Propriety

Dave and Dan -- as I've said before, I think propriety (Rothenberg) is a
better condition that maximal evenness for the melodic structure of a scale.
Propriety means that no "thirds" are smaller than the largest "second", no
"fourths" are smaller than the largest "third", etc. Many Indian scales are
not proper, but Carl Lumma said the drone effectively helps one find one's
melodic place the way propriety helps in the droneless case, or something
like that.

I think I'm starting to get the hang of the idea of
scale propriety. Could someone please tell me whether
the following is correct. Instead of creating an
interval chart of the type shown in Monz's tuning
dictionary, is it possible to determine propriety by
showing a scale with all of its modes, a la Dan
Stearns's method? For instance:

0 203 281 484 561 764 842 1045 1122
0 78 281 358 561 639 842 919 997
0 203 281 484 561 764 842 919 1122
0 78 281 358 561 639 716 919 997
0 203 281 484 561 639 842 919 1122
0 78 281 358 436 639 716 919 997
0 203 281 358 561 639 842 919 1122
0 78 155 358 436 639 716 919 997
0 78 281 358 561 639 842 919 1122

is Ring 9 of Golden horagram #8 of the Wilson Scale
Tree. Since some 1 step intervals @ 203 cents are
larger than one of the 2 step intervals @ 155 cents, and
so forth, that makes this an improper scale, right? And
if so, is this all that is needed to determine
propriety?

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html

--

David,

This is actually strictly proper.

From Scala:

"A "strictly proper" scale is one in which all melodic classes are
distinct and non-overlapping. In other words all 1-step intervals are
less than all 2-step intervals, all 2-step less than all 3 step, etc.
for all interval classes. Proper scales can have equal step-interval
classes. "Improper" scales have overlapping step-interval classes. An
example of a proper scale is the major scale in 12-tET. The interval
classes are non-overlapping, but the tritone occurs one time as a 3-
step and one time as a 4-step interval. A strictly proper scale is
for example the anhemitonic pentatonic scale."

JL

--- In tuning@y..., "David J. Finnamore" <daeron@b...> wrote:
> I think I'm starting to get the hang of the idea of
> scale propriety. Could someone please tell me whether
> the following is correct. Instead of creating an
> interval chart of the type shown in Monz's tuning
> dictionary, is it possible to determine propriety by
> showing a scale with all of its modes, a la Dan
> Stearns's method? For instance:
>
> 0 203 281 484 561 764 842 1045 1122
> 0 78 281 358 561 639 842 919 997
> 0 203 281 484 561 764 842 919 1122
> 0 78 281 358 561 639 716 919 997
> 0 203 281 484 561 639 842 919 1122
> 0 78 281 358 436 639 716 919 997
> 0 203 281 358 561 639 842 919 1122
> 0 78 155 358 436 639 716 919 997
> 0 78 281 358 561 639 842 919 1122
>
> is Ring 9 of Golden horagram #8 of the Wilson Scale
> Tree. Since some 1 step intervals @ 203 cents are
> larger than one of the 2 step intervals @ 155 cents, and
> so forth, that makes this an improper scale, right? And
> if so, is this all that is needed to determine
> propriety?
>
>
> --
> David J. Finnamore
> Nashville, TN, USA
> http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
>
> --

Hi Jacky and David,

The scale David gave is an improper scale.

By using modal rotations it's easy to see how the interval classes
line up. And in the Golden Horagram example David gave you can see
for example that the 203� seconds are significantly larger than the
155� thirds.

A classic example of an improper scale would be the Pythagorean
diatonic. Here the #4th is larger than the b5th. (Though this is
easier to see as meaningfully improper -- i.e., in a not trivial
sense -- in say 17-tET.)

And a classic example of a proper scale would be the 12-tET diatonic.
Here the #4th and the b5th are a shared interval.

And a classic example of a strictly proper scale would be the syntonic
diatonic. Here there are no overshooting or shared interval classes.

--Dan Stearns

Hi Monz,

I wonder if your propriety page may need some clarification
for mathematicians.

http://www.ixpres.com/interval/dict/proper.htm
"
Strictly Proper scales are those in which all intervallic size classes are distinct and
non-overlapping.
"

I took it in the set theoretic sense, that two classes overlap if they have members in common,
translating "overlap" as "set intersection".

