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We've seen this before: and it's likely wrong. He ended his experiment too soon at 24 divisions, but even a little googling should have told him to go to 31, w

We've seen this before: and it's likely wrong. He ended his experiment too soon at 24 divisions, but even a little googling should have told him to go to 31, which is more accurate than 12. The 12-note scale long predates the notion of just or equal temperament.

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ledauphin 4y ago
i don't think this post is making any arguments about _accuracy_ - the point is that 12 is the _simplest_ (smallest) number that gets reasonably close. Simplicity is incredibly powerful.
bryanlarsen 4y ago
According to https://news.ycombinator.com/item?id=32641527 https://news.ycombinator.com/item?id=32641527, 12 is substantially more accurate than 31. 41 is slightly more accurate, and 53 is a lot more accurate.
lynndotpy 4y ago
Even better, if we abandon the idea of an octave base of 2, we can get other scales. What divisions lie between, say, powers of 3 or 5? (You can find these naively by brute force!)
elihu 4y ago
For the intervals they look at in the article, the perfect 4th and 5th, 31-EDO is worse -- about 5 cents of error, versus about 2. What 31-EDO has is a major third that's almost dead-on, and a minor third that's a lot closer. https://en.wikipedia.org/wiki/31_equal_temperament#Interval_size https://en.wikipedia.org/wiki/31_equal_temperament#Interval_... 41-EDO though has a perfect 4th and 5th that are closer than 12-EDO, being off by about half a cent rather than about 2 cents. In fact, 41-EDO is better at every commonly-used interval than 12-EDO, plus it adds a lot of very good 7-limit intervals too (i.e. ratios with sevens in them like 7:4, which is way off in 12-EDO). By a weird set of mathematical coincidences, 41-EDO is actually quite playable on guitar with the right layout. The trick is to omit half the frets and tune the strings so that each string has the notes that the strings above and below it lack. Tuning by major 3rds, you get a whole lot of useful notes clustered where they're easy to play. There's a handful of us (in Portland mostly) trying to promote this idea: https://kiteguitar.com/ https://kiteguitar.com/

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ledauphin 4y ago

i don't think this post is making any arguments about _accuracy_ - the point is that 12 is the _simplest_ (smallest) number that gets reasonably close. Simplicity is incredibly powerful.

bryanlarsen 4y ago

According to https://news.ycombinator.com/item?id=32641527 https://news.ycombinator.com/item?id=32641527, 12 is substantially more accurate than 31. 41 is slightly more accurate, and 53 is a lot more accurate.

lynndotpy 4y ago

Even better, if we abandon the idea of an octave base of 2, we can get other scales. What divisions lie between, say, powers of 3 or 5? (You can find these naively by brute force!)

elihu 4y ago

For the intervals they look at in the article, the perfect 4th and 5th, 31-EDO is worse -- about 5 cents of error, versus about 2. What 31-EDO has is a major third that's almost dead-on, and a minor third that's a lot closer. https://en.wikipedia.org/wiki/31_equal_temperament#Interval_size https://en.wikipedia.org/wiki/31_equal_temperament#Interval_... 41-EDO though has a perfect 4th and 5th that are closer than 12-EDO, being off by about half a cent rather than about 2 cents. In fact, 41-EDO is better at every commonly-used interval than 12-EDO, plus it adds a lot of very good 7-limit intervals too (i.e. ratios with sevens in them like 7:4, which is way off in 12-EDO). By a weird set of mathematical coincidences, 41-EDO is actually quite playable on guitar with the right layout. The trick is to omit half the frets and tune the strings so that each string has the notes that the strings above and below it lack. Tuning by major 3rds, you get a whole lot of useful notes clustered where they're easy to play. There's a handful of us (in Portland mostly) trying to promote this idea: https://kiteguitar.com/ https://kiteguitar.com/