • The hammers were producing different sounds when struck, and Pythagoras realized that the sounds could be related to the weights of the hammers. However, historical accuracy of this story is debatable, and it serves more as a myth illustrating Pythagorean interests in harmony and proportion.

2. Harmonic Ratios:

   • Ratios and Intervals: Pythagoras is credited with discovering that musical intervals can be expressed as simple ratios of whole numbers. For example:
     • An octave corresponds to a 2:1 ratio.
     • A fifth corresponds to a 3:2 ratio.
     • A fourth corresponds to a 4:3 ratio.
   • These ratios were found to correspond to the lengths of strings that produce harmonious sounds when plucked.

3. Pythagorean Tuning:

   • Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2, known as the perfect fifth.
   • This tuning system was used to tune musical instruments in ancient Greece and influenced Western musical theory.

Influence and Legacy:

1. Mathematics and Music:

   • The discovery of harmonic ratios was significant because it demonstrated a fundamental relationship between mathematics and the physical world, particularly in the realm of acoustics and music.
   • It showed that musical harmony could be described mathematically, laying the groundwork for the field of music theory.

2. Philosophical Implications:

   • For the Pythagoreans, the harmony found in music was a reflection of the order and harmony of the cosmos. They believed that numerical relationships governed the universe and that understanding these relationships could lead to a deeper understanding of reality.
   • The phrase "harmony of the spheres" encapsulates this idea, suggesting that the movements of celestial bodies create a form of music based on mathematical ratios.

3. Scientific Development:

   • The study of harmonic ratios influenced later developments in science, particularly in the fields of acoustics, physics, and mathematics. The exploration of wave frequencies, resonance, and vibrations can trace its roots back to these early observations.

Conclusion:

The story of Pythagorean hammers, whether factual or mythological, serves to illustrate the profound insight that Pythagoras and his followers had into the relationship between mathematics and the natural world. The discovery of harmonic ratios demonstrated that mathematical principles could explain the phenomena of music and sound, laying the foundation for the scientific study of acoustics and influencing the philosophical understanding of the cosmos. The legacy of Pythagorean thought continues to resonate in both the fields of mathematics and music theory today.

Sources

Weiss, Piero, and Richard Taruskin, eds. Music in the Western World: A History in Documents. 2nd ed. N.p.: Thomson Schirmer, 1984. 3. ISBN 9780534585990.

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. ISBN 9780933999510.

Christensen, Thomas, ed. The Cambridge history of Western music theory. Cambridge: Cambridge University Press, 2002. 143. ISBN 9780521623711.

Burkert, Walter (1972). Lore and Science in Ancient Pythagoreanism, p.375. ISBN 9780674539181. Cited in Christensen 2002, p.143.

Barker (2004). Andrew, ed. Greek musical writings (1st pbk. ed.). Cambridge: Cambridge University Press. p. 30. ISBN 978-0-521-61697-3.

Lucas N.H. Bunt; Phillip S. Jones; Jack D. Bedient (1988). The historical roots of elementary mathematics (Reprint ed.). New York: Dover Publications. p. 72. ISBN 978-0-486-25563-7.

Christian, James. Philosophy An Introduction to the Art of Wondering. Wadsworth Pub Co. p. 517. ISBN 978-1-111-29808-1.

Waterfield, transl. with commentary by Robin (2000). The first philosophers : the Presocratics and Sophists (1. publ. as an Oxford world's classics paperback ed.). Oxford: Oxford Univ. Press. p. 103. ISBN 978-0-19-282454-7.