Yahoo Tuning Groups Ultimate Backup tuning-math Homometric sets

A finite set S of real numbers is "homometric" to another such set T,
if the distances between S's elements are the same multiset as the distanced between T's
elements. This is trivial if S and T are the same set up to translation
and reflection, nontrivial otherwise.

I classified all homometric pairs of N-element sets for N<=7 in 2010, and I now post the results without proof.

GAP NOTATION: [a,b,c] means the 4-element set {0, a, a+b, a+b+c};
the numbers in the square brackets are the consecutive gaps.

Theorem:
There are exactly two families of homometric 6-element sets.
They are
FAMILY I:
[ 2x-q, 2q-2x, x, 4x-3q, q-x ]
[ 3x-2q, 2q-2x, 4x-3q, q, q-x ]
FAMILY II:
[ x+a, 2x+4a, x, x+2a, a ]
[ x+2a, 2x+2a, x+3a, x, a ]
(And of course, the reversals of these also are trivially homometric.)

Theorem:
Every cardinality=7 homopair is either one of the
SIX 2-PARAMETER(a,b) FAMILIES:
FAMILY I: sum=a+11b
[3b, 2b, b, a, 4b, b]
[4b, b, a, 3b, 2b, b]

FAMILY II: sum=6a+6b which always is a multiple of 6
[a, b, a, a+2b, 3a+2b, b]
[a, 2a+3b, 2a+b, b, a, b]

FAMILY III: sum=7a+6b
[a+b, a, a+b, b, 3a+3b, a]
[2a+b, a+2b, 2a+2b, a, b, a]

FAMILY IV: sum=8a+b
[3a, b, 2a-b, a, b+a, a]
[2a+b, 2a-b, a, b, 2a, a]

FAMILY V: sum=8a+7b
[2a+b, 2a+3b, a, b, 2a+b, a+b]
[3a+2b, b, 2a+2b, a+b, a, a+b]

FAMILY VI: sum=10a+11b
[4b+4a, b, a, 3b+2a, 2b+2a, b+a]
[3b+3a, 2b+2a, b, a, 4b+3a, b+a]

or it arises as a multiple (each thus, really, is a 1-parameter family)
of one of the FIVE SPORADIC MIRACLES
span=12:
[ 2, 2, 1, 2, 4, 1]
[ 3, 2, 2, 3, 1, 1]
genfn: 1+x^2+x^4+x^5+x^7+x^11+x^12 = (x^7+x^6+x^5+x^4+x^3+x^2+1)*(x^5-x^3+1)

span=15(i):
[ 3, 1, 2, 2, 6, 1]
[ 4, 2, 5, 1, 2, 1]
genfn: 1+x^3+x^4+x^6+x^8+x^14+x^15 =
(x^12+x^11+x^10 - x^7 + x^4+x^3+x^2+x+1)*(x^3-x+1)

span=15(ii):
[ 3, 3, 1, 1, 5, 2]
[ 5, 1, 2, 4, 1, 2]
genfn: 1+x^3+x^6+x^7+x^8+x^13+x^15 =
(x^7-x^6+x^5-x^4+x^2-x+1)*(x^8+x^7+x^6+x^5+x^4+x+1)

span=16:
[ 3, 1, 2, 5, 4, 1]
[ 3, 2, 6, 1, 3, 1]
genfn: 1+x^3+x^4+x^6+x^11+x^15+x^16 =
(x^7+x^6+x^5+x^4+x^3+x^2+1)*(x^9-x^7+x^4-x^2+1)

span=17:
[ 4, 1, 3, 2, 5, 2]
[ 5, 1, 4, 3, 2, 2]
genfn:
1+x^4+x^5+x^8+x^10+x^15+x^17 =
(x^9-x^8+x^7-x^6+x^4-x^3+x^2-x+1)*(x^8+x^7+x^6+x^5+x^4+x+1)

🔗Mike Battaglia <battaglia01@gmail.com>

This is excellent work! The obvious application of this is to
Z-relations, so Paul H should take note. Keenan's been brewing over
sets of differences for weeks now in their applications to
understanding Fokker periodicity blocks, so this might have some
application there as well, and would finally provide the connection
I've had a hunch exists between Z-relations and periodicity blocks.

Since sets of differences and interval vectors are closely related,
this goes a long way in explaining some of the properties of
Z-relations and generalizing them as well. I've only run a quick test
so far, but from what I've seen, it looks to hold. For example:

On Fri, Oct 7, 2011 at 2:53 PM, WarrenS <warren.wds@gmail.com> wrote:
>
> Theorem:
> Every cardinality=7 homopair is either one of the
> SIX 2-PARAMETER(a,b) FAMILIES:
> FAMILY I: sum=a+11b
> [3b, 2b, b, a, 4b, b]
> [4b, b, a, 3b, 2b, b]

So if we plug in a=1 and b=1, we obtain the sets 0 4 5 6 9 11 12 and 0
3 5 6 7 11 12. These can be taken to correspond to positions along a
chain of meantone generators. If we do this, the first set represents
C E B F# D# E# B#, which when reduced to the octave is C D# E E# F# B
B# C. The second set is C A B F# C# E# B#, which when reduced to the
octave yields the 7-note pitch set C C# E# F# A B B#.

If we generalize the concept of an interval vector as I laid out here -

/tuning-math/message/19650

Then the intvec for both of these scales is <3 2 2 2 3 3 2 1 1 0 1 1>,
and so we now have our first rank-2 Z-related set. This "family" seems
so far to hold for all integer values of a and b I've tested.

I'll do a more thorough search when I have some more time, but this is
a pretty great development!

