HAPPY NEW YEAR to all my friends in UTC +0 Time Zone! 🥳
Thank you to everyone who tuned in to my #live#disco -themed show on #aNONradio earlier, esp. to all my DAMGUDCYBERCHATTER friendz on SDF COMMODE chat! 👏 If you missed the live show, pls check out the recording here 👇
Lmao, I started 2024 by seeing the opening cinematic of Doom 3 where all hell breaks loose. I hate this game (was scraping the barrel of things to do nye) but it ended up perfect. #newyear2024#videogames
I've been working so hard this weekend that even the pets are exhausted! I thought this was an interesting angle of Penguin. We feel blessed to be starting the New Year with her inside.
Dalek says reach for your dreams and/or something sparkly. Happy New Year from the Miao kitties and Miao cousins. May 2024 bring better things for all. 🧿
And so, with that beautiful smile that I love, with that optimism and enthusiasm, with that strength and courage and above all with a lot of love to share, I am ready for the adventures that await me this 2024.
Good morning/afternoon/evening, depending on where you are this New Year's Eve
Wherever you find yourself, I hope you spend it with the ones you love. Ring in the new year with those who helped you through the last one and who mean enough to you to carry into the next one!
You may have seen this Jedi math trick: take something that makes sense with real numbers and try it with complex numbers. Weird yet useful things happen.
For example it turns out that imaginary time is 1/temperature. Hawking used this to compute the temperature of black holes.
An easier example is that the exponential function applied to an imaginary number gives you the trig functions sine and cosine.
Indeed, if ω is any complex number, exp(iωt) is a function of time that oscillates at a frequency equal to the real part of ω, and decays exponentially at a rate equal to the imaginary part of ω. So we can think of ω as a complex frequency! Its real part is an ordinary frequency, while its imaginary part is a decay rate.
Thus, in music it makes sense to consider tuning systems where the frequency ratios are complex. I haven't yet found anything interesting to do with this thought. But it makes sense to have notes that oscillate but also decay.
Here's a dumb idea. Nobody knows Bach's original well-tempered scale. In 1977, Herbert Anton Kellner had a wacky suggestion: the beats in the major third (which is close to a frequency ratio of 5/4, but not quite) should have the same frequency as those of the perfect fifth (which is close to 3/2, but not quite).
This led him - the derivation is too long to fit in the margin of this post - to the 'Bach equation':
F⁴ + 2F - 8 = 0
where F is the frequency ratio of the perfect fifth. He got a solution
F ≈ 1.495953506
for the perfect fifth. But it also has a negative solution, and two complex solutions that aren't real. Do these mean anything?
Maybe you can come up with a better idea about complex tuning systems....
Inspired by @johncarlosbaez's post on complex frequencies, thinking about complex numbers:
If you are learning complex analysis, and want to really understand branch cuts, and live in the US, I recommend moving to east Asia.
No, really: I lived in South Korea for a few years, working at KAIST, and having to deal with the international date line -- the branch cut for the Earth -- forced me to understand that little bit of complex analysis.
This is especially relevant today, December 31st, because as I write this in the morning (US central time), it's already tomorrow -- January 1st, 2024 (or, apropos of my recent post, January 1st, 45² - 1) in parts of Australia. Happy #newyear2024.