EDO classification by families of scale patterns

Lately I've been toying with the idea of devising a classification
system for EDO's based on the scale shapes they support (as opposed to
their harmonic properties/vanishing commas, as has been thoroughly
done). Obviously there's not much to it, since the scale types a
given EDO would support is just a function of its cardinality, but I
think it would be very useful to have a reference chart of such data
(and appropriate terminology to refer to it), especially for the
polymicrotonal composer.

In case it's not obvious what I mean by "scale shapes", I'm referring
specifically to n-tone MOS scales with x number of large steps (l) and
y number of small steps (s), where n=x+y. Thus I'd group EDOs
together that all support, say, 7-note scales with 5(l)+2(s)...or
8-note scales with 3(l)+5(s) [written such to include all possible
modes/permutations of l's and s's]. For the moment I'm going to limit
myself to EDOs < 36, and scales of no more than 10 steps. Also, the
ratio of l:s shouldn't exceed 5:1, because most listeners would find
such scales rather lopsided and unpleasant (I think?).

Has this been done before? If so, where can I find information on it?
If not, then I have a question or two. Namely, how should I name
each family? For instance, there are 5 families of 5-tone(pentatonic)
scales: 5(l=s), 4(l)+1(s), 3(l)+2(s), 2(l)+3(s), and 1(l)+4(s). Can
anyone suggest a methodology to name them? Obviously the first one
could be something like "uniform pentatonic"; and perhaps the
2(l)+3(s) family should be given something to reflect that it is the
"standard" common practice shape?

Many thanks,

-Igliashon

Igliashon Jones wrote:

> In case it's not obvious what I mean by "scale shapes", I'm referring
> specifically to n-tone MOS scales with x number of large steps (l) and
> y number of small steps (s), where n=x+y. Thus I'd group EDOs
> together that all support, say, 7-note scales with 5(l)+2(s)...or
> 8-note scales with 3(l)+5(s) [written such to include all possible
> modes/permutations of l's and s's]. For the moment I'm going to limit
> myself to EDOs < 36, and scales of no more than 10 steps. Also, the
> ratio of l:s shouldn't exceed 5:1, because most listeners would find
> such scales rather lopsided and unpleasant (I think?).

So what are "l" and "s"? You only defined "x" and "y".

> Has this been done before? If so, where can I find information on it?
> If not, then I have a question or two. Namely, how should I name
> each family? For instance, there are 5 families of 5-tone(pentatonic)
> scales: 5(l=s), 4(l)+1(s), 3(l)+2(s), 2(l)+3(s), and 1(l)+4(s). Can
> anyone suggest a methodology to name them? Obviously the first one
> could be something like "uniform pentatonic"; and perhaps the
> 2(l)+3(s) family should be given something to reflect that it is the
> "standard" common practice shape?

It sounds like the Scale Tree. There'll be things about it at the Wilson Archives, and here's a mathematical introduction:

http://www.cut-the-knot.org/blue/Stern.shtml

You can name them with the numbers, although of course there are different ways to define that so make sure you explain it in the context. Many families now will have names from the temperaments they support, and if you aren't a purist you can always use those names.

It happens that the same EDO will appear multiple times on the chart because of the multiple contexts it can be used in. I don't think you can get round that.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> Igliashon Jones wrote:
>
> > In case it's not obvious what I mean by "scale shapes", I'm referring
> > specifically to n-tone MOS scales with x number of large steps (l) and
> > y number of small steps (s), where n=x+y. Thus I'd group EDOs
> > together that all support, say, 7-note scales with 5(l)+2(s)...or
> > 8-note scales with 3(l)+5(s) [written such to include all possible
> > modes/permutations of l's and s's]. For the moment I'm going to limit
> > myself to EDOs < 36, and scales of no more than 10 steps. Also, the
> > ratio of l:s shouldn't exceed 5:1, because most listeners would find
> > such scales rather lopsided and unpleasant (I think?).
>
> So what are "l" and "s"? You only defined "x" and "y".
>

l and s are the sizes of the large and the small step, in units of the smallest
step of the EDO, or not?

Well, if you take k as the subdivision of the EDO, the values x, y, l and s
fulfill

x*l + y*s = k

which is a linear Diophantine equation. It appears to me that this probably has
been done before - dunno though. Maybe I will try to do it...
In any case, you might check the archives in tuning-math for "diophantine".
I cross-posted this to tuning-math.

BTW, is "City of the Asleep" not on soundclick any more? I am missing it...

Hans Straub

These can all be seen as sets and subsets on the scale tree where they will take up whole bands of fractions.
another way to see these is via the horograms, since each one is the 'archetype' of all the scales that lie below it.
quicker to see visually

From: "Igliashon Jones" <igliashon@sbcglobal.net>
Subject: EDO classification by families of scale patterns

Lately I've been toying with the idea of devising a classification
system for EDO's based on the scale shapes they support (as opposed to
their harmonic properties/vanishing commas, as has been thoroughly
done). Obviously there's not much to it, since the scale types a
given EDO would support is just a function of its cardinality, but I
think it would be very useful to have a reference chart of such data
(and appropriate terminology to refer to it), especially for the
polymicrotonal composer.

In case it's not obvious what I mean by "scale shapes", I'm referring
specifically to n-tone MOS scales with x number of large steps (l) and
y number of small steps (s), where n=x+y. Thus I'd group EDOs
together that all support, say, 7-note scales with 5(l)+2(s)...or
8-note scales with 3(l)+5(s) [written such to include all possible
modes/permutations of l's and s's]. For the moment I'm going to limit
myself to EDOs < 36, and scales of no more than 10 steps. Also, the
ratio of l:s shouldn't exceed 5:1, because most listeners would find
such scales rather lopsided and unpleasant (I think?).

Has this been done before? If so, where can I find information on it?
If not, then I have a question or two. Namely, how should I name
each family? For instance, there are 5 families of 5-tone(pentatonic)
scales: 5(l=s), 4(l)+1(s), 3(l)+2(s), 2(l)+3(s), and 1(l)+4(s). Can
anyone suggest a methodology to name them? Obviously the first one
could be something like "uniform pentatonic"; and perhaps the
2(l)+3(s) family should be given something to reflect that it is the
"standard" common practice shape?

Many thanks,

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Yay! And ambivalence towards the potential of new terminology.

One distinction that could be helpful in naming or classifying these
is whether or not the small steps (s) outnumber the large steps (l).
My first guess is to call this fair/unfair; if there were a MOS-fight,
the small guys should at least hope to outnumber the big guys.

With this plus x and y and l:s, you can specify any of those scales
with little ambiguity. Except for mode.

I'd also like a name for MOSes that haven't hit the period and wrapped
around yet; e.g. a 6+1 5:2 scale, fair or unfair. Virgin MOS? I feel
like I've proposed this before. Apologies.

And a way to relate generators to one another by multiple. Which seems
important because they share notes. For example, doubling the 4/31
generator to 8/31 results in a new system in which half the notes
generated are shared among both. Of course in this example they'll hit
all the notes of 31 eventually...but locally...

Now, on to scales with three step sizes...

"Igliashon Jones" <igliashon@sbcglobal.net> writes:

> For the moment I'm going to limit
> myself to EDOs < 36, and scales of no more than 10 steps. Also, the
> ratio of l:s shouldn't exceed 5:1, because most listeners would find
> such scales rather lopsided and unpleasant (I think?).

You might also think carefully about scales where the ratio of l:s is
*small*, since such scales themselves are approximate EDOs, and the
distinction between n*l+m*s and m*l+n*s may not be particularly
useful. For instance, 32-EDO supports a scale of 2 steps of size 7
and 3 steps of size 6; 33-EDO supports a scale of 3 steps of size 7
and 2 steps of size 6. These are both approximate 5-EDOs, and
depending on your purposes it may or may not be useful to, say, put
the former in the same category as the 12-EDO pentatonic scale of 3
steps of size 2 and 2 steps of size 3 while putting the other in an
entirely different category.

- Rich Holmes

> So what are "l" and "s"? You only defined "x" and "y".

Oops. Knew I'd miss something. As Hans said, "l" and "s" are
essentially variables, the size of the large and small steps
(respectively) in units of EDO-steps relative to cardinality. Thus EDO
cardinality (C)=x(l)+y(s).

> It sounds like the Scale Tree. There'll be things about it at the
> Wilson Archives, and here's a mathematical introduction:
>

I see similarities (I think), but it's far from a comprehensive
classification of the EDO's.

>
> You can name them with the numbers, although of course there are
> different ways to define that so make sure you explain it in the
> context. Many families now will have names from the temperaments they
> support, and if you aren't a purist you can always use those names.

Well, as Paul Erlich explained to me, those temperament names are
based more on harmonic properties than scale shapes. For instance the
Orwell family produces a 9-note MOS of 4(l)+5(s), but while both
22-EDO and 13-EDO support that scale shape, only 22 is consider an
Orwell temperament. So rather than start confusing people by using
the same names but in different ways, I'd like a purer terminology
that is specifically descriptive of the attributes of the scale.

> It happens that the same EDO will appear multiple times on the chart
> because of the multiple contexts it can be used in. I don't think you
> can get round that.

Nor would I want to! That is precisely the point, to cluster EDO's
together in different ways to see which ones would share similar
melodic properties. That way, 12 17 19 22 24 26 27 29 an 31 would all
be in the family of 7-note scales of shape 5(l)+2(s), instead of being
broken up into different groups because of their differing harmonic
properties.

-Igs

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> These can all be seen as sets and subsets on the scale tree where
they will take up whole bands of fractions.
> another way to see these is via the horograms, since each one is
the 'archetype' of all the scales that lie below it.
> quicker to see visually

Can you post a link for me, Kraig? I thought that horograms were more
related to temperaments (i.e. they come from the repetition of certain
generator intervals). In fact what originally led me to this was
arguing with Paul Erlich about whether 13-EDO was an Orwell
temperament, because the Horagram he drew up of Orwell suggested to me
that 13 would be a member because of its shape, whereas he informed me
that it is NOT because it lacks the correct harmonic structures.
Likewise, there are plenty of different horagrams that produce the
same scale shapes. Is there a different sense in which horagrams are
use? Please educate me.

-Igs

distinction that could be helpful in naming or classifying these
> is whether or not the small steps (s) outnumber the large steps (l).
> My first guess is to call this fair/unfair; if there were a MOS-fight,
> the small guys should at least hope to outnumber the big guys.