Some random thoughts about things that I'd like to do - in a rush so
will have to come back later
1) Generalizing this to sets of n-tuples of integers, so we can
generalize to rank 3 and periodicity blocks - for some set S of
n-tuples of integers, we can define a function f(S,x) for 0<x<=n that
returns a new set S of integers, consisting of only the xth elements
of each n-tuple in S. Unless I'm missing something obvious, it should
be the case that any two sets S and T of n-tuples are homometric iff
f(S,x) is homometric to f(T,x) for any 0<x<=n.
2) Finding homometric sets in which all of the numbers in the set map
to a unique value mod x, where x is some position along the chain of
generators corresponding to the chroma of the scale (in the case of
meantone[7], x would just be 7). I'll devote some thought later on how
to do this.
3) Seeing if there are any 6- or 7-note rank-3 Fokker blocks which
fall into one of the above families, and seeing what the corresponding
Z-related Fokker blocks are

Great work!

-Mike

🔗Paul <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> This is excellent work! The obvious application of this is to
> Z-relations, so Paul H should take note. Keenan's been brewing over
> sets of differences for weeks now in their applications to
> understanding Fokker periodicity blocks, so this might have some
> application there as well, and would finally provide the connection
> I've had a hunch exists between Z-relations and periodicity blocks.
>
> Since sets of differences and interval vectors are closely related,
> this goes a long way in explaining some of the properties of
> Z-relations and generalizing them as well. I've only run a quick test
> so far, but from what I've seen, it looks to hold. For example:
>
> On Fri, Oct 7, 2011 at 2:53 PM, WarrenS <warren.wds@...> wrote:
> >
> > Theorem:
> > Every cardinality=7 homopair is either one of the
> > SIX 2-PARAMETER(a,b) FAMILIES:
> > FAMILY I: sum=a+11b
> > [3b, 2b, b, a, 4b, b]
> > [4b, b, a, 3b, 2b, b]
>
> So if we plug in a=1 and b=1, we obtain the sets 0 4 5 6 9 11 12 and 0
> 3 5 6 7 11 12. These can be taken to correspond to positions along a
> chain of meantone generators. If we do this, the first set represents
> C E B F# D# E# B#, which when reduced to the octave is C D# E E# F# B
> B# C. The second set is C A B F# C# E# B#, which when reduced to the
> octave yields the 7-note pitch set C C# E# F# A B B#.
>
> If we generalize the concept of an interval vector as I laid out here -
>
> /tuning-math/message/19650
>
> Then the intvec for both of these scales is <3 2 2 2 3 3 2 1 1 0 1 1>,
> and so we now have our first rank-2 Z-related set. This "family" seems
> so far to hold for all integer values of a and b I've tested.
>
> I'll do a more thorough search when I have some more time, but this is
> a pretty great development!
>
> Some random thoughts about things that I'd like to do - in a rush so
> will have to come back later
> 1) Generalizing this to sets of n-tuples of integers, so we can
> generalize to rank 3 and periodicity blocks - for some set S of
> n-tuples of integers, we can define a function f(S,x) for 0<x<=n that
> returns a new set S of integers, consisting of only the xth elements
> of each n-tuple in S. Unless I'm missing something obvious, it should
> be the case that any two sets S and T of n-tuples are homometric iff
> f(S,x) is homometric to f(T,x) for any 0<x<=n.
> 2) Finding homometric sets in which all of the numbers in the set map
> to a unique value mod x, where x is some position along the chain of
> generators corresponding to the chroma of the scale (in the case of
> meantone[7], x would just be 7). I'll devote some thought later on how
> to do this.
> 3) Seeing if there are any 6- or 7-note rank-3 Fokker blocks which
> fall into one of the above families, and seeing what the corresponding
> Z-related Fokker blocks are
>
> Great work!
>
> -Mike

Well you guys did it! It's a little strange by my way of doing things, but I have to admit it works. First off, the vector has 12 positions, instead of 11, and that is due to the B# and C enharmonicity. I actually prefer to call these isometric sets, but homometric is fine also.

This would not work if it were not for the enharmonicity, because if you collapse them
you get two hexads here (my N5 and D) which are not the same intvec at all. They are not even the same type.

And I found that it never works for just 12 pitches in a cycle, because, if the extended 11-place vectors are the same, then the types "must needs be" the same, so they would not be Z-related.

I have to think about this, it's blown away a lot of my theories. But I am excited to see
intvecs and isometric sets extended to higher ranks. I'm glad the Z-relation didn't have
to die (or poop out at rank-1)

I would like however to keep discussing how Z-relations in rank-1 "work" due to the reflections and such. This extends to Coxeter-Dynkin diagram theory, and maybe even
Lie Algebras (which are infinite, hence > rank-1.....) Perhaps this rank-2 homometric
pair exhibits some cool reflection symmetries, I will have to study it.

pgh

🔗Mike Battaglia <battaglia01@gmail.com>

Some quick afterthoughts -

On Fri, Oct 7, 2011 at 4:35 PM, Mike Battaglia <battaglia01@gmail.com>
wrote:
>
> Some random thoughts about things that I'd like to do - in a rush so
> will have to come back later
> 1) Generalizing this to sets of n-tuples of integers, so we can
> generalize to rank 3 and periodicity blocks - for some set S of
> n-tuples of integers, we can define a function f(S,x) for 0<x<=n that
> returns a new set S of integers, consisting of only the xth elements
> of each n-tuple in S. Unless I'm missing something obvious, it should
> be the case that any two sets S and T of n-tuples are homometric iff
> f(S,x) is homometric to f(T,x) for any 0<x<=n.

If using n-tuples is supposed to be one way to represent the lattice, we
might not want to just use unweighted n-tuples of reals. Most generally, an
n-tuple represents an interval's position on the lattice, but for us to get
the "distance" between that interval and another, we can define a (perhaps
weighted) norm over the space of intervals, and use that for the
corresponding multiset. The most direct generalization of z-relation would
imply that we use Tenney weighting with the L1 norm.

But who says we can't work with an unweighted L1 norm, or a weighted L2 norm
instead?

