Generalizing Z-relations to higher-rank temperaments

I've been fascinated with the "Z-relation" concept from music set
theory for a little bit now, ever since I managed to finally squeeze
out of Paul Hjelmstad what they are. I think that they turn out to be
very useful in understanding certain things about MODMOS's and/or what
I called "domes" of rank-3 scales earlier, although I'm not sure how.
The main problem with these sorts of things is that they've been
implemented exclusively for 12-equal, or more generally for equal
temperaments, which has lended to them having a certain atonal flair
about them. One way to start getting away from this is to generalize
them to higher rank temperaments.

Any two scales are "z-related" if they share the same "interval
vector." An interval vector is a thing telling you how many times each
interval in 12-equal occurs in the scale, up to 6\12. Intervals higher
than 6\12 aren't counted for the sake of not counting
octave-inversions of the same interval twice. Things like modal
permutations and scale inversions, which also share the same interval
vector as their parent scale, don't count.

Interval vectors come in the form <xxxxxx>, where each successive
position tells you how many times 1\12, 2\12, 3\12, 4\12, 5\12, and
6\12 respectively (and their octave inversions) appear in the scale.
For example, the interval vector for the 12-equal diatonic scale is
<254361>, and the interval vector for the 12-equal pentatonic scale is
<032140>.

Generalizing the interval vector to rank-2 is easy enough: get some
rank-2 scale, and arrange it in terms of (period, generator)
coordinates, and then just drop the period (assume a full-octave
period for now). See how many times each note in the chain, up to
octave-equivalence, appears in the scale. Only look at positive
generators for the sake of assuming octave-equivalence. More
precisely, arrange your scale out in terms of a generator chain,
obtain the set of differences of this this set, and then remove any
element from this set that is less than 1. The interval vector can be
constructed by going through the natural numbers in order, and writing
how many times each one appears in the final resulting set.

As an example, the intvec for meantone[7] is <654321000000000...>.
Since only a finite number of elements in this set are nonzero, we can
truncate the intvec at <654321...> and leave it at that. The intvec
for meantone harmonic minor is <434332011...>.

It's a bit trickier to generalize to half-octave periods, but not
impossible. Gotta jump off now, will be back later...

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I've been fascinated with the "Z-relation" concept from music set
> theory for a little bit now, ever since I managed to finally squeeze
> out of Paul Hjelmstad what they are. I think that they turn out to be
> very useful in understanding certain things about MODMOS's and/or what
> I called "domes" of rank-3 scales earlier, although I'm not sure how.
> The main problem with these sorts of things is that they've been
> implemented exclusively for 12-equal, or more generally for equal
> temperaments, which has lended to them having a certain atonal flair
> about them. One way to start getting away from this is to generalize
> them to higher rank temperaments.
>
> Any two scales are "z-related" if they share the same "interval
> vector." An interval vector is a thing telling you how many times each
> interval in 12-equal occurs in the scale, up to 6\12. Intervals higher
> than 6\12 aren't counted for the sake of not counting
> octave-inversions of the same interval twice. Things like modal
> permutations and scale inversions, which also share the same interval
> vector as their parent scale, don't count.
>
> Interval vectors come in the form <xxxxxx>, where each successive
> position tells you how many times 1\12, 2\12, 3\12, 4\12, 5\12, and
> 6\12 respectively (and their octave inversions) appear in the scale.
> For example, the interval vector for the 12-equal diatonic scale is
> <254361>, and the interval vector for the 12-equal pentatonic scale is
> <032140>.
>
> Generalizing the interval vector to rank-2 is easy enough: get some
> rank-2 scale, and arrange it in terms of (period, generator)
> coordinates, and then just drop the period (assume a full-octave
> period for now). See how many times each note in the chain, up to
> octave-equivalence, appears in the scale. Only look at positive
> generators for the sake of assuming octave-equivalence. More
> precisely, arrange your scale out in terms of a generator chain,
> obtain the set of differences of this this set, and then remove any
> element from this set that is less than 1. The interval vector can be
> constructed by going through the natural numbers in order, and writing
> how many times each one appears in the final resulting set.
>
> As an example, the intvec for meantone[7] is <654321000000000...>.
> Since only a finite number of elements in this set are nonzero, we can
> truncate the intvec at <654321...> and leave it at that. The intvec
> for meantone harmonic minor is <434332011...>.
>
> It's a bit trickier to generalize to half-octave periods, but not
> impossible. Gotta jump off now, will be back later...
>
> -Mike
>
So for meantone[7], 6 fifths, 5 major seconds, 4 minor thirds, 3 major thirds, 2 semitones, 1 tritone, (resulting from a generator of a fifth). and for meantone harmonic minor I get 4 fifths, 3 seconds, etc... apparently nothing at level 7. okay. Okay, lets compare two septads I know to be Z-related, how about (0,2,5,6,7,8,9) and (0,4,5,6,7,9,11)?

