Music 1649A/B Lecture Notes - Pythagorean Theorem, Tetractys, Reinforcement
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September 11 Readings/Class Notes - Theory
Pythagoras: Music and Number
–People aren't sure if he actually ever lived
–Sometime in the 6th century B.C. - ca. 570 – ca. 490
–Founder of the Pythagorean Theorem
–Figurehead of a religious brotherhood more than a mathematical/scientific
group
The Pythagoreans (ca. 530 BCE)
–Samos – journeys across the sea to Croton in Italy – sets up a colony
Tetractys (1:2:3:4)
–First 4 numbers had special meaning to the pythagoreans
–1 + 2 + 3 + 4 = 10
–Tetractys: a triangular figure consisting of 10 points organized into 4 rows
corresponding to the first four positive integers.
–Tetractys of multiplication
–“All things are number” - particles are all numerically-based arrangements –
tetractys of addition explicitly arranges to a triangle.
–6 and 3 would be triangular numbers because dots are arranged in a row
Prayer to the Tetractys – showed the cult-like organization that was the
Pythagoreans
Pythagorean Harmonics
2:1 – simplest ratio of all – octave has priority over all the other intervals due to it
being a foundational interval
Diapason = octave
3:2
Diapente = fifth
shorter string pitch produces a fifth
- shorter string vibrates 3 times for every 2 times the longer string vibrates
4:3
Diatessaron = fourth
Consonance and Dissonance - Classifying Intervals
Consonance
Perfect consonances
–P1, P8, P5, (P4)
- unison is not a true interval
Imperfect consonances
M3, m3, M6, m6
Dissonance
Dissonant intervals
M2, m2, M7, m7, (P4), aug and dim intervals
•perfect 4 is a variable interval – consonant normally, but often dissonant in
certain cases.