There is a serious mathematical problem with the tuning of certain musical instruments, a problem that even luminaries such as Galileo, Newton, and Euler tried to solve. This video is about this problem and about some of the ways to tackle it. It starts from the basic physics of sound, proves mathematically why some musical instruments can never be perfectly in tune, and then introduces the main solutions that were proposed to solve this problem, along with their upsides and downsides: Pythagorean tuning, Just intonation, the Meantone temperament, and finally – the equal temperament, which is the tuning system almost everybody uses today.
— Further Resources —
▶ To learn more about the connections between music and mathematics, I highly recommend the book “Music – A Mathematical Offering” by David Benson. This terrific book is a treasure trove of information, extremely well written, and its thorough discussion of temperaments is just one of the many topics it covers. The book can be downloaded legally and for free from here: https://homepages.abdn.ac.uk/d.j.bens…
▶ Sevish is a master of electronic microtonal music. His compositions, despite their ominous genre, sound light and fun. Check him out. https://sevish.com
▶ Paul Davis explaining how and why John Frusciante (of the Red Hot Chili Peppers) “mistuned” his guitar in the song “Scar Tissue”. https://youtu.be/Daw93bRHe4Y
▶ A Madrigal by Nicola Vicentino (1555), played on a 24-tone harpsichord tuned in meantone temperament, by Johannes Keller. https://youtu.be/0akGtDPVRxk
▶ A concise introduction to Arabic music. Pay attention especially to the Albayati, Alsaba, Alsard, and Ahuzaam maqams, with their intense microtonality. https://youtu.be/Rk8mY1cxKNI
▶ The Lumatone Isomorphic Keyboard is a cool interface to microtonal music. https://youtu.be/jyvQnAtlnek
— Thanks to —
▶ Yehezkel Raz, the Ableton wizard, for transforming me from a complete Ableton noob to a good-enough user in less than two hours. https://yehezkelraz.com
▶ Dina Lurie, a dear neighbor and a great violinist (in the Irish fiddle tradition!), for contributing two notes and one double glissando. https://www.facebook.com/dina.lurie
▶ Alon Schab, a musicologist, and also a friend, bandmate, and academic colleague, for advice on some musical and historical issues. https://haifa.academia.edu/AlonSchab
▶ The intro and outro music is “Snowfall Butterflies” by Asher Fulero (via YouTube Audio Library). https://asherfulero.com
▶ The flute and Qanun sound samples are from FreeSound. https://freesound.org
▶ Photo of the Antegnati Organ in Santa Barbara, Mantua (1565), courtesy of the organist Simon Lloyd. https://simon-lloyd.com
— Contents —
00:00 – Intro
00:44 – What is sound?
02:42 – Melodies
04:48 – Intervals
07:00 – Choosing frequencies
11:56 – Pythagorean Tuning
14:33 – Just Intonation
18:36 – Meantone Temperament
24:34 – Equal Temperament
29:50 – Other temperaments
31:09 – Outro
The sound samples were prepared with Ableton Live 11. To make them piano-like but still as accurate as possible, I used the physical-modelling Pianoteq plugin, with the unison width set to 0, octave stretching ratio set to 1, string length set to its maximum value (to minimize string inharmonicity), hammer noise set to 0.5, pedal noise set to 0, and the velocity-to-dynamics curve considerably lowered (ending at mezzo-piano).
Created for the 2022 “Summer of Math Exposition” (SoME2) competition, hosted by the one and only 3Blue1Brown (Grant Sanderson).