Yahoo Tuning Groups Ultimate Backup tuning Ums

>> No offense guys, but "um" sucks, really, really bad.
>> I do agree there should be terms for commas that are
>> tempered out vs. those that are not. I quite liked
>> Paul's original terms, used in The Forms of Tonality
>> and countless posts.
>
>I hope you don't mean "commatic unison vector" and
>"chromatic unison vector"!

I do!

>Why don't you write up a proposed definition(s)?

First, "vector" is apparently (according to mathworld) ok.
They say it's ok to use it to mean n-tuple, as computer
scientists do...

http://mathworld.wolfram.com/n-Tuple.html

But further, tonespace as we know it appears to qualify as
a "point lattice"....

http://mathworld.wolfram.com/PointLattice.html

...which are discrete subgroups of Euclidean space (which
is itself a vector space), which apparently means their
elements can be named by lists of integers and that they
don't require a field of scalars (which would also require
something stronger than integers). And point lattices are
often just called "lattices", fancy that.

So first define the p-limit lattice as a point lattice with
elements given by

{ p1*v1 + p2*v2 ... pn*vn }

where pn is the nth prime and vn are vectors chosen to give
the shape of lattice you want. Golly, I hope I'm doing
this right.

So a p-limit unison vector is an element of the p-limit
lattice such that, when added to any element Alpha in that
lattice, returns an element Beta with the same musical name
as Alpha.

Here a "musical name" is an element of an epimorphic scale,
and a musical naming scheme a mapping from the p-limit
lattice into a given epimorphic scale.

Then, a chromatic uv is a unison vector such that Alpha
and Beta have different pitches in the given tuning, while
a commatic uv is a unison vector such that Alpha and Beta
have identical pitches in that tuning.

Whaddya think?

-Carl

🔗monz <monz@tonalsoft.com>

hi Gene,

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

> Monz's defintion says that an um is reduced--that is,
> it is not a power.
> It also says that it is a unison-vector, which given his
> definition of unison vector is not a good idea. A
> unison vector is defined as a xenharmonic bridge of
> size small enough that it can be disregarded.
> However, if we temper 128/125 it is neither a
> xenharmonic bridge, nor is its size (41 cents) small
> enough to safely ignore.

that old defintion of "unison-vector" was really bad.
i've totally revised it, so you should have another look
now with what you wrote above in mind.

note that "xenharmonic-bridge" is actually a very
restricted type of unison-vector, in which one of the
two pitches/intervals connected by the xenharmonic-bridge
must be only the higher-prime-factor with an exponent of 1,
and the two ends of the xenharmonic-bridge therefore also
must be certain taxicab distances from 1/1.

> If "um" simply meant "reduced kernel element", that is,
> something which is not 1 and not a power and which is
> tempered out by some temperament, I could use
> the word. An alternative possibility would be to require
> than an um is greater than one.

not quite sure what you mean here ... a vum (formerly "um"
-- i've decided to go with the new suggestion by Robert Walker)
is always greater than 1, if you mean a ratio somewhere between
1 and the equivalence-interval (usually 2).

if that's not what you mean, then please explain.

> The definitions become more byzantine with "bium",

now "bivum"

> which isn't the wedge product of two ums, but the rank-two
> group they generate. I think this is heading towards serious
> confusion, particularly since he uses "unison vector" in the
> defintion, which contradicts the way "unison vector" is defined.

again, "unison-vector" now has a more accurate and correct
defition, so please reconsider.

> Since "bium" is intended to describe a kernel
> (the example given is the kernel for <12 19 28|) it seems
> to me we would be better off with a definition of the kernel
> of a temperament.

i've needed one in the Encyclopaedia for a long time.
thanks for providing it.

-monz

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> No offense guys, but "um" sucks, really, really bad.
> I do agree there should be terms for commas that are
> tempered out vs. those that are not. I quite liked
> Paul's original terms, used in The Forms of Tonality
> and countless posts. If we have to change those, so
> be it, but we can do better than "ums".
>
> -Carl

neither Paul nor i really cared for "um" either,
and after discussing some of the illogicalities
apparent in the way "monzo", "unison-vector", and
"um" were defined, we decided that the essential
quality of these "ums" is that the monzo really
does vanish.

for example, these are all equivalent as part of the "um":

2,3,5-monzo ratio
etc.
[ -4 4, -1 > * 3 = [-12 12, -3 > 531441/512000
[ -4 4, -1 > * 2 = [ -8 8, -2 > 6561/6400
[ -4 4, -1 > * 1 = [ -4 4, -1 > 81/80
[ -4 4, -1 > * 0 = [ 0 0, 0 > 1/1
[ -4 4, -1 > * -1 = [ 4 -4, 1 > 80/81
[ -4 4, -1 > * -2 = [ 8 -8, 2 > 6400/6561
[ -4 4, -1 > * -3 = [ 12 -12, -3 > 512000/531441
etc.

the only thing that's different about them in the
expanded notation is the integer coefficient. (and
even that really doesn't matter much since they're
all equivalent ... but anyway ...)

so if one considers the coefficients to be the only
thing that differentiates between the various members
of the set, then the monzo part, [-4 4, -1> in this case,
in a way really does "vanish".

so Paul and i decided to go with Robert Walker's
suggestion of "vum", derived from the acronym
for "vanishing unison monzo". i've renamed the
terms and the webpages.

still not really crazy about it, but at least its
etymology makes sense.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> >I hope you don't mean "commatic unison vector" and
> >"chromatic unison vector"!
>
> I do!

"Chromatic unison vector" may be the single most confusing and
misleading bit of terminology ever introduced into tuning theory.
It should have a stake driven through its heart, and never, ever, ever
be used, for any purpose.

> >Why don't you write up a proposed definition(s)?
>
> First, "vector" is apparently (according to mathworld) ok.
> They say it's ok to use it to mean n-tuple, as computer
> scientists do...

They say it can be interpreted to be a vector, which is correct, and
that a vector is an element of a vector space, which is the
mathematical definition. They don't say it is OK to use "vector" as a
synonym for "n-tuple", though of course people do, so it is
usage-sanctioned.

> http://mathworld.wolfram.com/n-Tuple.html
>
> But further, tonespace as we know it appears to qualify as
> a "point lattice"....