So, came to the preliminary, somewhat puzzling conclusion that you could have a scale
that is strictly proper, and also improper.

However here it means that A and B overlap if B has a member smaller than one of the members
of A.

Once I realised that, all became clear.

Robert

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:

/tuning/topicId_3822.html#21421

> Hi Monz,
>
> I wonder if your propriety page may need some clarification
> for mathematicians.
>
> http://www.ixpres.com/interval/dict/proper.htm
> "
> Strictly Proper scales are those in which all intervallic size
> classes are distinct and non-overlapping.
> "
>
> I took it in the set theoretic sense, that two classes overlap
> if they have members in common, translating "overlap" as "set
> intersection".
>
> So, came to the preliminary, somewhat puzzling conclusion that
> you could have a scale that is strictly proper, and also improper.
>
> However here it means that A and B overlap if B has a member
> smaller than one of the members of A.
>
> Once I realised that, all became clear.

Thanks much, Robert. The page has been updated with your
amendment.

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., "David J. Finnamore" <daeron@b...> wrote:

> Since some 1 step intervals @ 203 cents are
> larger than one of the 2 step intervals @ 155 cents, and
> so forth, that makes this an improper scale, right?

Right.

> And
> if so, is this all that is needed to determine
> propriety?

Yes.

Hi Monz,

Thanks!

Maybe won't puzzle many, but if anyone does happen
to take it the way I did, could save them some head scratching
and question asking.

Robert

[ I was not available at moment this thread was active . . .
I wish to use here the following discussion about "words"
as starting point to discuss about the "propriety" concept ]

By remarks of Paul Erlich I knowed the term "proper" was used in
microtonality but I didn't know exactly what was its meaning. I had only
changed my ancient use of the term "proper" for "sui generis" to avoid
misunderstanding.

The Robert Walker's post 21421 led me to the Joseph Monzo's propriety page.

I think that the use of the term "class" in that page could be
inappropriate. Since its use suggests a mathematical sense it seems it
might be used strictly in that sense. Assuming that, it would make no sense
to write

<< Scales in which two or more pairs of interval
classes contain identical members are Proper >>

since two classes by (mathematical) definition could never have an
identical member. The "Proper" category would be so empty.

Now I would like to add this comment : even if we change the term "classes"
by "columns in the interval matrix", the idea itself, as expressed, would
be contradictory.

Before to show that contradiction I first write what I presume could be a
mathematical formulation of that definition over the interval matrix where
i and j are column indexes :

Let x be an interval in column Ci and y be an interval in column Cj. Then
for any index i and j and, within the designated columns, any interval x
and y,

"Strictly proper" <==> "i < j implies x < y"

"Proper" <==> "i < j implies x < y or x = y"

"Improper" <==> "not Proper"

The precedent formulation was contradictory for we could have both in the
same scale one case where x = y and another case where x > y (overlapping),
while the definition appears only as being -- apparently a specification
from "Strictly proper" -- simply related to the existence of some x = y
cases : what is an illogism.

After this formulation, a scale could be both "Proper" and "Improper".
Proper for having a non-empty columns intersection (x = y case) and
Improper for having an overlapping (x > y case), since it is also written :

<< Scales with overlapping [..] are Improper >>

Since the contradiction appears at definition level, it would remain even
if there wouldn't exist any case where we have both overlapping (x > y) and
ambiguity (x = y). The definition would have to refer at a proof or to
mention minimally a conjecture about a presumed incompatibility between
overlapping and ambiguity.

The following example shows both overlapping and ambiguity :

0 4 8 9 12 16 20 21 = 4 4 1 3 4 4 1
0 4 5 8 12 16 17 21 = 4 1 3 4 4 1 4
0 1 4 8 12 13 17 21 = 1 3 4 4 1 4 4
0 3 7 11 12 16 20 21 = 3 4 4 1 4 4 1
0 4 8 9 13 17 18 21 = 4 4 1 4 4 1 3
0 4 5 9 13 14 17 21 = 4 1 4 4 1 3 4
0 1 5 9 10 13 17 21 = 1 4 4 1 3 4 4

The pseudo-classes C3 = {8,9,11} and C4 = {10,12,13} overlaps, thus it's
Improper. There exist also ambiguity since 8 belongs to C3 and C2 =
{4,5,7,8}. Is it also Proper?

