Some 9-tone 72-et scales

The first three numbers on the third row are number of edges in the 5, 7, and 11 limits, and the second connectivity in the 5, 7, and 11 limits.

[0, 7, 14, 21, 30, 37, 44, 56, 63]
[7, 7, 7, 9, 7, 7, 12, 7, 9]
12 22 29 1 4 6

[0, 7, 14, 26, 33, 40, 49, 56, 63]
[7, 7, 12, 7, 7, 9, 7, 7, 9]
12 21 29 1 3 6

[0, 7, 14, 21, 33, 40, 49, 56, 63]
[7, 7, 7, 12, 7, 9, 7, 7, 9]
11 20 28 1 3 6

[0, 7, 14, 21, 33, 42, 49, 56, 63]
[7, 7, 7, 12, 9, 7, 7, 7, 9]
10 20 28 1 3 5

[0, 7, 14, 21, 33, 40, 47, 56, 63]
[7, 7, 7, 12, 7, 7, 9, 7, 9]
9 17 28 0 2 5

[0, 7, 14, 21, 28, 37, 44, 56, 63]
[7, 7, 7, 7, 9, 7, 12, 7, 9]
10 21 27 0 3 5

[0, 7, 14, 21, 28, 37, 44, 51, 63]
[7, 7, 7, 7, 9, 7, 7, 12, 9]
9 20 27 0 2 5

[0, 7, 14, 21, 28, 40, 49, 56, 63]
[7, 7, 7, 7, 12, 9, 7, 7, 9]
8 18 26 0 2 5

[0, 7, 14, 21, 28, 40, 47, 56, 63]
[7, 7, 7, 7, 12, 7, 9, 7, 9]
7 16 26 0 2 5

[0, 7, 14, 21, 33, 40, 47, 54, 63]
[7, 7, 7, 12, 7, 7, 7, 9, 9]
6 12 26 0 1 5

[0, 7, 14, 21, 28, 35, 44, 51, 63]
[7, 7, 7, 7, 7, 9, 7, 12, 9]
7 18 25 0 2 4

In-Reply-To: <a0uhdq+onbm@eGroups.com>
Can you get your program to check for propriety of these things? All
these are strictly proper except for

> [0, 7, 14, 21, 33, 42, 49, 56, 63]
> [7, 7, 7, 12, 9, 7, 7, 7, 9]
> 10 20 28 1 3 5

> [0, 7, 14, 21, 28, 37, 44, 56, 63]
> [7, 7, 7, 7, 9, 7, 12, 7, 9]
> 10 21 27 0 3 5

> [0, 7, 14, 21, 28, 37, 44, 51, 63]
> [7, 7, 7, 7, 9, 7, 7, 12, 9]
> 9 20 27 0 2 5

> [0, 7, 14, 21, 28, 40, 49, 56, 63]
> [7, 7, 7, 7, 12, 9, 7, 7, 9]
> 8 18 26 0 2 5

> [0, 7, 14, 21, 28, 40, 47, 56, 63]
> [7, 7, 7, 7, 12, 7, 9, 7, 9]
> 7 16 26 0 2 5

which are proper but not strictly proper, and

> [0, 7, 14, 21, 28, 35, 44, 51, 63]
> [7, 7, 7, 7, 7, 9, 7, 12, 9]
> 7 18 25 0 2 4

which is improper. It's interesting that so many scales came out proper
when that wasn't a criterion in the search. All the 10-note 72= scales
are strictly proper.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a0uhdq+onbm@e...>

> Can you get your program to check for propriety of these things?

I'd need to write the code for it, and it isn't a graph property so I'm not going to start with any advantage from the Maple graph theory package. Paul did not think propriety was very important--what's your take on it?

> It's interesting that so many scales came out proper
> when that wasn't a criterion in the search. All the 10-note 72= scales
> are strictly proper.

It's also interesting that the best scores were all proper.

>>Can you get your program to check for propriety of these things?
>
>I'd need to write the code for it, and it isn't a graph property
>so I'm not going to start with any advantage from the Maple graph
>theory package. Paul did not think propriety was very important--
>what's your take on it?

You might like to read Rothenberg's original papers on the subject.
There's graph stuff in there that none of us have touched (propriety
was just a starting point for Rothenberg), plus a fancy algorithm
generating all the proper subsets of a scale.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> You might like to read Rothenberg's original papers on the subject.
> There's graph stuff in there that none of us have touched (propriety
> was just a starting point for Rothenberg), plus a fancy algorithm
> generating all the proper subsets of a scale.