I like it. Clever clever!

> With this plus x and y and l:s, you can specify any of those scales
> with little ambiguity. Except for mode.

Mode isn't really a worry to me, since all EDOs that share scale
shapes will necessarily contain all the same modes.

While it would be a simple matter to call the scales "x+y n-tonic",
that lacks elegance. It will do in a pinch, but I was hoping there
might (in the typical "english scientist" way) be some handy
descriptive names that I could string together. I have a few ideas
myself, I just wanted to see if there was a "proper" way of going
about it.

> I'd also like a name for MOSes that haven't hit the period and wrapped
> around yet; e.g. a 6+1 5:2 scale, fair or unfair. Virgin MOS? I feel
> like I've proposed this before. Apologies.

Whoa, you're getting ahead of me here!

> And a way to relate generators to one another by multiple. Which seems
> important because they share notes. For example, doubling the 4/31
> generator to 8/31 results in a new system in which half the notes
> generated are shared among both. Of course in this example they'll hit
> all the notes of 31 eventually...but locally...

I kind of want to keep generators out of this, because scales formed
from different generators can have the same (absolute) shape.
Consider meantone vs. superpyth. Both make 5(l)+2(s) heptatonic scales.

> Now, on to scales with three step sizes...

Soon enough!

-Igs

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> Igliashon Jones wrote:
>
> > In case it's not obvious what I mean by "scale shapes", I'm
referring
> > specifically to n-tone MOS scales with x number of large steps
(l) and
> > y number of small steps (s), where n=x+y. Thus I'd group EDOs
> > together that all support, say, 7-note scales with 5(l)+2(s)...or
> > 8-note scales with 3(l)+5(s) [written such to include all possible
> > modes/permutations of l's and s's]. For the moment I'm going to
limit
> > myself to EDOs < 36, and scales of no more than 10 steps. Also,
the
> > ratio of l:s shouldn't exceed 5:1, because most listeners would
find
> > such scales rather lopsided and unpleasant (I think?).
>
> So what are "l" and "s"? You only defined "x" and "y".
>
> > Has this been done before? If so, where can I find information
on it?
> > If not, then I have a question or two. Namely, how should I name
> > each family? For instance, there are 5 families of 5-tone
(pentatonic)
> > scales: 5(l=s), 4(l)+1(s), 3(l)+2(s), 2(l)+3(s), and 1(l)+4(s).
Can
> > anyone suggest a methodology to name them? Obviously the first
one
> > could be something like "uniform pentatonic"; and perhaps the
> > 2(l)+3(s) family should be given something to reflect that it is
the
> > "standard" common practice shape?
>
> It sounds like the Scale Tree. There'll be things about it at the
> Wilson Archives, and here's a mathematical introduction:
>
> http://www.cut-the-knot.org/blue/Stern.shtml
>
> You can name them with the numbers, although of course there are
> different ways to define that so make sure you explain it in the
> context. Many families now will have names from the temperaments
they
> support, and if you aren't a purist you can always use those names.

I don't see how the Scale Tree can deal with the families where the
period is a fraction of an octave, such as the Diminished, Augmented,
Srutal etc.

> It happens that the same EDO will appear multiple times on the
chart
> because of the multiple contexts it can be used in. I don't think
you
> can get round that.
>
>
> Graham

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> These can all be seen as sets and subsets on the scale tree where
>they will take up whole bands of fractions.
>
> another way to see these is via the horograms, since each one is
the 'archetype' of all the scales that lie below it.
> quicker to see visually

This would seem to miss out on many important families. Some specific
scale types include:

Augmented[6]: 3(l)+3(s)
Diminished[8]: 4(l)+4(s)
Srutal[10]: 2(l)+8(s)
Blackwood[10]: 5(l)+5(s)

And your most recent remarks seemed to indicate that, at least in their
most symmetrical arrangement, such scales would count as MOS.

Hi Igs, I'm so delighted that you're back!

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...>
wrote:
> > So what are "l" and "s"? You only defined "x" and "y".
>
> Oops. Knew I'd miss something. As Hans said, "l" and "s" are
> essentially variables, the size of the large and small steps
> (respectively) in units of EDO-steps relative to cardinality. Thus
EDO
> cardinality (C)=x(l)+y(s).
>
> > It sounds like the Scale Tree. There'll be things about it at
the
> > Wilson Archives, and here's a mathematical introduction:
> >
>
> I see similarities (I think), but it's far from a comprehensive
> classification of the EDO's.

It would be a comprehensive classification of the scale types you're
interested in, except that it's missing diminished, augmented,
blackwood, srutal etc. . . . it's missing all the cases where x and y
have a common divisor greater than 1.

> Nor would I want to! That is precisely the point, to cluster EDO's
> together in different ways to see which ones would share similar
> melodic properties. That way, 12 17 19 22 24 26 27 29 an 31 would
all
> be in the family of 7-note scales of shape 5(l)+2(s), instead of
being
> broken up into different groups because of their differing harmonic
> properties.

But each of these EDOs is capable of so many scale shapes! You'd
hardly expect this cluster to hold up as a cluster when looking at
almost any of the other scale shapes . . .

Best,
Paul

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> > These can all be seen as sets and subsets on the scale tree where
> they will take up whole bands of fractions.
> > another way to see these is via the horograms, since each one is
> the 'archetype' of all the scales that lie below it.
> > quicker to see visually
>
> Can you post a link for me, Kraig? I thought that horograms were more
> related to temperaments

Absolutely not -- they have nothing inherently to do with temperament.
Some 2D tuning systems may be derived through tempering a higher-
dimensional JI system (such as 5-limit JI, which is 3D). But it's
perfectly possible to specify 2D tuning systems any way you want,
without any reference to an underlying JI 'ideal' or anything like
that. And any 2D tuning system can be displayed as a horagram (you just
need to specify an interval of equivalence, usually the octave, which
occurs somewhere in the tuning system).

> (i.e. they come from the repetition of certain
> generator intervals).

All of the scales you're interested in can be constructed in
essentially this way, can't they? Perhaps you didn't realize that!

> In fact what originally led me to this was
> arguing with Paul Erlich about whether 13-EDO was an Orwell
> temperament, because the Horagram he drew up of Orwell suggested to me
> that 13 would be a member because of its shape, whereas he informed me
> that it is NOT because it lacks the correct harmonic structures.

Although I also sent you a pretty huge caveat to that judgment, if you
recall.

> Likewise, there are plenty of different horagrams that produce the
> same scale shapes.

Wilson's Golden Horagrams pretty much keep to l:s in the golden ratio,
wherever possible. So in a sense, there normally is only one Golden
Horagram for the scale shape you're interested in. One could add to
Wilson's collection by also allowing cases where x and y have a common
denominator greater than 1.

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:

> > And a way to relate generators to one another by multiple. Which
seems
> > important because they share notes. For example, doubling the 4/31
> > generator to 8/31 results in a new system in which half the notes
> > generated are shared among both. Of course in this example they'll
hit
> > all the notes of 31 eventually...but locally...
>
> I kind of want to keep generators out of this, because scales formed
> from different generators can have the same (absolute) shape.
> Consider meantone vs. superpyth. Both make 5(l)+2(s) heptatonic
>scales.

But Igs, they have the *same* generator and period in this sense! Both
systems have a generator of 3*l+s and a period of 5*l+2*s. If you're
talking about 'shape' in a qualitative sense (is that what you mean by
absolute?), then you should think about the generators in this way too.
I think you may be missing something pretty crucial here . . .

wallyesterpaulrus wrote:

> I don't see how the Scale Tree can deal with the families where the > period is a fraction of an octave, such as the Diminished, Augmented, > Srutal etc.

Use a different tree for each period. The smaller the period, the smaller the tree you need.

If you want everything in two dimensions, plot a function of the generator size against a function of the period size. This will give the same grouping as the scale tree in one direction. The period=octave case will be the Farey sequence. Other temperaments will probably have to be redundantly placed, so that diaschismic shows up as having a generator of a perfect fourth as well as a semitone for example.

Graham

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:
>
> Well, as Paul Erlich explained to me, those temperament names are
> based more on harmonic properties than scale shapes. For instance the
> Orwell family produces a 9-note MOS of 4(l)+5(s), but while both
> 22-EDO and 13-EDO support that scale shape, only 22 is consider an
> Orwell temperament.

But they both are Orwell MOS, and this is closely related. The
semiconvergents of the classic Orwell generator of 19/84 are

1/4, 1/5, 2/9, 3/13, 5/22, 7/31, 12/53, 19/84

Looking at denomiators, we see that 9 notes of 13, 22, 31, 53 and 84
are all related in this sense. Moreover, just from the information "9"
and "13", we can obtain an interval related to the two, by taking the
penultimate convergent. The convergents of 9/13 are

0, 1, 2/3, 9/13

so the penultimate convergent is 2/3. Taking the ratios of the
numerators and of the denominators gives us 2/9 and 3/13, which
therefore gives us the interval [2/9, 3/13], and we can classify any
rational number in this interval as belonging to 9 and 13. Taking the
Farey sequence up to denominator 84 for this interval, for instance,
leads to

2/9 17/76 15/67 13/58 11/49 9/40 16/71 7/31 19/84 12/53
17/75 5/22 18/79 13/57 8/35 19/83 11/48 14/61 17/74 3/13

This associated interval, and hence the associated set of rational
numbers in it, is a function of the two numbers, and is therefore
asociated in a canonical way to the idea of "9 out of 13". The
interval for one pair of numbers can be contained in the interval for
another (for example, "22 out of 31" is contained in "9 out of 13".)
We also might redefine things so that complementary intervals are
regarded as equivalent. For instance, "13 out of 22" leads to
a penultimate convergent of 10/17, and hence to [10/13, 17/22]. But
this interval, by taking 1-endpoints, is equivalent to [3/13, 5/22],
so we can also regard "13 out of 22" as contained in "9 out of 13",
and also "9 out of 22" as contained in "9 out of 13", for example.

So rather than start confusing people by using
> the same names but in different ways, I'd like a purer terminology
> that is specifically descriptive of the attributes of the scale.

> > It happens that the same EDO will appear multiple times on the chart
> > because of the multiple contexts it can be used in. I don't think
you
> > can get round that.
>
> Nor would I want to! That is precisely the point, to cluster EDO's
> together in different ways to see which ones would share similar
> melodic properties. That way, 12 17 19 22 24 26 27 29 an 31 would all
> be in the family of 7-note scales of shape 5(l)+2(s), instead of being
> broken up into different groups because of their differing harmonic
> properties.
>
> -Igs

Graham wrote...

