Music and Mathematics - Mathematician & Concert Pianist Eugenia Cheng

Eugenia Cheng, a mathematician who also is a concert pianist, describes how a mathematical breakthrough enabled Johann Sebastian Bach to write “The Well-Tempered Clavier” (1722). Cheng is a visiting senior lecturer in mathematics at the University of Chicago.
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[@vonhoother]

There are a lot of opinions about temperament. To name a few: some musicologists say that before equal temperament took hold, composers deliberately exploited the weird sound of keys remote from the more-in-tune C, F, and G major--and that part of an authentic performance of early music is NOT adjusting your intonation to make it sound more "in tune" by modern standards, but to go ahead and let the pitches go slightly sharp and flat as they may. Harry Partch and other composers held that there is much expressive potential in just intonation (based on the natural harmonic series) and other systems, and that Western music's turn toward equal temperament was a mistake. In Java and Bali, scales are generated not by mathematics but by taste (we might argue that they are produced by taste everywhere, and then justified by mathematics in some cultures). There are templates by which two distinct agreed-upon scales (slendro and pelog, in Java) are produced, and any musician can distinguish one from the other--but there is no precise set of frequencies or frequency ratios to build those scales. Back in Europe, in Bach's time, there were many systems of temperament that produced scales somewhere between the crunchiness of mean-tone intonation and the mathematical uniformity of equal temperament. Finally, despite what Cheng says, what Bach used in "The Well-Tempered Clavier" was not equal temperament but Andreas Werckmeister's "correct temperament" (Werckmeister III). It's based on tempered fifths and produces almost but not quite equal temperament: each key is in tune, pretty much, but C# major sounds noticeably different from C major, and so on. Bach was aware of this and exploited it; the evidence is there in the work itself.

[@dfhwze]

I would say we are less in tune with equal temperamet than we were before. We have traded a bit of quality for a lot of flexibility.

[@VoicesofMusic]

Bach did not compose in equal temperament, he composed in well temperament, which is really not the same thing. In addition, the famous tuning competition shows that Bach preferred to tune by ear. Bach would have been aware of the tuning competition between Nicolaus Bach of Jena and the mathematician Neidhardt. To appreciate Bach, I recommend you try one of the many unequal temperaments from the time of Bach, or even generate your own mathematical tuning from the Bach squiggle. Several mathematicians have worked on it!

[@bluesfiddle]

Asked about his family, because I think they are one line to understanding his intentions: "Accounts of Bach's own tuning practice are few and inexact. The three most cited sources are Forkel, Bach's first biographer; Friedrich Wilhelm Marpurg, who received information from Bach's sons and pupils; and Johann Kirnberger, one of those pupils." Also, saw some reports on at the Bach Museum in Leipzig about current research into Bach's students and their subsequent work, that might suggest some of his motivation.

[@erinmcnally7764]

I adore her. She makes me hate math a smidgen less. Her enthusiasm is so contagious. I love her.

[@always-stay-positive5187]

I love her more :-). She has to compose some songs, if she hasn't yet. Just getting into music at 47, but my PhD in applied math helps me with the guitar a little bit. Just 4 months into it everyday after work, so cant tell what's next.

[@andreasihlen6244]

@@always-stay-positive5187 Good for you! Did you continue with the guitar? I have been playing guitar since 2003 and have more recently got quite into music theory and the underlying maths and geometry. Have you considered the 12 notes of an octave as a clock diagram? I find that helps many of the pieces of the puzzle fit together. For example, chords and scales make specific shapes which can be rotated to easily show the notes in different keys.

[@MitchBoucherComposer]

What a unique video! Very interesting.

[@ChoBee333]

Amazing!

[@Ludwig1954]

The standard work linkong Music to Categories is: The Topos of Music Author: Guerino Mazzola

[@davidespinosa1910]

Highly recommended: William Sethares, Tuning, Timbre, Spectrum, Scale. Music theory before Sethares is like Biology before the discovery of DNA -- lots of classification, but no real understanding.

[@beback_]

Her Category Theory lectures are fantastic.

[@pony8136]

Just looking at this video and the comments again, I feel it's important to emphasise that *nobody knows* which temperament(s) Bach had in mind--that includes me, and Eugenia Cheng, and all the other commenters. Here's a relevant quote from David Schulenberg (an expert on Bach's keyboard music): "Equal temperament is only one of various types of so-called well-tempered (wohltemperierte) tuning systems that Bach would have known. There survive no unambiguous accounts as to how Bach tuned his instruments, and it is entirely possible that he used different systems at different times and for different types of music. By selective citation of individual pieces or other documents, one can adduce internal "evidence" that Bach used one sort of tuning or another. But in the absence of sufficient documentation for Bach's preferences, such claims invariably turn out to be little more than an expression of personal preference".

[@bluesfiddle]

Bach hat "well" auf Englisch nicht gesagt aber "Wohl" auf Deutch.