This does a simple tritone reflection thing across a wedge with a tritone fixed. Might these have the same interval vectors in the new sense? Well, if meantone, you are just measuring with a fifth instead of a semitone, which doesn't change the theory at all, really, but...the difference being that you are going beyond using only 6 fifths (I only go to 6 semitones)....If I look at this with semitones, (ordered) then you do obtain different vectors, which is to say forths and fifths are not the same, but they can be collapsed....

I too need to go....

pgh

Do we currently know of any example at all of a rank > 1 Z-relation?

Keenan

On Wed, Sep 28, 2011 at 7:22 PM, Paul <phjelmstad@msn.com> wrote:
>
> So for meantone[7], 6 fifths, 5 major seconds, 4 minor thirds, 3 major thirds, 2 semitones, 1 tritone, (resulting from a generator of a fifth). and for meantone harmonic minor I get 4 fifths, 3 seconds, etc... apparently nothing at level 7. okay. Okay, lets compare two septads I know to be Z-related, how about (0,2,5,6,7,8,9) and (0,4,5,6,7,9,11)?

In meantone, the spelling for these is ambiguous. I arbitrarily picked
the first to be C D F F# G G# A, although it could have been Gb and Ab
in there instead of F# and G#. I got an intvec of <443222211...> Then,
your second one I decided to spell C E F F# G A B; this has an intvec
of <5433321...>. I'll try some different spellings to see if any of
them sync up.

C D F Gb G Ab A

Gb-x-Ab-x-x-F-C-G-D-A

First septad (0,2,5,6,7,8,9)
C D F F# G G# A - [F C G D A xx xx F# xx G#] - <443222211...>
C D F Gb G Ab A - [Gb xx Ab xx xx F C G D A] - <443222211...>
C D F F# G Ab A - [Ab xx xx F C G D A xx xx F#] - <4343222001...>
C D F Gb G G# A - [Gb xx xx xx xx F C G D A xx xx xx xx G#] -
<43212222200001...>

Second septad (0,4,5,6,7,9,11)
C E F F# G A B - [F C G xx A E B F#] - <5433321…>
C E F Gb G A B - [Gb xx xx xx xx F C G xx A E B] - <43233210111...>

I don't want to play this game anymore, it's looking pretty clear to
me that these aren't z-related.

Paul, you know, one way to find z-relations - are there any z-related
sets in both 12-TET and also in 19-TET that share a meantone spelling?
That might be an interesting way to start.

-Mike

No, because I just worked this out today and haven't coded up any
brute force routines yet. One possibility is that we might find that
the only sets sharing the same intvec are inversions of one another;
e.g. things like harmonic minor and harmonic major share an intvec.
This is equivalent to stating that no other way to share an intvec can
occur for rank 2, which conjectures that z-relations themselves are a
property of working within a cyclic group.

That would suck. But, it might be true that z-relations can be
generalized in a meaningful way to higher ranks by relaxing the
restrictions a bit. This has happened before in set theory, a good
example being the "deep scale" property. A scale has the "deep scale
property" if every entry in its intvec is unique. For example, the
12-tet major scale's intvec is <254361>, which makes it a "deep
scale." The 12-tet pentatonic scale's intvec is <032140>, which is not
a "deep scale" because 0 appears twice. Likewise, the 12-tet major
scale no longer becomes "deep" if you just play it in 24-equal,
because your intvec will be <020504030601>, which is the original one
but upsampled.