> If you want everything in two dimensions, plot a function of the
> generator size against a function of the period size. This will
> give the same grouping as the scale tree in one direction. The
> period=octave case will be the Farey sequence. Other temperaments
> will probably have to be redundantly placed, so that diaschismic
> shows up as having a generator of a perfect fourth as well as a
> semitone for example.

I'd love to see this!
-Carl

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> wallyesterpaulrus wrote:
>
> > I don't see how the Scale Tree can deal with the families where
the
> > period is a fraction of an octave, such as the Diminished,
Augmented,
> > Srutal etc.
>
> Use a different tree for each period. The smaller the period, the
> smaller the tree you need.
>
> If you want everything in two dimensions, plot a function of the
> generator size against a function of the period size.

Which functions do you have in mind here?

> This will give
> the same grouping as the scale tree in one direction. The
period=octave
> case will be the Farey sequence. Other temperaments will probably
have
> to be redundantly placed, so that diaschismic shows up as having a
> generator of a perfect fourth as well as a semitone for example.

Hmm . . . I'm not sure if the "treeness" will be preserved here.

Rather that a 1-dimensional scale tree growing through an additional
dimension, I've proposed a 2-dimensional scale tree (that can be seen
to grow through a third dimension). This doesn't leave out systems
where the period is a fraction of the octave of equivalence.

/tuning/files/perlich/tree.gif

This was drawn with approximation to JI in mind but need not be seen
that way -- just ignore the grid, and zoom out to include more
numbers and more occurences of numbers.

I left out the branches, but you can see that each ET occurs at a
weighted midpoint along the line connecting several pairs of larger-
font ("closer"), smaller-number ETs, and each pair adds up to the ET.

For example, 12 occurs as 7+5 (which specifies the fourth and octave
as generator and period), 3+9 (which specifies the semitone and major
third as generator and period), and 4+8 (which specifies the semitone
and minor third as generator and period). [I used interval names to
mean specifically 12-equal intervals here.]

15 occurs as 7+8 (which specifies 2/15 oct. and 1 oct. as generator
and period), 12+3 (which specifies 1/15 oct. and 5/15 = 1/3 oct. as
generator and period), and 10+5 (which specifies 1/15 oct. and 3/15 =
1/5 oct. as generator and period).

19 occurs as 12+7 (which specifies 8/19 oct. and 1 oct. as generator
and period), 10+9 (which specifies 2/19 oct. and 1 oct. as generator
and period), and 16+3 (which specifies 6/19 oct. and 1 oct. as
generator and period).

A point slightly away from 12 along any of the three lines I
mentioned in connection with 12 above would correspond to what Igs is
asking about, a scale with 7 steps of one size and 5 steps of another
size, or 3 steps of one size and 9 steps of another size, or 4 steps
of one size and 8 steps of another size, respectively. If you're on
the line between 7 and 5, being much closer to 7 means 7 large, 5
small; being much closer to 5 means 5 large, 7 small. Each 'branch'
of the tree would correspond to one of Igs' "scale shapes", and any
ET lying 'over' this branch would support the relevant scale shape.

It would be nice to have a rotatable, 3D model of this tree, complete
with branches.

It seems a single, 3-dimensional tree structure could truly represent
and meaningfully connect all these scale types.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> > Nor would I want to! That is precisely the point, to cluster EDO's
> > together in different ways to see which ones would share similar
> > melodic properties.

> > That way, 12 17 19 22 24 26 27 29 an 31 would all
> > be in the family of 7-note scales of shape 5(l)+2(s)...

I posted my response by accident before finishing. I was thinking of
adding how the "7 out of n" intervals I defined work by containment,
which could be used to make a tree structure. Using "<" to indicate
containment of corresponding intervals, we get:

26<19<12, 29<22, 24<27, 24<31

7 out of 31 isn't in the 26<19<12 chain since the proceedure gives the
9/31 generator MOS instead, associated to the semififths temperament.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> 7 out of 31 isn't in the 26<19<12 chain since the proceedure gives the
> 9/31 generator MOS instead, associated to the semififths temperament.

The reason for that is simple: the MOS is more regular. We get
5,4,5,4,5,4,4 instead of 5,5,5,3,5,5,3 for the diatonic scale. This
approach always homes in on the MOS with l/s closest to 1. However, we
get 31<19<12 by looking at 12 out of 19 instead.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> > 7 out of 31 isn't in the 26<19<12 chain since the proceedure gives
the
> > 9/31 generator MOS instead, associated to the semififths
temperament.
>
> The reason for that is simple: the MOS is more regular. We get
> 5,4,5,4,5,4,4 instead of 5,5,5,3,5,5,3 for the diatonic scale. This
> approach always homes in on the MOS with l/s closest to 1. However, we
> get 31<19<12 by looking at 12 out of 19 instead.

In academic musical "set-theory" parlance, the maximally even 7-out-of-
31 is not the diatonic scale, but the second-order maximally even 7-out-
of-12-out-of-31 is.

> But each of these EDOs is capable of so many scale shapes! You'd
> hardly expect this cluster to hold up as a cluster when looking at
> almost any of the other scale shapes . . .

That's the point! The point isn't to establish universal clusters at
all, it's to name scale families and then make a table of which EDOs
contain them. That way you would have a handy reference table for all
EDOs that allowed, say, heptatonic scales of 5(s)+2(l), or
?nonatonic?(9-note) scales of 5(s)+4(l). The idea is that if you had
a particular melody that was composed in an MOS scale (or I guess
really any scale of only 2 step sizes), and wanted to try out some
variations on it, you could just look at the table and see what other
EDO's would contain similar shapes (but with different sizes) or maybe
the opposite scale shape, or some other such purpose.

It will probably make more sense when I actually DO it.

-Igs

> Absolutely not -- they have nothing inherently to do with temperament.

Okay.

> All of the scales you're interested in can be constructed in
> essentially this way, can't they? Perhaps you didn't realize that!

Not true! Meantone and Superpyth both have different generators
(flattened vs. sharpened 3:2) but both create heptatonic scales of
5(l)+2(s)! It is in this sense that I consider 22-EDO, 12-EDO,
19-EDO, 17-EDO, 26-EDO, 27-EDO etc etc to be related, inspite of the
fact that their harmonic capabilities are vastly different, and it is
to this purpose that I have started this excercise (I feel there
should be some simple easy way of showing their similarities).

> Wilson's Golden Horagrams pretty much keep to l:s in the golden ratio,
> wherever possible. So in a sense, there normally is only one Golden
> Horagram for the scale shape you're interested in. One could add to
> Wilson's collection by also allowing cases where x and y have a common
> denominator greater than 1.

But I'm not interested in Golden Horagrams for this purpose (though I
am generally fascinated by them), since scales with l:s equal to phi
are only approximated by EDOs, and I'm not considering these scales or
the EDOs that contain them to be approximations to anything.

-Igs

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
> But Igs, they have the *same* generator and period in this sense! Both
> systems have a generator of 3*l+s and a period of 5*l+2*s. If you're
> talking about 'shape' in a qualitative sense (is that what you mean by
> absolute?), then you should think about the generators in this way too.
> I think you may be missing something pretty crucial here . . .

Ooooohhhhhh...I getcha. That certainly isn't a familiar usage of the
term "generator" to me, though. I thought it referred to an absolute
pitch? Or is it context-dependent? Anyway, I don't think it
necessary or helpful to include period and generator, since those will
only produce certain permutations of l's and s's, leaving out others.
Not that that matters in reality, since it wouldn't change the family
groupings, but I don't really see how including them would improve
anything.

-Igs

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:
>
> > But each of these EDOs is capable of so many scale shapes! You'd
> > hardly expect this cluster to hold up as a cluster when looking at
> > almost any of the other scale shapes . . .
>
> That's the point! The point isn't to establish universal clusters at
> all,

Thank goodness!

> it's to name scale families

Since you don't want to refer to temperament, a good naming system
might be the a+b nomenclature that came up on tuning-math.

> and then make a table of which EDOs
> contain them. That way you would have a handy reference table for all
> EDOs that allowed, say, heptatonic scales of 5(s)+2(l),

2+5: I might call these pelogic or mavila or something.

> or
> ?nonatonic?(9-note) scales of 5(s)+4(l).

4+5.

The "3D" tree that I illustrated allows you to do just that. For
example, consider the traditional diatonic case of 5(l)+2(s), or 5+2.
You'd look at the line that connects 5 and 5+2=7 on the chart, and
*all* the numbers on the chart correspond to ETs which support such
scales.

/tuning/files/perlich/tree.gif

Font size is inversely related to the cardinality of the ET here, so
you're just missing out on large-cardinality ETs if you can't read the
small numbers. I can see:

(7), 40, 33, 26, 45, 19, 50, 31, 43, 55, 12, 17, (5)

All of these ETs support 5(l)+2(s) scales. Actually, this particular
chard leaves off a lot of ET-points because it was geared toward
approximating 5-limit harmony. I should make another chart for you that
drops this restriction, allowing for a fuller answer to all your
questions all on one single chart.

> The idea is that if you had
> a particular melody that was composed in an MOS scale (or I guess
> really any scale of only 2 step sizes), and wanted to try out some
> variations on it, you could just look at the table and see what other
> EDO's would contain similar shapes (but with different sizes) or maybe
> the opposite scale shape, or some other such purpose.

Sure.

> It will probably make more sense when I actually DO it.

Let me know if you think I'm on the right track, and I'll make a more
general (fuller, less geared toward 5-limit approximations) chart for
you.

> Use a different tree for each period. The smaller the period, the
> smaller the tree you need.
>
> If you want everything in two dimensions, plot a function of the
> generator size against a function of the period size.

This is far more mathematical and complex than I wanted to go. What
I'm interested in primarily (to stick to the subject) is NAMING.
NAMING families of scales. How you derive them, how you arrange them,
how you display them...I don't really care. I'm interested in simply
naming, so that one could say 12, 17, 19, 22, 24, 26, 27, 29,
31...etc. all belong to the "such-and-such" scale family, because they
all contain heptatonic scales of 5(l)+2(s).

Capice?

-igs

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...>
wrote:
>
> > Absolutely not -- they have nothing inherently to do with
temperament.
>
> Okay.
>
> > All of the scales you're interested in can be constructed in
> > essentially this way, can't they? Perhaps you didn't realize that!
>
> Not true! Meantone and Superpyth both have different generators
> (flattened vs. sharpened 3:2) but both create heptatonic scales of
> 5(l)+2(s)!