[@voidisyinyangvoidisyinyang885]

Professor Cheng Eugenia is neglecting to point out the truth of noncommutative phase. Fields Medal math professor Alain Connes emphasizes that in fact the empirical truth of music is 2, 3, infinity from noncommutative phase since 2 does not go into 3. So the truth is that if C is 1 and 2 is C as the octave then G is 3 as the overtone Perfect Fifth. She gives the harmonic series but neglects to mention that 4/3, Perfect 4th is not part of the harmonic series since the root tonic 1 with octave 2 does not divide into 3 evenly - and so in fact C to F as Perfect 4th is called the Ghost Tonic since it changes the value of the root tonic. The noncommutative phase then is C to F as 2/3 subharmonic and this got covered up ever since Plato and Archytas creating the first symmetric math equation - Perfect Fifth plus Perfect Fourth = Octave. So Professor Cheng presents it as a problem that was "fixed" when in fact it's quite the opposite as math professor Luigi Borzacchini points out in his 2007 essay on incommensurability and music and the continuum. He says this secret music origin of Western math got covered up and so is "really astonishing" and "shocking" and is a "pre-established deep disharmony" that guides WEstern science. The truth is rather what Daoism teaches - as did Pythagorean philosophy of the Tetraktys and the three gunas of India - the noncommutative phase is an infinity spiral of fifths as complementary opposites of time-frequency energy due to noncommutative phase as the 5th dimension that is quantum nonlocal entanglement - or "yuan qi" energy - the hidden momentum of light as relativistic mass, as Louis de Broglie figured out with his Law of Phase Harmony.

[@bluesfiddle]

Put that on KZbin video, please, so we can understand it more completely...

[@voidisyinyangvoidisyinyang885]

@@bluesfiddle if u don't click on my channel then my uploads don't exist. pretty clear logic.

[@danielwoodwardcomposer2040]

This post should be re-titled the Mathematics of frequency, not Music.

[@brightx1729]

Superb mam

[@theultimatereductionist7592]

Are you the same mathematician I saw give a talk online applying math / game theory to social justice? I have so much to discuss about that!

[@Hegerridadorno]

I thought that until quite recently the general assumption was that Bach had equal temperament (or something very close to it) in mind, and that the current scholarly consensus was that we simply don't know what exact tuning Bach had in mind, but that (some approximation of) equal temperament was still a candidate. So it seems like a bit of a stretch to call this video "totally misinformed". Here are some relevant quotes: 1. "The function of The Well-tempered Clavier is, according to the title-page, to demonstrate the possibility of writing and playing in all 24 major and minor keys. The tuning of the instrument was a means to that end. The term "well tempered" does not in itself imply a specific tuning, any more than "clavier" implies a specific instrument. It means no more than a tuning in which it is possible to play tolerably in all keys. Much hangs on the word "tolerably" since a variety of tunings have been associated with that term." (from "Bach's Well-tempered Clavier: The 48 Preludes and Fugues", David Ledbetter). In fact, the entire second chapter of this book is a must-read for anyone interested in this topic. 2. "...there is no direct evidence of the exact kind of temperament that Bach himself used." (from www.juilliard.edu/journal/ongoing-quest-bachs-temperament)

[@weewilly2007]

Pyg Ma Lion - The fare lady

[@asherwade]

Σχcεεδinglγ lovely 💕 || in another one of your YT videos (Photo of you with violin) You are analyzing and demonstrating the frequency of the note [we call] “D” on an oscillator {and I notice that you have, `ahem, an oldish model iPad Pro {as I also δο}. I was wondering if you would share with your YT audience [re: ‘mε’] the different apps you have on your iPad used for Music analysis (I play piano, violin & viola). Would greatly appreciate it.

[@jasoncarrizales]

Cool! 👍 🇺🇸 🌈😎🤓🌈

[@markcerisano4268]

Notwithstanding all the negative comments, take heart that even a university professor in mathematics can learn something. There is a type of personality that speaks as though their knowledge is fact but don't be fooled. These people speak what they until they are shown that their knowledge is faulty. You negative posters have probably encouraged her to research the subject a bit more and advance her own understanding. Her next video will probably have all of the things you criticized corrected, if she is interested in knowledge and not just showcasing her university. To recap, Bach may have been a tuner as well and he may have found a way to tune a piano that produced intervals that were less dissonant than the previous tuning styles that favoured common keys. When he discovered this, he decided to write music in all 12 keys to test it out. On the original manuscript there is a curious diagram that many believe to be instructions on how to tune the piano so that no one key is unplayable. He called the book the well tempered clavier and well temperament means no one key is unplayable but all keys are not equally dissonant. So these musicians, not mathematicians, were searching, and found, a way to tune the piano so all the keys are acceptable. There is great debate that true equal temperament has destroyed music but I think that's a bit excessive. The vast majority of people don't care and couldn't tell the difference between equal temperament and a Victorian temperament for example. What the math did was allow programmers to code software to help people tune a piano, that's it.

[@DavidMcbrady]

wow wow wow wow !!

[@Bob-yl9pm]

No? It has to do with what sounds good!

[@user-xd2tr6xc3x]

Bach literally wrote this to oppose equal temperament not to support it what the hell

[@bluesfiddle]

He told you that? Amazing! Did his family know?

[@user-xd2tr6xc3x]

@@bluesfiddle Ok David

[@samspianos]

Crikey!

[@Bob-yl9pm]

Ok? could this happen to coincide with our amazing Cochlea? Look? just play your amazing music! It's not rocket science!

[@voidisyinyangvoidisyinyang885]

So the Octave is the Emptiness, Yang is the Perfect Fifth and Yin is the Perfect Fourth. Or Sattva is the Octave, Raja is the Perfect Fifth and Tamas is the Perfect Fourth. Western math shut off the truth of noncommutative phase found in nonwestern music as infinite time-frequency complementary opposites energy.

[@always-stay-positive5187]

One hot woman you got there as a professor.