The connections between mathematics and music are many. For example, the differential equations of vibrating strings and surfaces help us understand harmonics and tuning systems, rhythm analysis tells us the ways a measure can be divided into beats, and the study of symmetry relates to the translations in time and pitch that occur in a fugue or canon.

This video explores a less well-known connection. It turns out that musical chords naturally inhabit various topological spaces, which show all the possible paths that a composer can use to move between chords. Surprisingly, the space of two-note chords is a *Möbius *strip, and the space of three-note chords is a kind of twisted triangular torus.

For a thorough presentation of the ideas introduced here, suitable for both mathematicians and musicians, see “A Geometry of Music” by Dmitri Tymoczko.

The “Umbilic Torus” sculpture shown at the end of the video was created by Helaman Ferguson.

Related:

More videos from the Mathematical Impressions series.

Bunch of nonsense. Mathermaticians trying to be music theorists but not realizing that 12tet is a tempered scale and in real life enharmonics MATTER. A G# is not an Ab etc so it does not make a moebius strip etc.

blah

Marcel is correct that there are many types of scales other than the well-tempered chromatic scale discussed here. The sentence near the beginning of the video about how we conventionally discretize the continuous pitch space into a scale with twelve well-tempered semitones was an attempt to make clear what model I am assuming within this rich domain. But the many other types of scales are beyond the scope of this short video.

It is not about the scale. It is about how we perceive music.

We perceive music according to a chain of fifths which is not closed in how we assign function / identity to intervals. It does not matter if we then play the music in a tuning with a closed circle of fifths (like 12 tone equal temperament) as we will still hear an Ab as a functionally different note than a G#, even if we tune them to the exact same pitch.

An augmented fourth is not a diminished fifth for instance, they have a completely different place in music and lead to different tones. Same for a diminished fourth vs major third, minor third vs augmented second, etc. All are made the same in thjs Moebius strip.

The Moebius strip idea may have an application for some forms of atonal or serial music, but for music as we hear every day and as it is taught in schools it does not fit.

I just realized my first comment was not in a very respectful manner. I apologize for this.

And it is an absolutely beautiful video.

I hope my comment can be helpful in some way.

My main point is that the Moebius strip does not follow conventional notation. It is therefor a “lossy” conversion. For instance, you cannot input a composition by Bach or Beethoven, and then convert back to the original score as the enharmonic information is lost.

And as I said before, this information is very important to how the music actually functions / how we perceive it.

And for study in harmony, couterpoint, composition and analysis all this information matters. Without it these fields would no longer work. There would have been no Beethoven or Mozart etc without theory that makes the enharmonic difference. All the great composers of the past studied this in depth and relied heavily on it.

As for the true structure of music, it is as I previously said based on a chain of octaves and perfect fifths. These fifths continue indefinitely in theory, though in practice music does not modulate very far. Something like Gbb to Ax will cover almost all common practice period music.

You can use octave equivalence as you did, and make the fold so C-E and E-C are the same, but the rolling to make a closed circle of 12 fifths is not correct.

However, it just occurred to me that for all practical purpose it will work if one uses 53 tone equal temperament. You can make a Moebius strip with 53tet as then the equivalences will be so remote that they do not matter in practice, and for instance a major third and a diminished fourth will be correctly displayed as different intervals.

Intervals which would become equivalent would be a triple diminished sixth and a quadruple augmented prime, not likely to ever be used in music or understood as such by our brain 😉

Btw, as one additional piece of information that may be helpful.

A tuning of a scale is not linked in a direct way to our perception of this scale. One can tune for instance a major triad to 1/4 comma meantone, or Pythagorean, or 12tet, and they will all be interpreted as a major triad by our brain. One can even play a diminished fourth in Pythagorean or meantone instead of the major third in that triad, and while the tuning will be different, if the diminished fourth is no prepared and resolved properly, on other words not properly indicated by the music to be a diminished fourth, then our brain will still interpret it as a major third. Therefore a lot of tuning differences merely give a different “color” to the timbres but are not functionally different. Our brain is very adaptive, and luckily so as otherwise very small tuning differences would result in completely different notes which would be very impractical for the performing musicians

Kind regards,

Marcel

Marcel, I understand your point. This video is an oversimplification of music and there are many things not taken into account which are actually very important in real life. Now, that said, this video tries to illustrate a mathematical model. It is supposed to be as simple as possible in order to be short and easy to understand. This is common to most (if not all) sciences. When you first learn the laws of physics, the teacher will tell you that an object falls with a given constant gravitational acceleration. Now, eventually you’ll learn that the shape of the object matters, as well as air resistance, the location on the earth and so on. For some things you need a model that is more precise and for others a simple model works fine. This musical approximation is not intended to insult musicians, it is not trying to show that music is something basic and simple and that things can be ignored. The video is just illustrating that there are connections between music and mathematics using a very simple example.

Marcel, the Mobius strip is a continuous space and you can embed *any* octave repeating scale inside it. This is discussed extensively in *A Geometry of Music*, chapter 4. In particular, you can embed the 19-tone equal tempered scale, in which there are enharmonic distinctions are available. (I also address enharmonic distinctions in terms of scalar embeddings; see section 4.5 in GOM.) This is all well-known stuff, and if you’re interested you can read about it.

The bit about “lossy representation” is also well known — this space eliminates octave and order information so it is *intentionally* lossy; the goal is to represent certain (but not all) relationships perspicuously. Don’t let your desire for (unattainable) theoretical perfection become the enemy of the good.

