At a fundamental level, music may be described as a train of vibrations in the air. It can be further reduced to numbers — a string of binary digits (bits) that can be stored on a CD or sent around the world in a split second. But music carries enormous emotional content and can stir … Continue reading Music and Maths from Bach to Bacherach
Tag: Music
Bach and Euler chat in Frederick’s Court
Frederick the Great of Prussia, a devoted patron of the arts, had a particular interest in music, and admired the music of Johann Sebastian Bach. In 1747, Bach visited Potsdam, where his son Carl Philipp Emanuel was the Kapellmeister in Frederick’s court. When Frederick learned of this, he summoned ‘Old Bach’ to the palace and … Continue reading Bach and Euler chat in Frederick’s Court
Don’t be Phased by Waveform Distortions
For many years there has been an ongoing debate about the importance of phase changes in music. Some people claim that we cannot hear the effects of phase errors, others claim that we can. Who is right? The figure below shows a waveform of a perfect fifth, with components in the ratio $latex {3 : … Continue reading Don’t be Phased by Waveform Distortions
Johannes Kepler and the Song of the Earth
Johannes Kepler, German mathematician and astronomer, sought to explain the solar system in terms of divine harmony. His goal was to find a system of the world that was mathematically correct and harmonically pleasing. His methodology was scientific in that his hypotheses were inspired by and confirmed by observations. However, his theological training and astrological … Continue reading Johannes Kepler and the Song of the Earth
Tom Lehrer: Comical Musical Mathematical Genius
Tom Lehrer, mathematician, singer, songwriter and satirist, was born in New York ninety years ago. He was active in public performance for about 25 years from 1945 to 1970. He is most renowned for his hilarious satirical songs, many of which he recorded and which are available today on YouTube [see TM147, or search for “thatsmaths” … Continue reading Tom Lehrer: Comical Musical Mathematical Genius
Euler’s “Degree of Agreeableness” for Musical Chords
The links between music and mathematics stretch back to Pythagoras and many leading mathematicians have studied the theory of music. Music and mathematics were pillars of the Quadrivium, the four-fold way that formed the basis of higher education for thousands of years. Music was a central theme for Johannes Kepler in his Harmonices Mundi – … Continue reading Euler’s “Degree of Agreeableness” for Musical Chords
Motifs: Molecules of Music
Motif: A short musical unit, usually just few notes, used again and again. A recurrent short phrase that is developed in the course of a composition. A motif in music is a small group of notes encapsulating an idea or theme. It often contains the essence of the composition. For example, the opening four notes … Continue reading Motifs: Molecules of Music
Doughnuts and Tonnetze
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. The … Continue reading Doughnuts and Tonnetze
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 … Continue reading Quadrivium: The Noble Fourfold Way
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 … Continue reading Beautiful Patterns in Maths and Music
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 … Continue reading Hearing Harmony, Seeing Symmetry
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 … Continue reading Peano Music
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 … Continue reading Entropy Piano Tuning
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 … Continue reading Temperamental Tuning
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