If someone prefers an historical example, the greek enharmonic genus
corresponds to (1 1 8 4 1 1 8) in 24-tet. It's there an "extremely"
Improper genus while its 24-tet version presents a slight ambiguity : 12
belongs both to

C2 = {2,5,9,12} and C5 = {12,15,19,22}

In particular, we have by rotation these (Aristoxenian) scales

0 8 (12) 13 14 22 23 24 = 8 4 1 1 8 1 1
0 1 2 10 11 (12) 20 24 = 1 1 8 1 1 8 4

Naturally, JI versions or any coherent version avoid this ambiguity. I
already shown three coherent JI versions of the enharmonic genus in

http://www.aei.ca/~plamothe/gammes-gsp.htm

That leads to a content discussion . . .

---------------

Forgetting the precedent remarks which concern the formulation uniquely,
rather than the content, I would like to comment now about the concepts.

While the "Interval classes" are possibly compatible with "Improper" scales
but strictly incompatible with "Simply Proper" ones -- in sense of "Proper
but not Strictly Proper" -- it is implied in the "Strictly Proper" concept
and in any "unambiguous" scale discussed by Robert Walker in post 21420 :

<< Here I'm coining a word "unambiguous"
for these types of improper scales >>

In fact, any scale being not "Simply proper" implies a partition of S, the
set of elements in the interval matrix, and consequently implies the
existence on that set of an equivalence relation R.

Expressing that relation R with the formalism

x = y mod R

or simply by

x == y

meaning "x is equivalent to y modulo R", I recall that the equivalence
relation is defined by the following axioms :

Reflexivity or (x == x)
Symmetry or (x == y) implies (y == x)
Transitivity or (x == y) and (y == z) implies (x == z)

The classes S/R exist only for any "non-simply-proper" scale and correspond
to the columns of its interval matrix.

That is important. The main property of a coherent musical system appears
to me the congruence relation between intervals and degrees, what may be
expressed by the existence of a (degree) function obeying to the following
relation :

D(x * y) = D(x) + D(y)

-- where * is multiplication with ratios and
addition with log (ratios) like cents --

what is common to many approachs and particularly explicit in Z-module
context. An equivalence relation is not forcely a congruence relation but
it is implied by the congruence.

A Proper scale which is not a Strictly Proper scale cannot be part of any
coherent musical system (in mathematical sense). There are only Strictly
Proper scales and "unambiguous" scales defined by Robert Walker having
chance to be part of such coherent musical systems.

However we find often as Simply Proper scales some degenerated N-tet
versions of coherent systems. A degenerated version is never an algebraic
structure but only an approximative image of the primeval set without the
primeval relations giving the algebraic structure.

The 53-tet versions of the Zarlino and Pythagore scales

0 9 17 22 31 39 48 53 = 9 8 5 9 8 9 5
0 9 18 22 31 40 49 53 = 9 9 4 9 9 9 4

are part of coherent systems which have all the algebraic properties of
their corresponding JI version while their common 12-tet version

0 2 4 5 7 9 11 12 = 2 2 1 2 2 2 1

is a degenerated image having not in itself an algebraic consistensy. The
problem is not simply the ambiguity

augmented fouth = diminued fifth = 6

in 12-tet while the corresponding values 26 and 27 are distinct in 53-tet,
what we could write in term of degree function

D(26) = 4
D(27) = 5

What would be D(6) in 12-tet?

However it's there only something revealing a much deeper inconsistency.
For instance, in the 12-tet version, there is no explicit reason for which
we cannot split the tone 2 in two half-tone 1, since 1 and 2 exist in the
set and

1 + 1 = 2

but explicitely the value 4 + 4 don't exist in the t53-Pythagore set and
the value 5 + 5 don't exist in the t53-Zarlino set.

There exist an algebraic structure in the t12 set : the cyclic group Z/12.
Simply that structure don't correspond to the presumed major scale
structure having to be epimorph to Z/7.

---------------

Now about the Strictly Proper sense.

When I wrote about 15 years ago an empirical algorithm permitting to
produce an ordered list of the gammier structures I had implicitely
introduced the "Strictly Proper" property. Subsequently I sought to give an
axiomatic formulation at the gammier structure. I found about (what I can
name now) the "Strictly Proper" property that it was not resulting from
more fundamental properties (which I retained as axioms). That property
would have to be itself an axiom : what it appears to me unjustifiable at
the paradigmatic level.