If you relax this definition a bit, and state that a scale is "deep"
if every -nonzero- entry in its intvec is unique, then all of the
above are now deep scales, and in fact you'll find that you've just
discovered MOS, which can be generalized to rank 2 and higher (as
we're working on now).

So it might be that z-relations are likewise too harshly defined, and
defined in such a way that only makes sense with respect to ETs, but
that there's a looser definition which can be more readily
generalized. We'll see...

-Mike

On Wed, Sep 28, 2011 at 9:00 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> Do we currently know of any example at all of a rank > 1 Z-relation?
>
> Keenan

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Wed, Sep 28, 2011 at 7:22 PM, Paul <phjelmstad@...> wrote:
> >
> > So for meantone[7], 6 fifths, 5 major seconds, 4 minor thirds, 3 major thirds, 2 semitones, 1 tritone, (resulting from a generator of a fifth). and for meantone harmonic minor I get 4 fifths, 3 seconds, etc... apparently nothing at level 7. okay. Okay, lets compare two septads I know to be Z-related, how about (0,2,5,6,7,8,9) and (0,4,5,6,7,9,11)?
>
> In meantone, the spelling for these is ambiguous. I arbitrarily picked
> the first to be C D F F# G G# A, although it could have been Gb and Ab
> in there instead of F# and G#. I got an intvec of <443222211...> Then,
> your second one I decided to spell C E F F# G A B; this has an intvec
> of <5433321...>. I'll try some different spellings to see if any of
> them sync up.
>
> C D F Gb G Ab A
>
> Gb-x-Ab-x-x-F-C-G-D-A
>
> First septad (0,2,5,6,7,8,9)
> C D F F# G G# A - [F C G D A xx xx F# xx G#] - <443222211...>
> C D F Gb G Ab A - [Gb xx Ab xx xx F C G D A] - <443222211...>
> C D F F# G Ab A - [Ab xx xx F C G D A xx xx F#] - <4343222001...>
> C D F Gb G G# A - [Gb xx xx xx xx F C G D A xx xx xx xx G#] -
> <43212222200001...>
>
> Second septad (0,4,5,6,7,9,11)
> C E F F# G A B - [F C G xx A E B F#] - <5433321…>
> C E F Gb G A B - [Gb xx xx xx xx F C G xx A E B] - <43233210111...>
>
> I don't want to play this game anymore, it's looking pretty clear to
> me that these aren't z-related.
>
> Paul, you know, one way to find z-relations - are there any z-related
> sets in both 12-TET and also in 19-TET that share a meantone spelling?
> That might be an interesting way to start.
>
> -Mike
>

I think computer help is in order, you don't want to do all this work by hand. Remember,
the underlying aspect of the Z-relation is the mechanism that sends one set to the other.
I have found they all reflect across a hyperplane and/or wedge of notes and also keep
say, a tritone, or two, or a triune fixed. So that is the guiding principle. In rank-2 you
have to ask yourself (or me myself) what is happening here ---- For example with
pentads in 12-tET (Which btw are less cumbersome to work with and complements of
septad scales) this would merely be

0 1 3 5 6 -> 0 3 5 6 7 (complements of the scales above) and this

sends 1 to 7 across a missing hyperplane of 4 and a wedge of 3 5, and 0 6 is fixed.

So therefore, in this case ------- how do you want to spell them? The main thing is
7 : 6 must be the same as 1 : 0 in terms of scale degrees. 0 6 must be fixed. 3 5 is also
fixed. 1 -> 7 must be the same distance as 0 -> 6 (tritone).

Let's start with this one. Z-relations in 19-tET don't have much going for them because
there is no axis of symmetry (however they could be even more interesting, they are
not created by the same mechanism). And in like 14-tET and 22-tET you have the
doubleflid 2p thing going on with the different cycles (M1, M3, M5....)