So what? That doesn't prevent all the scales you're interested in to
be expressed in terms of some period and generator.

Plus, *all* evenly-distributed heptatonic scales of 5(l)+2(s) can be
constructed using a generator of 3*l+s and a period of 5*l+2*s. In
this very important (for you!) sense, Meantone and Superpyth have the
*same* generators.

> It is in this sense that I consider 22-EDO, 12-EDO,
> 19-EDO, 17-EDO, 26-EDO, 27-EDO etc etc to be related, inspite of the
> fact that their harmonic capabilities are vastly different, and it
is
> to this purpose that I have started this excercise (I feel there
> should be some simple easy way of showing their similarities).

Yes, there is. They all fall along the line connecting 5 and 7 on the
chart I proposed (similar to the one I've displayed and have been
giving links to).

> But I'm not interested in Golden Horagrams for this purpose (though
I
> am generally fascinated by them), since scales with l:s equal to phi
> are only approximated by EDOs, and I'm not considering these scales
or
> the EDOs that contain them to be approximations to anything.

Excellent. Then my charts with all the ETs on them should be exactly
what you need.

> I posted my response by accident before finishing. I was thinking of
> adding how the "7 out of n" intervals I defined work by containment,
> which could be used to make a tree structure. Using "<" to indicate
> containment of corresponding intervals, we get:
>
> 26<19<12, 29<22, 24<27, 24<31
>
> 7 out of 31 isn't in the 26<19<12 chain since the proceedure gives the
> 9/31 generator MOS instead, associated to the semififths temperament.

That is all well and good, but there are still naming problems. I
can't use the temperament names because I am grouping different
temperaments into same families. For instance, "superpyth" and
"meantone" (among others) would both be in the family of 5(l)+2(s)
heptatonic scales. Really, there's no real reason even to be getting
mathematical here. It's a straightforward task of assigning names,
and I was really just hunting for appropriate terms to use.

-Igs

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...>
wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> > But Igs, they have the *same* generator and period in this sense!
Both
> > systems have a generator of 3*l+s and a period of 5*l+2*s. If
you're
> > talking about 'shape' in a qualitative sense (is that what you
mean by
> > absolute?), then you should think about the generators in this
way too.
> > I think you may be missing something pretty crucial here . . .
>
> Ooooohhhhhh...I getcha. That certainly isn't a familiar usage of
the
> term "generator" to me, though.

It's the same usage -- the interval which, when stacked upon itself
repeatedly modulo the period, creates all the notes of the scale and
no others.

> I thought it referred to an absolute
> pitch?

A-440 is an absolute pitch, but it's not a generator. A generator is
normally an interval.

> Or is it context-dependent?

Sure -- there's also that generator that you plug your appliances
into when the power goes out :)

>Anyway, I don't think it
> necessary or helpful to include period and generator, since those
will
> only produce certain permutations of l's and s's, leaving out
others.
> Not that that matters in reality, since it wouldn't change the
family
> groupings, but I don't really see how including them would improve
> anything.

It's of mathematical interest in understanding your problem but I
think we've gone past that straight to the solution anyway: the
charts I'm talking about.

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:
>
> > Use a different tree for each period. The smaller the period, the
> > smaller the tree you need.
> >
> > If you want everything in two dimensions, plot a function of the
> > generator size against a function of the period size.
>
> This is far more mathematical and complex than I wanted to go. What
> I'm interested in primarily (to stick to the subject) is NAMING.

a+b.

> NAMING families of scales. How you derive them, how you arrange them,
> how you display them...I don't really care.

Isn't it of interest to you to be able to display *all* the families,
and see at a glance which ETs belong to each of them, on one single
wall chart?

> I'm interested in simply
> naming, so that one could say 12, 17, 19, 22, 24, 26, 27, 29,
> 31...etc. all belong to the "such-and-such" scale family, because they
> all contain heptatonic scales of 5(l)+2(s).
>
> Capice?

This has been called the 5+2 family. I think that's quite appropriate
given your specific questions here. And what's great about the chart
I'm talking about is that all you have to do to get a list of the ETs
that support this is to look at the chart along the line connecting 5
with 5+2=7, and there they all are.

> Let me know if you think I'm on the right track, and I'll make a more
> general (fuller, less geared toward 5-limit approximations) chart for
> you.

Yeah. You got it. I suppose the a+b approach is the most practical,
but I was kinda hoping to get more, I dunno, linguistic about it (word
names rather than number names). I mean, I'm really being
uberpractical here, not concerning myself with anything greater than a
10-note scale, given Rothenberg's prediction that anything much
greater than a 9-note scale will not be heard as one cohesive scale by
most listeners. Not to mention the fact that I'm currently only
considering EDOs less than 36, since anything higher doesn't really
interest me as a musician (even 36 is pushing it).

I'd like to see a fuller chart, whenever you have sufficient free time
to do it up.

_igs

> A-440 is an absolute pitch, but it's not a generator. A generator is
> normally an interval.

Hee hee, I guess my language skills are a bit rusty. Absolute
interval, I meant. As quantified in cents.

> Isn't it of interest to you to be able to display *all* the families,
> and see at a glance which ETs belong to each of them, on one single
> wall chart?

Not if the chart requires intense scrutiny to read ;-> I was thinking
more of a table, actually. I have one mocked up on which cardinality
is on the vertical and scale shape is on the horizontal, and each
cell is filled with the step sizes of l and s (respectively) for which
the EDO contains the scale. Of course there are lots of empty cells,
so I'm trying to think of a way to streamline it. Ideally I'd like
the chart/table to reflect not just which EDOs are members of which
family, but what values of l and s plug in to MAKE the EDOs belong to
the family. If you can think of a way to do this that is more
effecient than my current method, I would be very excited.

-Igs

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...>
wrote:
>
> > Let me know if you think I'm on the right track, and I'll make a
more
> > general (fuller, less geared toward 5-limit approximations) chart
for
> > you.
>
> Yeah. You got it. I suppose the a+b approach is the most
practical,
> but I was kinda hoping to get more, I dunno, linguistic about it
(word
> names rather than number names). I mean, I'm really being
> uberpractical here, not concerning myself with anything greater
than a
> 10-note scale, given Rothenberg's prediction that anything much
> greater than a 9-note scale will not be heard as one cohesive scale
by
> most listeners. Not to mention the fact that I'm currently only
> considering EDOs less than 36, since anything higher doesn't really
> interest me as a musician (even 36 is pushing it).
>
> I'd like to see a fuller chart, whenever you have sufficient free
time
> to do it up.
>
> _igs

It's the first thing I'll do when I return here (the office) next
week (I presume you noticed that 22, 27, and 32 were 'missing' along
the line from 5 to 7 on the chart from Oct. 16 '02 that I posted a
link to).

What's the *smallest* number of notes in a scale you'd be interested
in?

Have a nice weekend, I'll be gigging in Scarsdale, NY tomorrow!

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:
>
> > A-440 is an absolute pitch, but it's not a generator. A generator
is
> > normally an interval.
>
> Hee hee, I guess my language skills are a bit rusty. Absolute
> interval, I meant. As quantified in cents.

Much academic music theory deals with abstract, qualitative scale
structures, specifically *not* quantified in cents, and includes
concepts such as this 'generator' of which we're speaking.

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:
> > Isn't it of interest to you to be able to display *all* the
families,
> > and see at a glance which ETs belong to each of them, on one single
> > wall chart?
>
> Not if the chart requires intense scrutiny to read ;->

Eliminating EDOs > 36 should solve that problem.

> I was thinking
> more of a table, actually. I have one mocked up on which cardinality
> is on the vertical and scale shape is on the horizontal, and each
> cell is filled with the step sizes of l and s (respectively) for which
> the EDO contains the scale. Of course there are lots of empty cells,
> so I'm trying to think of a way to streamline it. Ideally I'd like
> the chart/table to reflect not just which EDOs are members of which
> family, but what values of l and s plug in to MAKE the EDOs belong to
> the family. If you can think of a way to do this that is more
> effecient than my current method, I would be very excited.

Yes, I think my chart may serve this purpose more efficiently,
especially if you can do a couple of 1-digit sums in your head. To find
the values of l and s to plug in, you follow the zigs and zags in
the "third dimension" of the tree (imagining the larger-font numbers to
be closer to you, connected to the further-away numbers that they sum
up to). For example, let's say you see 45 on the line for the 5+2
family and you want to know what l and s should be. Each zig or zag
will take you to a higher-cardinality EDO, and with each step you add
the relevant quantities to l and s. For 5, l=1 and s=0, and for 7, l=1
and s=1. That's all you need to start with. Each time you add one of
these numbers to the EDO cardinality, you add the relevant quantities
to l or s. Since you're not going past 36-EDO, these calculations will
always be quick and doable in your head.

You can see that 45 arises on the tree as follows:

7 . . . . . . s=1, l=1
7+5=12 . . . . s=1+0=1, l=1+1=2
12+7=19 . . . . s=1+1=2, l=2+1=3
19+7=26 . . . . s=2+1=3, l=3+1=4
26+19=45 . . . . s=3+2=5, l=4+3=7 (+2 and +3 come from the 19 row above)

Check: 2*5 + 5*7 = 45!

Since 45 > 36, this is harder than what you'd need to do in practice.

Graham Breed wrote:
> Igliashon Jones wrote:
>>Has this been done before? If so, where can I find information on it?
>> If not, then I have a question or two. Namely, how should I name
>>each family? For instance, there are 5 families of 5-tone(pentatonic)
>>scales: 5(l=s), 4(l)+1(s), 3(l)+2(s), 2(l)+3(s), and 1(l)+4(s). Can
>>anyone suggest a methodology to name them? Obviously the first one
>>could be something like "uniform pentatonic"; and perhaps the
>>2(l)+3(s) family should be given something to reflect that it is the
>>"standard" common practice shape?
> > > It sounds like the Scale Tree. There'll be things about it at the > Wilson Archives, and here's a mathematical introduction:
> > http://www.cut-the-knot.org/blue/Stern.shtml
> > You can name them with the numbers, although of course there are > different ways to define that so make sure you explain it in the > context. Many families now will have names from the temperaments they > support, and if you aren't a purist you can always use those names.