Where might Rothenberg's papers be found?

In-Reply-To: <a1173r+ob05@eGroups.com>
gene wrote:

> I'd need to write the code for it, and it isn't a graph property so I'm
> not going to start with any advantage from the Maple graph theory
> package. Paul did not think propriety was very important--what's your
> take on it?

I don't have a definitive answer. It, or something like it, may be
important for modality. Especially for subsets of "comprehensible" ETs.
The Pythagorean diatonic works fine despite being slightly improper, so
you shouldn't be over-strict. For a scale with three step sizes to be
proper shows that it has a certain level of cohesion. The extremely
improper Magic subsets you gave composers, performers and listeners are
likely to expect the large gaps to be filled in by more notes. This is a
general problem with Magic scales of between 3 and 19 notes.

The Decimal MOS has the opposite problem -- its step sizes are so closely
equal that it doesn't have any shape. So it's great as a basis for
notation, but doesn't have any sense of tonal center. Your top 10 note
scale might solve this problem, because it's largest interval is almost
twice the size of its smallest. And that single 5/72 step could be
extremely important for leading to the tonic.

10 notes still seems like a lot for a mode. Perhaps the 6 note [12, 14,
9, 14, 9, 14] would work. How is it in terms of connectedness?

Graham

> > It's interesting that so many scales came out proper
> > when that wasn't a criterion in the search. All the 10-note 72=
> > scales are strictly proper.
>
> It's also interesting that the best scores were all proper.
>
>
>
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>>You might like to read Rothenberg's original papers on the subject.
>>There's graph stuff in there that none of us have touched (propriety
>>was just a starting point for Rothenberg), plus a fancy algorithm
>>generating all the proper subsets of a scale.
>
>Where might Rothenberg's papers be found?

Tuning bibliography at:

http://www.xs4all.nl/~huygensf/doc/bib.html

The papers:

Rothenberg, David. "A Model for Pattern Perception with Musical
Applications. Part I: Pitch Structures as Order-Preserving Maps",
Mathematical Systems Theory vol. 11, 1978, pp. 199-234.

Rothenberg, David. "A Model for Pattern Perception with Musical
Applications Part II: The Information Content of Pitch structures",
Mathematical Systems Theory vol. 11, 1978, pp. 353-372.

Rothenberg, David. "A Model for Pattern Perception with Musical
Applications Part III: The Graph Embedding of Pitch Structures",
Mathematical Systems Theory vol. 12, 1978, pp. 73-101.

I'd send you copies, but my copies are locked away in Montana.

-Carl

>I don't have a definitive answer. It, or something like it, may be
>important for modality. Especially for subsets of "comprehensible"
>ETs.

Needless to say, I think it's very important. There's nothing
against improper scales -- it isn't that kind of criterion. But
R. shows that certain musical devices rely on proper scales. These
effects are still available on proper subsets of improper scales,
but if you want some of the effects of traditional diatonic music,
where the entire pitch set is involved with these effects, then you
need propriety over the whole scale.

>The Pythagorean diatonic works fine despite being slightly
>improper, so you shouldn't be over-strict.

Right, which is why R. never uses propriety -- he uses stability.

>For a scale with three step sizes to be proper shows that it has a
>certain level of cohesion.

I'll take this opportunity to stamp out the myth of the importance
of 2nds. The variety of all the classes of scale intervals are
equally important. So you need the variety of 2nds, and the
ordering. To speak of only the variety is scale voodoo. IMO, the
best measure is simply R.'s mean variety.

>The Decimal MOS has the opposite problem -- its step sizes are so
>closely equal that it doesn't have any shape.

That's an issue of efficiency, not propriety.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> >The Decimal MOS has the opposite problem -- its step sizes are so
> >closely equal that it doesn't have any shape.
>
> That's an issue of efficiency, not propriety.

I don't know what "efficiency" means in this connection, but it sounds to me like what you were talking about--variety. I haven't jumped on the Miracle bandwagon precisely because the pudding-like sameness of the Decimal put me off; but the 72-et scales I've been cooking up are just the sort of thing I like.