This is of course more than being a cycling group, it has to do with the multiplicative
modulo group and the ring of units (numbers cototient to 2p).....But we don't need that yet.

pgh

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > On Wed, Sep 28, 2011 at 7:22 PM, Paul <phjelmstad@> wrote:
> > >
> > > So for meantone[7], 6 fifths, 5 major seconds, 4 minor thirds, 3 major thirds, 2 semitones, 1 tritone, (resulting from a generator of a fifth). and for meantone harmonic minor I get 4 fifths, 3 seconds, etc... apparently nothing at level 7. okay. Okay, lets compare two septads I know to be Z-related, how about (0,2,5,6,7,8,9) and (0,4,5,6,7,9,11)?
> >
> > In meantone, the spelling for these is ambiguous. I arbitrarily picked
> > the first to be C D F F# G G# A, although it could have been Gb and Ab
> > in there instead of F# and G#. I got an intvec of <443222211...> Then,
> > your second one I decided to spell C E F F# G A B; this has an intvec
> > of <5433321...>. I'll try some different spellings to see if any of
> > them sync up.
> >
> > C D F Gb G Ab A
> >
> > Gb-x-Ab-x-x-F-C-G-D-A
> >
> > First septad (0,2,5,6,7,8,9)
> > C D F F# G G# A - [F C G D A xx xx F# xx G#] - <443222211...>
> > C D F Gb G Ab A - [Gb xx Ab xx xx F C G D A] - <443222211...>
> > C D F F# G Ab A - [Ab xx xx F C G D A xx xx F#] - <4343222001...>
> > C D F Gb G G# A - [Gb xx xx xx xx F C G D A xx xx xx xx G#] -
> > <43212222200001...>
> >
> > Second septad (0,4,5,6,7,9,11)
> > C E F F# G A B - [F C G xx A E B F#] - <5433321…>
> > C E F Gb G A B - [Gb xx xx xx xx F C G xx A E B] - <43233210111...>
> >
> > I don't want to play this game anymore, it's looking pretty clear to
> > me that these aren't z-related.
> >

Flashing Neon LIghts Here * * * *

f you collapse the vector so (1 & 11, 2 & 10, 3 & 9, 4 & 8, 5 & 7, 6) these are z-related.
And then the intvec merely swaps 1 and 5th positions. But that is not rank-2 like.
I must admit. * * * * * *

> > Paul, you know, one way to find z-relations - are there any z-related
> > sets in both 12-TET and also in 19-TET that share a meantone spelling?
> > That might be an interesting way to start.
> >
> > -Mike
> >
>
> I think computer help is in order, you don't want to do all this work by hand. Remember,
> the underlying aspect of the Z-relation is the mechanism that sends one set to the other.
> I have found they all reflect across a hyperplane and/or wedge of notes and also keep
> say, a tritone, or two, or a triune fixed. So that is the guiding principle. In rank-2 you
> have to ask yourself (or me myself) what is happening here ---- For example with
> pentads in 12-tET (Which btw are less cumbersome to work with and complements of
> septad scales) this would merely be
>
> 0 1 3 5 6 -> 0 3 5 6 7 (complements of the scales above) and this
>
> sends 1 to 7 across a missing hyperplane of 4 and a wedge of 3 5, and 0 6 is fixed.
>
> So therefore, in this case ------- how do you want to spell them? The main thing is
> 7 : 6 must be the same as 1 : 0 in terms of scale degrees. 0 6 must be fixed. 3 5 is also
> fixed. 1 -> 7 must be the same distance as 0 -> 6 (tritone).
>
> Let's start with this one. Z-relations in 19-tET don't have much going for them because
> there is no axis of symmetry (however they could be even more interesting, they are
> not created by the same mechanism). And in like 14-tET and 22-tET you have the
> doubleflid 2p thing going on with the different cycles (M1, M3, M5....)
>
> This is of course more than being a cycling group, it has to do with the multiplicative
> modulo group and the ring of units (numbers cototient to 2p).....But we don't need that yet.
>
> pgh
>