The Scale Tree is a good reference if you keep in mind that you can divide the octave into 2, 3, 4, or more parts and use those fractions of the octave as your period. For instance, the octatonic scale in 12-ET is 4L+4s, which repeats every 1/4 of an octave.

The main issue with using the temperament names for the scale structures is that a temperament like "meantone" can be associated with 2+3, 5+2, or even 7+5, for instance. Additionally, some scale structures are associated with more than one temperament. Here's a partial list of scale structures corresponding to named temperaments:

1L+2s: dicot
2L+1s: father
1L+3s: bug
3L+1s: dicot
2L+3s: meantone, mavila
3L+2s: father
4L+1s: bug
1L+5s: porcupine
3L+3s: augmented
1L+6s: porcupine
2L+5s: mavila
3L+4s: dicot
4L+3s: hanson
5L+2s: meantone
1L+7s: negri
2L+6s: srutal
4L+4s: diminished
7L+1s: porcupine
1L+8s: negri
3L+6s: augmented
7L+2s: mavila
1L+9s: ripple
2L+8s: diaschismic
3L+7s: magic
5L+5s: blackwood
6L+4s: lemba
7L+3s: dicot
9L+1s: negri
1L+10s: ripple, passion
4L+7s: hanson
8L+3s: semisixths
1L+11s: passion
3L+9s: augmented
5L+7s: schismic
7L+5s: meantone
8L+4s: diminished
9L+3s: augmented
10L+2s: srutal
11L+1s: ripple

> What's the *smallest* number of notes in a scale you'd be interested
> in?

Five. Anything less I consider more of a chord than a scale.

> Have a nice weekend, I'll be gigging in Scarsdale, NY tomorrow!

Good luck! Er, I mean Break a string!

-Igs

> Much academic music theory deals with abstract, qualitative scale
> structures, specifically *not* quantified in cents, and includes
> concepts such as this 'generator' of which we're speaking.

I'll take your word on it. I just haven't previously encountered any,
which is why I was confused.

> Yes, I think my chart may serve this purpose more efficiently,
> especially if you can do a couple of 1-digit sums in your head. To find
> the values of l and s to plug in, you follow the zigs and zags in
> the "third dimension" of the tree (imagining the larger-font numbers to
> be closer to you, connected to the further-away numbers that they sum
> up to).

Sounds clever, but...

> 7 . . . . . . s=1, l=1
> 7+5=12 . . . . s=1+0=1, l=1+1=2
> 12+7=19 . . . . s=1+1=2, l=2+1=3
> 19+7=26 . . . . s=2+1=3, l=3+1=4
> 26+19=45 . . . . s=3+2=5, l=4+3=7 (+2 and +3 come from the 19 row
> above)

I'm going to have to have you explain this more thoroughly once the
final chart is made.

-Igs

> The main issue with using the temperament names for the scale
structures
> is that a temperament like "meantone" can be associated with 2+3, 5+2,
> or even 7+5, for instance.

That wouldn't be a problem, since I would propose a two-word naming
system, where the first word described the shape and the second gave
number of notes.

> Additionally, some scale structures are
> associated with more than one temperament.

That is exactly one of the two main reasons why I don't want to use
temperament names. 5+2 is both meantone AND superpyth.

Here's a partial list of
> scale structures corresponding to named temperaments:
>
> 1L+2s: dicot
> 2L+1s: father
> 1L+3s: bug
> 3L+1s: dicot
> 2L+3s: meantone, mavila
> 3L+2s: father
> 4L+1s: bug
> 1L+5s: porcupine
> 3L+3s: augmented
> 1L+6s: porcupine
> 2L+5s: mavila
> 3L+4s: dicot
> 4L+3s: hanson
> 5L+2s: meantone
> 1L+7s: negri
> 2L+6s: srutal
> 4L+4s: diminished
> 7L+1s: porcupine
> 1L+8s: negri
> 3L+6s: augmented
> 7L+2s: mavila
> 1L+9s: ripple
> 2L+8s: diaschismic
> 3L+7s: magic
> 5L+5s: blackwood
> 6L+4s: lemba
> 7L+3s: dicot
> 9L+1s: negri
> 1L+10s: ripple, passion
> 4L+7s: hanson
> 8L+3s: semisixths
> 1L+11s: passion
> 3L+9s: augmented
> 5L+7s: schismic
> 7L+5s: meantone
> 8L+4s: diminished
> 9L+3s: augmented
> 10L+2s: srutal
> 11L+1s: ripple

This is the OTHER reason why I wouldn't want to use temperament names.
That is an INTIMIDATING list, with very little internal logic.
Ideally, the name of the shape should tell you something ABOUT the
shape; it should be descriptive, and for the most part these
temperament-names say nothing about the shape. Without numbers
attached, no one outside of the tuning list would have any idea what
sort of scale a Dicot Decatonic or a Hanson Heptatonic would be.

_Igs

To give a better idea of what I'm after, let me present my current
ideas (with Jacob Barton's "fair/unfair" distinction) for
criticism/modification/etc. Hopefully the internal logic behind the
naming is apparent:

0+5 uniform pentatonic
1+4 narrow pentatonic
2+3 fair pentatonic/western pentatonic
3+2 unfair pentatonic/inverse pentatonic
4+1 wide pentatonic

0+6 uniform hexatonic
1+5 narrow hexatonic
2+4 fair hexatonic
3+3 even hexatonic
4+2 unfair hexatonic
5+1 wide hexatonic

0+7 uniform heptatonic
1+6 narrow heptatonic
2+5 western heptatonic
3+4 fair heptatonic
4+3 unfair heptatonic
5+2 inverse heptatonic
6+1 wide heptatonic

0+8 uniform octatonic
1+7 narrow octatonic
2+6 ?
3+5 fair octatonic
4+4 even octatonic
5+3 unfair octatonic
6+2 ?
7+1 wide octatonic

As you can see, I ran out of ideas at the octatonics. But you all
should get the idea of what I'm attempting. If you think it's a bad
idea, I'd like to know why. If you think I should change some words,
I'd like suggestions. And if you have any ideas how I could fill in
the gaps in the octatonics (as well as those that will arise in the
nonatonics and decatonics), I'd like to hear them.

_Igs

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> Font size is inversely related to the cardinality of the ET here, so
> you're just missing out on large-cardinality ETs if you can't read the
> small numbers. I can see:
>
> (7), 40, 33, 26, 45, 19, 50, 31, 43, 55, 12, 17, (5)
>
> All of these ETs support 5(l)+2(s) scales.

Rather than squinting your eyes, you could recognize that what you
want are the denominators of a portion of the 55 row of the Farey
sequence, between 4/7 and 3/5, which is easy to calculate (though if
someone wants to discuss how to do that, I suggest tuning-math.)

The complete list is:

4/7, 31/54, 27/47, 23/40, 19/33, 15/26, 26/45, 11/19, 29/50, 18/31,
25/43, 32/55, 7/12, 31/53, 24/41, 17/29, 27/46, 10/17, 23/39, 13/22,
29/49, 16/27, 19/32, 22/37, 25/42, 28/47, 31/52, 3/5

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:

> This is far more mathematical and complex than I wanted to go. What
> I'm interested in primarily (to stick to the subject) is NAMING.
> NAMING families of scales. How you derive them, how you arrange them,
> how you display them...I don't really care. I'm interested in simply
> naming, so that one could say 12, 17, 19, 22, 24, 26, 27, 29,
> 31...etc. all belong to the "such-and-such" scale family, because they
> all contain heptatonic scales of 5(l)+2(s).

You could just name it [2/5, 3/7], which is an interval between two
adjacent fractions, or equivalently "5 out of 7".

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:

> That is all well and good, but there are still naming problems. I
> can't use the temperament names because I am grouping different
> temperaments into same families. For instance, "superpyth" and
> "meantone" (among others) would both be in the family of 5(l)+2(s)
> heptatonic scales. Really, there's no real reason even to be getting
> mathematical here. It's a straightforward task of assigning names,
> and I was really just hunting for appropriate terms to use.

I think there's a good reason to get mathematical, in that what you
are defining is equivalent to an interval between two adjacent Farey
fractions. Therefore, the interval itself is a possible name.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> This has been called the 5+2 family. I think that's quite appropriate
> given your specific questions here. And what's great about the chart
> I'm talking about is that all you have to do to get a list of the ETs
> that support this is to look at the chart along the line connecting 5
> with 5+2=7, and there they all are.

The "n+m" gives us "n out of n+m", which gives the interval between
adjacent Farey fractions by the definition involving convergents of a
continued fraction I discussed before.

Carl Lumma wrote:
> Graham wrote...
> > >>If you want everything in two dimensions, plot a function of the >>generator size against a function of the period size. This will
>>give the same grouping as the scale tree in one direction. The
>>period=octave case will be the Farey sequence. Other temperaments
>>will probably have to be redundantly placed, so that diaschismic
>>shows up as having a generator of a perfect fourth as well as a
>>semitone for example.
> > I'd love to see this!

Here's one I rustled up. I left in the tree shapes so that it makes the structure clear. The vertical axis is determined by the usual, primary scale tree. It's not pretty, but you should be able to work out what's going on. Set a fixed width font by whatever means.

|0/1 0/2 0/3 0/4
| 1/7 1/8 1/9 1/8
| 1/6 1/6 1/6 2/12
| 2/11 3/16 4/21 3/16
| 1/5 2/10 3/15 4/20
| 3/14 3/14 5/24 5/24
| 2/9 4/18 2/9
| 3/13 5/22 5/21
| 1/4 1/4 3/12 1/4
| 4/15 7/26 4/15
| 3/11 6/22 9/33
| 5/18 5/18 5/18 9/32
| 2/7 4/14 6/21 8/28
| 5/17 7/24 7/24 7/24
| 3/10 3/10 8/27 6/20
| 4/13 5/16 5/16
| 1/3 2/6 1/3 4/12
| 5/14 5/14 7/20
| 4/11 8/22 11/30
| 7/19 11/30 10/27
| 3/8 3/8 9/24 3/8
| 8/21 13/34 8/21
| 5/13 10/26 15/39
| 7/18 7/18 7/18 11/28
| 2/5 4/10 6/15 8/20
| 7/17 9/22 11/27 13/32
| 5/12 5/12 5/12 5/12
| 8/19 11/26 14/33
| 3/7 6/14 9/21 12/28
| 7/16 7/16 13/30 7/16
| 4/9 8/18 4/9 16/36
| 5/11 9/20 7/15 9/20
|1/2 1/2 3/6 2/4

Graham

Igliashon Jones wrote:
> To give a better idea of what I'm after, let me present my current
> ideas (with Jacob Barton's "fair/unfair" distinction) for
> criticism/modification/etc. Hopefully the internal logic behind the
> naming is apparent:

I've tried to set your names to the scale trees. I don't know if I got them all the right way round -- you didn't say which is the L and which the s, and I can't make sense of it either way. On the scale tree, the nearer you get to a number the more it looks like that equal temperament, so you can work out from that which step size is greater. Because you want to distinguish large from small steps, the number associated with the scale isn't the same as the number it sits next to.