It's why I released my algorithm from that constraint. Comparing then the
corrected list of gammiers with the ancient, I found at rank 8 just before
the Zarlino gammier, a new gammier which is precisely the japanese gammier
described in the precedent link.

---------------

I can't say more for I never read about the Rothenberg concept something
else that few recent posts in this thread and the propriety page in Monzo
site. I have simply a light scepticism about its potentiality to explain a
categoric perception. Since important historical systems as the japanese
one and the Aristoxenian enharmonic genus don't use it, I presume that it
could serve as a beacon in the actual babel tower where so much numbers and
idioms are in use.

Pierre Lamothe

Hi Pierre,

Maybe a mathematical definition of propriety may help.

For a scale S, define A(s,1) = set of all the single step intervals of s.
Define A(s,2) = set of all two step intervals, and so on.

E.g. for s={1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1}

A(s,1) = {25/24, 135/128, 16/15, 27/25, 2/1}
A(s,2) = {10/9, 9/8, 256/225}
A(s,3) = {75/64, 32/27, 6/5}
etc.

A scale is strictly proper if we can define a linear ordering < by

A(s,i) < A(s,j) <-> x < y for any choice of x in A(s,i) and y in A(s,j)

and if this linear ordering has the additional property that
i<j -> A(s,i)<A(s,j).

In this case the implication will go both ways,
i<j <-> A(s,i)<A(s,j).
and so we have an isomorphism between the < on the integers and the
< on the sets.

It is proper if we can define a linear ordering <2 on these sets by

A(s,i) <2 A(s,j) <-> x <= y for any choice of x in A(s,i) and y in A(s,j)

and if this ordering satisfies the additional property that
i<j -> A(s,i)<2 A(s,j).

This is no longer necessarily an isomorphism.

Both of these orderings are clearly partial orderings
- reflexive, anti-symmetric and transitive.

However, either of them can fail to be a linear ordering if there is an
i and j with A(s,i) and A(s,j) in no order relation with
each other - i.e. if it isn't total.

Now consider the set I(s) of all the intervals between elements of s.

Then my definition of an unambiguous scale s is one in which every element of
I(s) belongs to only one of the A(s,i)

So a strictly proper scale is an unambiguous proper scale.

In this case, the property of belonging to the same class is an equivalence
relation between interval sizes, and the classes provide a partition of
I(s) into equivalence classes. So the word class here is appropriate
mathematically, if you think of it as being used like "equivalence class".

A proper scale is ambiguous, and the property of belonging to the same
class is no longer an equivalence relation. For example, in 12-tet,
the tritone belongs to two different classes, and the other members
of those classes belong to one of them only, so the relation of
belonging to the same class isn't transitive.

An improper scale, if we define it as meaning "not proper"
can be either ambiguous or unambiguous.

One could have an unambiguous improper scale if the A(s,i) provide a
partition of I(s) into equivalence classes, but one that doesn't
preserve the ordering on the i.

In this case, the ordering A(s,i) < A(s,j) is still a linear ordering,
but it is no longer isomorphic to the ordering on the i,j.

One could also have an ambiguous improper scale which can still be ordered
using the ordering <2, as a total linear ordering. This then would be
somewhat analagous to a proper scale.

Or, one could have an ambiguous proper scale for which neither
< nor <= is total.

I.e. with sets A(s,i) and A(s,j) not related to each other by either
of the orderings - e.g. if A(s,i) has some elements greater than some
of A(s,j), and some elements smaller than its elements.

Robert

Hi Pierre,

Correction:

I see I hadn't taken equality into account properly for the
defintion of a linear ordering.

Define A(s,k) = set of all the k step intervals of s.

Define:
A(s,i) = A(s,j) if they are equal as sets.

A(s,i) < A(s,j) <-> x < y for any choice of x in A(s,i) and y in A(s,j)

A(s,i) <2 A(s,j) <-> x <= y for any choice of x in A(s,i) and y in A(s,j)
and A(s,i) != A(s,j)

A(s,i) <= A(s,j) if A(s,i) < A(s,j) or A(s,i) = A(s,j)
A(s,i) <=2 A(s,j) if A(s,i) <2 A(s,j) or A(s,i) = A(s,j)

Then here are the corrected definitions:

A scale is strictly proper if the ordering <= is total
and if i<j -> A(s,i)<A(s,j).

