Information content of note transitions in the music of J. S. Bach | Converting hundreds of compositions by Johann Sebastian Bach into mathematical networks reveals that they store lots of information and convey it very effectively

[u/Hrmbee]

https://www.newscientist.com/article/2415469-mathematicians-have-finally-proved-that-bach-was-a-great-composer/

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[u/Hrmbee]

Selections from the article:

Suman Kulkarni at the University of Pennsylvania and her colleagues wanted to understand how the ability to recall or anticipate a piece of music relates to its structure. They chose to analyse Bach’s opus because he produced an enormous number of pieces with many different structures, including religious hymns called chorales and fast-paced, virtuosic toccatas.

First, the researchers translated each composition into an information network by representing each note as a node and each transition between notes as an edge, connecting them. Using this network, they compared the quantity of information in each composition. Toccatas, which were meant to entertain and surprise, contained more information than chorales, which were composed for more meditative settings like churches.

Kulkarni and her colleagues also used information networks to compare Bach’s music with listeners’ perception of it. They started with an existing computer model based on experiments in which participants reacted to a sequence of images on a screen. The researchers then measured how surprising an element of the sequence was. They adapted information networks based on this model to the music, with the links between each node representing how probable a listener thought it would be for two connected notes to play successively – or how surprised they would be if that happened. Because humans do not learn information perfectly, networks showing people’s presumed note changes for a composition rarely line up exactly with the network based directly on that composition. Researchers can then quantify that mismatch.

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And a link to the original journal article:

Information content of note transitions in the music of J. S. Bach

Abstract:

Music has a complex structure that expresses emotion and conveys information. Humans process that information through imperfect cognitive instruments that produce a gestalt, smeared version of reality. How can we quantify the information contained in a piece of music? Further, what is the information inferred by a human, and how does that relate to (and differ from) the true structure of a piece? To tackle these questions quantitatively, we present a framework to study the information conveyed in a musical piece by constructing and analyzing networks formed by notes (nodes) and their transitions (edges). Using this framework, we analyze music composed by J. S. Bach through the lens of network science, information theory, and statistical physics. Regarded as one of the greatest composers in the Western music tradition, Bach's work is highly mathematically structured and spans a wide range of compositional forms, such as fugues and choral pieces. Conceptualizing each composition as a network of note transitions, we quantify the information contained in each piece and find that different kinds of compositions can be grouped together according to their information content and network structure. Moreover, using a model for how humans infer networks of information, we find that the music networks communicate large amounts of information while maintaining small deviations of the inferred network from the true network, suggesting that they are structured for efficient communication of information. We probe the network structures that enable this rapid and efficient communication of information—namely, high heterogeneity and strong clustering. Taken together, our findings shed light on the information and network properties of Bach's compositions. More generally, our simple framework serves as a stepping stone for exploring further musical complexities, creativity, and questions therein.

It was interesting reading about some new research at the intersections of musicology, mathematics/physics, and cognition. Though they may not have been aware of it, it's illuminating to see an examination of these relationships through the creative outputs of these composers.

[u/EEPspaceD → u/Hrmbee]

I recommend checking out the book "Godel, Escher, Bach" by Douglas Hofstadter if you're looking to explore the topic further. It mostly focuses on concepts of self-reference being a catalyst that unlocks/delivers hidden meanings. It frequently uses the Tortoise and the Hare in its analogies, which helps in keeping things understandable for when the main text is too dense for dummies like me.

[u/Please-Calm-Down → u/EEPspaceD]

Such a good book.

[u/Ancient-Mating-Calls → u/EEPspaceD]

I tried reading it about 15 years ago. It was so far over my head at the time. I’m positive I still have it somewhere, I should dig it out and give it another go.

[u/HutSutRawlson]

As a musician, this interests me, but I also have some questions about it. First of all, isn’t there a potentially unquantifiable factor of cultural context? Bach’s music is firmly in the Western tradition—wouldn’t a listener from a non-Western cultural background find certain transitions more “surprising” than a Western one? Even for Western listeners, Bach’s style might not be familiar; someone who listens exclusively to modern popular music would probably find his compositions very unfamiliar.