"Tonespace" as I know it is meaningless unless defined.

> http://mathworld.wolfram.com/PointLattice.html
>
> ...which are discrete subgroups of Euclidean space (which
> is itself a vector space), which apparently means their
> elements can be named by lists of integers and that they
> don't require a field of scalars (which would also require
> something stronger than integers).

The defintion "discrete subgroup of a Euclidean space" is a correct
mathematical definition, but for our purposes we would prefer the more
general definition, "discrete subgroup of a normed real vector space".
This is because eg a Tenney space is *not* Euclidean.

but that does not mean the same thing as a square mesh, which is a
special case.

And point lattices are
> often just called "lattices", fancy that.

Do that and Paul may shoot you, since the definition they give
excludes his favorite lattices. Anyway, the business about points is
actually to keep this kind of lattice separate from the partial order
kind.

> So first define the p-limit lattice as a point lattice with
> elements given by
>
> { p1*v1 + p2*v2 ... pn*vn }
>
> where pn is the nth prime and vn are vectors chosen to give
> the shape of lattice you want. Golly, I hope I'm doing
> this right.

What space do these vectors live in, and what is the norm on that space?

> So a p-limit unison vector is an element of the p-limit
> lattice such that, when added to any element Alpha in that
> lattice, returns an element Beta with the same musical name
> as Alpha.

You are introducing new and uncessary complications with this
"musicial name" terminology. You could define "musical name" as the
image under some mapping defined by a list of reduced and linearly
independent vals (an "icon"), I suppose, but we don't need any
additional burdens to overload our brain-pans.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

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

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > not quite sure what you mean here ... a vum (formerly "um"
> > -- i've decided to go with the new suggestion by Robert Walker)
> > is always greater than 1, if you mean a ratio somewhere between
> > 1 and the equivalence-interval (usually 2).
>
> You don't have anything about it being less than 2, and
> you shouldn't, so that's good.

glad i did something mathematically correct, for a change.

> "Vum" sounds better than "um" and this definition is
> looking good.

Paul really did provide lots of help.

> The bivum or bium really should go, I think--it is too
> easy to confuse with bival and bimonzo, and it suggests
> it is more like a vum than it actally is. Moreover, it
> isn't needed; we don't need to specify the dimension of
> the kernel, which is the codimension of the corresponding
> temperament and not something we have to refer to
> as a general rule.

hmm. i had originally written up the nice detailed 12-et
example as part of the "vum" page, then decided that it
was complicated enough that i should just make a separate
page for "bivum", then i was going to do one for "trivum"
as soon as i made a nice 7-limit example.

if you really think it's superfluous, i could put the
example back into the "vum" page and get rid of "bivum".

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

>> http://mathworld.wolfram.com/n-Tuple.html
>>
>> But further, tonespace as we know it appears to qualify as
>> a "point lattice"....
>
>"Tonespace" as I know it is meaningless unless defined.

Any of the lattices ever discussed here could be called
tonespaces.

>> http://mathworld.wolfram.com/PointLattice.html
>>
>> ...which are discrete subgroups of Euclidean space (which
>> is itself a vector space), which apparently means their
>> elements can be named by lists of integers and that they
>> don't require a field of scalars (which would also require
>> something stronger than integers).
>
>The defintion "discrete subgroup of a Euclidean space" is a correct
>mathematical definition, but for our purposes we would prefer the more
>general definition, "discrete subgroup of a normed real vector space".
>This is because eg a Tenney space is *not* Euclidean.

Don't worry about Tenney space! It's utterly unrequired for
defining unison vectors, which is what you asked me to do.

>but that does not mean the same thing as a square mesh, which is a
>special case.
>
> And point lattices are
>> often just called "lattices", fancy that.
>
>Do that and Paul may shoot you, since the definition they give
>excludes his favorite lattices.

Don't worry about it!

>> So first define the p-limit lattice as a point lattice with
>> elements given by
>>
>> { p1*v1 + p2*v2 ... pn*vn }
>>
>> where pn is the nth prime and vn are vectors chosen to give
>> the shape of lattice you want. Golly, I hope I'm doing
>> this right.
>
>What space do these vectors live in, and what is the norm on
>that space?

Euclidean, I think. Do we need a norm? We're just defining
a subgroup of R.

>> So a p-limit unison vector is an element of the p-limit
>> lattice such that, when added to any element Alpha in that
>> lattice, returns an element Beta with the same musical name
>> as Alpha.
>
>You are introducing new and uncessary complications with this
>"musicial name" terminology.

Howabout "note name"? This is about music, after all. And
the concept is utterly fundamental to the notion of chromatic
and commatic unison vectors, which you asked me to explain
and which are terms that some find useful.

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> http://mathworld.wolfram.com/n-Tuple.html
> >>
> >> But further, tonespace as we know it appears to qualify as
> >> a "point lattice"....
> >
> >"Tonespace" as I know it is meaningless unless defined.
>
> Any of the lattices ever discussed here could be called
> tonespaces.

Is the lattice a tone space, or the space the lattice lives in a tone
space? Does the space have a way of measuring distance, or not? If it
does, what is it?

> Don't worry about Tenney space! It's utterly unrequired for
> defining unison vectors, which is what you asked me to do.

Then why are we talking about lattices and vectors at all? My point of
view has been very much along the don't worry about it line--I simply
want an abelian group, nothing more. If I do talk about a space, I'll
say what space.

Over on tuning-math it seems to have emerged that Monz's intuitive
notion of what this space should be is that it is a topological vector
space with no norm on it, so here's another contender for what a
"Tonespace" might possibly be.

> Euclidean, I think. Do we need a norm? We're just defining
> a subgroup of R.

We don't need a norm to define a subgroup of R^n, but then we don't
need to define a subgroup of R^n. All we need is the abelian
group--the bells, whistles, and duck-quacks people add are extra and
in many contexts of no use, so why add them unless we *do*, in fact,
want a lattice geometry?

> >> So a p-limit unison vector is an element of the p-limit
> >> lattice such that, when added to any element Alpha in that
> >> lattice, returns an element Beta with the same musical name
> >> as Alpha.
> >
> >You are introducing new and uncessary complications with this
> >"musicial name" terminology.
>
> Howabout "note name"? This is about music, after all. And
> the concept is utterly fundamental to the notion of chromatic
> and commatic unison vectors, which you asked me to explain
> and which are terms that some find useful.

