I've used a measure which is related to the idea of consistency

proposed here, which I would like to explain.

Let w be an odd number, and let p_d <= w be the largest prime less

than or equal to w, and suppose that there are d primes p_i less than

or equal to l. Let h:G_p_d ->> Z be a homomorphism (more precisely

epimorphism, meaning onto) giving a p_d-limit et. Let {q_i} be the

set of rational numbers q_i >= 1 which are ratios of any two odd

numbers less than or equal to w, and let n = h(2). We define the w-

consistent goodness measure of h as

n^(1/d) * max(abs(n*log_2(q_i) - h(q_i))

In the most interesting cases, the homomorphism h will simply be the

homomorphism h_n obtained by rounding n*log_2(p_i) to the nearest

integer; this lets us define the w-consistent goodness of an integer

n, by setting

cons(w, n) = n^(1/d) * max(abs(n*log_2(q_i)) - h_n(q_i))

The d-th root of n is introduced because it is the appropriate

multiplier according to theorems relating to the simultaneous

Diophantine approximation of d independent numbers.

Here are two tables by way of example:

From 1 to 10000, 5-consistent measure cons(5, n) < 1

1 .736965595

2 .7439736471

3 .4245472985

4 .679700008

5 .8728704449

7 .6706891205

12 .5418300757

15 .8735997285

19 .5083949041

31 .8578063580

34 .6488389972

53 .4527427539

65 .7839449193

118 .4134352529

171 .6499654470

289 .9207676

441 .6622791

559 .9976240155

612 .5676032129

730 .6113208564

1171 .7597497149

1783 .5008376597

2513 .5396476355

4296 .2748910262

6809 .7979361504

8592 .7776946207

From 1 to 10000, 9-consistent measure cons(9, n) < 1.25

1 1.152003094

2 .9136193505

4 1.078956495

5 .7303891055

7 1.151607929

10 1.120372697

12 .8032252875

19 .9065721690

22 .9640922367

27 1.236074910

31 .9044163694

41 .8004237371

46 1.181094402

53 1.023130352

72 .9759757458

99 .8207791216

130 1.213753821

171 .3202177291

270 .7772103754

342 .8068974155

441 .8113651760

612 .9093921787

935 1.231274556

1106 1.219566982

1277 1.192780688

1848 1.206088177

2954 1.055468357

3125 .6018359509

3296 1.116123065

3566 1.147361792

6691 .9930572626

8539 1.219825812

--- In tuning-math@y..., genewardsmith@j... wrote:

> Let w be an odd number, and let p_d <= w be the largest prime less

> than or equal to w, and suppose that there are d primes p_i less

than

> or equal to l.

Sorry, an ascii "l" looks like the number "1", so I changed the l's

to w's but missed this one. It should be p_i less than or equal to w.

I'll have to look at this later, Gene. Sounds very interesting and

not unlike some things I've mentioned on the tuning list about 5

years ago when it was on the Mills server.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I'll have to look at this later, Gene. Sounds very interesting and

> not unlike some things I've mentioned on the tuning list about 5

> years ago when it was on the Mills server.

Incidentally, if you or anyone else is interested in Maple routines

for functions such as cons(w, n) I could post them.