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4ef8245d | 1 years ago (raw) | root | parent | reply | flag 
 @b4c50e1b This was beautifully explained - thank you for sharing!
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John Carlos Baez | 1 years ago (raw) | root | parent | reply | flag 
 @2c8b8664 - thanks! The subject is inherently beautiful, but a lot of explanations manage to bury it in clutter and jargon.
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ariels | 1 years ago (raw) | root | parent | reply | flag 
 @b4c50e1b @2c8b8664 Thanks for the explanation!  It's beautiful even before the amazing diagram.  Something confuses me about the diagram: it would make more sense to my eyes if the labels "64/729" and "729/64" swapped places.  All arrows except the two arrows to and from these nodes multiply by 3/2. The arrow from 729/64 to 32/243 multiplies by 2/3, as does the arrow from 243/32 to 64/729.

(And especially sorry if I am "that guy".)
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John Carlos Baez | 1 years ago (raw) | root | parent | reply | flag 
 @6f0f8100 - Yeah, I screwed up the diagram.   Each arrow should point to a frequency that's 3/2 times as high.   

Thanks - I'll fix it!   You're apparently the first person who actually looked at it.

To draw the diagram correctly it's also important to know which is bigger: 729/64 or the appropriate power of 2 (namely 128) time 64/729.  That's what tells us which circle to draw to the left, and which to the right!

I think I got that part right, but just drew the arrows wrong.

@2c8b8664
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John Carlos Baez | 1 years ago (raw) | root | parent | reply | flag 
 @6f0f8100 -by the way, part of my confusion was due to the fact that the augmented 4th is higher, not lower, than the diminished 5th.   

If you don't get that, never mind - it's just somewhat confusing stuff!

@2c8b8664