Your primary sorting is by the number of notes, and so the scale tree will change that drastically. Also, if you think of scales as occupying regions, some of the big ones sit inside the smaller ones.

Some of your scales aren't MOS because one step size is zero. Hence they can't be shown on a diagram of scales with two step sizes.

Primary Scale Tree

1
wide octatonic
8 narrow heptatonic
narrow octatonic
7 wide hexatonic
wide heptatonic
6 wide pentatonic

narrow hexatonic

5

narrow pentatonic

4

unfair heptatonic

7

fair heptatonic

3

unfair octatonic

unfair/inverse pentatonic

fair octatonic

5

17

western heptatonic?

19

7 fair/western pentatonic

16

inverse heptatonic?

11

2

Secondary Scale Tree

2
unfair anonymous octatonic
8 unfair hexatonic?
fair anonymous octatonic
6

fair hexatonic?

4

Tertiary Scale Tree

3
even hexatonic
6

Quaternary Scale Tree

4
even octatonic
8

Unclassified

all uniform scales

> 0+5 uniform pentatonic
> 1+4 narrow pentatonic
> 2+3 fair pentatonic/western pentatonic
> 3+2 unfair pentatonic/inverse pentatonic
> 4+1 wide pentatonic
> > > 0+6 uniform hexatonic
> 1+5 narrow hexatonic
> 2+4 fair hexatonic
> 3+3 even hexatonic
> 4+2 unfair hexatonic
> 5+1 wide hexatonic
> > > 0+7 uniform heptatonic
> 1+6 narrow heptatonic
> 2+5 western heptatonic
> 3+4 fair heptatonic
> 4+3 unfair heptatonic
> 5+2 inverse heptatonic
> 6+1 wide heptatonic
> > > 0+8 uniform octatonic
> 1+7 narrow octatonic
> 2+6 ?
> 3+5 fair octatonic
> 4+4 even octatonic
> 5+3 unfair octatonic
> 6+2 ?
> 7+1 wide octatonic
> > As you can see, I ran out of ideas at the octatonics. But you all
> should get the idea of what I'm attempting. If you think it's a bad
> idea, I'd like to know why. If you think I should change some words,
> I'd like suggestions. And if you have any ideas how I could fill in
> the gaps in the octatonics (as well as those that will arise in the
> nonatonics and decatonics), I'd like to hear them.

Your octatonics are related to your unfair hexatonics, so perhaps they could inherit the name. Except if "unfair hexatonic" is the name that'll just get confusing.

As you're leaving the tuning out of these names, they can also be defined by the generator and the number of notes in the scale. Where the period divides the octave, you choose a generator with the correct common factor. Your anonymous octatonic pair have the generator of 3, so they could be "fair octatonic 3" and "unfair octatonic 3" or some such. Not very romantic, but that would cover all scales.

You can always find the generator by following down the scale tree. (I think I got this one right.) It's also an inverse modulo operation. That's useful to know if you want to calculate it automatically.

Graham

Me:
>>If you want everything in two dimensions, plot a function of the >>generator size against a function of the period size.

Paul:
> Which functions do you have in mind here?

Reciprocals would be interesting. They'd make integer divisions of the octave equally spaced, and stop the 1/n scales bunching together when n gets large.

>>This will give >>the same grouping as the scale tree in one direction. The > period=octave >>case will be the Farey sequence. Other temperaments will probably > have >>to be redundantly placed, so that diaschismic shows up as having a >>generator of a perfect fourth as well as a semitone for example.
> > Hmm . . . I'm not sure if the "treeness" will be preserved here.

Maybe not, but it would show the generator size and allow non-octave tunings to be placed as well.

> Rather that a 1-dimensional scale tree growing through an additional > dimension, I've proposed a 2-dimensional scale tree (that can be seen > to grow through a third dimension). This doesn't leave out systems > where the period is a fraction of the octave of equivalence.
> > /tuning/files/perlich/tree.gif
> > This was drawn with approximation to JI in mind but need not be seen > that way -- just ignore the grid, and zoom out to include more > numbers and more occurences of numbers.

I can see the logic, but I'm not sure what it "means". How do you seed the "uber-tree" that contains all 2-D scales? It's a different idea to the period/generator map, anyway, and there's room in the world for both. Could they even be projections of each other?

It makes sense to me that different octave-dividing periods lead to drastically different structures. Making them clearly separate (on one diagram or otherwise) naturally follows from that.

Graham

> >>If you want everything in two dimensions, plot a function of the
> >>generator size against a function of the period size. This will
> >>give the same grouping as the scale tree in one direction. The
> >>period=octave case will be the Farey sequence. Other temperaments
> >>will probably have to be redundantly placed, so that diaschismic
> >>shows up as having a generator of a perfect fourth as well as a
> >>semitone for example.
> >
> > I'd love to see this!
>
> Here's one I rustled up. I left in the tree shapes so that it
> makes the structure clear. The vertical axis is determined by the
> usual, primary scale tree. It's not pretty, but you should be
> able to work out what's going on. Set a fixed width font by
> whatever means.
>
> |0/1 0/2 0/3 0/4
> | 1/7 1/8 1/9 1/8
> | 1/6 1/6 1/6 2/12
> | 2/11 3/16 4/21 3/16
> | 1/5 2/10 3/15 4/20
> | 3/14 3/14 5/24 5/24
> | 2/9 4/18 2/9
> | 3/13 5/22 5/21
> | 1/4 1/4 3/12 1/4
> | 4/15 7/26 4/15
> | 3/11 6/22 9/33
> | 5/18 5/18 5/18 9/32
> | 2/7 4/14 6/21 8/28
> | 5/17 7/24 7/24 7/24
> | 3/10 3/10 8/27 6/20
> | 4/13 5/16 5/16
> | 1/3 2/6 1/3 4/12
> | 5/14 5/14 7/20
> | 4/11 8/22 11/30
> | 7/19 11/30 10/27
> | 3/8 3/8 9/24 3/8
> | 8/21 13/34 8/21
> | 5/13 10/26 15/39
> | 7/18 7/18 7/18 11/28
> | 2/5 4/10 6/15 8/20
> | 7/17 9/22 11/27 13/32
> | 5/12 5/12 5/12 5/12
> | 8/19 11/26 14/33
> | 3/7 6/14 9/21 12/28
> | 7/16 7/16 13/30 7/16
> | 4/9 8/18 4/9 16/36
> | 5/11 9/20 7/15 9/20
> |1/2 1/2 3/6 2/4
>
>
> Graham

Kyool...

-Carl

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:
> > What's the *smallest* number of notes in a scale you'd be
interested
> > in?
>
> Five. Anything less I consider more of a chord than a scale.

OK, I'll keep that in mind when I try to make a wall chart for you.

> > Have a nice weekend, I'll be gigging in Scarsdale, NY tomorrow!
>
> Good luck! Er, I mean Break a string!

Thanks. Luckily, I didn't! :)

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:

> > The main issue with using the temperament names for the scale
> structures
> > is that a temperament like "meantone" can be associated with 2+3,
5+2,
> > or even 7+5, for instance.
>
> That wouldn't be a problem, since I would propose a two-word naming
> system, where the first word described the shape and the second gave
> number of notes.

I don't understand. Can you elaborate?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...>
wrote:
>
> > This is far more mathematical and complex than I wanted to go. What
> > I'm interested in primarily (to stick to the subject) is NAMING.
> > NAMING families of scales. How you derive them, how you arrange
them,
> > how you display them...I don't really care. I'm interested in
simply
> > naming, so that one could say 12, 17, 19, 22, 24, 26, 27, 29,
> > 31...etc. all belong to the "such-and-such" scale family, because
they
> > all contain heptatonic scales of 5(l)+2(s).
>
> You could just name it [2/5, 3/7], which is an interval between two
> adjacent fractions, or equivalently "5 out of 7".

How would you name the family of 2(l)+8(s) scales using this notation?

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> I can see the logic, but I'm not sure what it "means". How do you
seed
> the "uber-tree" that contains all 2-D scales?

A bit triangle, with 1 in each corner.

> It makes sense to me that different octave-dividing periods lead to
> drastically different structures. Making them clearly separate (on
one
> diagram or otherwise) naturally follows from that.

They're not separate on my diagram, nor should they be (I feel).

> > > The main issue with using the temperament names for the scale
> > structures
> > > is that a temperament like "meantone" can be associated with 2+3,
> 5+2,
> > > or even 7+5, for instance.
> >
> > That wouldn't be a problem, since I would propose a two-word naming
> > system, where the first word described the shape and the second gave
> > number of notes.
>
> I don't understand. Can you elaborate?

Well, the 3 would be differentiated as "meantone pentatonic, meantone
heptatonic, and meantone dodecatonic". Make sense?

-Igs

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...>
wrote:
>
> > > > The main issue with using the temperament names for the scale
> > > structures
> > > > is that a temperament like "meantone" can be associated with
2+3,
> > 5+2,
> > > > or even 7+5, for instance.
> > >
> > > That wouldn't be a problem, since I would propose a two-word
naming
> > > system, where the first word described the shape and the second
gave
> > > number of notes.
> >
> > I don't understand. Can you elaborate?
>
> Well, the 3 would be differentiated as "meantone pentatonic,
meantone
> heptatonic, and meantone dodecatonic". Make sense?
>
> -Igs

I see a problem with that, in that you're now going to have 3(s)+2(L)
scales which are "meantone", and others which aren't (like those from
mavila, which doesn't have a dodecatonic with 2 step sizes at all --
so your term "meantone" couldn't reasonably apply to them (?)) But
wasn't the point to give all 3(s)+2(L) scales the same 'name',
essentially?