Can you please say where the musical phrase comes from that is played at the beginning and end of the video?

Reading the name Hart together with music and Möbius strip I wondered if you were inspired by this:

https://www.youtube.com/watch?v=WkmPDOq2WfA

rob

To Michael’s question: I wrote the musical example because I wanted a short two-voice tune illustrating different parallel motions and a contrary motion. For the ending, after the mention of 3-note chords, I could include a bass line and when played on the organ like that, I think of it as “Hymn to Geometry.”

To Rob’s: The apple doesn’t fall far from the tree; we both enjoy math and music and inspire each other with cool ideas.

the shape of the sculpture at the end reminds me of the cochlea. It makes sense that the cochlea is shaped to make sense of chords by firing of cilial nerves located at specific locations on this structure.

This is a very cool representation of classes of two-note pitch sequences. In a similar vein, Dmitri Tymoczko has written about a similarly compelling geometric representation of three-note pitch sequences under additional equivalence relations. Dmitri was instrumental in constructing an interactive musical sculpture, in the shape of an orbifold, on display now at the National Museum of Mathematics. Read some of Dmitri’s work here http://www.sciencemag.org/content/320/5874/346.abstract?ijkey=oKFBI.P9CHzfI&keytype=ref&siteid=sci

So why aren’t the enharmonic notes on here? They are, they just aren’t shown. This visualization is discretized for understanding, when in fact there an infinite number of note combinations that make up the mobius strip. To show them all would be to show an evenly coated mobius strip, with no significant points that the viewer can see to make reference of. Same with the 3-dimensional one with tri-notes. Every single point within it (3 dimensionally) is a different tri-note combination. In reality, this representation is lossless. It’s a ‘flaw’ in the visualization that is to blame for any confusion this may have caused.

Hi, I’m not a musician or mathmetician but am a quilt designer. This is a beautiful introduction of something wonderfully complex. Thank you Mr. Hart.

that’s pretty fascinating academically – i really like how it takes something that people normally see in a two-dimensional plane and, after an adjustment in thinking, is a simpler means of displaying relevant pitch information.

As someone who is going to analyze a piece of music, i could see this being useful, but i’m not sure where the practical is as a composer of today other than those that are dominantly serial composers in which pitch-class analysis is key. Although you can add more levels of depth to the notes to get more than just two pitches, the schematic, as you say, starts to get pretty gnarly and thus more impractical. it’s just as easy if not easier for me to reference notes on the page or a more standard pitch-class notation or harmonic notation for my material than trying to conceptualize my building materials in the context of a three-dimensional shape.

still, there something interesting there, a potential for creating and exploring correlations and connections in a new way.

Why don’t the lines you draw on the axes have a gradient of 1? Surely this is what should be drawn for the same interval, increasing its pitch.

He intented to explain the chromatic scale only in a continuous space of the M-strip…and the perception is beautifully geometrical!

This is a really cool video and concept. If I understand it correctly, the scale you choose should not make any difference in the shape of the mapping, just the granularity of it.

One thought that crossed my mind is that in a just tempered space, you can create a 3rd pitch as the sum of the two waveforms (tartini tone) which would cause a 3 tone chord, and would seemingly transform the mobius strip into a different shape. (the shape described in the statue at the end of the video.)

If you have thoughts on this, I would love to hear them.

Thanks for sharing!

Vi

Intriguing presentation of a topological representation of a musical space. Has the notion of a musical space been explored from a group theoretic perspective? This seems like an obvious line of inquiry, given well-known and well-understood cyclic groups and alternating groups, and their relations to various geometries and topologies; for instance, finite groups and their relations to regular solids. But I have little background in music, so do not know if there’s something I’m missing.

I’m afraid Marcel de Velde’s first comment is a bunch of nonsense – and no advance of any kind (whether in music or math or science or anything) would be possible if this kind of closed mind is the norm. All too often, this kind of mind is, indeed, the norm.

I observe that Mr de Velde has later tried to make some kind of amends for his initial remarks, but to me it appears that the mind is as closed as ever.

I strongly commend George Hart’s thoughts on relating maths to music – even if he hasn’t been completely successful according to musical ‘rigorists’ like Mr Marcel de Velde. Not to worry about such closed minds at all!

In his response, George Hart might have also pointed to some of his astonishing ‘mathematical sculptures’ (see http://www.georgehart.com/sculpture/sculpture.html) practically all of which are profoundly insightful.

GSC

Despite the comments listed here, I don’t think of Mozart or Bach as mathematicians. The examples using the Möbius strip and their musical works, just works. Their understanding of music tone scale and composition was beyond pure math.

http://youtu.be/xUHQ2ybTejU

It just works. The music they created behind is left for us to enjoy. No solving required

Bach had complimenting melodies throughout his compositions.

Mozart had an understanding of music arrangement, tones and composition most cannot comprehend. And he did it in one take.

Regardless of whether it makes mathematical

sense, the sounds themselves, the sum of the parts, add up to perfectly.

Hello,

if find the point with the “endless-rising-notes-illusion” very interesting but don’t really understand how to do it. Maybe someone could explain it differently or in some more detail.

I’m aware of Shepard, but this seems to be a completely different approach.

Would be great if someone could help!

Thanks,

Mono

Ok, beautiful, but what is this tool for? There’s nothing interesting for who know music theory. It’s all obvious, nothing new!

PS

Sorry for my horrible english

Thanks for the nice video !

in case anyone is interested:

https://en.wikipedia.org/wiki/Tonnetz

https://en.wikipedia.org/wiki/Torus#Configuration_space