Interval of equivalence and difference tones

Hi.

I have a question on intervals [i] with a specific property: a chain of
[i] should yield combination tones (in the form f2-f1, or even 2*f1-f2
and/or 3*f1-2*f2) belonging to the chain itself. Such intervals may be
useful as interval of equivalence.

The set (chain) of intervals of octave i=2, (i^m) is:
1 2 4 8 16 32 64 ...
The successive differences (i^n - i^(n-1), i.e. f2-f1) are:
1 2 4 8 16 32 ...

For the golden proportion i=1.61803398874989, (i^m) is:
1.00000000000000
1.61803398874989
2.61803398874989
4.23606797749979
6.85410196624969
11.09016994374948
17.94427190999916 ...
The successive differences are:
0.618033988749895
1.000000000000000
1.618033988749895
2.618033988749896
4.236067977499791
6.854101966249686 ...

Note that only the first difference is not in i^m.
However, please note that 1/0.618033988749895 = 1.618033988749895, thus
~0.618 belongs to i^m if [m] can be -1. This way, all difference tones
in the form f2-f1 (for successive tones) don't bring any different (new)
intervals.

*************
Then, I want know values for [i] so that [i^m] is a subset of [i^n -
i^(n-1)] for integers m,n. How to obtain such values for [i]?
*************

Still for ~1.618, if we let [m] have negative values, more combinations
tones coincide with the original set:

i^m, for [m] in -6 to 6:
0.0557280900008412
0.0901699437494742
0.1458980337503154
0.2360679774997897
0.3819660112501052
0.6180339887498948
1.0000000000000000
1.6180339887498949
2.6180339887498949
4.2360679774997898
6.8541019662496856
11.0901699437494763
17.9442719099991628

The main combination tones are (f1 is the lower frequency of a pair, f2
is the higher frequency):

f2 - f1:
0.0344418537486330
0.0557280900008412
0.0901699437494742
0.1458980337503155
0.2360679774997896
0.3819660112501052
0.6180339887498949
1.0000000000000000
1.6180339887498949
2.6180339887498958
4.2360679774997907
6.8541019662496865

2*f1 - f2:
0.0212862362522082
0.0344418537486330
0.0557280900008412
0.0901699437494742
0.1458980337503155
0.2360679774997896
0.3819660112501051
0.6180339887498949
1.0000000000000000
1.6180339887498940
2.6180339887498949
4.2360679774997898

3*f1 - 2*f2:
-0.0131556174964248
-0.0212862362522082
-0.0344418537486330
-0.0557280900008413
-0.0901699437494741
-0.1458980337503157
-0.2360679774997898
-0.3819660112501051
-0.6180339887498949
-1.0000000000000018
-1.6180339887498967
-2.6180339887498931

BTW, I've made some experiments with spectra based on the interval
1.618. The combination tones (like 0.618) are very audible, thus the
pitch of the sound is very ambiguous. (They sound like ``generalized''
Shepard sounds.)

Here are two test .ogg files:
http://geocities.yahoo.com.br/hfmlacerda/1.618.zip

Regards,
Hudson

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