Organizing and classifying scales

There seems to be quite a good number of proper scales, and even the strictly proper scales are common in larger ETs. But it's clear that there are similarities between some of these scales. Porcupine[7] is a scale that exists in 15-ET as well as 22-ET (and also happens to be strictly proper). Could there be other 22-ET scales with 15-ET counterparts? Well, take the interval structure of a scale like the second one on the list (identified as "narada"), 4 3 3 3 3 4 2. If you subtract 1 from each step, you get a 15-ET scale: 3 2 2 2 2 3 1. As it happens, this is also strictly proper and has the same harmonic structure.

! 22-ET version 15-ET version
! (10) (07)
! 16 07 20 11 05 14
! 00 13 04 00 09 03
! 10 07

But you could also add 1 to both of the large steps and end up with a 24-ET scale of the same basic structure, 5 3 3 3 3 5 2. If you check the harmonic structure, though, you can see that it's not the same as the 15-ET and 22-ET versions.

! 11
! 08--22\
! 05 / \ / 17
! \00--14
! (11)

It seems like it would be useful to categorize scales according to the sequence of steps of different sizes. So all of these would end up as L M M M M L S. The harmonic structure depends on the temperament mapping, and there may be more than one for a particular ET, so these could be subcategories of the basic scale.

Each of these scale structures can have a number of different modes, so for convenience it would be nice to have a common label for the set. One way would be to order the modes in a sort of "alphabetical" order, with small steps before medium steps, etc.

S L M M M M L
M M M M L S L
M M M L S L M
M M L S L M M
M L S L M M M
L S L M M M M
L M M M M L S

Then the set of modes as a whole could be labeled something like

[S L M M M M L]

Or would it be better to just use numbers for the step sizes?

[1 3 2 2 2 2 3]

There could get to be a lot of different sizes of steps between small, medium, and large.

On 6/12/06, Herman Miller <hmiller@io.com> wrote:
> There seems to be quite a good number of proper scales, and even the
> strictly proper scales are common in larger ETs. But it's clear that
> there are similarities between some of these scales. Porcupine[7] is a
> scale that exists in 15-ET as well as 22-ET (and also happens to be
> strictly proper). Could there be other 22-ET scales with 15-ET
> counterparts? Well, take the interval structure of a scale like the
> second one on the list (identified as "narada"), 4 3 3 3 3 4 2. If you
> subtract 1 from each step, you get a 15-ET scale: 3 2 2 2 2 3 1. As it
> happens, this is also strictly proper and has the same harmonic structure.

I would analyse this as a non-contiguous subset of porcupine, just as
I analyse harmonic minor as a non-contiguous subset of meantone.

Keenan

>Or would it be better to just use numbers for the step sizes?
>
>[1 3 2 2 2 2 3]
>
>There could get to be a lot of different sizes of steps between small,
>medium, and large.

You'd wind up with what Rothenberg calls the rank order matrix
("show /ranking intervals" in Scala). According to R, listeners
categorize scales this way.

-Carl

Carl Lumma wrote:
>> Or would it be better to just use numbers for the step sizes?
>>
>> [1 3 2 2 2 2 3]
>>
>> There could get to be a lot of different sizes of steps between small, >> medium, and large.
> > You'd wind up with what Rothenberg calls the rank order matrix
> ("show /ranking intervals" in Scala).

How so? I get this when I try that command:

1/1 : 3 5 7 9 11 14 15
218.2 : 2 4 6 8 11 12 15
381.8 : 2 4 6 9 10 13 15
545.5 : 2 4 7 8 11 13 15
709.1 : 2 5 6 9 11 13 15
872.7 : 3 4 7 9 11 13 15
1090.9: 1 4 6 8 10 12 15

Maybe there's some other option in Scala which does what I've described; I'm not familiar with the many options. On the other hand, you can derive "1 3 2 2 2 2 3" from "1 4 6 8 10 12 15" by simply taking the difference between adjacent steps, so it might be just as well to use what's conveniently available.

