Posts Tagged 'Music'

Quadrivium: The Noble Fourfold Way

According to Plato, a core of mathematical knowledge – later known as the Quadrivium – was essential for an understanding of the Universe. The curriculum was outlined in Plato’s Republic. The name Quadrivium means four ways, but this term was not used until the time of Boethius in the 6th century AD [see TM119 or search for “thatsmaths” at irishtimes.com].

Quadrivium-Book

Image from here.

It is said that an inscription over the entrance to Plato’s Academy read “Let None But Geometers Enter Here”. This indicated that the Quadrivium was a prerequisite for the study of philosophy in ancient Greece.

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Beautiful Patterns in Maths and Music

The numerous connections between mathematics and music have long intrigued practitioners of both. For centuries scholars and musicians have used maths to analyze music and also to create it. Many of the great composers had a deep understanding of the mathematical principles underlying music. Johann Sebastian Bach was the grand master of structural innovation and invention in music. While his compositions are the free creations of a genius, they have a fundamentally mathematical basis [See TM116 or search for “thatsmaths” at irishtimes.com].

J-S-Bach

Johann Sebastian Bach, the grand master of structural innovation and invention in music.

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Hearing Harmony, Seeing Symmetry

Musical notes that are simply related to each other have a pleasing effect when sounded together. Each tone has a characteristic rate of oscillation, or frequency. For example, Middle C on the piano oscillates 264 times per second or has a frequency of 264 Hz (Hertz). If the frequencies of two notes have a ratio of two small whole numbers, the notes are harmonically related and sound pleasant when played together.

Lissajous-Interval-16-17-SemiTone

Beats from two notes close in pitch.

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Peano Music

The links between mathematics and music are manifold. Mathematics can be set to music in a simple but surprising manner. For the award ceremony of the Gödel Medal in 2014, a musical interpretation of Gödel’s incompleteness Theorems was written by Danish composer Niels Marthinsen. It encodes the basic axioms of number theory that form the focus of Gödel’s Theorems.

Peano-Axioms

The Peano Axioms in symbolic form.

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Entropy Piano Tuning

An ingenious method of tuning pianos, based on the concept of entropy, has recently been devised by Haye Hinrichsen of Würzburg University. Entropy, which first appeared in the mid-nineteenth century in thermodynamics and later in statistical mechanics, is a measure of disorder. Around 1948 Claude Shannon developed a mathematical theory of communications and used entropy as an indicator of information content [TM084, or search for “thatsmaths” at irishtimes.com].

EPT-StretchCurve.jpg

Tuning curve showing the stretch for high and low notes (Image: Wikimedia Commons).

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Temperamental Tuning

Every pure musical tone has a frequency, the number of oscillations per second in the sound wave. Doubling the frequency corresponds to moving up one octave. A musical note consists of a base frequency or pitch, called the fundamental together with a series of harmonics, or oscillations whose frequencies are whole-number multiples of the fundamental frequency.

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