It's looking more and more to me like the music set-theoretic concept of a
"Z-relation" as applied to 12-equal can be viewed as a subset of a more
fundamental thing with more free parameters (basis lattice and norm, for
starters). Once it's fleshed out exactly what that is, set-theoretic
Z-relations in 12-equal will likely end up looking like instantiations of
that more fundamental thing, but with those parameters tweaked to properties
that are 12-centric rather than auditory system-centric. This seems to be a
common theme in academia, and I wouldn't be surprised if we run into it
again here too.

One more obvious question comes to mind: consider the hexad represented by
4:5:6:7:9:11. I don't think that there's going to be any isometric set to
this in JI except for the 11-limit utonality and the translations of all
these. However, what temperaments are there that map this hexad to one of
Warren's families? Such a temperament would contain the 11-limit otonality,
the 11-limit utonality, and a third choice, Z-related to the other two,
which contains only individual dyads up to the 11-odd-limit.

(It would probably be best to limit a search for this to the rank-2, period
1 case for now, until the issues I fleshed out above are worked out.)

-Mike

🔗Paul <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Some quick afterthoughts -
>
> On Fri, Oct 7, 2011 at 4:35 PM, Mike Battaglia <battaglia01@...>
> wrote:
> >
> > Some random thoughts about things that I'd like to do - in a rush so
> > will have to come back later
> > 1) Generalizing this to sets of n-tuples of integers, so we can
> > generalize to rank 3 and periodicity blocks - for some set S of
> > n-tuples of integers, we can define a function f(S,x) for 0<x<=n that
> > returns a new set S of integers, consisting of only the xth elements
> > of each n-tuple in S. Unless I'm missing something obvious, it should
> > be the case that any two sets S and T of n-tuples are homometric iff
> > f(S,x) is homometric to f(T,x) for any 0<x<=n.
>
> If using n-tuples is supposed to be one way to represent the lattice, we
> might not want to just use unweighted n-tuples of reals. Most generally, an
> n-tuple represents an interval's position on the lattice, but for us to get
> the "distance" between that interval and another, we can define a (perhaps
> weighted) norm over the space of intervals, and use that for the
> corresponding multiset. The most direct generalization of z-relation would
> imply that we use Tenney weighting with the L1 norm.
>
> But who says we can't work with an unweighted L1 norm, or a weighted L2 norm
> instead?
>
> It's looking more and more to me like the music set-theoretic concept of a
> "Z-relation" as applied to 12-equal can be viewed as a subset of a more
> fundamental thing with more free parameters (basis lattice and norm, for
> starters). Once it's fleshed out exactly what that is, set-theoretic
> Z-relations in 12-equal will likely end up looking like instantiations of
> that more fundamental thing, but with those parameters tweaked to properties
> that are 12-centric rather than auditory system-centric. This seems to be a
> common theme in academia, and I wouldn't be surprised if we run into it
> again here too.
>
> One more obvious question comes to mind: consider the hexad represented by
> 4:5:6:7:9:11. I don't think that there's going to be any isometric set to
> this in JI except for the 11-limit utonality and the translations of all
> these. However, what temperaments are there that map this hexad to one of
> Warren's families? Such a temperament would contain the 11-limit otonality,
> the 11-limit utonality, and a third choice, Z-related to the other two,
> which contains only individual dyads up to the 11-odd-limit.
>
> (It would probably be best to limit a search for this to the rank-2, period
> 1 case for now, until the issues I fleshed out above are worked out.)
>
> -Mike
>
Yes, stick to rank-2, its a rich enough environment. I am still troubled by the
fact that it is not cyclic, it doesn't connect at the endpoints, and seems to have
nothing whatsoever to do with rank-1 Z-relations, but I can adjust. I have to study
this more thoroughly but if rank-1 is a subset of the higher ranks, do we just
throw reflection theory out the window?

