Re: Proprety

Hi David,

Yes, that's an improper scale.

I've been doing some computer searches for an idea Dan
has, and this interesting point came up when asking him
for clarification:

Using Dan's notation:

Rotations of 2 2 1 2 2 2 1 in 12-tet
(i.e. showing it as a mode of 12-tet)
0 2 4 5 7 9 11 12 = 2 2 1 2 2 2 1
0 1 3 5 6 8 10 12 = 1 2 2 1 2 2 2
0 2 3 5 7 8 10 12 = 2 1 2 2 1 2 2
0 2 4 5 7 9 10 12 = 2 2 1 2 2 1 2
0 2 4 6 7 9 11 12 = 2 2 2 1 2 2 1
0 1 3 5 7 8 10 12 = 1 2 2 2 1 2 2
0 2 3 5 7 9 10 12 = 2 1 2 2 2 1 2

then you have 1s and 2s in first col.,
3s and 4s, 5s and 6s, then
in fourth col. you have 7s and 6s again
- since an augmented fourth is the same as
a diminished fifth.

This scale is proper, but not strictly proper.

It's proper because a larger number of steps always
makes an interval which is at least as large as
a smaller number of steps.

However, if one has good relative pitch and hears the interval
of 600 cents, without any other context you don't know
if it is a four step or a five step interval.

So it has a level of ambiguity, and isn't strictly proper.

Rotations of 3 3 1 3 3 3 1

0 3 6 7 10 13 16 17 = 3 3 1 3 3 3 1
0 1 4 7 8 12 14 17 = 1 3 3 1 3 3 3
0 3 4 7 10 12 14 17 = 3 1 3 3 1 3 3
0 3 6 7 10 13 14 17 = 3 3 1 3 3 1 3
0 3 6 9 10 13 16 17 = 3 3 3 1 3 3 1
0 1 6 7 10 12 14 17 = 1 3 3 3 1 3 3
0 3 6 7 10 13 14 17 = 3 1 3 3 3 1 3

So that one is improper (the diatonic scale in 17-tet)

The 4 step interval n(8/17) (my notation for 9 out of 17-tet)
is smaller than the 3 step interval n(9/17)

However although improper, it still has an interesting
property, because the only n(8/17) interval is the 4 step one.

So in a sense it is unambiguous. You can always in principle
tell the number of steps in an interval heard, without
any more context.

Then a scale is strictly proper if it is proper and
unambiguous.

Rotations of 3 3 2 3 3 3 2

0 3 6 8 11 14 16 19 = 3 3 2 3 3 3 2
0 2 5 8 10 13 16 19 = 2 3 3 2 3 3 3
0 3 5 8 11 13 16 19 = 3 2 3 3 2 3 3
0 3 6 8 11 14 16 19 = 3 3 2 3 3 2 3
0 3 6 9 11 14 17 19 = 3 3 3 2 3 3 1
0 2 5 8 11 13 16 19 = 2 3 3 3 2 3 3
0 3 6 8 11 14 16 19 = 3 2 3 3 3 2 3

This one is strictly proper.

A larger number of steps will always make a larger interval,
and also, there is no ambiguity, if one hears a particular
interval size in complete isolation, you know immmediately
how many steps there are in it (supposing your sense of relative pitch
is up to it - mine isn't unfortunately!)

In Dan's notation you also easily see another interesting
property of this scale, that it has different numbers of
all the interval sizes - one 9, two 2s, three 5s, four 6s,
five 3s, and 6 8s (similarly of course for the complements
10, 17, 14, 13, 16, 11).

Manuel pointed this out to me once, but I can't remember
what such a scale is called.

It seems to me however that maybe there is something interesting
about scales that are improper but also unambiguous.

Maybe some compositional use can be made of the property.

17-tet diatonic is a very pleasant scale I think.

The computer search turned up lots of unambiguous improper
scales, so they are rather common.

Here I'm coining a word "unambiguous" for these types of
improper scales, and would be interested to know if they
already have a name.

Robert

Proprety is tehft.