At 05:39 PM 6/13/2006, you wrote:
>Carl Lumma wrote:
>>> Or would it be better to just use numbers for the step sizes?
>>>
>>> [1 3 2 2 2 2 3]
>>>
>>> There could get to be a lot of different sizes of steps between small,
>>> medium, and large.
>>
>> You'd wind up with what Rothenberg calls the rank order matrix
>> ("show /ranking intervals" in Scala).
>
>How so? I get this when I try that command:
>
> 1/1 : 3 5 7 9 11 14 15
> 218.2 : 2 4 6 8 11 12 15
> 381.8 : 2 4 6 9 10 13 15
> 545.5 : 2 4 7 8 11 13 15
> 709.1 : 2 5 6 9 11 13 15
> 872.7 : 3 4 7 9 11 13 15
> 1090.9: 1 4 6 8 10 12 15
>
>Maybe there's some other option in Scala which does what I've described;
>I'm not familiar with the many options. On the other hand, you can
>derive "1 3 2 2 2 2 3" from "1 4 6 8 10 12 15" by simply taking the
>difference between adjacent steps,

You don't have to -- it's the first column. Or am I missing
something?

-Carl

Hi Herman,

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...>
wrote:
>
> There seems to be quite a good number of proper scales,
> and even the strictly proper scales are common in larger
> ETs. But it's clear that there are similarities between
> some of these scales. Porcupine[7] is a scale that exists
> in 15-ET as well as 22-ET (and also happens to be strictly
> proper). Could there be other 22-ET scales with 15-ET
> counterparts? Well, take the interval structure of a scale
> like the second one on the list (identified as "narada"),
> 4 3 3 3 3 4 2. If you subtract 1 from each step, you get a
> 15-ET scale: 3 2 2 2 2 3 1. As it happens, this is also
> strictly proper and has the same harmonic structure.
>
> ! 22-ET version 15-ET version
> ! (10) (07)
> ! 16 07 20 11 05 14
> ! 00 13 04 00 09 03
> ! 10 07

I don't get the meaning of these numbers.

> But you could also add 1 to both of the large steps and
> end up with a 24-ET scale of the same basic structure,
> 5 3 3 3 3 5 2. If you check the harmonic structure,
> though, you can see that it's not the same as the 15-ET
> and 22-ET versions.
>
> ! 11
> ! 08--22\
> ! 05 / \ / 17
> ! \00--14
> ! (11)
>
> It seems like it would be useful to categorize scales
> according to the sequence of steps of different sizes.
> So all of these would end up as L M M M M L S.

Yes. A while ago (on one of these lists) I posted some
scale examples that require more than two types of step,
but I thought from the (lack of) response that no-one
wanted to know about them. Between "two-step" scales
and JI scales, there's quite a bit of territory to cover,
with three-step and four-step scales being the logical
next candidates for exploration, IMO.

> The
> harmonic structure depends on the temperament mapping,
> and there may be more than one for a particular ET, so
> these could be subcategories of the basic scale.

It probably also depends to a large degree on the way
you partition the octave [equivalent] into steps.

I'm thinking that in a moderately high-numbered ET - say
anything from 50-ET to 150-ET, the number of different
ways of assigning an integer to each of the 2, 3, 4 ...
step sizes can be quite large. The combinatorics of
partitions surely come in here. Whenever there are enough
steps available so that you can redistribute 1 unit of the
EDO from all of the steps of a given size among all those
of the next lower size without affecting their relative
sizes, you have at least two ways of partitioning the n
units of n-EDO over the different steps.

Eg 99-EDO, 7 steps, pattern L M M M M L S.
Average step size = 99/7
Try M = [average] = 14.
Then L = M+1 = 15 ==> S = 99 - 4*14 - 2*15 = 13.
And L = M+2 = 16 ==> S = 99 - 4*14 - 2*16 = 11, etc.

Denoting the solution set with M=14 by <L,14,S>
= {<15, 14, 13>, <16, 14, 11>, <17, 14, 9>,
<18, 14, 7>, <19, 14, 5>, <20, 14, 3>, <21, 14, 1>}
and the solution set with general M by <L,M,S>
= <L,Mmax,S> U ... U <L,14,S> U <L,13,S>
U <L,12,S> U ... U <L,Mmin,S>
(obviously there's a minimum and maximum M that
preserves the ordering L>M>S)
- you can see we have quite a lot of different possible
structures. The harmonic possiblilties of the solutions
<15, 14, 13> and <21, 14, 1> will be quite radically
different.