pgh

🔗Paul <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > This is excellent work! The obvious application of this is to
> > Z-relations, so Paul H should take note. Keenan's been brewing over
> > sets of differences for weeks now in their applications to
> > understanding Fokker periodicity blocks, so this might have some
> > application there as well, and would finally provide the connection
> > I've had a hunch exists between Z-relations and periodicity blocks.
> >
> > Since sets of differences and interval vectors are closely related,
> > this goes a long way in explaining some of the properties of
> > Z-relations and generalizing them as well. I've only run a quick test
> > so far, but from what I've seen, it looks to hold. For example:
> >
> > On Fri, Oct 7, 2011 at 2:53 PM, WarrenS <warren.wds@> wrote:
> > >
> > > Theorem:
> > > Every cardinality=7 homopair is either one of the
> > > SIX 2-PARAMETER(a,b) FAMILIES:
> > > FAMILY I: sum=a+11b
> > > [3b, 2b, b, a, 4b, b]
> > > [4b, b, a, 3b, 2b, b]
> >
> > So if we plug in a=1 and b=1, we obtain the sets 0 4 5 6 9 11 12 and 0
> > 3 5 6 7 11 12. These can be taken to correspond to positions along a
> > chain of meantone generators. If we do this, the first set represents
> > C E B F# D# E# B#, which when reduced to the octave is C D# E E# F# B
> > B# C. The second set is C A B F# C# E# B#, which when reduced to the
> > octave yields the 7-note pitch set C C# E# F# A B B#.
> >
> > If we generalize the concept of an interval vector as I laid out here -
> >
> > /tuning-math/message/19650
> >
> > Then the intvec for both of these scales is <3 2 2 2 3 3 2 1 1 0 1 1>,
> > and so we now have our first rank-2 Z-related set. This "family" seems
> > so far to hold for all integer values of a and b I've tested.
> >
> > I'll do a more thorough search when I have some more time, but this is
> > a pretty great development!
> >
> > Some random thoughts about things that I'd like to do - in a rush so
> > will have to come back later
> > 1) Generalizing this to sets of n-tuples of integers, so we can
> > generalize to rank 3 and periodicity blocks - for some set S of
> > n-tuples of integers, we can define a function f(S,x) for 0<x<=n that
> > returns a new set S of integers, consisting of only the xth elements
> > of each n-tuple in S. Unless I'm missing something obvious, it should
> > be the case that any two sets S and T of n-tuples are homometric iff
> > f(S,x) is homometric to f(T,x) for any 0<x<=n.
> > 2) Finding homometric sets in which all of the numbers in the set map
> > to a unique value mod x, where x is some position along the chain of
> > generators corresponding to the chroma of the scale (in the case of
> > meantone[7], x would just be 7). I'll devote some thought later on how
> > to do this.
> > 3) Seeing if there are any 6- or 7-note rank-3 Fokker blocks which
> > fall into one of the above families, and seeing what the corresponding
> > Z-related Fokker blocks are
> >
> > Great work!
> >
> > -Mike
>
> Well you guys did it! It's a little strange by my way of doing things, but I have to admit it works. First off, the vector has 12 positions, instead of 11, and that is due to the B# and C enharmonicity. I actually prefer to call these isometric sets, but homometric is fine also.
>
> This would not work if it were not for the enharmonicity, because if you collapse them
> you get two hexads here (my N5 and D) which are not the same intvec at all. They are not even the same type.
>
> And I found that it never works for just 12 pitches in a cycle, because, if the extended 11-place vectors are the same, then the types "must needs be" the same, so they would not be Z-related.
>
> I have to think about this, it's blown away a lot of my theories. But I am excited to see
> intvecs and isometric sets extended to higher ranks. I'm glad the Z-relation didn't have
> to die (or poop out at rank-1)
>
> I would like however to keep discussing how Z-relations in rank-1 "work" due to the reflections and such. This extends to Coxeter-Dynkin diagram theory, and maybe even
> Lie Algebras (which are infinite, hence > rank-1.....) Perhaps this rank-2 homometric
> pair exhibits some cool reflection symmetries, I will have to study it.
>
> pgh
>

Errata: They correspond to my hexads "I" which has three tritones and "O5" which has
two. These have collapsed rank-1 intvecs <322242> and <224223> A fun fact as
you can see is that the intvecs are scrambles of each other. (1<->6, 3<->5). I played
with that permutation a bit but it is not as common as my rank-1 1<->5 and 2<->4.

pgh

🔗Paul <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
> A finite set S of real numbers is "homometric" to another such set T,
> if the distances between S's elements are the same multiset as the distanced between T's
> elements. This is trivial if S and T are the same set up to translation
> and reflection, nontrivial otherwise.
>
> I classified all homometric pairs of N-element sets for N<=7 in 2010, and I now post the results without proof.
>
> GAP NOTATION: [a,b,c] means the 4-element set {0, a, a+b, a+b+c};
> the numbers in the square brackets are the consecutive gaps.
>
> Theorem:
> There are exactly two families of homometric 6-element sets.
> They are
> FAMILY I:
> [ 2x-q, 2q-2x, x, 4x-3q, q-x ]
> [ 3x-2q, 2q-2x, 4x-3q, q, q-x ]
> FAMILY II:
> [ x+a, 2x+4a, x, x+2a, a ]
> [ x+2a, 2x+2a, x+3a, x, a ]
> (And of course, the reversals of these also are trivially homometric.)
>
> Theorem:
> Every cardinality=7 homopair is either one of the
> SIX 2-PARAMETER(a,b) FAMILIES:
> FAMILY I: sum=a+11b
> [3b, 2b, b, a, 4b, b]
> [4b, b, a, 3b, 2b, b]
>
> FAMILY II: sum=6a+6b which always is a multiple of 6
> [a, b, a, a+2b, 3a+2b, b]
> [a, 2a+3b, 2a+b, b, a, b]
>
> FAMILY III: sum=7a+6b
> [a+b, a, a+b, b, 3a+3b, a]
> [2a+b, a+2b, 2a+2b, a, b, a]
>
> FAMILY IV: sum=8a+b
> [3a, b, 2a-b, a, b+a, a]
> [2a+b, 2a-b, a, b, 2a, a]
>
> FAMILY V: sum=8a+7b
> [2a+b, 2a+3b, a, b, 2a+b, a+b]
> [3a+2b, b, 2a+2b, a+b, a, a+b]
>
> FAMILY VI: sum=10a+11b
> [4b+4a, b, a, 3b+2a, 2b+2a, b+a]
> [3b+3a, 2b+2a, b, a, 4b+3a, b+a]
>
> or it arises as a multiple (each thus, really, is a 1-parameter family)
> of one of the FIVE SPORADIC MIRACLES
> span=12:
> [ 2, 2, 1, 2, 4, 1]
> [ 3, 2, 2, 3, 1, 1]
> genfn: 1+x^2+x^4+x^5+x^7+x^11+x^12 = (x^7+x^6+x^5+x^4+x^3+x^2+1)*(x^5-x^3+1)
>
> span=15(i):
> [ 3, 1, 2, 2, 6, 1]
> [ 4, 2, 5, 1, 2, 1]
> genfn: 1+x^3+x^4+x^6+x^8+x^14+x^15 =
> (x^12+x^11+x^10 - x^7 + x^4+x^3+x^2+x+1)*(x^3-x+1)
>
> span=15(ii):
> [ 3, 3, 1, 1, 5, 2]
> [ 5, 1, 2, 4, 1, 2]
> genfn: 1+x^3+x^6+x^7+x^8+x^13+x^15 =
> (x^7-x^6+x^5-x^4+x^2-x+1)*(x^8+x^7+x^6+x^5+x^4+x+1)
>
> span=16:
> [ 3, 1, 2, 5, 4, 1]
> [ 3, 2, 6, 1, 3, 1]
> genfn: 1+x^3+x^4+x^6+x^11+x^15+x^16 =
> (x^7+x^6+x^5+x^4+x^3+x^2+1)*(x^9-x^7+x^4-x^2+1)
>
> span=17:
> [ 4, 1, 3, 2, 5, 2]
> [ 5, 1, 4, 3, 2, 2]
> genfn:
> 1+x^4+x^5+x^8+x^10+x^15+x^17 =
> (x^9-x^8+x^7-x^6+x^4-x^3+x^2-x+1)*(x^8+x^7+x^6+x^5+x^4+x+1)
>
Warren, could you explain the generating function, is it like a cycle index (Polya's)?