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
> >
> >
> >
> > --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > On Wed, Sep 28, 2011 at 7:22 PM, Paul <phjelmstad@> wrote:
> > > >
> > > > So for meantone[7], 6 fifths, 5 major seconds, 4 minor thirds, 3 major thirds, 2 semitones, 1 tritone, (resulting from a generator of a fifth). and for meantone harmonic minor I get 4 fifths, 3 seconds, etc... apparently nothing at level 7. okay. Okay, lets compare two septads I know to be Z-related, how about (0,2,5,6,7,8,9) and (0,4,5,6,7,9,11)?
> > >
> > > In meantone, the spelling for these is ambiguous. I arbitrarily picked
> > > the first to be C D F F# G G# A, although it could have been Gb and Ab
> > > in there instead of F# and G#. I got an intvec of <443222211...> Then,
> > > your second one I decided to spell C E F F# G A B; this has an intvec
> > > of <5433321...>. I'll try some different spellings to see if any of
> > > them sync up.
> > >
> > > C D F Gb G Ab A
> > >
> > > Gb-x-Ab-x-x-F-C-G-D-A
> > >
> > > First septad (0,2,5,6,7,8,9)
> > > C D F F# G G# A - [F C G D A xx xx F# xx G#] - <443222211...>
> > > C D F Gb G Ab A - [Gb xx Ab xx xx F C G D A] - <443222211...>
> > > C D F F# G Ab A - [Ab xx xx F C G D A xx xx F#] - <4343222001...>
> > > C D F Gb G G# A - [Gb xx xx xx xx F C G D A xx xx xx xx G#] -
> > > <43212222200001...>
> > >
> > > Second septad (0,4,5,6,7,9,11)
> > > C E F F# G A B - [F C G xx A E B F#] - <5433321…>
> > > C E F Gb G A B - [Gb xx xx xx xx F C G xx A E B] - <43233210111...>
> > >
> > > I don't want to play this game anymore, it's looking pretty clear to
> > > me that these aren't z-related.
> > >
>
> Flashing Neon LIghts Here * * * *
>
> f you collapse the vector so (1 & 11, 2 & 10, 3 & 9, 4 & 8, 5 & 7, 6) these are z-related.
> And then the intvec merely swaps 1 and 5th positions. But that is not rank-2 like.
> I must admit. * * * * * *

If you use the outrageous spellings ---

F C G xx A E B Gb and
Gb xx xx xx xx Gbb Dbb Abb xx Bbb Fb Cb

It will work, if you collapse the vectors.

Yikes.

You get <5 4 3 3 3 4 2> for both.

Of course, the second one could just as easily commence on C

C xx xx xx xx Cb Gb Db xx Eb Bb F

Now of course your fifth-generated vector is merely with 1 and 5 swapped from
the standard rank-1 semitone vector, but the main thing is both sets have the
same vector now so are Z-related (but is this really rank-2?)

pgh

>
> > > Paul, you know, one way to find z-relations - are there any z-related
> > > sets in both 12-TET and also in 19-TET that share a meantone spelling?
> > > That might be an interesting way to start.
> > >
> > > -Mike
> > >
> >
> > I think computer help is in order, you don't want to do all this work by hand. Remember,
> > the underlying aspect of the Z-relation is the mechanism that sends one set to the other.
> > I have found they all reflect across a hyperplane and/or wedge of notes and also keep
> > say, a tritone, or two, or a triune fixed. So that is the guiding principle. In rank-2 you
> > have to ask yourself (or me myself) what is happening here ---- For example with
> > pentads in 12-tET (Which btw are less cumbersome to work with and complements of
> > septad scales) this would merely be
> >
> > 0 1 3 5 6 -> 0 3 5 6 7 (complements of the scales above) and this
> >
> > sends 1 to 7 across a missing hyperplane of 4 and a wedge of 3 5, and 0 6 is fixed.
> >
> > So therefore, in this case ------- how do you want to spell them? The main thing is
> > 7 : 6 must be the same as 1 : 0 in terms of scale degrees. 0 6 must be fixed. 3 5 is also
> > fixed. 1 -> 7 must be the same distance as 0 -> 6 (tritone).
> >
> > Let's start with this one. Z-relations in 19-tET don't have much going for them because
> > there is no axis of symmetry (however they could be even more interesting, they are
> > not created by the same mechanism). And in like 14-tET and 22-tET you have the
> > doubleflid 2p thing going on with the different cycles (M1, M3, M5....)
> >
> > This is of course more than being a cycling group, it has to do with the multiplicative
> > modulo group and the ring of units (numbers cototient to 2p).....But we don't need that yet.
> >
> > pgh
> >
>