Robert Walker wrote:

> Rotations of 3 3 1 3 3 3 1
>
> 0 3 6 7 10 13 16 17 = 3 3 1 3 3 3 1
> 0 1 4 7 8 12 14 17 = 1 3 3 1 3 3 3
> 0 3 4 7 10 12 14 17 = 3 1 3 3 1 3 3
> 0 3 6 7 10 13 14 17 = 3 3 1 3 3 1 3
> 0 3 6 9 10 13 16 17 = 3 3 3 1 3 3 1
> 0 1 6 7 10 12 14 17 = 1 3 3 3 1 3 3
> 0 3 6 7 10 13 14 17 = 3 1 3 3 3 1 3
>
> So that one is improper (the diatonic scale in 17-tet)
>
> The 4 step interval n(8/17) (my notation for 9 out of 17-tet)
> is smaller than the 3 step interval n(9/17)
>
> However although improper, it still has an interesting
> property, because the only n(8/17) interval is the 4 step one.

Are there any singly positive scales for which the diatonics (on both
the Pythagorean and JI model) aren't like this? I think it's always
the tritones that make them improper, and they can never be ambiguous,
can they? Except for 12 and 22.

We'll assume that 17-equal only has the one kind of diatonic.

> So in a sense it is unambiguous. You can always in principle
> tell the number of steps in an interval heard, without
> any more context.

I principle, yes, but to enforce this distinction in general would
assume hearing worked to mathematical precision.

> Then a scale is strictly proper if it is proper and
> unambiguous.

Yes, that follows.

> It seems to me however that maybe there is something interesting
> about scales that are improper but also unambiguous.
>
> Maybe some compositional use can be made of the property.
>
> 17-tet diatonic is a very pleasant scale I think.

I don't think the ambiguity of tritones, even when they are ambiguous,
is that important. You could call all these diatonics "near enough
proper". I feel the tritones being rare, and functioning as
dissonances, is more important than learning which set to place them
in. Except for cases where the ambiguity is a good thing, I suppose.

Maybe propriety in general isn't that important.

> The computer search turned up lots of unambiguous improper
> scales, so they are rather common.
>
> Here I'm coining a word "unambiguous" for these types of
> improper scales, and would be interested to know if they
> already have a name.

Do you have any ambiguous improper scales? ISTM that the ambiguity
will naturally lead to propriety. You can probably find some
exceptions if you try, but are they important?

Graham

Hi Graham,

Yes, in fact, the ambiguous improper scales were by far the most
common ones in this particular search.

> > So in a sense it is unambiguous. You can always in principle
> > tell the number of steps in an interval heard, without
> > any more context.

> I principle, yes, but to enforce this distinction in general would
> assume hearing worked to mathematical precision.

Yes, I imagine so in general, especially when you get unambiguous
improper scales with the steps as ratios rather than n-tet - I expect
many rational scales are unambiguous and improper, bur may involve
fine distinctions.

However, the search also turned up some unambiguous improper scales
with large distinctions of step size, and perhaps one could make use of the
unambiguity compositionally?

> I don't think the ambiguity of tritones, even when they are ambiguous,
> is that important. You could call all these diatonics "near enough
> proper". I feel the tritones being rare, and functioning as
> dissonances, is more important than learning which set to place them
> in. Except for cases where the ambiguity is a good thing, I suppose.

Yes, maybe also could be interesting to investigate maximally ambiguous
scales!

You can get a 7 note scale with 17 ambiguous intervals.

No idea whether or not it is maximal - as it was just a bi-product of the
search.

Other scales found were: 4 notes, 4 amb., 5 notes + 7 amb., 6 notes + 12 amb.
8 notes + 24 amb. and 10 + 25 ambig. (but I set a time out on the search
for each n-tet and number of notes, and was timing out early for 10 note
scales - might have found ones that were even more ambiguous).

Here the question is, what is the maximum number of ambiguous intervals
you can have in a scale of n notes.

Another approach is to try to find a scale in n-tet with the maximal number
of ambiguous intervals. Since the smallest possible step size and its complement
are inevitably unambiguous, you can have at most n-2 ambiguous intervals in
n-tet.

There is a "trivial" solution for proper scales:
1 1 1 ... 1 2

Not so easy to make a proper maximally ambiguous scale with less than
n-1 notes in n-tet.

Then, can you make a maximally ambiguous improper scale?

Answer is yes, and the search turned up a few.

E.g. a 15-tet improper scale of 6 notes with 13 ambiguous intervals, and only 1
and 14 unambiguous.

May be just a curiosity, or I wonder if maximally ambiguous improper scales are
of any other interest?

I leave the problem of a maximally ambiguous 6 note 15-tet scale as a
puzzle for now.

Robert