It is proper if the ordering <=2 is total and i<j -> A(s,i)<2 A(s,j).

I.e. as before, but using <= instead of <, as the definition of a linear
ordering is defined in terms of <= type orderings rather than < type orderings,
and also one should add the extra condition A(s,i) != A(s,j) for <2.

Robert

Hi Robert,

I like very much discussion about definitions. That permits to open new
doors. I shall begin with short remarks before to talk about main subjects.

--1--

I forgot to note in my precedent post I was very glad to see that Joseph
Monzo used the interval matrix with scales. That will facilitate
eventually, as reference, the presentation of my algebraic structures.

--2--

You wrote

<< Now consider the set I(s) of all the intervals between
elements of s. Then my definition of an unambiguous scale s
is one in which every element of I(s) belongs to only one
of the A(s,i) >>

I note the term "unambiguous" is not used only with improper scales. It is
much more simple. Thus the unambiguous scales are synonymous with scales
having "equivalence classes" (so potentially "congruence classes"). That
replaces the barbarism "non-simply-proper" scales I was using.

--3--

I would like first to signal I made an error in my precedent post. When I
wrote

<< even if we change the term "classes"
by "columns in the interval matrix" >>

I had in head your precedent post using the Dan Stearn's presentation of
the scales obtained by rotation. Since I was refering to interval matrix
used by Joseph Monzo rather than the Dan Stearn's presentation I would have
had to talk about diagonals rather than columns.

Besides, the interval matrix applied to a scale corresponds to a torus
where the classes -- or pseudo-classes if ambiguous scales -- are circles
on the torus : what is related to diagonals in the matrix.

--- Use of ordering ---

You wrote

<< For a scale S, define A(s,1) = set of all the single
step intervals of s. Define A(s,2) = set of all two step
intervals, and so on. >>

and briefly

<< Define A(s,k) = set of all the k step intervals of s. >>

As long as your subsets A(s,k) refer directly to a scale s

-- rather than uniquely to the generated intervals I(s)

there exist a canonical total ordering =< on the subsets which may be
simply defined by the total ordering =< on the indexes

A(s,i) =< A(s,j) <--> i =< j

-- what implies the expression << k step interval >>
is well defined (I discuss that forward) --

This well ordering has the following property : min A(s,k) and max A(s,k)
respect the standard ordering =< on intervals

min A(s,i) =< min A(s,j) <--> i =< j
max A(s,i) =< max A(s,j) <--> i =< j

So I think there is no need to use another ordering if the goal is simply
to define "Strictly proper" and "Proper". Besides it may be hard for most
people to manipulate partial ordering lattices of subsets.

Let x be an interval of A(s,i) and y be an interval of A(s,j) where

A(s,i) < A(s,j) -- what is equivalent to i < j --

then the scale s in "strictly proper" if (for any x and any y) we have

x < y

and the scale s is "proper" if (for any x and any y) we don't NEVER have

x > y

--- Classes on set I ---

It's a more sophisticated problem to define the classes in a set I(s) if s
itself is unknown. We cannot use the steps in s and the expression

<< k step interval >>

is no more a well defined object. We have to use another concept I name
"atom" which corresponds to "step" only if the set is consistent.

Let I be an ordered set of intervals in the first octave. By definition, an
"atom" of I is any interval k greater than unison that cannot be splitted
in two other intervals x and y greater than unison, where x < k and y < k.

For instance, the sets I(s) corresponding to

s1 = <1 9/8 5/4 4/3 3/2 5/3 15/8>

s2 = [0 9 17 22 31 39 48]

s3 = [0 2 4 5 7 9 11]

are I(s1) I(s2) I(s3) = '<s1\s1> '[s2\s2] '[s3\s3] =

<1 16/15 10/9 9/8 32/27 6/5 5/4 4/3 27/20 45/32 .. (reverse) >

[0 5 8 9 13 14 17 22 23 26 27 31 36 39 40 44 45 48]

[0 1 2 3 4 5 6 7 8 9 10 11]

and the atoms 'a'(s\s) are

<16/15 10/9 9/8>

[5 8 9]

[1]

So the interval sets I(s1) and I(s2) are consistent relatively to the
scales s1 and s2 for the atoms of I(s1) and I(s2) correspond respectively
to the steps in s1 and s2. The third interval set I(s3) is not consistent
relatively to the scale s3 for the unique atom [1] of I(s3) don't
correspond to the steps [1 2] in S3. However the set I(s3) is consistent
relatively to the dodecatonic scale.

