Stepwise harmonizing property

I was implicitly assuming that one of the chords harmonized to the
root, which doesn't make a lot of sense to assume. Dropping that makes
the analysis far easier--steps have this property iff they are
products (or ratios, but that adds nothing) of consonant intervals
(including 1 as a consonant interval.) This is obvious enough if you
think about it; the situation no longer depends on fine distinctions.
You get one consonant interval from the unison to the common note in
one chord, and from the common note to another interval in the next
chord; that interval, by definition one reachable by a chord with a
common note, is therefore a product of consonances of the system.

In the 7-limit, this has the effect of adding 50/49 (= (10/7)(5/7)) to
the list of harmonizable intervals. I did, and also extended the size
up to 8/7, getting a considerably larger list this time, and including
some six and seven note scales for Carl.

1 [50/49, 49/48, 36/35, 16/15] [5, 8, 6, 4] 23
2 [50/49, 49/48, 36/35, 10/9] [1, 4, 6, 4] 15
3 [50/49, 49/48, 36/35, 28/25] [5, 4, 2, 4] 15
4 [50/49, 49/48, 36/35, 8/7] [1, 4, 2, 4] 11
5 [50/49, 49/48, 21/20, 16/15] [5, 2, 6, 4] 17
6 [50/49, 49/48, 21/20, 28/25] [5, 2, 2, 4] 13
7 [50/49, 49/48, 21/20, 8/7] [1, 2, 2, 4] 9
8 [50/49, 49/48, 16/15, 9/8] [2, 2, 4, 3] 11
9 [50/49, 49/48, 15/14, 28/25] [3, 2, 2, 4] 11
10 [50/49, 49/48, 28/25, 9/8] [4, 2, 4, 1] 11
11 [50/49, 36/35, 25/24, 28/25] [1, 2, 4, 4] 11
12 [50/49, 36/35, 21/20, 10/9] [1, 2, 4, 4] 11
13 [50/49, 36/35, 16/15, 35/32] [1, 2, 4, 4] 11
14 [50/49, 25/24, 21/20, 16/15] [3, 2, 6, 4] 15
15 [50/49, 25/24, 21/20, 28/25] [3, 2, 2, 4] 11
16 [50/49, 25/24, 15/14, 28/25] [1, 2, 2, 4] 9
17 [50/49, 25/24, 28/25, 9/8] [2, 2, 4, 1] 9
18 [50/49, 21/20, 16/15, 35/32] [3, 4, 4, 2] 13
19 [50/49, 21/20, 16/15, 10/9] [3, 6, 2, 2] 13
20 [50/49, 21/20, 10/9, 28/25] [3, 4, 2, 2] 11
21 [50/49, 21/20, 10/9, 8/7] [1, 4, 2, 2] 9
22 [50/49, 16/15, 35/32, 9/8] [1, 4, 2, 2] 9
23 [50/49, 10/9, 28/25, 9/8] [1, 2, 2, 2] 7
24 [49/48, 36/35, 25/24, 16/15] [3, 6, 5, 4] 18
25 [49/48, 36/35, 25/24, 10/9] [3, 6, 1, 4] 14
26 [49/48, 36/35, 25/24, 8/7] [3, 2, 1, 4] 10
27 [49/48, 36/35, 16/15, 15/14] [3, 1, 4, 5] 13
28 [49/48, 36/35, 15/14, 10/9] [3, 5, 1, 4] 13
29 [49/48, 36/35, 15/14, 8/7] [3, 1, 1, 4] 9
30 [49/48, 36/35, 35/32, 10/9] [2, 5, 1, 4] 12
31 [49/48, 36/35, 35/32, 8/7] [2, 1, 1, 4] 8
32 [49/48, 36/35, 10/9, 9/8] [2, 4, 4, 1] 11
33 [49/48, 25/24, 21/20, 8/7] [1, 1, 2, 4] 8
34 [49/48, 21/20, 16/15, 15/14] [2, 1, 4, 5] 12
35 [49/48, 21/20, 15/14, 8/7] [2, 1, 1, 4] 8
36 [49/48, 21/20, 35/32, 8/7] [1, 1, 1, 4] 7
37 [49/48, 16/15, 15/14, 28/25] [2, 3, 5, 1] 11
38 [49/48, 16/15, 15/14, 9/8] [2, 4, 4, 1] 11
40 [49/48, 15/14, 28/25, 8/7] [2, 2, 1, 3] 8
41 [36/35, 25/24, 21/20, 16/15] [3, 5, 3, 4] 15
42 [36/35, 25/24, 21/20, 10/9] [3, 1, 3, 4] 11
43 [36/35, 25/24, 16/15, 35/32] [3, 2, 4, 3] 12
44 [36/35, 25/24, 16/15, 28/25] [3, 5, 1, 3] 12
45 [36/35, 25/24, 10/9, 28/25] [3, 4, 1, 3] 11
46 [36/35, 25/24, 28/25, 8/7] [2, 4, 3, 1] 10
47 [36/35, 21/20, 15/14, 10/9] [2, 3, 1, 4] 10
48 [36/35, 21/20, 35/32, 10/9] [3, 2, 1, 4] 10
49 [36/35, 21/20, 10/9, 9/8] [2, 2, 4, 1] 9
50 [36/35, 16/15, 15/14, 35/32] [1, 4, 2, 3] 10
51 [36/35, 16/15, 35/32, 10/9] [3, 2, 3, 2] 10
52 [36/35, 16/15, 35/32, 8/7] [1, 2, 3, 2] 8
53 [36/35, 35/32, 10/9, 28/25] [3, 2, 3, 1] 9
54 [36/35, 10/9, 28/25, 9/8] [1, 3, 1, 2] 7
55 [25/24, 21/20, 16/15, 15/14] [2, 3, 4, 3] 12
56 [25/24, 21/20, 16/15, 8/7] [2, 3, 1, 3] 9
57 [25/24, 21/20, 10/9, 8/7] [1, 3, 1, 3] 8
58 [25/24, 21/20, 28/25, 8/7] [2, 2, 1, 3] 8
59 [25/24, 16/15, 15/14, 28/25] [2, 1, 3, 3] 9
60 [25/24, 15/14, 10/9, 28/25] [1, 3, 1, 3] 8
61 [25/24, 15/14, 28/25, 8/7] [2, 2, 3, 1] 8
62 [25/24, 28/25, 9/8, 8/7] [2, 2, 1, 2] 7
63 [21/20, 16/15, 15/14, 35/32] [1, 4, 3, 2] 10
64 [21/20, 16/15, 15/14, 10/9] [3, 2, 3, 2] 10
65 [21/20, 16/15, 35/32, 8/7] [1, 1, 2, 3] 7
66 [21/20, 15/14, 10/9, 28/25] [1, 3, 2, 2] 8
67 [21/20, 15/14, 10/9, 8/7] [3, 1, 2, 2] 8
68 [21/20, 35/32, 10/9, 8/7] [2, 1, 1, 3] 7
69 [21/20, 10/9, 9/8, 8/7] [2, 2, 1, 2] 7
70 [16/15, 15/14, 35/32, 28/25] [3, 3, 2, 1] 9
71 [16/15, 15/14, 35/32, 9/8] [4, 2, 2, 1] 9
72 [16/15, 35/32, 9/8, 8/7] [2, 2, 1, 2] 7
74 [15/14, 10/9, 28/25, 9/8] [2, 2, 2, 1] 7
75 [10/9, 28/25, 9/8, 8/7] [2, 1, 2, 1] 6