One could borrow terms from the Pythagoreans and add no new terms at all.
if the Large outnumber the S you can call them superabundant, if the same perfect, if less deficient or you could reverse this depending on you bias for the small.
a good starting point at least

Message: 15 Date: Mon, 12 Sep 2005 20:13:20 -0000
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
Subject: Re: EDO classification by families of scale patterns

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:

>>> > The main issue with using the temperament names for the scale
>> >>
>> structures > >
>>> > is that a temperament like "meantone" can be associated with 2+3, >> >>
5+2, >>> > or even 7+5, for instance. >> >>
>> >> That wouldn't be a problem, since I would propose a two-word naming
>> system, where the first word described the shape and the second gave
>> number of notes.
> >

I don't understand. Can you elaborate?

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:

> One could borrow terms from the Pythagoreans and add no new terms at
all.
> if the Large outnumber the S you can call them superabundant,
> if the same perfect, if less deficient or you could reverse this
depending on you bias for the small.
> a good starting point at least

Fantastic. I will go with this. But I will need additional modifiers
for 8, 9, and 10-note scales to signify degree of
superabundance/deficience. Any ideas there?

Thank you thank you very much!

-Igs

> Message: 15
> Date: Mon, 12 Sep 2005 20:13:20 -0000
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> Subject: Re: EDO classification by families of scale patterns
>
> --- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:
>
>
> >>> > The main issue with using the temperament names for the scale
> >>
> >>
> >> structures
> >
> >
> >>> > is that a temperament like "meantone" can be associated with 2+3,
> >>
> >>
> 5+2,
>
> >>> > or even 7+5, for instance.
> >>
> >>
> >>
> >> That wouldn't be a problem, since I would propose a two-word naming
> >> system, where the first word described the shape and the second gave
> >> number of notes.
> >
> >
>
> I don't understand. Can you elaborate?
>
>
>
>
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> I see a problem with that, in that you're now going to have 3(s)+2(L)
> scales which are "meantone", and others which aren't (like those from
> mavila, which doesn't have a dodecatonic with 2 step sizes at all --
> so your term "meantone" couldn't reasonably apply to them (?))

Yep. Hit it right on the head. Thus the problem with using
temperament names: they connote certain things, certain commonalities,
which aren't relevant on mere "scale shape" categorization. To put a
scale into one family wouldn't mean that ALL scales it contained of
various note numbers belonged in that family. Using an amended
version of my proposed terminology (to include pythagorean terms as
suggested by Kraig Grady), an EDO might be superabundant (unfair, more
L's than s's) in the pentatonics but "perfect" (even, equal L's and
s's) in the octatonics. In fact many EDOs would belong to multiple
families in the heptatonics and up.

The idea is not give each EDO one name. It is to give each scale
shape one name (irrespective of EDOs that contain it), and then list
EDOs that are compatible.

Does that make sense?

-Igs

Igliashon Jones wrote:

> Fantastic. I will go with this. But I will need additional modifiers
> for 8, 9, and 10-note scales to signify degree of
> superabundance/deficience. Any ideas there?
> > Thank you thank you very much!
> > -Igs

One thing you could do with those (but not with 11-note scales or other higher prime numbers) is to single out the scales that divide the octave evenly into 2 or more parts. So for instance a 6L+2s scale would be superabundant half-octave octatonic, 3L+6s deficient third-octave enneatonic, and so on. That at least would limit the number of additional modifiers you'd need.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > How would you name the family of 2(l)+8(s) scales using this notation?
>
> [0, 1/10]

Eh, no good. Maybe [1, 1/5]/2 instead.

> One thing you could do with those (but not with 11-note scales or other
> higher prime numbers) is to single out the scales that divide the
octave
> evenly into 2 or more parts. So for instance a 6L+2s scale would be
> superabundant half-octave octatonic, 3L+6s deficient third-octave
> enneatonic, and so on. That at least would limit the number of
> additional modifiers you'd need.

Good idea. I'll try messing around with that, see how it fits with
everything. Thanks, Mr. Miller!

-Igs

i think you might get stuck using the number names although while using greek ideas, you might go with octotonic, I can't remember nine off hand ( and are close to passing out here) , dekatoniic.

>> One could borrow terms from the Pythagoreans and add no new terms at
> >
all.

>> if the Large outnumber the S you can call them superabundant, >> if the same perfect, if less deficient or you could reverse this
> >
depending on you bias for the small.

>> a good starting point at least
> >
Fantastic. I will go with this. But I will need additional modifiers for 8, 9, and 10-note scales to signify degree of superabundance/deficience. Any ideas there? Thank you thank you very much! -Igs

> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, "hstraub64@t..." <hstraub64@t...>
wrote:
> > Igliashon Jones wrote:
> >
> > > For the moment I'm going to limit
> > > myself to EDOs < 36, and scales of no more than 10 steps. Also,
> > > the ratio of l:s shouldn't exceed 5:1, because most listeners
> > >would find such scales rather lopsided and unpleasant (I think?).
> >

>
> Well, if you take k as the subdivision of the EDO, the values x, y,
l and s
> fulfill
>
> x*l + y*s = k
>
> which is a linear Diophantine equation. It appears to me that this
> probably has been done before - dunno though. Maybe I will try to do
> it...

I did some calculations, meanwhile. I have now the complete list of
pentatonic MOS scales for all EDOs from 5 to 36. Interested, anyone?
--
Hans Straub

--- In tuning@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:

> I did some calculations, meanwhile. I have now the complete list of
> pentatonic MOS scales for all EDOs from 5 to 36. Interested, anyone?
> --
> Hans Straub

I actually already have lists of all scales of 2 or fewer step-sizes
of 5 to 10 notes for all EDOs from 5 to 36, though I didn't do it by
equation, I did it visually by chart. I could post these charts if
anyone wants, to see the simple patterns that emerge.

-Igs

> > I did some calculations, meanwhile. I have now the complete list
> > of pentatonic MOS scales for all EDOs from 5 to 36. Interested,
> > anyone?
> > --
> > Hans Straub
>
> I actually already have lists of all scales of 2 or fewer step-sizes
> of 5 to 10 notes for all EDOs from 5 to 36, though I didn't do it by
> equation, I did it visually by chart. I could post these charts if
> anyone wants, to see the simple patterns that emerge.

Please don't ask; please post the charts! :)

Hans, if you have a text file, I'd like to see that too.

-Carl

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...>
wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > I see a problem with that, in that you're now going to have 3(s)+2
(L)
> > scales which are "meantone", and others which aren't (like those
from
> > mavila, which doesn't have a dodecatonic with 2 step sizes at
all --
> > so your term "meantone" couldn't reasonably apply to them (?))
>
> Yep. Hit it right on the head. Thus the problem with using
> temperament names: they connote certain things, certain
commonalities,
> which aren't relevant on mere "scale shape" categorization. To put
a
> scale into one family wouldn't mean that ALL scales it contained of
> various note numbers belonged in that family. Using an amended
> version of my proposed terminology (to include pythagorean terms as
> suggested by Kraig Grady), an EDO might be superabundant (unfair,
more
> L's than s's) in the pentatonics

9-EDO has both 3(s)+2(L) and 4(L)+1(s) pentatonics -- is it
superabundant or deficient in the pentatonics?

> but "perfect" (even, equal L's and
> s's) in the octatonics. In fact many EDOs would belong to multiple
> families in the heptatonics and up.

And down too.

> The idea is not give each EDO one name. It is to give each scale
> shape one name (irrespective of EDOs that contain it), and then list
> EDOs that are compatible.
>
> Does that make sense?

Sure does -- that's why I suggested the a+b notation (or whatever
Herman Miller and Dave Keenan were using).

Gene:
>>You could just name it [2/5, 3/7], which is an interval between two
>>adjacent fractions, or equivalently "5 out of 7".

Paul:
> How would you name the family of 2(l)+8(s) scales using this notation?

The "n out of m" notation uniquely identifies a node of a scale tree, and so is isomorphic to "a+b" and "n/d" (where the n and d have to be carefully chosen to have the right common factors). You'd need some convention (super-duper abundant?) for scales that aren't maximally even like 31-equal's meantone heptatonic.

The 2L+8s family would be 10 from 12, wouldn't it? What am I missing?

Graham

Igliashon Jones wrote:

> Yep. Hit it right on the head. Thus the problem with using
> temperament names: they connote certain things, certain commonalities,
> which aren't relevant on mere "scale shape" categorization. To put a
> scale into one family wouldn't mean that ALL scales it contained of
> various note numbers belonged in that family. Using an amended
> version of my proposed terminology (to include pythagorean terms as
> suggested by Kraig Grady), an EDO might be superabundant (unfair, more
> L's than s's) in the pentatonics but "perfect" (even, equal L's and
> s's) in the octatonics. In fact many EDOs would belong to multiple
> families in the heptatonics and up.
> > The idea is not give each EDO one name. It is to give each scale
> shape one name (irrespective of EDOs that contain it), and then list
> EDOs that are compatible. > > Does that make sense?

It depends on what you mean by "scale shape". You said the meantone heptatonic and pentatonic scales have the same shape. If you follow that idea to its logical conclusion, all scales with an octave period would have the same shape. So you have to draw the line somewhere. My idea would be to literally draw lines on the scale trees, and say which scales have which shapes. Then you can deal with naming each branch within a given shape.

In fact, with that method, each shape would be an m+n scale tree.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> Gene:
> >>You could just name it [2/5, 3/7], which is an interval between two
> >>adjacent fractions, or equivalently "5 out of 7".
>
> Paul:
> > How would you name the family of 2(l)+8(s) scales using this
notation?
>
> The "n out of m" notation uniquely identifies a node of a scale tree,
> and so is isomorphic to "a+b" and "n/d" (where the n and d have to be
> carefully chosen to have the right common factors). You'd need some
> convention (super-duper abundant?) for scales that aren't maximally
even
> like 31-equal's meantone heptatonic.

That's still 5+2 though, even though it's in 31-equal, so I'm not sure
why you need a different convention here.

> It depends on what you mean by "scale shape". You said the meantone
> heptatonic and pentatonic scales have the same shape.

I said no such thing!! By shape I simply mean the number of L's and
s's (which, if I'm not mistaken, is how it works out with branches on
the scale trees).

-Igs

--- In tuning@yahoogroups.com, "wallyesterpaulrus"

> 9-EDO has both 3(s)+2(L) and 4(L)+1(s) pentatonics -- is it
> superabundant or deficient in the pentatonics?

Both. Like I said, every EDO will belong to multiple families, many
even within the same "n-tonic" group.