Thanks

pgh

🔗Mike Battaglia <battaglia01@gmail.com>

🔗Paul <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Oct 7, 2011 at 5:18 PM, Paul <phjelmstad@...> wrote:
> > I am still troubled by the
> > fact that it is not cyclic, it doesn't connect at the endpoints, and seems to have
> > nothing whatsoever to do with rank-1 Z-relations, but I can adjust.
>
> It has everything to do with rank-1 Z-relations, because it tells you
> how many instance of each interval in the temperament, up to octave
> equivalence, exists in a certain scale or set.
>
> > I have to study
> > this more thoroughly but if rank-1 is a subset of the higher ranks, do we just
> > throw reflection theory out the window?
>
> What is "reflection theory?"
>
> -Mike
>
In rank-1, I have classified all Z-relations in 12-tET (and in many other temperaments) based on reflection theory. For example,
in 12-tET, hexads (which are the same anyway due to complementation)
there are 5 species as I mentioned in a previous post, but for example, take a pair like 0,1,2,3,5,9 and it's complement 4,6,7,8,10,11 you can get from one type to the other by reflecting
5 across the cluster 0,1,2 to go to 9 (1 being the hyperplane) and keep 3,9 tritone fixed. Therefore, you are reflecting across a fixed cluster, so the intvec of 0,1,2,5 and 9,0,1,2 are the same, the reflected node's relationship to the cluster is the same, and to the fixed 3,9 because of the motion by 6. It's just invariance theory,
so if that applies to rank-2 and up I am happy with that. It also
ties into Coxeter-Dykin diagrams, I suspect, and Lie Algebras, and so forth.

I am looking at the example mentioned, and it's like some kind of voodoo! These sets have nothing to do with each other, at first glance, so I am trying to figure out the mechanism, which is far more complicated than rank-1, I have a question to Warren as you see.

Since you do not wrap around in a cyclic way and use reduced vectors
its is quite different. In rank-1, position n and position 12-n are always the same value (so the count for 5-steps = 7-steps, e.g.)

pgh

🔗WarrenS <warren.wds@gmail.com>

uh, I don't even understand what you guys are talking about, for
the most part. Nor am I sure how important this all is for music.
This music professor named Ozzard got on my case 1-2 years back
and nagged me to do the
classification I just posted. I'd sent him it before.

Anyhow, in this paper in about 1990 by Lemke, Skiena, and Smith (me)
in some order, we discussed the theory of the problem of reconstructing sets
from their distances, and theory of homometric sets. There have been various
versions of that paper floating around the internet ever since, which I daresay you can quickly find. Skiena made an updated version some years later, though I'm not sure if anything good ever happened to it. There have also been other papers by other authors that build on or cite S+S+L.

One thing in there is the generating functions technique and the theorem there are no homopairs for sets with <=5 elements. Read it.

That paper also had big tables of homometric sets, though Skiena made me cut those tables down a lot :( For music on 12-tone scale I think you'd mainly be interested in sets with computations made in mod-12 or maybe mod-24 arithmetic (we called the problem with circular "wraparound" such as mod-12, the "beltway problem" as opposed to the line problem the "turnpike problem"). It is an easy matter to find all homopairs mod 12
since there are only 2^12 = 2048 subsets of {0,1,2,3,...,11} and your computer
can find them all in a fraction of a second and find which pairs among them are homometric.

If I really wanted to I could probably extend the classification further, I went to sets of size 7, but it probably would be feasible to do, say, 8 and 9. However, this would not be easy and it would require writing some pretty serious software to get the computer to do it. My
size 6 and 7 cases were done partly by computer and partly by hand, and the software
needed was not very sophisticated. But to go larger you'd have to get considerably more serious about computerization, and you might find 100 families.

🔗WarrenS <warren.wds@gmail.com>

🔗Paul <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
> uh, I don't even understand what you guys are talking about, for
> the most part. Nor am I sure how important this all is for music.
> This music professor named Ozzard got on my case 1-2 years back
> and nagged me to do the
> classification I just posted. I'd sent him it before.
>
> Anyhow, in this paper in about 1990 by Lemke, Skiena, and Smith (me)
> in some order, we discussed the theory of the problem of reconstructing sets
> from their distances, and theory of homometric sets. There have been various
> versions of that paper floating around the internet ever since, which I daresay you can quickly find. Skiena made an updated version some years later, though I'm not sure if anything good ever happened to it. There have also been other papers by other authors that build on or cite S+S+L.
>
> One thing in there is the generating functions technique and the theorem there are no homopairs for sets with <=5 elements. Read it.
>
> That paper also had big tables of homometric sets, though Skiena made me cut those tables down a lot :( For music on 12-tone scale I think you'd mainly be interested in sets with computations made in mod-12 or maybe mod-24 arithmetic (we called the problem with circular "wraparound" such as mod-12, the "beltway problem" as opposed to the line problem the "turnpike problem"). It is an easy matter to find all homopairs mod 12
> since there are only 2^12 = 2048 subsets of {0,1,2,3,...,11} and your computer
> can find them all in a fraction of a second and find which pairs among them are homometric.
>
> If I really wanted to I could probably extend the classification further, I went to sets of size 7, but it probably would be feasible to do, say, 8 and 9. However, this would not be easy and it would require writing some pretty serious software to get the computer to do it. My
> size 6 and 7 cases were done partly by computer and partly by hand, and the software
> needed was not very sophisticated. But to go larger you'd have to get considerably more serious about computerization, and you might find 100 families.
>

Actually, I read that paper. Using Fourier Transforms IIRC. Actually, I have all the Z-relations for all 2048 sets of 12-tET. There is actually a pair of tetrads (4), and three pairs of pentads (5). But perhaps you didn't mean Z-relations in the rank-1 sense.