Carl, I spent months getting what they meant wrong, and every time I
used them, Paul would say I was using it wrong and it meant something
else. Some days I would think a chromatic "unison vector" was an
interval mapped to one degree by some val which mapped all the
commatic "unison vectors" to 0, and other days I would think it was an
interval which was mapped to 0 mod n, but not to 0, where n is the
number of notes in a MOS for some linear temperament that we wanted to
detemper, so that the linear temperament together with the chromatic
unison vector gave us a val for n-equal. Whether or not a "chromatic
unison vector" was, in fact, a "unison vector", and for what, was a
problem. Were there perhaps two vals, defining a linear temperament,
and it was in the kernel of one but not the other? What is it and what
good is it?

Useful? I hadn't noticed.

🔗Carl Lumma <ekin@lumma.org>

>> >> tonespace as we know it appears to qualify as
>> >> a "point lattice"....
>> >
>> >"Tonespace" as I know it is meaningless unless defined.
>>
>> Any of the lattices ever discussed here could be called
>> tonespaces.
>
>Is the lattice a tone space, or the space the lattice lives
>in a tone space?

The latter, most likely.

>Does the space have a way of measuring distance, or not? If it
>does, what is it?

I'd say not.

>> Don't worry about Tenney space! It's utterly unrequired for
>> defining unison vectors, which is what you asked me to do.
>
>Then why are we talking about lattices and vectors at all?

Because we can define periodicity blocks with them.

>> Euclidean, I think. Do we need a norm? We're just defining
>> a subgroup of R.
>
>We don't need a norm to define a subgroup of R^n, but then we don't
>need to define a subgroup of R^n. All we need is the abelian
>group--the bells, whistles, and duck-quacks people add are extra and
>in many contexts of no use, so why add them unless we *do*, in fact,
>want a lattice geometry?

Abelian groups aren't hard to come by. According to mathworld all
you need is commutativity, and to be a group you must have closure
under an associative operation.

>> >> So a p-limit unison vector is an element of the p-limit
>> >> lattice such that, when added to any element Alpha in that
>> >> lattice, returns an element Beta with the same musical name
>> >> as Alpha.
>> >
>> >You are introducing new and uncessary complications with this
>> >"musicial name" terminology.
>>
>> Howabout "note name"? This is about music, after all. And
>> the concept is utterly fundamental to the notion of chromatic
>> and commatic unison vectors, which you asked me to explain
>> and which are terms that some find useful.
>
>Carl, I spent months getting what they meant wrong, and every time I
>used them, Paul would say I was using it wrong and it meant something
>else. Some days I would think a chromatic "unison vector" was an
>interval mapped to one degree by some val which mapped all the
>commatic "unison vectors" to 0, and other days I would think it was an
>interval which was mapped to 0 mod n, but not to 0, where n is the
>number of notes in a MOS for some linear temperament that we wanted to
>detemper, so that the linear temperament together with the chromatic
>unison vector gave us a val for n-equal. Whether or not a "chromatic
>unison vector" was, in fact, a "unison vector", and for what, was a
>problem. Were there perhaps two vals, defining a linear temperament,
>and it was in the kernel of one but not the other? What is it and what
>good is it?
>
>Useful? I hadn't noticed.

I guess even the brightest of us have trouble understanding things
sometimes. In my case these terms really helped bring everything
together. Chromatic unison vectors are unison vectors, definitely.
For example, if one played Mozart in 5-limit JI, one would call
81:80 a chromatic uv because pairs of notes an 81:80 apart would
have the same name in the score yet would be at different pitches.
In meantone the 81:80 would become a commatic uv.

-Carl

🔗monz <monz@tonalsoft.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> The example driven home in the Forms of Tonality is the diatonic
> scale, where 81:80 is a commatic uv and 25:24 is a chromatic uv.
>
> I like these terms because they are already in use.

I would like them a lot better if they actually meant something, but
they don't seem to. From the above example, you might guess the
important fact is that 81/80 is sent to 0 by 12-et, and 25/24 is sent
to 1, but that's wrong. You might guess the important fact is that
81/80 is sent to 0 and 25/24 to 7 by meantone, but that's wrong also.
You might guess a lot of things, and I spent months guessing.
Everything is wrong. I suspect it has no real meaning.

> ...with standard music terminology. In the case of the diatonic
> scale, it's 25:24 that gives you chromatic alterations.

And what does 16/15 give you--pentatonic chromatic alterations? Is it
a pentatonic chromatic unison vector? How in hell do you know what a
so-called "chromatic unison vector" actually is?

> Another interesting tidbit is that in a MOS, the chromatic uv
> is always the accidental you need to notate its modulations.

So 25/24 isn't a "chromatic unison vector" in any general sense, it is
something *specifically connected* to 7-note MOS for meantone, and
hence you should say so! I ended up doing that with "chroma", so I
could use a word which I knew actually meant something. An interval
which is sent to 0 by 7-et and to 7 by meantone means something, and
you can call it a chroma for the 7-note MOS of meantone. "Chromatic
unison vector" is a confusing mumble unless it is defined. It is also
simply bad terminology, since it is not a unison vector for meantone,
which happens to be what we are talking about. The fact that a
so-called "chromatic unison vector" for a given temperament is not, in
fact, a unison vector for that temperament was to me a source of major
confusion.

🔗Graham Breed <graham@microtonal.co.uk>

🔗Carl Lumma <ekin@lumma.org>

>> The example driven home in the Forms of Tonality is the diatonic
>> scale, where 81:80 is a commatic uv and 25:24 is a chromatic uv.
>>
>> I like these terms because they are already in use.
>
>I would like them a lot better if they actually meant something,
>but they don't seem to.

I asked you for comments on my definition, and those you gave
didn't seem in any way damning.

>From the above example, you might guess the
>important fact is that 81/80 is sent to 0 by 12-et, and 25/24 is
>sent to 1, but that's wrong. You might guess the important fact
>is that 81/80 is sent to 0 and 25/24 to 7 by meantone, but that's
>wrong also. You might guess a lot of things, and I spent months
>guessing. Everything is wrong. I suspect it has no real meaning.