> > but "perfect" (even, equal L's and
> > s's) in the octatonics. In fact many EDOs would belong to multiple
> > families in the heptatonics and up.
>
> And down too.
>

> Sure does -- that's why I suggested the a+b notation (or whatever
> Herman Miller and Dave Keenan were using).

Yeah, I know that that would be the most effecient and logical way of
going about it, rather than coming up with new terminologies. It's
just that I find words so much more appealing than numbers, especially
when there are already so many numbers to be dealt with. And if
temperaments are worthy of naming, surely scale families are as well?

me:
>>It depends on what you mean by "scale shape". You said the meantone >>heptatonic and pentatonic scales have the same shape. Igliashon:
> I said no such thing!! By shape I simply mean the number of L's and
> s's (which, if I'm not mistaken, is how it works out with branches on
> the scale trees).

I was referring to your message of September 12th at 15:20 PDT:

"""
> > > The main issue with using the temperament names for the scale
> > structures
> > > is that a temperament like "meantone" can be associated with 2+3,
> 5+2,
> > > or even 7+5, for instance.
> >
> > That wouldn't be a problem, since I would propose a two-word naming
> > system, where the first word described the shape and the second gave
> > number of notes.
>
> I don't understand. Can you elaborate?

Well, the 3 would be differentiated as "meantone pentatonic, meantone
heptatonic, and meantone dodecatonic". Make sense?
"""

I still think you were saying what I thought you said there.

The scale tree shows rational numbers, and how you relate those to scales is up to you. The simplest way is that the number refers to the ratio of generator to period. In that case there's nothing to say which of the scale steps is the larger. You can get the scale tree to distinguish the L and s, as I showed before. I'm thinking now that it might make sense for the lines connecting the nodes to be the L and s scales.

A branch of the scale tree has a lot of different nodes on it. The scale tree shows relationships between scales, not only the scales themselves. It wouldn't be very interesting otherwise, would it? You don't need the diagram to show what a 7+5 scale is.

The 2+3 branch you said no such thing about above would be (use fixed width font, etc):

| 2 3
|
| 5
| 7 8
| 9 12 13 11
| 11 16 19 17 18 21 19 14

Now, that certainly includes everything that could sensibly be called "meantone". It also includes the mavila family (which was a shape before it was a temperament). And it includes a whole load of other stuff that looks nothing at all like meantone.

But you can always draw a line around the ones you think are meantone, and then think of other names for the other groups. Maybe it's already disappeared under your SEP field, but you might have to decide if 17-equal is a meantone or not.

Graham

I have always been the same way. as soon as i have a tuning , the first thing i want are letter names, even if i associate numbers with them, it gives them more identity, like people

From: "Igliashon Jones" <igliashon@sbcglobal.net>
Subject: Re: EDO classification by families of scale patterns

It's
just that I find words so much more appealing than numbers, especially
when there are already so many numbers to be dealt with. -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

wallyesterpaulrus wrote:

> That's still 5+2 though, even though it's in 31-equal, so I'm not sure > why you need a different convention here.

3*2 + 5*5 = 8*2 + 3*5 = 13*2 + 1*5 = 31

There are three different 5+2 scales in 31-equal. You need to say which one you mean.

Not that this has much to do with what I was originally talking about.

Graham

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> >
> > I actually already have lists of all scales of 2 or fewer
step-sizes
> > of 5 to 10 notes for all EDOs from 5 to 36, though I didn't do it
by
> > equation, I did it visually by chart. I could post these charts
if
> > anyone wants, to see the simple patterns that emerge.
>
> Please don't ask; please post the charts! :)
>
>
> Hans, if you have a text file, I'd like to see that too.
>

I have no text file at the moment, just a few sheets of paper...
But here are the formulae I derived.

Bad news: for each scale type we need a separate formula.
Good news: The same formulae work for all EDOs.

For a scale in n-EDO with 1 large step (size L) and 4 small steps
(size s), L and s are of the form

L = 5n - 4t
s = -n + t

where t is an integer with n < t < 6n/5. If n is such that there is no
integer fulfilling this condition, there is no scale of this type in
n-EDO. (This one is still quite trivial...)

Dito for a scale with 2 large and 3 small steps:

L = -n + 3t
s = n - 2t

with 2n/5 < t < n/2

For a scale with 3 large and 2 small steps:

L = n - 2t
s = -n + 3t

with n/3 < t < 2n/5

For a scale with 4 large and 1 small step:

L = -n + t
s = 5n - 4t

with 6n/5 < t < 5n/4

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> wallyesterpaulrus wrote:
>
> > That's still 5+2 though, even though it's in 31-equal, so I'm not
sure
> > why you need a different convention here.
>
> 3*2 + 5*5 = 8*2 + 3*5 = 13*2 + 1*5 = 31
>
> There are three different 5+2 scales in 31-equal.

Or perhaps two 2+5 scales and one 5+2 scale.

> You need to say which
> one you mean.

Interesting twist, Igs: some EDOs may have more than one instance of
the same "scale shape". Those would still occur in different places on
the wall chart I'm planning to make for you . . .

Hi everyine
I wrote am mail but somehow it seems not to have gone
through so I'll write it again:

I play the shakuhachi. Since about the last 100 years
western music has been taught in favur of traditional
music in Japan. The result seems that now in Japan
equal temperament is the standard. I have been unable
to find any chart listing the exact pitches of the
scale as used in Japan prior to that. Does anyone know
of such information? Also for China would be
interesting. All that I seem to find on the net is
nice comparisons between different European tunings,
but not for Japan or China.
Thanks
Best wishes
Justin.

__________________________________________________
Do You Yahoo!?
Tired of spam? Yahoo! Mail has the best spam protection around
http://mail.yahoo.com

>> Well, the 3 would be differentiated as "meantone pentatonic,
>> meantone heptatonic, and meantone dodecatonic". Make sense?
>> """
>
> I still think you were saying what I thought you said there.

No. You are confusing "related shapes" with "same shape". It is
obviously absurd to say that a pentatonic scale has the same shape as
a heptatonic scale! Perhaps my use of the word "shape" is too
ambiguous. Allow me to clarify it as "a specific number of
equal-sized large steps (l; plural l's) and equal-sized small steps
(s; plural s's), that when summed together are equal to the number of
notes in the scale." So an n-tonic scale shape would be defined as
l+s=n. The name would tell you both the number of l's or s's and the
number of notes n. The example I gave is misleading, I would not use
the term "meantone" even to denote meantone-esque scales (such as
5l+2s or 2l+3s).

>
> The scale tree shows rational numbers, and how you relate those to
> scales is up to you. The simplest way is that the number refers to the
> ratio of generator to period. In that case there's nothing to say
which
> of the scale steps is the larger. You can get the scale tree to
> distinguish the L and s, as I showed before. I'm thinking now that it
> might make sense for the lines connecting the nodes to be the L and s
> scales.
>
> A branch of the scale tree has a lot of different nodes on it. The
> scale tree shows relationships between scales, not only the scales
> themselves. It wouldn't be very interesting otherwise, would it? You
> don't need the diagram to show what a 7+5 scale is.
>
> The 2+3 branch you said no such thing about above would be (use fixed
> width font, etc):
>
> | 2 3
> |
> | 5
> | 7 8
> | 9 12 13 11
> | 11 16 19 17 18 21 19 14
>
> Now, that certainly includes everything that could sensibly be called
> "meantone". It also includes the mavila family (which was a shape
> before it was a temperament). And it includes a whole load of other
> stuff that looks nothing at all like meantone.
>
> But you can always draw a line around the ones you think are meantone,
> and then think of other names for the other groups. Maybe it's already
> disappeared under your SEP field, but you might have to decide if
> 17-equal is a meantone or not.

This is all well and good, perhaps would make a good wall chart, but
it has nothing to do with what I originally asked, which was for
advice on NOMENCLATURE, what to CALL the families. Fortunately, I
think all my questions have been answered and I now have a workable
solution, which I will post shortly.

Regards,

-Igs

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:

> Allow me to clarify it as "a specific number of
> equal-sized large steps (l; plural l's) and equal-sized small steps
> (s; plural s's), that when summed together are equal to the number of
> notes in the scale."

And does distributional eveness come in here? If it does, you are
classifying MOS, and the a+b notation seems excellent.

> Interesting twist, Igs: some EDOs may have more than one instance of
> the same "scale shape". Those would still occur in different places on
> the wall chart I'm planning to make for you . . .

Oh, I am well aware of this. I do not think it is a problem, as the
specification of which values of L and s make an EDO a member of a
particular family is not a part of the family name.

-Igs

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> I have always been the same way. as soon as i have a tuning , the
> first thing i want are letter names, even if i associate numbers
with > them, it gives them more identity, like people

ha! Yes. Definitely. Since I always get into tunings before I have
instruments capable of playing them, to bide my time until the
instrument is finished I always play around with naming.

-Igs

> I have no text file at the moment, just a few sheets of paper...
> But here are the formulae I derived.
>
> Bad news: for each scale type we need a separate formula.
> Good news: The same formulae work for all EDOs.
>
> For a scale in n-EDO with 1 large step (size L) and 4 small steps
> (size s), L and s are of the form
>
> L = 5n - 4t
> s = -n + t
>
> where t is an integer with n < t < 6n/5. If n is such that there is no
> integer fulfilling this condition, there is no scale of this type in
> n-EDO. (This one is still quite trivial...)
>
> Dito for a scale with 2 large and 3 small steps:
>
> L = -n + 3t
> s = n - 2t
>
> with 2n/5 < t < n/2
>
> For a scale with 3 large and 2 small steps:
>
> L = n - 2t
> s = -n + 3t
>
> with n/3 < t < 2n/5
>
> For a scale with 4 large and 1 small step:
>
> L = -n + t
> s = 5n - 4t
>
> with 6n/5 < t < 5n/4

Wow. Your mathematical skills amaze me. It would certainly have
taken me weeks to work that out on my own.

--- In tuning@yahoogroups.com, "Igliashon Jones" <igliashon@s...> wrote:

>
> Wow. Your mathematical skills amaze me. It would certainly have
> taken me weeks to work that out on my own.

Well, ti did take me more than one day, too... And my mans skill may
have been to identify that the equation is a standard problem that has
been solved, and finding the solution...

But I am just thinking: more constraints might make sense - as you
already wrote, the difference between L and s should not be to big,
and also (as Rich wrote) not to small.

Would L/s < 5 and L/s > 7/6 be constraints making sense?