3 ms·

Music and Measure Theory (2015) [video]

56 points by ogogmad 6y ago

dia80 6y ago
It's a fun personal discovery in mathematics when you realise there are different degrees of infinites.
- daviddaviddavid 6y ago
  Totally agree! When I was in school, I took a logic class that used Boolos & Jeffrey's "Computability and Logic", which is a really great book. Chapter 2 is a very clear presentation of Cantor's diagonalization proof that there exist uncountable sets. It blew my mind.
  - deleted 6y ago
    [deleted]
- ogogmad 6y ago
  The fact that all countable subsets of the real numbers have Lebesgue measure zero (as proved in the video) implies that the real numbers are uncountable. Otherwise the real numbers would have Lebesgue measure zero, which is absurd. This approach to proving R uncountable is carried out more formally here (proof A.3): https://link.springer.com/content/pdf/bbm%3A978-1-4614-8854-5%2F1.pdf https://link.springer.com/content/pdf/bbm%3A978-1-4614-8854-...
  - shitgoose 6y ago
    Speaking of absurd. I remember when I found out that Cantor set (uncountable) has measure 0, my mind was blown. No wonder Cantor ended up in an institution (as I heard).
Kye 6y ago
This is one of the few people who connects math and music in a way that's both useful and interesting. I followed this channel back when I found the video on Fourier transforms[1], but never looked at other videos. I've never really thought about the math of consonance and dissonance, but I use it all the time without realizing it. For example: a hypersaw. That's 7 saws slightly detuned played at the same time. The more you detune them, the worse it sounds. If you own Serum, you can hear this in realtime if you make a 7 voice saw and play with the detune knob. You can use an LFO to modulate this and get a sound that's a little uneasy while still somehow being harmonious. It's perfect for dark/spooky music. [1] https://www.youtube.com/watch?v=spUNpyF58BY https://www.youtube.com/watch?v=spUNpyF58BY
- nicetryguy 6y ago
  A little detuning goes a long way! It gives it that full warbly phase sound. 5-10 cents down is usually my sweet spot for synth plugins. I even detune one string on each key on my piano about 5-10 cents down, it gives it some life! Natural phasing sounds awesome.
diiq 6y ago
I love 3B1B, but it's a little disingenuous to attribute small integer ratios being are consonant to our brains/taste because of the simple polyrhythms. Our brains never receive the waveforms in that way; the ear performs a mechanical fourier transform, first. Small integer ratios mean many harmonics overlap, so the frequency-space signal from the ears is also relatively simple. The simpler the picture in freq-space, the more consonant (not necessarily more beautiful or pleasant, but specifically consonant) Unfortunately, that reality is a very poor setup for the math he wants to demonstrate -- he needs to imagine someone for whom all ratios are consonant to motivate the measure theory question. So I'm not sure music was the best road to the goal.

56 points by ogogmad 6y ago

dia80 6y ago

It's a fun personal discovery in mathematics when you realise there are different degrees of infinites.

daviddaviddavid 6y ago
Totally agree! When I was in school, I took a logic class that used Boolos & Jeffrey's "Computability and Logic", which is a really great book. Chapter 2 is a very clear presentation of Cantor's diagonalization proof that there exist uncountable sets. It blew my mind.
- deleted 6y ago
  [deleted]
ogogmad 6y ago
The fact that all countable subsets of the real numbers have Lebesgue measure zero (as proved in the video) implies that the real numbers are uncountable. Otherwise the real numbers would have Lebesgue measure zero, which is absurd. This approach to proving R uncountable is carried out more formally here (proof A.3): https://link.springer.com/content/pdf/bbm%3A978-1-4614-8854-5%2F1.pdf https://link.springer.com/content/pdf/bbm%3A978-1-4614-8854-...
- shitgoose 6y ago
  Speaking of absurd. I remember when I found out that Cantor set (uncountable) has measure 0, my mind was blown. No wonder Cantor ended up in an institution (as I heard).

Kye 6y ago

This is one of the few people who connects math and music in a way that's both useful and interesting. I followed this channel back when I found the video on Fourier transforms[1], but never looked at other videos. I've never really thought about the math of consonance and dissonance, but I use it all the time without realizing it. For example: a hypersaw. That's 7 saws slightly detuned played at the same time. The more you detune them, the worse it sounds. If you own Serum, you can hear this in realtime if you make a 7 voice saw and play with the detune knob. You can use an LFO to modulate this and get a sound that's a little uneasy while still somehow being harmonious. It's perfect for dark/spooky music. [1] https://www.youtube.com/watch?v=spUNpyF58BY https://www.youtube.com/watch?v=spUNpyF58BY

nicetryguy 6y ago
A little detuning goes a long way! It gives it that full warbly phase sound. 5-10 cents down is usually my sweet spot for synth plugins. I even detune one string on each key on my piano about 5-10 cents down, it gives it some life! Natural phasing sounds awesome.

diiq 6y ago

I love 3B1B, but it's a little disingenuous to attribute small integer ratios being are consonant to our brains/taste because of the simple polyrhythms. Our brains never receive the waveforms in that way; the ear performs a mechanical fourier transform, first. Small integer ratios mean many harmonics overlap, so the frequency-space signal from the ears is also relatively simple. The simpler the picture in freq-space, the more consonant (not necessarily more beautiful or pleasant, but specifically consonant) Unfortunately, that reality is a very poor setup for the math he wants to demonstrate -- he needs to imagine someone for whom all ratios are consonant to motivate the measure theory question. So I'm not sure music was the best road to the goal.