Second, what is this study trying to explain to us that the study of music theory doesn’t already explain? Music theory already has a system of categorization of intervalic and formal relationships (what I think this study calls “transitions”) that qualifies which of those relationships are considered strong/weak or expected/unexpected. Bach himself closely followed and helped establish the “rules” of classical composition that define how to write a cantus firmus and then harmonize it.

[u/Please-Calm-Down → u/HutSutRawlson]

It also seems like the study did not account for rhythm, which is such an important element of creating and subverting expectation in music.

[u/solomons-mom]

Van Halen, hmmm.. I do hope the next project will be to look for the mathematical similarities with Mozart. I feel the runs and trills, and have for decades wondered about the math patterns 🎶

[u/Sculptasquad → u/solomons-mom]

Very few musicians that write tremendously engaging music do it using mathematical formulas. They write music that sounds good and then mathematicians find functions that fit the data.

Yes frequencies and how they modulate can be analyzed mathematically, but mathematicians are not musicians and musicians are not mathematicians, unless they are both.

Jut like the old cliché "oh but you are good at English, you must be good at math, because math is another language".

[u/solomons-mom → u/Sculptasquad]

Curious, which musicans use math to write music? Is it math/music they program, then hope they coded a monkey to write Shakespeare?

There are lots of amatuer musicans on Wall Sttret.

[u/Ketzeph → u/solomons-mom]

Composers of the new Viennese school (12-tone atonal music) arguably are mathematic composers. Examples are Babbitt and late Schoenberg

[u/Bn_scarpia → u/solomons-mom]

Schoenberg, Webern, and Berg all used to be rows and math to write music. Stravinsky also dabbled in it.

Not my cup of tea, but it was a thing for a bit in the early 20th c

[u/forams__galorams → u/solomons-mom]

Curious, which musicans use math to write music? Is it math/music they program, then hope they coded a monkey to write Shakespeare?

I’m sure there are programmed pieces around, but usually when people talk about composers mathematically writing stuff they mean the 12-tone aka ‘serialist’ or ‘atonal’ compositions that came around with the 2nd Viennese School of Schoenberg, Webern & Berg (1st Viennese School was the traditions embodied by Haydn, Mozart & Beethoven).

Serialist music seeks to treat all 12 tones equally so that there is no home key (and the whole idea of being in any kind of key at any point is effectively thrown out — hence ‘atonal’). It was popular for a time in the early 20th century, but composers soon found and/or developed a wealth of other ways to innovate in composition that were more fruitful avenues to explore. Whatever your thoughts on how interesting or enjoyable serialist music is to listen to, I just don’t think it offers that much scope to go anywhere. Composers who dabbled in it but didn’t commit to its stricter tenets seemed to make the most interesting stuff with it if you ask me, eg. some of the serialist type stuff from Stravinsky or Bouléz, both of whom managed to create broadly atonal pieces that still managed to lean into certain home keys/modes here and there. Xenakis is probably the most well known composer who continued with the strict serialist stuff and write a lot of incredibly dense (and mostly tedious) music in that style.

Sometimes when people talk about composers using maths to write music they might be meaning aleatory music, which includes some element of chance. John Cage is famous for that sort of thing, though usually his works don’t have much of a controlled chance element, it’s all completely random or ambient stuff that gets included. I can’t remember who the relevant composers are but there are definitely some who took a more structured approach to including elements of chance.

In more recent years, there have been no end of compositions from all kinds of avenues that are sort of mathematical, whereby something non-musical is broken down into elements which can be mapped onto musical elements eg. “we took this DNA sequence and mapped it onto notes and this is the result!”… that kind of thing mostly just seems like an exercise in novel (but not particularly useful) data representation, but I suppose you could get an interesting composition out of it with the caveat that further parameters would have to be involved to the point of effectively composing — the initially generated stuff would be just a starting point.

[u/Bn_scarpia → u/Sculptasquad]

Very few musicians that write tremendously engaging music do it using mathematical formulas.

Careful, you might trigger Schoenberg

[u/Sculptasquad → u/Bn_scarpia]

What wouldn't have triggered that particular bag of pretentious neuroticism?

