The Mathematical Problem with Music, and How to Solve It

i always wanted someone to explain this to me, so clearly and eloquently. great work. thanks Yuval

Fun fact : In professional orchestras, musician change the tuning of each notes dynamically to get "just intonation" sounds, in particular they will slightly decrease a third and slightly increase a fifth.

Fun fact: violinists frequently use all three systems: pythagorean, just, and equal temperament. We tune the instrument's strings according to the perfect 3:2 pythagorean fifth and usually play single lines without accompaniment wholly in the pythagorean system, as the whole steps are wider and the half steps are narrower, giving melodic lines more direction. When playing chords, or playing with other instruments which also use pythagorean tuning (like other stringed instruments), we often will adjust certain notes to just intonation to avoid clashing. We try to avoid adjusting melodic notes this way, instead preferring to adjust only the harmony notes. When playing with equal temperament instruments like piano, if there are any long sustained notes where the intonation difference and resultant clash will be clearly noticeable, we occasionally adjust to equal temperament for just a moment to avoid this. Performance is an art of compromise!

I'm a piano tuner. I've read and seen many explanations on this subject. This is the best and most intuitive description of tunings and why I've seen yet. The math concepts here have made the separate tunings including equal temperment so much more understandable. Thank you so much!

Over the years, I've had several high school senior math students try to write their final math explorations on music intervals and tunings. This is exactly what I had in my mind that I wanted them to do. Sadly, they rarely came close. It's so much harder than it seems to make the math and musical terminology accessible to everyone. You do an absolutely fantastic job!

My teacher says that music is not an invention, music is a discovery. There is music in nature, in law of nature . We can understand it, see it, hear it with mathematics 🥰this is just melting my heart ridiculously🫶

In Indian classical music, we have something called "Shruti", which are 22 in number. 12 of them are picked as the main notes. In some prices, some shrutis are used, giving it a different mood. A musician can pick the notes as per his compositions and needs.

Outstanding. I was a music math/physics/electrical engineering student in school. This is the confluence of so much of my fields of interest. OUTSTANDING job sir. I salute you.

This is genuinely one of the best explanations of musical temperaments I have ever seen. Amazing!

Great video. Just one suggestion which I think would be really useful is to show the wave interference when the ratios don't quite work. A visualisation of the 'messy' waveforms really helps explain a lot as to why we hear the dissonance and feel it as so jarring.

Thank you for not just talking about these differences, but mostly for actually doing the math with us and showing the results. I hare read several books on the subject over the decades. I had a "flavor" of what they meant, but now I see and hear the differences. Some things can't be described by words. You need to just do them. You are a great teacher as well as a theoretician.

When I was 12 years old I tried to tune our piano. It took me weeks and I finally gave up. Thanks to your discussion I now understand why. Thank you so much. Hal

Fun fact: The beep that's used to censor swear words is exactly 1,000 Hz, the pitch played at 1:33. Great video btw

I played the slide trombone so I could make any note I wanted. When I play the piano, I have no choices. This was a great explanation, thanks!

FINALLY YT recommended me what I've been looking for for years! Great explanation! Thank you.

Incredibly clear and rigorous. I love how you've rigorously combined math with musical fundamentals. I had seen several videos on this topic and I never quite understood it. Now yes. Thank you very much!!

I was always waiting for a video that combined my favourite subjects of maths and music but never thought it would be done as spectacularly as this. Thank you!

I don't think I've seen such an elegant explanation of these concepts. I really enjoyed your explanation of the rationale of each tuning system, and their benefits and shortcomings. I often find people become too preferential, glossing over the problems of their favorite system to make it seem better. This was a lovely video to watch, thank you!

Amazingly clearly-presented! Thanks. This topic I’ve been into since 1977, and it’s gloriously intriguing! A couple minor “nits” need to be mentioned though: 11:50 — Equating a Temperament with a Tuning is a common mistake, but in fact not quite correct! Temperaments a subset of Tunings ; all Temperaments are Tunings but not all Tunings are Temperaments! A temperament is a scheme for adjusting pitches from their exact-integer-ratios. So, Pythagorean and Just Intonation are Tunings, but they are not Temperaments, because they use exact integer ratios. Equal-Temperaments, Meantone Temperaments, and Well-Temperaments are temperaments. They have deliberately and systematically adjusted their pitches away from exact whole-number ratios. 18:37 — Minor Historical nit: Meantone Temperaments were much more common in the mid-late Renaissance than in the Baroque, by which time Well-Temperaments began to take over (and persisted into the mid-late 1800s, BTW — longer than most people realize).

Wow. THANK you! I have this natural tendancy to struggle with things that I cannot understand the underlying reason for. You just completely opened music up for me, and I am already a musician. And, as an experimental musician w/ a tendancy towards the technical side... you just fed my imagination with enough ideas to try to play with for many YEARS to come! Liked/subscribed/bell'ed/commented, based solely on this one video. If the rest of your content is even only 1/5th (heh) as great as this, I will benefit.