>>>The Decimal MOS has the opposite problem -- its step sizes are so
>>>closely equal that it doesn't have any shape.
>>
>>That's an issue of efficiency, not propriety.
>
>I don't know what "efficiency" means in this connection, but it
>sounds to me like what you were talking about--variety. I haven't
>jumped on the Miracle bandwagon precisely because the pudding-like
>sameness of the Decimal put me off; but the 72-et scales I've been
>cooking up are just the sort of thing I like.

Try message number 4044 on the main list for a general overview
of and quotes from the Rothenberg papers, including the formula
for efficiency.

-Carl

>>Try message number 4044 on the main list for a general overview
>>of and quotes from the Rothenberg papers, including the formula
>>for efficiency.
>
>Yahoo has removed the tabs from my message, and least on
>the web, so you can tell where the citations start and stop..
>Egregious.

That's supposed to be _can't_ tell.

-Carl

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a1173r+ob05@e...>
> gene wrote:
>
> > I'd need to write the code for it, and it isn't a graph property
so I'm
> > not going to start with any advantage from the Maple graph theory
> > package. Paul did not think propriety was very important--what's
your
> > take on it?
>
> I don't have a definitive answer. It, or something like it, may be
> important for modality. Especially for subsets of "comprehensible"
ETs.
> The Pythagorean diatonic works fine despite being slightly
improper, so
> you shouldn't be over-strict.

The 22-tET "Pythagorean diatonic" works exceptionally well.

>>It's interesting that so many scales came out proper
>>when that wasn't a criterion in the search. All the
>>10-note 72= scales are strictly proper.
>
>It's also interesting that the best scores were all proper.

The max connectedness c of a scale of cardinality k is k-1,
right? To actually get in octave-equivalent scales at
odd-limit n, you have to have k <= (n+1)/2, right?

For scales with a high c for k > n, some n-limit interval(s)
will have to appear between more than one pair of scale
members. Can we get propriety from this?

It seems that any scale with one interval class always
consonant will be connected. Given that the sizes of the
consonances are fairly well distributed across the octave. . .

Oh well. Something interesting is going on here, that's
for sure.

-Carl

In-Reply-To: <a13hji+jr57@eGroups.com>
Me:
> > I don't have a definitive answer. It, or something like it, may be
> > important for modality. Especially for subsets of "comprehensible"
> ETs.
> > The Pythagorean diatonic works fine despite being slightly
> improper, so
> > you shouldn't be over-strict.

Paul:
> The 22-tET "Pythagorean diatonic" works exceptionally well.

You mean 4 4 1 4 4 4 1 ? Isn't it proper, and only one interval away
from being strictly proper?

Graham

--- In tuning-math@y..., graham@m... wrote:

> Paul:
> > The 22-tET "Pythagorean diatonic" works exceptionally well.
>
> You mean 4 4 1 4 4 4 1 ?

Yes.

> Isn't it proper

No: 4 + 4 + 4 > 1 + 4 + 4 + 1.

>> Isn't it proper
>
> No: 4 + 4 + 4 > 1 + 4 + 4 + 1.

The point non-strict propriety for the diatonic scale is 12-tET.
It will be improper in any tuning with pos. fifths (22-tET), and
strictly proper in any tuning with negative fifths (meantone).
None of this matters too much, because you still have all the
other properties of the diatonic scale, namely:

() Pitch set under Miller limit of 7-9.

Pitch tracking possible. Other properties may be applied
to entire scale.

() Low mean variety.
() Tetrachordal.

Singable.

() One interval class gives the same consonance mosts modes (5th).

As easy to locate yourself in a mode as it is in the entire scale
(efficiency).

() One int. class gives different consonances every mode (3rd).

Possible to harmonize with consecutive scale degrees without
timbre-fusing. Parallel fifths don't work because it's the same
interval every time, and you don't hear two parts. To put it
another way, you can write harmony parts that themselves are
shaped like the melody.

And since you've already learned the proper map for the diatonic
scale, it's easy to put the 22-tET diatonic in the same map.
Remember, propriety refers to a mental object, _not_ a scale.
Applying it to a scale assumes the listener has a matching map.
When I hear a new scale, I almost always hear subsets of a diatonic
scale. I've only recently learned to hear the wholetone and
pentatonic scales, and am still working on the diminished scale.

If you read R.'s papers, he _never_ applies the concept of
propriety like you guys tried to here.

-Carl