> Each of these scale structures can have a number of
> different modes, so for convenience it would be nice
> to have a common label for the set. One way would be to
> order the modes in a sort of "alphabetical" order, with
> small steps before medium steps, etc.

At least for three-step scales, the normal alphabetic
ordering reverses this, so it may be better to sort them
purely aphabetically, L before M before S.

> S L M M M M L
> M M M M L S L
> M M M L S L M
> M M L S L M M
> M L S L M M M
> L S L M M M M
> L M M M M L S
>
> Then the set of modes as a whole could be labeled something like
>
> [S L M M M M L]

or [L M M M M L S]

> Or would it be better to just use numbers for the step sizes?
>
> [1 3 2 2 2 2 3]

Probably would add to the confusion, since numbers
already have 2 (oops, too) many uses in tuning theory ...
I'd suggest instead using an otherwise rarely-used segment
of the alphabet for these step-size ordinals; explicitly,
how about LMNPQRS? This sequence has the implications
L (Largest)
M (2nd-largest, or Middle of 3)
N (3rd-largest)
...
R (2nd-smallest)
S (Smallest)

(I avoid using O since it's confusable with zero.)

> There could get to be a lot of different sizes of steps
> between small, medium, and large.

But would you need more than 7 step sizes in a scale?
Even three step sizes is quite a shift from our usual
two-step thinking. And music traditions that have
historically distingushed fine intervals still mostly
manage with seven scale degrees per octave, which suggests
it's unlikely you'd need more than seven step sizes.

That's why I think the LMNPQRS labelling above would be
adequate. If you did need more step sizes, they could
pass beyond S into the Tiny, Ultra-small region TUVWXYZ.

Regards,
Yahya

Carl Lumma wrote:
> At 05:39 PM 6/13/2006, you wrote:
>> Carl Lumma wrote:
>>>> Or would it be better to just use numbers for the step sizes?
>>>>
>>>> [1 3 2 2 2 2 3]
>>>>
>>>> There could get to be a lot of different sizes of steps between small, >>>> medium, and large.
>>> You'd wind up with what Rothenberg calls the rank order matrix
>>> ("show /ranking intervals" in Scala).
>> How so? I get this when I try that command:
>>
>> 1/1 : 3 5 7 9 11 14 15
>> 218.2 : 2 4 6 8 11 12 15
>> 381.8 : 2 4 6 9 10 13 15
>> 545.5 : 2 4 7 8 11 13 15
>> 709.1 : 2 5 6 9 11 13 15
>> 872.7 : 3 4 7 9 11 13 15
>> 1090.9: 1 4 6 8 10 12 15
>>
>> Maybe there's some other option in Scala which does what I've described; >> I'm not familiar with the many options. On the other hand, you can >> derive "1 3 2 2 2 2 3" from "1 4 6 8 10 12 15" by simply taking the >> difference between adjacent steps,
> > You don't have to -- it's the first column. Or am I missing
> something?

The symmetry of this scale confuses things; it looks like you're reading from bottom to top. Take for instance an arbitrary 12-ET scale, 1/1 200.0 300.0 500.0 700.0 1100.0 2/1: the step sizes in 12-ET are 2, 1, 2, 2, 4, 1. "Show /ranking intervals" gives me this:

1/1 : 2 3 5 7 11 12
200.0 : 1 3 5 9 10 12
300.0 : 2 4 8 9 11 12
500.0 : 2 6 7 9 10 12
700.0 : 4 5 7 8 10 12
1100.0: 1 3 4 6 8 12

What I want is the sequence of steps of different sizes (small, medium, large), which is 2 1 2 2 3 1. Then, considering all the modes of this scale, pick the one that comes earliest in the "alphabetical" order:

[1 2 1 2 2 3]