You need a scale and a tuning. Both are sent to 0 in the scale;
only commatic uvs are sent to 0 in the tuning.

>> ...with standard music terminology. In the case of the diatonic
>> scale, it's 25:24 that gives you chromatic alterations.
>
>And what does 16/15 give you--pentatonic chromatic alterations? Is it
>a pentatonic chromatic unison vector? How in hell do you know what a
>so-called "chromatic unison vector" actually is?

How do I know what a pentatonic scale actually is? If it can
be expressed as a Fokker block involving 16:15 and 16:15 isn't
tempered out then 16:15 is a chromatic uv for it.

>> Another interesting tidbit is that in a MOS, the chromatic uv
>> is always the accidental you need to notate its modulations.
>
>So 25/24 isn't a "chromatic unison vector" in any general sense, it
>is something *specifically connected* to 7-note MOS for meantone,
>and hence you should say so!

Yes, you need a scale. In particular you need an epimorphic scale,
and I did say that in my definition.

Once again, I assume

Fokker block <=> epimorphic scale

though I'm not certain my perpetual asking about this was ever
answered. The tonalsoft entry for epimorphic doesn't mention
Fokker blocks, and the entry on Fokker blocks is broken at the
moment.

>I ended up doing that with "chroma", so I could use a word which
>I knew actually meant something.

The word itself is immaterial. Though I like chroma as a word,
and I would adopt it if it would quell you. How do you define
it?

>"Chromatic unison vector" is a confusing mumble unless it is
>defined.

I did define it.

>It is also simply bad terminology, since it is not a unison vector
>for meantone, which happens to be what we are talking about.

You have to use the proper definition of unison vector.

>The fact that a so-called "chromatic unison vector" for a given
>temperament is not, in fact, a unison vector for that temperament
>was to me a source of major confusion.

Yes, the interface between temperaments and scales has been a
major source of confusion all-around.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> You need a scale and a tuning. Both are sent to 0 in the scale;
> only commatic uvs are sent to 0 in the tuning.

What you should say, I think, is that you need a linear temperament
and a val; both are sent to 0 by the val, but only the commatic to 0
by the temperament.

> Yes, you need a scale. In particular you need an epimorphic scale,
> and I did say that in my definition.

No, you most certainly do NOT need a scale. What you need is a val; an
epimorphic scale is useful precisely and only because it gives a val.

> Once again, I assume
>
> Fokker block <=> epimorphic scale
>
> though I'm not certain my perpetual asking about this was ever
> answered.

No, they are not the same thing.

The tonalsoft entry for epimorphic doesn't mention
> Fokker blocks, and the entry on Fokker blocks is broken at the
> moment.
>
> >I ended up doing that with "chroma", so I could use a word which
> >I knew actually meant something.
>
> The word itself is immaterial. Though I like chroma as a word,
> and I would adopt it if it would quell you. How do you define
> it?

A "chroma" for an n-note chain scale of a temperament is an interval
which when wedged with (the complement of) a linear temperament wedgie
gives (complement again) a val v such that v[1]=+-n, which is an
n-equal val supported by the temperament. So it's only a chroma
relative to a specific size of chain (MOS, without the DE assumption,)
and more precisely only relative to a particular val.

Another way to say it is that the chroma is mapped to the difference
between the start and end of the chain by the temperament. 25/24 would
be a chroma for 7 notes of meantone, but so would 78125000/78121827.
Chromas which shift consonances, particularly the generator, presuming
it is consonant, to some other consonance are particularly interesting
from the point of view of circulation.

For example, 40/39 is a chroma for the 13-limit meantone from the
standard vals for 31 and 50. The val you get wedging it with the
wedgie is <12 19 28 34 41 45|, which is one version but hardly the
only one for 12-et in the 13-limit. A nice thing about it is that
(3/2)*(40/39) = 20/13, a 13-limit consonance; another nice thing is
that 36/35 is a chroma with the exact same val, and (5/4)*(36/35)=9/7.

> >"Chromatic unison vector" is a confusing mumble unless it is
> >defined.
>
> I did define it.

Not in a way which made sense to me; you brought in notation, and then
didn't explain precisely what we were talking about.

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> >> ... Chromatic unison vectors are unison vectors,
> >> definitely. For example, if one played Mozart in
> >> 5-limit JI, one would call 81:80 a chromatic uv
> >> because pairs of notes an 81:80 apart would
> >> have the same name in the score yet would be
> >> at different pitches.
> >> In meantone the 81:80 would become a commatic uv.
> >
> > thanks, Carl!
> >
> > that's the first time i've ever finally fully understood
> > the difference between those two terms.
>
> The example driven home in the Forms of Tonality is
> the diatonic scale, where 81:80 is a commatic uv and
> 25:24 is a chromatic uv.

and the only thing that makes 25:24 a unison-vector
at all is the fact that it uses the same nominal as
another note in the scale.

this points out how intimately the unison-vector concept
is tied to notation. and indeed, an entire section of
Fokker's paper is titled "The use of unison vectors to
simplify notation in practice".

> I like these terms because they are already in use.
> I also like them because they build on a term coined
> by Fokker, who started this entire field of inquiry,
>
> http://www.xs4all.nl/~huygensf/doc/fokkerpb.html
>
> (this was apparently written in English)
>
> ...with standard music terminology. In the case of
> the diatonic scale, it's 25:24 that gives you
> chromatic alterations.

that's true in the 5-limit JI diatonic major
"basic scale" posited by Rameau, Schoenberg, Ben Johnston,
et al. i don't see anything like this in Fokker's paper,
but it certainly is in the explanation given by Paul
("The Forms of Tonality").

but historically, the chromatic alterations of sharp/flat
arose within pythagorean tuning. so the oldest recorded
tuning of the "diatonic scale" was generated as
3^(-4...+2) thus:

. -4 .. -3 .. -2 .. -1 .. 0 .. 1 .. 2 = 3^x
.. F ... C ... G ... D .. A .. E .. B

the next power of 3, 3^3, would be F#. thus, the
chromatic alteration is given by 3^(3-(-4)) = 3^7.
its full 2,3-monzo is [-11 7,> = ratio 2187:2048 =
~ 113.6850061 cents.

