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Evaluating Sagittal accidentals for regular temperaments

🔗Herman Miller <hmiller@IO.COM>

12/9/2007 2:42:54 PM

I wrote a little program to automate some of the work of evaluating Sagittal accidentals. Given a generator mapping for a rank 2 temperament (the general idea can be applied to other temperaments, but those are the ones I've currently been looking at), and values for the period and generator size in cents, it calculates the tempered size of each Sagittal accidental and compares it with the untempered size. Here's a sample of what I get when I run it with the mapping for pajara temperament [<2, 3, 5, 6], <0, 1, -2, -2]> and the TOP-RMS sizes for the generator and period: P = 598.859347, G = 106.844183.

( +2, -11) ~|( B -0.661 15.861 [0, -5, 1, 2>
( -1, +6) )| -4.433 30.002 [-15, 8, 1>
( -1, +6) ~|( A -1.613 13.943 [1, 2, -3, 1>
( -1, +6) /| -0.973 12.662 [-4, 4, -1>
( -1, +6) '|) )|) -0.531 23.639 [-9, 6, 1, -1>
( -1, +6) .//| ~|\\ -0.040 13.966 [7, 0, -3>
( -1, +6) /|) 0.209 10.299 [2, 2, -1, -1>
( -1, +6) )/|\ 0.420 17.276 [-3, 4, 1, -2>
( +1, -5) ~|) -0.857 11.200 [-4, -1, 0, 2>
( +1, -5) .(|\ |\\ -0.038 9.562 [2, -3, 0, 1>
( +1, -5) )||( 0.129 9.229 [-3, -1, 2>

(This is just the first part of the result.) On the left column is what the accidental maps to in generators and periods of the temperament. Note that the two different alternative 7-limit values of ~|( are represented; if ~|( is used in a 7-limit temperament, which one it represents will have to be learned explicitly for each temperament. In this case, there's no alternative for (+2, -11), at least without adding accented symbols into the list. The few accented symbols I've got currently in the list are the ones with unaccented equivalents.

The first number represents how close the tempered symbol is to its untempered equivalent: (log(untempered size) - log(tempered size)) / log(2). So, numbers close to zero represent better size matches. Using TOP-MAX instead of TOP-RMS gives slightly different values. The second number is a measure of complexity, weighted by the base 2 logarithms of the prime factors.

What this says is that .//| or ~|\\ is the closest match as a Sagittal symbol for (-1, +6), but /|) represents a simpler (and likely a more familiar) interval. It's interesting that neither of these is the one I originally selected for pajara notation, which is /| . This is because I was looking at the notation of the system as a whole rather than the accidentals. If D is selected as the 1/1 pitch, F/| represents 6/5 and C/| is 9/5. So this looks like a good example of a case where you can't just look at the size of the intervals if the notation needs to make sense in a tonal context.

Still, a list like this makes it easy to compare the possible options, and at least rule out the least suitable ones.