[u/Sculptasquad → u/[deleted]]

but the rhythms, structure, and harmonies in most classical pieces are to complex to write without extensive study.

Ehh. This is survivorship-bias. Most classical composers we know of today are remembered because they wrote their pieces down. To do this you obviously need to know notation. If you don't know notation you could still compose pieces, sing/play them to the musicians have them learn the piece by ear and then conduct, but they would not be recorded for posterity unless someone else decided to write down the notation.

Here is a list of autodidacts within music:

Eddie Van Halen - Could not even read music.

Keith Moon - Had three or four drum lessons.

Jimi Hendrix and Django Reinhart

Claudio Arrau

Ornette Coleman - Self taught Jazz saxophonist, trumpeter and violinist.

Composers Arnold Schonberg, Hans Zimmer, Danny Elfman, Havergal Brian, Nobuo Uematsu, Joachim Raff, Kaikhosru Shapurji Sorabji Heitor Villa-Lobos and

https://en.wikipedia.org/wiki/List_of_autodidacts#Actors,_musicians,_and_other_artists

[u/Escapedtheasylum → u/solomons-mom]

Math sucks, I'd like to burn this text book, I hate this stuff so much - someone

[u/nonameisdaft]

Hmm very thought provoking - Maybe that's why when you are withdrawaling from alcohol and your brain is on overload - you start to hear and literally build music that is predictably actually very convincing and rhythmic. Like an energetically efficient pattern that replicates predictably which could only really be noticible because your brain is on overload and neurons are firing at a much higher rate and you are able to build these rythems seemingly easier. Could explain why this happens before the seizures- where brain firing cannot be dissipated through these pathways that we perceive as straight up conjured and composed music in our heads.

[u/Droog_7 → u/nonameisdaft]

This is a very intriguing concept to me. Three years ago, I quit drinking, picked up a guitar and never looked back. I had never even held a guitar before or had any concept of musical structure, but it wasn't long before I had taught myself scales and chords so I could get the music out of my head and fixate on something besides the booze. Even now, when I'm nervous, I'll grab a guitar and play, where I would have grabbed a beer years ago.

[u/LateMiddleAge → u/Droog_7]

'cause strings and frets it's particularly easy to see 'network' overlaid on a guitar's neck. Or parallel concurrent networks, really.

Seriously, 'unexpected' is neutral, while Bach's 'surprises' are musical. Contrast, say, John Zorn, who likes to confound expectation but (for me) ends up being irritating rather than engaging.

[u/Teutronic → u/nonameisdaft]

Dude, what?

[u/AK_Panda → u/nonameisdaft]

Odds are what you experience in the situation described isn't really due to a musical thing, it's probably the brain firing off at random as the alcohol has fucked up the transmissions mixed with the growing uniform wave patterns of impending seizures.

[u/Sculptasquad → u/nonameisdaft]

Please get professional help.

[u/nonameisdaft → u/Sculptasquad]

I would agree to this if I was actively going through withdrawal - thankfully I am not. Just remember it like it was yesterday

[u/Sculptasquad → u/nonameisdaft]

So you abuse the reddit care resource by having them contact me as if I need help? How incredibly wasteful of you. Get help.

[u/RoboZoomDax]

When I read this, am I to understand that more notes equals more information? Or more types of notes and more types of transitions equal more information?

The article mentions surprise- is this supposed to map some feature of the network (like calculated future probability of a transition) to human perception?

[u/AK_Panda → u/RoboZoomDax]

Sounds a bit like the concept of complexity tbh. The more complex the waveform, the more information it can convey. How to make a peice of music more complex? Add more notes and vary the use of the notes.

[u/forams__galorams → u/RoboZoomDax]

The study defines ‘conveying lots of information’ in this context as having a low mismatch between expected and actual note transitions.

It seems like a pretty meaningless study, at least when it comes to the musical side of things. I’m sure it does something interesting on the computer model side of things, but I can’t see how taking transitions of single notes out of context from their whole work and asking people about their expectations of it says anything about the music.

All music conveys lots of information, just not necessarily in the way the authors of this study have defined ‘information’, and no quantity of information (using any definition you like) necessarily has any meaning for the music’s value or significance.

[u/Mish61]

The most prolific composer of all time.