25:24 is exactly 2 syntonic-commas narrower than this:

[-11 7, 0> - ([-4 4, -1> * 2) = [-3 -1, 2> = 25:24

one could also use 135:128 as a "chromatic unison-vector":

[-11 7, 0> - ([-4 4, -1> * 1) = [-7 3, 1 > = 135:128

but the 25:24 is the one closest to 1/1, so it helps
to provide a compact lattice basis.

-monz

🔗Carl Lumma <ekin@lumma.org>

>> You need a scale and a tuning. Both are sent to 0 in the scale;
>> only commatic uvs are sent to 0 in the tuning.
>
>What you should say, I think, is that you need a linear temperament
>and a val; both are sent to 0 by the val, but only the commatic to
>0 by the temperament.

This sounds good, except it isn't just for linear temperaments.

>> Yes, you need a scale. In particular you need an epimorphic
>> scale, and I did say that in my definition.
>
>No, you most certainly do NOT need a scale. What you need is
>a val; an epimorphic scale is useful precisely and only because
>it gives a val.

Yes, that seems better.

>> Once again, I assume
>>
>> Fokker block <=> epimorphic scale
>>
>> though I'm not certain my perpetual asking about this was ever
>> answered.
>
>No, they are not the same thing.

Oh. Then maybe I need to look at it more. But you have said
that epimorphic is closely related to CS. And well-formed
Fokker blocks are CS according to Paul...

>> I like chroma as a word,
>> and I would adopt it if it would quell you. How do you define
>> it?
>
>A "chroma" for an n-note chain scale of a temperament is an
>interval which when wedged with (the complement of) a linear
>temperament wedgie gives (complement again) a val v such
>that v[1]=+-n, which is an n-equal val supported by the
>temperament. So it's only a chroma relative to a specific
>size of chain (MOS, without the DE assumption,) and more
>precisely only relative to a particular val.

What's a chain scale? What's a wedge, what's a complement?
What's a val? What's an n-equal val? How does a temperament
support a val?

You can see that this definition has its drawbacks too.

>Another way to say it is that the chroma is mapped to the
>difference between the start and end of the chain by the
>temperament. 25/24 would be a chroma for 7 notes of meantone,
>but so would 78125000/78121827.

I think this is the same thing I was saying.

>Chromas which shift consonances, particularly the generator,
>presuming it is consonant, to some other consonance are
>particularly interesting from the point of view of
>circulation.

Not to mention chord content.

>For example, 40/39 is a chroma for the 13-limit meantone from
>the standard vals for 31 and 50. The val you get wedging it
>with the wedgie is <12 19 28 34 41 45|, which is one version
>but hardly the only one for 12-et in the 13-limit. A nice
>thing about it is that (3/2)*(40/39) = 20/13, a 13-limit
>consonance; another nice thing is that 36/35 is a chroma with
>the exact same val, and (5/4)*(36/35)=9/7.

Giving an example is always good, but in this case we have
to know how to read wedgies.

>> >"Chromatic unison vector" is a confusing mumble unless it
>> >is defined.
>>
>> I did define it.
>
>Not in a way which made sense to me; you brought in notation,
>and then didn't explain precisely what we were talking about.

Imagine how difficult it is for me to understand you!
Understanding is difficult even in the best of circumstances.
Trying to do music theory over e-mail is not the best of
circumstances.

-Carl

🔗Carl Lumma <ekin@lumma.org>

Hi monz,

>> The example driven home in the Forms of Tonality is
>> the diatonic scale, where 81:80 is a commatic uv and
>> 25:24 is a chromatic uv.
>
>and the only thing that makes 25:24 a unison-vector
>at all is the fact that it uses the same nominal as
>another note in the scale.

Right.

>this points out how intimately the unison-vector concept
>is tied to notation. and indeed, an entire section of
>Fokker's paper is titled "The use of unison vectors to
>simplify notation in practice".

Cool. I'm embarrassed to say I haven't read it
carefully.

>but historically, the chromatic alterations of sharp/flat
>arose within pythagorean tuning. so the oldest recorded
>tuning of the "diatonic scale" was generated as
>3^(-4...+2) thus:
>
>. -4 .. -3 .. -2 .. -1 .. 0 .. 1 .. 2 = 3^x
>.. F ... C ... G ... D .. A .. E .. B
>
>the next power of 3, 3^3, would be F#. thus, the
>chromatic alteration is given by 3^(3-(-4)) = 3^7.
>its full 2,3-monzo is [-11 7,> = ratio 2187:2048 =
>~ 113.6850061 cents.
>
>25:24 is exactly 2 syntonic-commas narrower than this:
>
>[-11 7, 0> - ([-4 4, -1> * 2) = [-3 -1, 2> = 25:24

In my definition I say "p-limit unison vector" because
it has to be defined with respect to a given limit.

The traditional diatonic scale is a different scale
than the 5-limit diatonic scale in Paul's paper. But
it is also a periodicity block, and it also has a
notation based on its chromatic unison vector.

>one could also use 135:128 as a "chromatic unison-vector":

That's 81:80 + 25:24. 81:80 vanishes so, yes, 135:128
is also 'the' chromatic uv.

>but the 25:24 is the one closest to 1/1, so it helps
>to provide a compact lattice basis.

We're not actually using it here to provide a basis
for the lattice, but you're right that its lower
complexity makes it a more natural choice for the
title of "the chromatic uv".

-Carl

🔗Robert Walker <robertwalker@ntlworld.com>

Hi Carl,

Thanks for explaining chromatic unison vector.

Can I give a try at a newbie style definition?

First some observations. Notes aren't
usually notated using the scale
degrees.

So for instance, the 12-et val of C#
maps it to 1, since it is at scale degree 1.
However it is notated as C plus an accidental.

C to C# is a unison vector in the 7-tone
scale but not in the twelve tone one.

So the notation system is based on
two periodicity blocks one within another
and not just one - here the seven tone
periodicity block within the twelve tone
one.

The chromatic and commatic unison vector then is connected
with that.

So need a new definition (newbie style here but the
maths version is easily derived from it):

Notation periodicity block:
A periodicity block used to define a notation
system on another periodicity block

So the the five limit seven tone scale is used
as the notation periodicity block for twelve
tone scales - it maps notes to
F C G D A E depending on the position
of the note in the seven tone scale.
Notes that aren't in the same notation
periodicity block are notated using
accidentals.