--- Property of atoms ---

I signal here an interesting property concerning atoms and chordic
generators in consistent systems. The atoms of an interval set S generated
by a chordic generator G (which may be a scale) are the atoms of the finite
differences on the cyclical-ordered generator. What simplifies not only
calculation with computer but permits to design systems easily without
computer. For instance about the gammier 9 containing the Zarlino mode :

G = <1 3 5 9 15>

S = '<G\G> = <1 16/15 10/9 9/8 6/5 5/4 4/3 .. (reverse) >

A(S) = a <S> = <16/15 10/9 9/8>

A(G) = 'ad' <1 3 5 9 15>
'ad <1 9 5 3 15>
'a <9/8 10/9 6/5 5/4 16/15>
' <9/8 10/9 16/15>

<16/15 10/9 9/8>

--- Congruity axiom ---

I already shown how to define the degree function in the context of
infinite lattices (Z-modules). The problem, generally bad addressed by the
tenants of the infinite approach (say Hellegouarch), is to well determine
the finite sets for which the class algebra is optimally reflected in a
non-closed interval algebra.

The congruence expressed by the degree function may be obtained in
non-closed finite sets by obeying the congruity axiom which stands that all
complete factorization* of a same interval (in atoms of the set) have the
same number of atoms. It is the way I introduce unambiguouness in finite
context.

(*) These factorizations are represented by parenthesed
expressions or binary trees since the composition law
is only partial in finite sets.

--- Regularity axiom ---

I terminate by this remark : the only axiom I retained concerning the
ordering in the gammier theory is the regularity axiom which stands that
for any atom x we have to obey at x < 2/x. That eliminates torsion in
structures.

Regards,

Pierre

Hi Pierre,

Thanks, I also like working on definitions, in fact in a way that was
my area of research when I was at university - research into a new
axiom system for set theory.

"
I note the term "unambiguous" is not used only with improper scales. It is
much more simple. Thus the unambiguous scales are synonymous with scales
having "equivalence classes" (so potentially "congruence classes"). That
replaces the barbarism "non-simply-proper" scales I was using.
"

Yes indeed, that's how I'm using the word.

"
As long as your subsets A(s,k) refer directly to a scale s

-- rather than uniquely to the generated intervals I(s)

there exist a canonical total ordering =< on the subsets which may be
simply defined by the total ordering =< on the indexes

A(s,i) =< A(s,j) <--> i =< j

-- what implies the expression << k step interval >>
is well defined (I discuss that forward) --

This well ordering has the following property : min A(s,k) and max A(s,k)
respect the standard ordering =< on intervals

min A(s,i) =< min A(s,j) <--> i =< j
max A(s,i) =< max A(s,j) <--> i =< j

So I think there is no need to use another ordering if the goal is simply
to define "Strictly proper" and "Proper". Besides it may be hard for most
people to manipulate partial ordering lattices of subsets.

Let x be an interval of A(s,i) and y be an interval of A(s,j) where

A(s,i) < A(s,j) -- what is equivalent to i < j --

then the scale s in "strictly proper" if (for any x and any y) we have

x < y

and the scale s is "proper" if (for any x and any y) we don't NEVER have

x > y
"

Yes, that's a much simpler way of setting about it, and will prob. be
easier for most to use.

One can then introduce the idea of an alternate non isomporphic ordering
on the A(s,i) later if one wants.

"
It's a more sophisticated problem to define the classes in a set I(s) if s
itself is unknown. We cannot use the steps in s and the expression

<< k step interval >>

is no more a well defined object. We have to use another concept I name
"atom" which corresponds to "step" only if the set is consistent.

Let I be an ordered set of intervals in the first octave. By definition, an
"atom" of I is any interval k greater than unison that cannot be splitted
in two other intervals x and y greater than unison, where x < k and y < k.
"

Useful definition, thanks.

Robert