Post B1Zbloo9vBShmf54S0 by johncarlosbaez@mathstodon.xyz
 (DIR) More posts by johncarlosbaez@mathstodon.xyz
 (DIR) Post #B1ZblmCVcC9nhJRyUK by johncarlosbaez@mathstodon.xyz
       2025-12-24T11:56:41Z
       
       0 likes, 1 repeats
       
       Merry Xmas Eve!   I'm not Christian, I'm a Mathematician, but I love Xmas, Ymas and Zmas.I'm giving a talk on the math of tuning systems and I thought I'd share it here as a kind of present to y'all.   I'll start with the basics today and do the more advanced part tomorrow.  But I hope even the basics contain a few twists that not everyone knows.(1/n)
       
 (DIR) Post #B1ZblnQ15SYXTUmHBI by johncarlosbaez@mathstodon.xyz
       2025-12-24T12:04:39Z
       
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       If you look at a piano keyboard you'll see groups of 2 black notes alternating with groups of 3.  So the pattern repeats after 5 black notes, but if you count you'll see there are also 7 white notes in this repetitive pattern.   So: the pattern repeats each 12 notes.Some people who never play the piano claim it would be easier if had all white keys, or simply white alternating with black.  But in fact the pattern makes it easier to keep track of where you are - and it's not arbitrary, it's musically significant.(2/n)
       
 (DIR) Post #B1Zbloo9vBShmf54S0 by johncarlosbaez@mathstodon.xyz
       2025-12-24T12:10:47Z
       
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       Starting at any note and going up 12 notes, we reach a note whose frequency is almost exactly double the one we started with.   Other spacings correspond to other frequency ratios.  I don't want to overwhelm you with numbers.  So I'm only showing you a few of the simplest and most important ratios.  These are really worth remembering.(3/ n)
       
 (DIR) Post #B1ZblpYx7CtO7myRO4 by johncarlosbaez@mathstodon.xyz
       2025-12-24T12:18:41Z
       
       1 likes, 0 repeats
       
       We give the notes letter names.  This goes back at least to Boethius, the Roman famous for writing The Consolations of Philosophy before he was tortured and killed at the order of Theodoric the Great.  Yeah, "Great".  He was a counselor to Theodoric, but he really would have done better to stay out of politics.  Theodoric may be the reason the lowest note on the piano is called A.   We now repeat the names of the white notes as shown in the picture: seven white notes A,B,C,D,E,F,G and then it repeats.[Whoop, Lisa is making me get up, make breakfast and go to the gym.  I'll continue this later.](4/n)