Standard periodicity block notation system:

Notation sytem for a periodicity block scale
which uses another smaller periodicity block
as the notation periodicity block.

Notes that are in the same position in the
equivalence set up by the notation
periodicity block are given the same
note name. Ones that aren't use an accidental
to modify the name. An accidental
unison vector of the notation periodicity
block needs to be given as part
of the notation system. Accidentals
are then worked out in reference
to the notation periodicity block
scale.

Example, in twelve tone five limit,
the scale periodicity block is generated
by 81/80 and 125/128, and the
notation periodicity block is
generated by 81/80 and 25/24.
A suitable notation periodicity
block scale would be:

1/1 9/8 5/4 4/3 3/2 8/5 16/9 2/1

Then 25/24 is chosen as the
accidental inducing unison
vector of the periodicity block
- known as a chromatic unison vector.

Note that the particular choice of the scale
here affects the number of accidentals
shown in the notation system, e.g.
choice of 5/3 instead of 8/5
would change the notation of
8/5 from A to Ab

commatic unison vector:

Unison vector for a periodicity block
scale. It maps notes to
ones with the same note name in
a standard notation system for a Tenney lattice.
Any unison vector of the periodicity
block used to define the scale is
a commatic unison vector.

Example, 81/80 for twelve tone scales
as notes an 81/80 apart are treated
as the same note in the seven tone
periodicity block. Or 125/128
too.

chromatic unison vector:

Notation system unison vector used to define accidental
mapping in a standard periodicity block
notation. .

Example 25/24 for the seven tone
standard notation system for
twelve tone five limit scales.

It's a unison vector for
the seven tone scale used
for the notation system - but
not one for the periodicity block
scale it is used to notate.

However, you can choose different
vectors for your chromatic unison
vector - any unison vector
for the notation periodicity blcok
scale is suitable. So you could use
135:128 instead of 25:24.
Which one you use needs to be specified
as part of the standard notation system.

The tie in with vals then is that
the notation system uses the
a val for the notation periodicity
block e.g. if you ue the val for the
notation periodicity block to find
the position of 5/4 in the scale
it will give you 2 and so E as
desired. Applied to 81/64 then
it will also give an E as desired.
Applied to 6/5 then it will give E again.

To make it clear how it all ties together
lets do a completely worked out example
and find out the notation for 6/5
using the notation periodicity block
scale
1/1 9/8 5/4 4/3 3/2 8/5 16/9 2/1
and the chromatic unison vector
25/24 for a twelve tone five limit
periodicity block generated using
81/80 and 125/128.

First we need the val for the
notation periodicity block:

Well just cheating and making the
periodicity blocks in SCALA for
81/80 and 25/24 ignoring primes
2, 3 and 5 gives
7, 11 and 16 notes, so 2/1 is
at scale degree 7, 3/1 is at degree
11 and 5/1 at 16.

Or one could work it out using the wedge product:
81/80
[-4 4, -1>
25/24:
[-3 -1, 2>
bimonzo
(-4 e1 + 4e2 -e3) * (-3 e1 - e2 + 2 e3)
= - 7 e32 + 11 e31 + 16 e21
(just multiplying out using the rules that e11 = e22 = e33 = 0
and e21 = - e12 etc)

val
[7 11, 16>

Or one could just inspect the scale and note the
scale degrees for 2/1 3/1 and 5/1 in the scale
1/1 9/8 5/4 4/3 3/2 8/5 16/9 2/1 9/4 4/2 2/3 3/1 16/5 4/1 9/4 5/1
1 7 11 16

All those methods are equally valid. Anyway we have now found
the val by whichever method, and so know the scale degrees
for 2/1, 3/1 and 5/1.

Now, using those,

5/4 = 5 / 2^2 * 5 is at 16 - 7*2 = 2.
81/64 = 3^4 / 2^ 6 is similarly at 11*4 - 7 * 6 = 44 - 42 = 2 again.

So is 6/5 at:
6/5 = 2*3 / 5 -> 7 + 11 - 16 = 18 - 6 = 2 again.

So all use the same basic note name in the notation
periodicity block.

Then to find the sharp or flat
you need to use the val for the twelve
tone block.

So, 6/5 then using the 12 tone val of
[12 19, 28>
is at
6/5 = 2*3 / 5 -> 12 + 19 - 28 = 3 as expected since
it is scale degree 3.
5/4 = 5 / 2^2 * 5 is at 28 - 19*2 = 4.
again as expected.

So the accidental is then the difference of these
two so 3 - 4 = -1, so it is Eb.

So it all ties up with ones intuitive
notion of what should be the note name
for 6/5 using
1/1 9/8 5/4 4/3 3/2 8/5 16/9 2/1
C D E F G A B C'
as ones note naem system.
and 25/24 as the chromatic unison
vector

I know this may seem a bit long winded
but you need to do this in order to figure
out how to extend it to more complex
situations. With the basic idea made
clear then one can now go ahead and
work out notation system periodicity
blocks for any periodicity block scale
including more complex situations such
as multiple levels of accidentals,
and so maybe this may help
one to derive new and useful
notation systems :-).

Robert

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

>Hi Carl,
>
>Thanks for explaining chromatic unison vector.
>
>Can I give a try at a newbie style definition?

Of course!

>First some observations. Notes aren't
>usually notated using the scale
>degrees.

That was the reason for the phrase "musical
name", which I then described in terms of
scale degrees for Gene's benefit, though he
still balked.

>So for instance, the 12-et val of C#
>maps it to 1, since it is at scale degree 1.
>However it is notated as C plus an accidental.
>C to C# is a unison vector in the 7-tone
>scale but not in the twelve tone one.

Yep.

>So the notation system is based on
>two periodicity blocks one within another
>and not just one - here the seven tone
>periodicity block within the twelve tone
>one.

Or, as Gene says, a val and a temperament.

>Notation periodicity block:
>A periodicity block used to define a notation
>system on another periodicity block

Hmm....

