The circle of fifths is a remarkably useful diagram for the analysis of music. It shows the twelve notes of the chromatic scale arranged in a circle, with notes that are harmonically related (like C and G) being close together and notes that are discordant (like C and C♯) more distant from each other.

## Posts Tagged 'Music'

### Quadrivium: The Noble Fourfold Way

Published July 20, 2017 Irish Times Leave a CommentTags: Arithmetic, Astronomy, Geometry, History, Music

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 6^{th} century AD [see TM119 or search for “thatsmaths” at irishtimes.com].

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.

### Beautiful Patterns in Maths and Music

Published June 1, 2017 Irish Times Leave a CommentTags: 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].

### Hearing Harmony, Seeing Symmetry

Published May 11, 2017 Occasional Leave a CommentTags: Geometry, Music

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.

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.

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].

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.