I will see if I can figure out what the generating function polynomial for the sporadic
septads is all about. Thanks pgh.

PS Of the 2048 Subsets of 12-tET, exactly 12*72=864 are involved in Z-relating.
This leaves 1184 that are not, so it's about 42 percent. In terms of types, its 72/352 types

🔗Mike Battaglia <battaglia01@gmail.com>

On Fri, Oct 7, 2011 at 7:14 PM, WarrenS <warren.wds@gmail.com> wrote:
>
> uh, I don't even understand what you guys are talking about, for
> the most part. Nor am I sure how important this all is for music.
> This music professor named Ozzard got on my case 1-2 years back
> and nagged me to do the
> classification I just posted. I'd sent him it before.

It's important to an obscure branch of music theory called "music set
theory," which treats chords and scales as "sets," given by their
position on the 12-equal generator chain. Another way to put it is
that it works within the Z/12Z cyclic group, such that the note "C" is
mapped to 0 mod 12, "C#" is mapped to 1 mod 12, etc. It's mod 12
because all "C"'s are considered to be equal, in that they share the
same "pitch class." If you ditch this requirement, then it's easy to
see that we're going to instead be working within Z.

In this paradigm, 1 step out of 12 equal is the generating interval
for the 12-equal group, and we're simply drawing an isomorphism
between steps in 12-equal and the integers.

We're more interested in looking at groups like Z^2, sometimes free
abelian groups of even higher rank, which consist of more than one
linearly independent chain of generators. For example, consider
"meantone temperament," in which one generator is a slightly flattened
3/2 interval, and the other is the octave (or 2/1), and the stack of
3/2's does -NOT- close at some power of the octave or ever close at
all. This tuning consists of two linearly independent generators,
which could be said to describe Z^2. (There's a whole lot more to this
than that, but perhaps this will help you see what we're talking
about.)

Anyway, the thing that you wrote about is something that music set
theorists have been focused on some while - they've coined the phrase
"Z-relation" to describe scales that are homometric. For example, the
"musical sets" {0, 1, 4, 6} and {0, 1, 3, 7} are Z-related, because
they're also homometric. In "music set theory" they correspond to the
tetrads C-C#-E-F# and C-C#-Eb-G, respectively.

You may be wondering what auditory significance this has. As it's used
in 12-equal, I doubt that it has any significance at all. Both of
those sets, when played, sound like atonal mush to me, and if there's
some obscure way to hear them as sounding like related sets of atonal
mush, I don't really care too much what it is. However, many of the
vague 12-centric atonal mushlike entities that academia's thrown often
have a slightly generalized counterpart nearby, which makes a lot more
sense, and can be viewed within a psychoacoustic-centric paradigm
instead, more suitable for actually writing consonant music. I'm
interested in Z-relations/homometric sets because I'm curious to see
if any such equivalent exists here.

-Mike

🔗Paul <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Oct 7, 2011 at 7:14 PM, WarrenS <warren.wds@...> wrote:
> >
> > uh, I don't even understand what you guys are talking about, for
> > the most part. Nor am I sure how important this all is for music.
> > This music professor named Ozzard got on my case 1-2 years back
> > and nagged me to do the
> > classification I just posted. I'd sent him it before.
>
> It's important to an obscure branch of music theory called "music set
> theory," which treats chords and scales as "sets," given by their
> position on the 12-equal generator chain. Another way to put it is
> that it works within the Z/12Z cyclic group, such that the note "C" is
> mapped to 0 mod 12, "C#" is mapped to 1 mod 12, etc. It's mod 12
> because all "C"'s are considered to be equal, in that they share the
> same "pitch class." If you ditch this requirement, then it's easy to
> see that we're going to instead be working within Z.
>
> In this paradigm, 1 step out of 12 equal is the generating interval
> for the 12-equal group, and we're simply drawing an isomorphism
> between steps in 12-equal and the integers.
>
> We're more interested in looking at groups like Z^2, sometimes free
> abelian groups of even higher rank, which consist of more than one
> linearly independent chain of generators. For example, consider
> "meantone temperament," in which one generator is a slightly flattened
> 3/2 interval, and the other is the octave (or 2/1), and the stack of
> 3/2's does -NOT- close at some power of the octave or ever close at
> all. This tuning consists of two linearly independent generators,
> which could be said to describe Z^2. (There's a whole lot more to this
> than that, but perhaps this will help you see what we're talking
> about.)
>
> Anyway, the thing that you wrote about is something that music set
> theorists have been focused on some while - they've coined the phrase
> "Z-relation" to describe scales that are homometric. For example, the
> "musical sets" {0, 1, 4, 6} and {0, 1, 3, 7} are Z-related, because
> they're also homometric. In "music set theory" they correspond to the
> tetrads C-C#-E-F# and C-C#-Eb-G, respectively.
>
> You may be wondering what auditory significance this has. As it's used
> in 12-equal, I doubt that it has any significance at all. Both of
> those sets, when played, sound like atonal mush to me, and if there's
> some obscure way to hear them as sounding like related sets of atonal
> mush, I don't really care too much what it is. However, many of the
> vague 12-centric atonal mushlike entities that academia's thrown often
> have a slightly generalized counterpart nearby, which makes a lot more
> sense, and can be viewed within a psychoacoustic-centric paradigm
> instead, more suitable for actually writing consonant music. I'm
> interested in Z-relations/homometric sets because I'm curious to see
> if any such equivalent exists here.
>
> -Mike
>

Well put. However, I don't agree it's all for atonal music. The key thing in all the 72 Z-related sets of 12-tET is the tritone (except for that one pair using a triune). I do see these
patterns used in the music of Stravinsky, and Barber too, and of course Schoenberg.