>So the the five limit seven tone scale is used
>as the notation periodicity block for twelve
>tone scales - it maps notes to
>F C G D A E depending on the position
>of the note in the seven tone scale.
>Notes that aren't in the same notation
>periodicity block are notated using
>accidentals.
>
>Standard periodicity block notation system:
>
>Notation sytem for a periodicity block scale
>which uses another smaller periodicity block
>as the notation periodicity block.
>
>Notes that are in the same position in the
>equivalence set up by the notation
>periodicity block are given the same
>note name. Ones that aren't use an accidental
>to modify the name. An accidental
>unison vector of the notation periodicity
>block needs to be given as part
>of the notation system. Accidentals
>are then worked out in reference
>to the notation periodicity block
>scale.
>
>Example, in twelve tone five limit,
>the scale periodicity block is generated
>by 81/80 and 125/128, and the
>notation periodicity block is
>generated by 81/80 and 25/24.
>A suitable notation periodicity
>block scale would be:
>
>1/1 9/8 5/4 4/3 3/2 8/5 16/9 2/1
>
>Then 25/24 is chosen as the
>accidental inducing unison
>vector of the periodicity block
>- known as a chromatic unison vector.
>
>Note that the particular choice of the scale
>here affects the number of accidentals
>shown in the notation system, e.g.
>choice of 5/3 instead of 8/5
>would change the notation of
>8/5 from A to Ab
>
>commatic unison vector:
>
>Unison vector for a periodicity block
>scale. It maps notes to
>ones with the same note name in
>a standard notation system for a Tenney lattice.
>Any unison vector of the periodicity
>block used to define the scale is
>a commatic unison vector.
>
>Example, 81/80 for twelve tone scales
>as notes an 81/80 apart are treated
>as the same note in the seven tone
>periodicity block. Or 125/128
>too.
>
>chromatic unison vector:
>
>Notation system unison vector used to define accidental
>mapping in a standard periodicity block
>notation. .
>
>Example 25/24 for the seven tone
>standard notation system for
>twelve tone five limit scales.
>
>It's a unison vector for
>the seven tone scale used
>for the notation system - but
>not one for the periodicity block
>scale it is used to notate.
>
>However, you can choose different
>vectors for your chromatic unison
>vector - any unison vector
>for the notation periodicity blcok
>scale is suitable. So you could use
>135:128 instead of 25:24.
>Which one you use needs to be specified
>as part of the standard notation system.
>
>The tie in with vals then is that
>the notation system uses the
>a val for the notation periodicity
>block e.g. if you ue the val for the
>notation periodicity block to find
>the position of 5/4 in the scale
>it will give you 2 and so E as
>desired. Applied to 81/64 then
>it will also give an E as desired.
>Applied to 6/5 then it will give E again.
>
>To make it clear how it all ties together
>lets do a completely worked out example
>and find out the notation for 6/5
>using the notation periodicity block
>scale
>1/1 9/8 5/4 4/3 3/2 8/5 16/9 2/1
>and the chromatic unison vector
>25/24 for a twelve tone five limit
>periodicity block generated using
>81/80 and 125/128.
>
>First we need the val for the
>notation periodicity block:
>
>Well just cheating and making the
>periodicity blocks in SCALA for
>81/80 and 25/24 ignoring primes
>2, 3 and 5 gives
>7, 11 and 16 notes, so 2/1 is
>at scale degree 7, 3/1 is at degree
>11 and 5/1 at 16.
>
>Or one could work it out using the wedge product:
>81/80
>[-4 4, -1>
>25/24:
>[-3 -1, 2>
>bimonzo
>(-4 e1 + 4e2 -e3) * (-3 e1 - e2 + 2 e3)
>= - 7 e32 + 11 e31 + 16 e21
>(just multiplying out using the rules that e11 = e22 = e33 = 0
>and e21 = - e12 etc)
>
>val
>[7 11, 16>
>
>Or one could just inspect the scale and note the
>scale degrees for 2/1 3/1 and 5/1 in the scale
>1/1 9/8 5/4 4/3 3/2 8/5 16/9 2/1 9/4 4/2 2/3 3/1 16/5 4/1 9/4 5/1
>1 7 11 16
>
>
>All those methods are equally valid. Anyway we have now found
>the val by whichever method, and so know the scale degrees
>for 2/1, 3/1 and 5/1.
>
>Now, using those,
>
>5/4 = 5 / 2^2 * 5 is at 16 - 7*2 = 2.
>81/64 = 3^4 / 2^ 6 is similarly at 11*4 - 7 * 6 = 44 - 42 = 2 again.
>
>So is 6/5 at:
>6/5 = 2*3 / 5 -> 7 + 11 - 16 = 18 - 6 = 2 again.
>
>So all use the same basic note name in the notation
>periodicity block.
>
>Then to find the sharp or flat
>you need to use the val for the twelve
>tone block.
>
>So, 6/5 then using the 12 tone val of
>[12 19, 28>
>is at
>6/5 = 2*3 / 5 -> 12 + 19 - 28 = 3 as expected since
>it is scale degree 3.
>5/4 = 5 / 2^2 * 5 is at 28 - 19*2 = 4.
>again as expected.
>
>So the accidental is then the difference of these
>two so 3 - 4 = -1, so it is Eb.
>
>So it all ties up with ones intuitive
>notion of what should be the note name
>for 6/5 using
>1/1 9/8 5/4 4/3 3/2 8/5 16/9 2/1
>C D E F G A B C'
>as ones note naem system.
>and 25/24 as the chromatic unison
>vector
>
>I know this may seem a bit long winded
>but you need to do this in order to figure
>out how to extend it to more complex
>situations. With the basic idea made
>clear then one can now go ahead and
>work out notation system periodicity
>blocks for any periodicity block scale
>including more complex situations such
>as multiple levels of accidentals,
>and so maybe this may help
>one to derive new and useful
>notation systems :-).

Looks cool! I'll file this for when I might
need to work out a notation.

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗monz <monz@tonalsoft.com>

hi Gene and Carl,

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

> --- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
>
> ...
>
> > The tonalsoft entry for epimorphic doesn't mention
> > Fokker blocks,

Gene, could you please say more about "epimorphic",
and also provide a few examples?

> > and the entry on Fokker blocks is broken at the moment.

oops ... my bad. thanks for letting me know.
it's returned now, safe and healthy.