You can usually pick this out easily just by looking, but a more foolproof way would be to list all the possible rotations:

2 1 2 2 3 1
1 2 2 3 1 2
2 2 3 1 2 1
2 3 1 2 1 2
3 1 2 1 2 2
1 2 1 2 2 3

then sort the list and take the first one to identify the whole group:

1 2 1 2 2 3
1 2 2 3 1 2
2 1 2 2 3 1
2 2 3 1 2 1
2 3 1 2 1 2
3 1 2 1 2 2

And then label it in some sort of distinctive way so that we can identify ~1 2 1 2 2 3~, or ^a b a b b c^, or whatever, as referring to this whole class of related scales: "x y x y y z" and all of its rotations, where x, y, and z are positive integers, x<y, and y<z.

--- In tuning-math@yahoogroups.com, "yahya_melb" <yahya@...> wrote:

> At least for three-step scales, the normal alphabetic
> ordering reverses this, so it may be better to sort them
> purely aphabetically, L before M before S.

The trouble with this is that L, M, and S can have widely varying
sizes which can lead to scales which are really quite different. My
idea was somewhat along Herman's line; by classifying n-edo scales,
you can compare them to m-edo scales, which immediately locates them
in terms of a rank-two temperament.

> Take for instance an arbitrary 12-ET scale, 1/1
> 200.0 300.0 500.0 700.0 1100.0 2/1: the step sizes in
> 12-ET are 2, 1, 2, 2, 4, 1. "Show /ranking intervals"
> gives me this:
>
> 1/1 : 2 3 5 7 11 12
> 200.0 : 1 3 5 9 10 12
> 300.0 : 2 4 8 9 11 12
> 500.0 : 2 6 7 9 10 12
> 700.0 : 4 5 7 8 10 12
> 1100.0: 1 3 4 6 8 12
>
> What I want is the sequence of steps of different
> sizes (small, medium, large), which is 2 1 2 2 3 1.

As what? A way to categorize scales? Do you think
all scales sharing this pattern of 2nds will have
something in common?

-Carl

Carl Lumma wrote:
>> Take for instance an arbitrary 12-ET scale, 1/1 >> 200.0 300.0 500.0 700.0 1100.0 2/1: the step sizes in
>> 12-ET are 2, 1, 2, 2, 4, 1. "Show /ranking intervals"
>> gives me this:
>>
>> 1/1 : 2 3 5 7 11 12
>> 200.0 : 1 3 5 9 10 12
>> 300.0 : 2 4 8 9 11 12
>> 500.0 : 2 6 7 9 10 12
>> 700.0 : 4 5 7 8 10 12
>> 1100.0: 1 3 4 6 8 12
>>
>> What I want is the sequence of steps of different
>> sizes (small, medium, large), which is 2 1 2 2 3 1.
> > As what? A way to categorize scales? Do you think
> all scales sharing this pattern of 2nds will have
> something in common?

In the case of proper and strictly proper scales, they ought to have similar melodic properties at least. As I said in the original message, there is clearly a relationship between scales like 4 3 3 3 3 4 2 (22-ET) and 3 2 2 2 2 3 1 (15-ET). On the other hand, other scales with a similar pattern, such as 5 3 3 3 3 5 2 (24-ET) can have a different harmonic structure.

If you've got a list of scales that includes 4 3 3 3 3 4 2 and, for instance, 2 3 1 3 2 2 2, in no particular order, it may be hard to notice any relationship at all between the two scales. If they're notated as 0 4 7 10 13 16 20 and 0 2 5 6 9 11 13, the resemblance is even harder to notice. But if they're both listed as members of the [1 3 2 2 2 2 3] category, the resemblance is more likely to be noticed.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> If you've got a list of scales that includes 4 3 3 3 3 4 2 and, for
> instance, 2 3 1 3 2 2 2, in no particular order, it may be hard to
> notice any relationship at all between the two scales. If they're
> notated as 0 4 7 10 13 16 20 and 0 2 5 6 9 11 13, the resemblance is
> even harder to notice. But if they're both listed as members of the
[1 3
> 2 2 2 2 3] category, the resemblance is more likely to be noticed.