Try playing around with these chords (if you are interested), the hexachord I call "G"

G#BDFCE and G#BDCEbGb breaking them apart in different ways and looking at the relationships. The first is just A harmonic minor with no A. The second is like two different Hungarian scales abutted together. I guess its like those jazz artists who use parts
of Petroushka, its probably mostly trial and error but there is something to it. And of course its not the most tonal music in the world.

The other reason to study the Z-relation is merely to understand the "fabric" of musical
set theory, and where the "deformations" are, how that affects intervallic structure.
In 12-tET hexachords for example, we have the fourfold symmetry (or eightfold if
you want to go deeper) of 80 + 20 + 32 + 8 / 4 = 35 which is sets, symmetrical sets,
reverse complementable sets plus forwards complementable sets / 4 = 35 types.

Its fun here because the 32 + 8 part / 2 = 20 sets are those that are NOT Z-related (So its as much of talking about non-Z-related sets also). Kinda like fractals.

pgh

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗WarrenS <warren.wds@gmail.com>

If you had two generators for tuning, that is two fundamental ratios
instead of just one (the 12th root of 2)
then you'd generate from them via integer linear combinations
a DENSE set of frequencies,
instead of a discrete set.

So for practical purposes it'd just be the real numbers:
all frequencies allowed.

Which might be fine -- maybe music would be more interesting if you
could do that. You might say being pinned to the 12-note scale is very constraining
and very limiting for music, compared to the infinity of possible frequencies out there.

HOWEVER.... my debugging theory suggests that the human mind will be opposed to too much such freedom. Theory says human mind wants "frequency landmarks"
and does not want to drift off into the frequency wilderness where it risks
losing track of where the landmarks are and thus losing calibration benefits.

So I'd predict that scales are actually a good idea, but that doesn't mean adding some
excursions into the wilderness might not be an even better idea. For example,
some glissando occasionally could expose you to new debugging functionality which you'd never see if pinned to the 12-note scale. And indeed Gershwin's "Rhapsody in Blue"
has a big glissando near the beginning which produces a very cool effect.
There was another good one in some nameless music I heard in a car commercial.

I would however predict a piece made entirely of glissandos and random-frequency notes
would not sound good, and the best music will be mostly staying on a fixed discrete set of
notes -- which might not be an equally-tempered scale, but will be approximately so.

I personally think it'd be rather nice if those with more music ability and resources than me were to start exploring the debugging theory by creating all sorts of musical experiments
related to it. That was just one example of such a prediction, there are a huge number
of pretty obvious experiments that could be tried.

The homometric set classifications I devised are valid over the full real numbers.

I knew about Z-relations already, but I'd tried a few musical experiments
with homometric sets as chords and never saw anything musically impressive coming out of it, so I'd never been convinced that all that will ever have much impact on music.
Never been convinced YET, anyhow. You might be able to pick real numbers in those homopair families to generate chords that actually don't sound too bad, then listen to them. That might be interesting. All the experiments I had
done with them had been using piano scale and not taking advantage of the full freedom of the real numbers. You could simply program all my homopairs
into a computer and try all possible real numbers as the input (at some very fine grain)
regarding the points as lying in log(frequency) space --
and have it spit out the homopairs that result, that happen to enjoy a lot of
near-integer frequency ratios so that they're likely to "sound good." In
other words, you should now be able to systematically find ALL homopair chords
(using unrestricted real number frequencies) that "sound good."

🔗Mike Battaglia <battaglia01@gmail.com>

On Sat, Oct 8, 2011 at 12:50 AM, WarrenS <warren.wds@gmail.com> wrote:
>
> HOWEVER.... my debugging theory suggests that the human mind will be opposed to too much such freedom. Theory says human mind wants "frequency landmarks"
> and does not want to drift off into the frequency wilderness where it risks
> losing track of where the landmarks are and thus losing calibration benefits.

Well, you obviously just don't pick any pitch out of the lattice at
random and use that. The basic idea is to assign one of the generators
the special role of "period," meaning that it goes on forever. The
other generator, just called the "generator," is only stacked a finite
amount of times, and then reduced within the period (or tiled forever
with respect to the period, your choice).

The most common way to use these scales is to stack the generator
until you arrive at a scale that, when reduced within the period has
only two step sizes. If you do this, you'll find that all of the other
generic interval types - the "thirds," "fourths," etc in the scale,
also come in two sizes - a small (or "minor") one, or a large (or
"major") one. These scales are called "Moment of Symmetry" or MOS
scales, although they've been called at times an assortment of names
in the literature - "distributionally even" scales, "deep" scales,
scales with "Myhill's property," etc.

If you do this with the meantone generator, you'll find that MOS's are
generated at 5, 7, 12, 19, etc notes, corresponding to the pentatonic,
diatonic, chromatic, and enharmonic scales, respectively. However, if
your generator is sharper than 700 cents (but less than 720 cents),
rather than generating an MOS at 19 iterations of the generator, it'll
end up hitting 17 instead.

Infinite lattices like this can easily be made much more
comprehensible this way. But the real takehome point is that MOS
scales "exist" in equal temperaments - just like the diatonic scale,
derived ultimately from meantone temperament, exists in 12-equal, so
do the MOS's of other temperaments exist in other equal temperaments.
Porcupine temperament is a good example, existing in 22-equal, and if
you don't mind your intervals having a bit more roughness, mavila
temperament is also beautiful, in 16-equal.

