# 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.