>I was implicitly assuming that one of the chords harmonized to the
>root, which doesn't make a lot of sense to assume. Dropping that makes
>the analysis far easier--steps have this property iff they are
>products (or ratios, but that adds nothing) of consonant intervals
>(including 1 as a consonant interval.)

How is this any different than a symmetric lattice distance of 2,
which is what I thought you used in the first place.

>This is obvious enough if you
>think about it; the situation no longer depends on fine distinctions.
>You get one consonant interval from the unison to the common note in
>one chord, and from the common note to another interval in the next
>chord; that interval, by definition one reachable by a chord with a
>common note, is therefore a product of consonances of the system.

I can't parse this.

>In the 7-limit, this has the effect of adding 50/49 (= (10/7)(5/7)) to
>the list of harmonizable intervals. I did, and also extended the size
>up to 8/7, getting a considerably larger list this time, and including
>some six and seven note scales for Carl.

Well this is cool, but since many classic scales contain steps up to a
minor third apart, perhaps 6/5 should be the cutoff.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I was implicitly assuming that one of the chords harmonized to the
> >root, which doesn't make a lot of sense to assume. Dropping that makes
> >the analysis far easier--steps have this property iff they are
> >products (or ratios, but that adds nothing) of consonant intervals
> >(including 1 as a consonant interval.)
>
> How is this any different than a symmetric lattice distance of 2,
> which is what I thought you used in the first place.

50/49 has a distance of exactly 2; I said less than 2.

> Well this is cool, but since many classic scales contain steps up to a
> minor third apart, perhaps 6/5 should be the cutoff.

Hmmm. If you want to temper the results, you should bound the steps
pairwise if you are going to go this big--meaning the ratio of the
biggest to second-biggest is not allowed to be too large.