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Portent pentads, you say?

Indeed I do. I'm trying to find a nice way of characterizing the relationships of a set of two 7-limit pentads, two 11-limit pentads, four keenanismic (385/384) pentands, and four werckismic (441/440) pentads, along the lines of the slick method I used for the keenanismic tablet with its nice geometry. But I may have to settle for something cruder.

Are they on the wiki anywhere?

Not as yet.

Which pentads
> are they, if I might ask?

7-limit

9/8-5/4-3/2-7/4-2
9/8-9/7-3/2-9/5-2

11-limit

9/8-11/8-3/2-7/4-2
9/8-9/7-3/2-18/11-2

385/384 tempered

6/5-11/8-3/2-7/4-2
6/5-11/8-3/2-12/7-2
12/11-5/4-3/2-7/4-2
12/11-5/4-3/2-12/7-2

441/440 tempered

9/8-5/4-10/7-7/4-2
9/8-11/8-11/7-7/4-2
10/9-5/4-10/7-7/4-2
11/9-11/8-11/7-7/4-2

Modulo the 11-limit 5et patent val these are 1-2-3-4-5, and both 385/384 and 441/440 are tempered out by 5et.

🔗Paul <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > Portent pentads, you say?
>
> Indeed I do. I'm trying to find a nice way of characterizing the relationships of a set of two 7-limit pentads, two 11-limit pentads, four keenanismic (385/384) pentands, and four werckismic (441/440) pentads, along the lines of the slick method I used for the keenanismic tablet with its nice geometry. But I may have to settle for something cruder.
>
> Are they on the wiki anywhere?
>
> Not as yet.
>
> Which pentads
> > are they, if I might ask?
>
> 7-limit
>
> 9/8-5/4-3/2-7/4-2
> 9/8-9/7-3/2-9/5-2
>
> 11-limit
>
> 9/8-11/8-3/2-7/4-2
> 9/8-9/7-3/2-18/11-2
>
> 385/384 tempered
>
> 6/5-11/8-3/2-7/4-2
> 6/5-11/8-3/2-12/7-2
> 12/11-5/4-3/2-7/4-2
> 12/11-5/4-3/2-12/7-2
>
> 441/440 tempered
>
> 9/8-5/4-10/7-7/4-2
> 9/8-11/8-11/7-7/4-2
> 10/9-5/4-10/7-7/4-2
> 11/9-11/8-11/7-7/4-2
>
> Modulo the 11-limit 5et patent val these are 1-2-3-4-5, and both 385/384 and 441/440 are tempered out by 5et.
>

Do you think of 5-tET literally or is it just a means to an end? pgh

🔗Paul <phjelmstad@msn.com>

Here's my latest sales pitch. Instead of thinking of these clusters as "atonal mush" how about thinking about them as junctures of for example, different changes coming together,
or similarly,as parts of scales. For example, think of G melodic minor and then these sets are two fragments

F# G Bb C and
C E F# G

C F# and G are fixed and Bb -> E moves a tritone, across G, so the of course you get
111111 for each intvec --- certainly this has some value in the understanding of
tuning and music beyond "atonal mush" and "related sets of atonal mush"

I guess what I am saying is that atonal music theory is/was valuable to writing and
studying tonal music --- (Bartok, Stravinsky etc and anyone studying composition
in grad school --- that is before the Extended Common Practice (haha))

pgh

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Paul <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@> wrote:
> >
> > uh, I don't even understand what you guys are talking about, for
> > the most part. Nor am I sure how important this all is for music.
> > This music professor named Ozzard got on my case 1-2 years back
> > and nagged me to do the
> > classification I just posted. I'd sent him it before.
> >
> > Anyhow, in this paper in about 1990 by Lemke, Skiena, and Smith (me)
> > in some order, we discussed the theory of the problem of reconstructing sets
> > from their distances, and theory of homometric sets. There have been various
> > versions of that paper floating around the internet ever since, which I daresay you can quickly find. Skiena made an updated version some years later, though I'm not sure if anything good ever happened to it. There have also been other papers by other authors that build on or cite S+S+L.
> >
> > One thing in there is the generating functions technique and the theorem there are no homopairs for sets with <=5 elements. Read it.
> >
> > That paper also had big tables of homometric sets, though Skiena made me cut those tables down a lot :( For music on 12-tone scale I think you'd mainly be interested in sets with computations made in mod-12 or maybe mod-24 arithmetic (we called the problem with circular "wraparound" such as mod-12, the "beltway problem" as opposed to the line problem the "turnpike problem"). It is an easy matter to find all homopairs mod 12
> > since there are only 2^12 = 2048 subsets of {0,1,2,3,...,11} and your computer
> > can find them all in a fraction of a second and find which pairs among them are homometric.
> >
> > If I really wanted to I could probably extend the classification further, I went to sets of size 7, but it probably would be feasible to do, say, 8 and 9. However, this would not be easy and it would require writing some pretty serious software to get the computer to do it. My
> > size 6 and 7 cases were done partly by computer and partly by hand, and the software
> > needed was not very sophisticated. But to go larger you'd have to get considerably more serious about computerization, and you might find 100 families.
> >
>
> Actually, I read that paper. Using Fourier Transforms IIRC. Actually, I have all the Z-relations for all 2048 sets of 12-tET. There is actually a pair of tetrads (4), and three pairs of pentads (5). But perhaps you didn't mean Z-relations in the rank-1 sense.
>
> I will see if I can figure out what the generating function polynomial for the sporadic
> septads is all about. Thanks pgh.
>
> PS Of the 2048 Subsets of 12-tET, exactly 12*72=864 are involved in Z-relating.
> This leaves 1184 that are not, so it's about 42 percent. In terms of types, its 72/352 types
>

Correction: It's 4096 total so it's 4096-864= 3232 which are not.