I'm going through the 20 "unambiguously meantone" 31-et proper scales.
By "unambiguously meantone" I mean they have a more compact
representation in terms of meantone than any other 31-et linear
temperament. Because of this, I can put the scales in a general form,
in terms of generators, which immediately relates them to other
meantone tunings. I can also put them into standard notation, but I
won't be able to do that with the unambiguously miracle scales, etc,
which I hope to get to. Anyway, as a result, because of the key place
of 31, I think I'll end up covering a lot of the basic territory of
7-note proper scales. How does this plan strike you? I've also looked
at 12, 19, and 22 some, and plan on using 24 as a way to explore the
Arabic-type scale world.

The first five of these are the same five from 12-et (diatonic,
melodic minor, harmonic minor, harmonic major and major locrian) which
illustrates the point about generality. Then you get fun stuff like:

Diminished diatonic: C Db E Fb G Ab Bb
Enharmonic major: C D E F G Bbb B
Enharmonic mixolydian: C D E F G Bbb Bb
Enharmonic major-minor: C D E E# G Ab Bb

> >> Take for instance an arbitrary 12-ET scale, 1/1
> >> 200.0 300.0 500.0 700.0 1100.0 2/1: the step sizes in
> >> 12-ET are 2, 1, 2, 2, 4, 1. "Show /ranking intervals"
> >> gives me this:
> >>
> >> 1/1 : 2 3 5 7 11 12
> >> 200.0 : 1 3 5 9 10 12
> >> 300.0 : 2 4 8 9 11 12
> >> 500.0 : 2 6 7 9 10 12
> >> 700.0 : 4 5 7 8 10 12
> >> 1100.0: 1 3 4 6 8 12
> >>
> >> What I want is the sequence of steps of different
> >> sizes (small, medium, large), which is 2 1 2 2 3 1.
> >
> > As what? A way to categorize scales? Do you think
> > all scales sharing this pattern of 2nds will have
> > something in common?
>
> In the case of proper and strictly proper scales, they
> ought to have similar melodic properties at least.

I'm sure there's a similarity, but as Gene points out,
the tuning differences still allowed among scales
sharing a 'rank 2nds profile' is probably significant.
It would be interesting to try it out vs. Rothenberg's
more complete tables, but my guess is I'd prefer the
latter.

-Carl

Carl Lumma wrote:

> I'm sure there's a similarity, but as Gene points out,
> the tuning differences still allowed among scales
> sharing a 'rank 2nds profile' is probably significant.
> It would be interesting to try it out vs. Rothenberg's
> more complete tables, but my guess is I'd prefer the
> latter.

I'm not familiar with "Rothenberg's more complete tables"; could you give a brief description? Are the example scales (0 3 5 7 9 11 14 of 15-ET, 0 4 7 10 13 16 20 of 22-ET) included in his list in some form, and do the tables make it apparent that they have a similar structure? What about modes (rotations) of these scales, like 0 2 5 6 9 11 13 of 15-ET?

It looks like I can get what I'm looking for with "show /ranking intervals" if I sort the table by the first column, then by the second column and so on if there's more than one row that sorts to the top. But I find the "modenam.par" format useful for showing where the different size steps are without having to do mental arithmetic.

Basically what I want is something to get a better handle on a scale than a Scala file with tuning in cents for each note. Otherwise it's going to be impossible to keep track of these. If there's already a classification system for these, that's great, but I'm not familiar with the tuning literature. I do try to keep up with this list most of the time, but I often get to the point where I'm too busy to pay attention to every thread, and if this has come up before, I missed it.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> In the case of proper and strictly proper scales, they ought to have
> similar melodic properties at least.

One thing to keep an eye on is multiplicative transformation,
especially interesting for prime number divisions. If I have a scale
which is proper and I think it belongs unambiguously to generator i,
what about the scale you get by using generator k? Sometimes that
might even preserve some harmonic meaning.

