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Patent Val on "General Theory" page
I see that this has been added to the General Theory page, but I feel as though it might still be a bit too technical for the General Theory crowd. If I had my druthers, the way I'd explain it to them is more or less the way I wrote it in my initial abstract, which I've copied here:
"The patent val for some EDO is the val that you obtain by simply finding the closest rounded-off approximation to each prime in the tuning. For example, the patent val for 17-EDO is <17 27 39|, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal. This val can be extended to the case where the number of steps in an octave is a real number rather than an integer; for instance the 7-limit patent val for 16.9 is <17 27 39 47|, since 16.9 * log2(7) = 47.444, which rounds down to 47.
You may prefer to use the <17 27 40| val for as the 5-limit 17-equal instead, which val rather than <17 27 39|; this treats 424 cents as 5/4 - and indeed this val has lower Tenney-Euclidean error than the 17-EDO patent val. However, while <17 27 39| may not necessarily be the "best" val for 17-equal for all purposes, it is the obvious, or "patent" val, that you get by naively rounding primes off within the EDO and taking no further considerations into account. However, <17 27 40| is the patent val for 17.1, since 17.1 * log2(5) = 39.705, which rounds up to 40."
How should we handle this? It's a bit too short to spin off into its own article. However, it's not really that long, and it's technically accurate. Perhaps we should add it or something like it back in as a non-technical intro section? Then it'd be more appropriate for the General Theory page.
- mbattaglia1 September 13, 2011, 09:10:43 AM UTC-0700
(Even if this is a very late reply:) Your explanation is much easier to understand. I'd suggest to (at least) integrate it into this page. Especially your 17-EDO example is helpful.
- xenwolf February 16, 2016, 12:34:51 AM UTC-0800
monzo incorrect compilation
"...81 = 3*3*3*3. This can also be written as a power of a prime -- 3^4 -- or as a monzo -- | 0 3 >.
80 = 2*2*2*2*5. This can also be written as a product of powers of primes -- (2^4)*(5^1) -- or as a monzo -- | 2 0 1 >. "
ehm... these monzos aren't respectively
| 0 4> and | 4 0 1 > ?
- tetraF September 04, 2011, 07:12:26 AM UTC-0700
That's definitely true, so I corrected it.
- keenanpepper September 04, 2011, 05:50:16 PM UTC-0700
- tetraF September 05, 2011, 11:32:59 AM UTC-0700
Don't know where my brain was when I wrote that. Thanks for the correction.
- jdfreivald September 06, 2011, 10:40:26 AM UTC-0700