complexity-scaling.md

How complexity scales with...

	damage	prime limit
JI	Ω(1)	Ω(π(p))
rank 1	Ω(1/d)	Ω(π(p))
rank 2	Ω(1/d)	Ω(π(p)2)

In the prime limit column we're trying to answer: the complexity of the simplest scale that falls within a given TOP damage bin goes like ____ with increasing prime limit.

In the damage column we're trying to answer: for a given prime limit, to reduce damage by a factor f, we need to increase scale size by at least ____.

Here's an example investigating prime limit scaling for rank 1.
I took the data from something I computed for a different purpose, so they're not actually correct.

     5  7 11 13 17 p-limit
< 4                damage bound
    12 19 22 27 27 complexity of simplest val
    /3 /4 /5 /6 /7 pi(p)
     4  5  4  5  4 constant?
< 3
    19 27 31 39 39
    /3 /4 /5 /6 /7
     6  7  6  7  6

Here are some results from Graham's finder http://x31eq.com/temper/pregular.html

The "target error" concept is a bit fuzzy, so they're not too telling...

limit	error	ET	r2	r2comp	r3	r3comp
5	8	12	meantone	0.7	NA	NA
5	4	12	meantone	0.7	NA	NA
5	2	12	meantone	0.7	NA	NA
5	1	53	helmholtz	1.8	NA	NA
7	8	12	meantone	1.4	marvel	0.1
7	4	19	meantone	1.4	marvel	0.1
7	2	31	meantone	1.4	marvel	0.1
7	1	99	miracle	2.4	breed	0.2
11	8	12	augene	1.4	minerva	0.1
11	4	31	orwell	2.1	marvel	0.2
11	2	31	miracle	2.8	marvel	0.2
11	1	72	miracle	2.8	portent	0.2
13	8	12	negri	1.2	krypton	0.1
13	4	31	meantone	1.9	thrush	0.2
13	2	72	harry	4.9	jovis	0.3
13	1	72	harry	4.9	history	0.3
17	8	10	negroni+	1.7	krypton+	0.1
17	4	46	sensus+	2.7	demeter	0.2
17	2	72	hendec+	4.1	ominous+	0.3
17	1	72	hendec+	4.1	140+72+46	0.3