The (19,9,4) difference set scale

David Bowen wrote on tuning-math:

<<By coincidence, the April 2002 issue of the Mathematics Magazine
arrived at
my house yesterday and the lead article discusses the many applications
of
the 7-et set. One of the first theorems inthe article is that if p is a
prime of the form 4n+3, then the squares mod p will give you a set of
2n+1 elements where each difference occurs n times. So for 19 we have the
set {1, 4, 5, 6, 7, 9, 11, 16, 17} where each difference occurs 4 times
and
for 31 we have the set {1, 2, 4, 5, 7, 8, 9, 10, 14, 16, 18, 19, 20, 25,
28}
where each difference occurs 7 times.>>

The scale consisting of the 9 quadradic residues mod 19 seemed worth
investigating.
This is [1, 4, 5, 6, 7, 9, 11, 16, 17]; the characteristic polynomials
for the odd limits to 11 are given below; the x^7 term gives the number
of edges, and the x^6 twice the number of triads. If this experiment
works, you should be able to see graphs of the scale in the various
limits in the attachments.

p03 x^9-4*x^7+4*x^5-x^3
p05 x^9-12*x^7-6*x^6+40*x^5+30*x^4-38*x^3-32*x^2+7*x+6
p07 x^9-20*x^7-28*x^6+53*x^5+100*x^4-6*x^3-66*x^2-24*x
p09 x^9-28*x^7-74*x^6-35*x^5+54*x^4+42*x^3
p11 x^9-32*x^7-116*x^6-160*x^5-70*x^4+39*x^3+44*x^2+10*x