One of the temperaments I have yet to get to is hemimiracle, which is
generated by 14 steps out of 31 (an 11/8.) The wedgie is
<<12 -14 -4 -1 -50 -40 -43 30 46 11||. Take the OE part of that,
change signs and reverse the order and you get [1 4 14 -12], which
starts out looking like [1 4 10 -13]. So transformations between the
scales might be interesting. The MOS is not proper in 31-et, but it
has the same pattern of larger and small steps, and there are some
proper scales associated to it.

Hi Herman,

--- In tuning-math@yahoogroups.com, Herman Miller wrote:
[snip]
> ... Are the example scales (0 3 5 7 9 11 14 of
> 15-ET, 0 4 7 10 13 16 20 of 22-ET) included in his list in
> some form, and do the tables make it apparent that they have
> a similar structure?
> What about modes (rotations) of these scales, like
> 0 2 5 6 9 11 13 of 15-ET?
>
[snip]
>
> Basically what I want is something to get a better handle on
> a scale than a Scala file with tuning in cents for each note.
> Otherwise it's going to be impossible to keep track of these.
> If there's already a classification system for these, that's
> great ...
[snip]

What you're looking for is a classification of
scales as patterns of relative step sizes, right?

I don't believe I've ever seen anything more
pertinent than my suggestion to use the letters
LMNPQRSTUVWXYZ for decreasing relative sizes,
always starting with L for the largest and using
just as many letters as you have distinct sizes;
then grouping all rotations ("modes") of one scale
(sequence of relative stepsize letters), and using
the letter sequence for the mode which sorts first
lexically to represent the "melodic code" of the
scale.

Eg The pattern LPMMLNMM is an eight-step pattern,
with relative sizes L > M > N > P. Its eight
rotations are:
LPMMLNMM
PMMLNMML
MMLNMMLP
MLNMMLPM
LNMMLPMM
NMMLPMML
MMLPMMLN
MLPMMLNM

Sorted lexically (alphabetically), they are:
LNMMLPMM
LPMMLNMM
MLNMMLPM
MLPMMLNM
MMLNMMLP
MMLPMMLN
NMMLPMML
PMMLNMML

The first of these is LNMMLPMM, which becomes
the "melodic code" for the scale.

By convention, these letters represent (the
relatives sizes of) the ascending scale steps.
For scales which have different ascending and
descending forms, I guess it makes sense to
list both sequences, separated by, say, a pipe
character |.

Thus the rotation LNMMLPMM could be more fully
described as LNMMLPMM|MMPLMMNL, although the
second half could be left of for conciseness
when both ascending and descending forms are
the same.

I note that this "melodic code" convention
implies nothing particular about harmonic uses
of the scale.

Is there a more useful way of classifying scales
*melodically*? (Naming a scale, although more
concise, is not as informative.)

Regards,
Yahya

> > I'm sure there's a similarity, but as Gene points out,
> > the tuning differences still allowed among scales
> > sharing a 'rank 2nds profile' is probably significant.
> > It would be interesting to try it out vs. Rothenberg's
> > more complete tables, but my guess is I'd prefer the
> > latter.
>
> I'm not familiar with "Rothenberg's more complete tables";

I was just referring to the rank-order matrices we've
been discussing.

> Are the example scales (0 3 5 7 9 11 14 of
> 15-ET, 0 4 7 10 13 16 20 of 22-ET) included in his list in
> some form, and do the tables make it apparent that they
> have a similar structure?

Try 'em in Scala and see. Manuel, how do you enter ET
subsets into Scala? I've never known a convenient way.

> What about modes (rotations) of these scales, like
> 0 2 5 6 9 11 13 of 15-ET?

Rothenberg's technique gives the same answer for
any rotation.

> Basically what I want is something to get a better handle
> on a scale than a Scala file with tuning in cents for each
> note. Otherwise it's going to be impossible to keep track
> of these.

The rank-order matrix might not be what you want, but at
least scales sharing a rank-order matrix are supposed to
sound the same. I'm pretty sure I could come up with an
example of two scales that were completely different but
had the same notation you've described, if I wasn't late
for brunch.

> If there's already a
> classification system for these, that's great,

I don't know what scale types you're referring to, but
I've always thought someone should initiate a program to
catalog scales by rank-order matrix.

-Carl