> A "chroma" for an n-note chain scale of a temperament
> is an interval which when wedged with (the complement of)
> a linear temperament wedgie gives (complement again) a
> val v such that v[1]=+-n, which is an n-equal val supported
> by the temperament. So it's only a chroma relative to a
> specific size of chain (MOS, without the DE assumption,)
> and more precisely only relative to a particular val.
>
> Another way to say it is that the chroma is mapped to the
> difference between the start and end of the chain by the
> temperament. 25/24 would be a chroma for 7 notes of meantone,
> but so would 78125000/78121827.
> Chromas which shift consonances, particularly the generator,
> presuming it is consonant, to some other consonance are
> particularly interesting from the point of view of
> circulation.
>
> For example, 40/39 is a chroma for the 13-limit meantone
> from the standard vals for 31 and 50. The val you get
> wedging it with the wedgie is <12 19 28 34 41 45|, which
> is one version but hardly the only one for 12-et in the
> 13-limit. A nice thing about it is that (3/2)*(40/39) =
> 20/13, a 13-limit consonance; another nice thing is
> that 36/35 is a chroma with the exact same val, and
> (5/4)*(36/35)=9/7.

i've put all of that into the Encyclopaedia "chroma"
definition ... but if what you say here does not agree
with the definition which i give at the top, please say so,
and perhaps coin a new term.

(i know ... adding again to the jargon explosion ...)

-monz

🔗monz <monz@tonalsoft.com>

hi Robert,

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:

> So the notation system is based on
> two periodicity blocks one within another
> and not just one - here the seven tone
> periodicity block within the twelve tone
> one.

i think that's an excellent way to look at it.
a small notation periodicity-block embedded within
a larger tuning periodicity-block.

> commatic unison vector:
>
> Unison vector for a periodicity block
> scale. It maps notes to
> ones with the same note name in
> a standard notation system for a Tenney lattice.
> Any unison vector of the periodicity
> block used to define the scale is
> a commatic unison vector.
>
> Example, 81/80 for twelve tone scales
> as notes an 81/80 apart are treated
> as the same note in the seven tone
> periodicity block. Or 125/128
> too.
>
> chromatic unison vector:
>
> Notation system unison vector used to define accidental
> mapping in a standard periodicity block
> notation. .
>
> Example 25/24 for the seven tone
> standard notation system for
> twelve tone five limit scales.
>
> It's a unison vector for
> the seven tone scale used
> for the notation system - but
> not one for the periodicity block
> scale it is used to notate.

hmm ... that's how i always understood
the difference between "chromatic" and
"commatic" unison-vectors ... but that's
not what Carl just wrote about it.

his example said that in JI 81/80 is a
chromatic unison-vector -- which i interpreted
to mean that a chromatic unison-vector actually
exists -- and that in meantone 81/80 is a
commatic unison-vector, because it actually
does vanish.

> I know this may seem a bit long winded
> but you need to do this in order to figure
> out how to extend it to more complex
> situations. With the basic idea made
> clear then one can now go ahead and
> work out notation system periodicity
> blocks for any periodicity block scale
> including more complex situations such
> as multiple levels of accidentals,
> and so maybe this may help
> one to derive new and useful
> notation systems :-).

i think this is great.

Paul presents exactly the same two examples
in part 2 of his "Gentle Introduction to
Fokker Periodicity Blocks"

http://tonalsoft.com/td/erlich/intropblock2.htm

7-tone 5-limit diatonic
unison-vectors:
.. 25:24 , monzo [-3 -1, 2>
.. 81:80 , monzo [-4 4, -1>

12-tone 5-limit chromatic
unison-vectors:
.. 128:125 , monzo [7 0, -3>
.. 81:80 , monzo [-4 4, -1>

-monz

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗monz <monz@tonalsoft.com>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >> ... Chromatic unison vectors are unison vectors, definitely.
> >> For example, if one played Mozart in 5-limit JI, one would call
> >> 81:80 a chromatic uv because pairs of notes an 81:80 apart would
> >> have the same name in the score yet would be at different
pitches.
> >> In meantone the 81:80 would become a commatic uv.
> >
> >thanks, Carl!
> >
> >that's the first time i've ever finally fully understood
> >the difference between those two terms.
>
> The example driven home in the Forms of Tonality is the diatonic
> scale, where 81:80 is a commatic uv and 25:24 is a chromatic uv.
>
> I like these terms because they are already in use. I also like
> them because they build on a term coined by Fokker, who started
> this entire field of inquiry,
>
> http://www.xs4all.nl/~huygensf/doc/fokkerpb.html

The terms "unison vector" and "chromatic" are fine, but I believe
you are misusing the term "commatic" here. By normal conventions of
English "commatic" should mean merely "relating to a comma or
commas". You are using it to mean more specifically "vanishing"
or "tempered out" or "not notated". If you look at the very article
you quote above, you will find that Fokker uses the term "commatic"
in relation to commas which do _not_ vanish and which he in fact
_does_ notate.

Isn't "unison vector" almost a synonym for "comma"? At least
shouldn't we be able to speak of chromatic versus vanishing commas,
just as we can speak of chromatic versus vanishing unison vectors?
In that case, doesn't it sound silly to speak of "commatic commas".

🔗Carl Lumma <ekin@lumma.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗klaus schmirler <KSchmir@z.zgs.de>

Carl Lumma schrieb:
>>>http://www.xs4all.nl/~huygensf/doc/fokkerpb.html
>>
>>
>>The terms "unison vector" and "chromatic" are fine, but I believe >>you are misusing the term "commatic" here. By normal conventions of >>English "commatic" should mean merely "relating to a comma or >>commas". You are using it to mean more specifically "vanishing" >>or "tempered out" or "not notated". If you look at the very article >>you quote above, you will find that Fokker uses the term "commatic" >>in relation to commas which do _not_ vanish and which he in fact >>_does_ notate.
> > > But on these lists we've usually said that such-and-such ratio
> was a "comma" of such-and-such temperament it if vanished in that
> temperament.

The way I see it, this doesn't matter too much. Temperaments are tuning systems with vanishing commas, and it makes more sense in the context of temperaments to talk about the vanishing ones that define them. Anybody who has been able to follow the temperament making on this list should also be able to infer that. No special term needed.

> > >>Isn't "unison vector" almost a synonym for "comma"? At least >>shouldn't we be able to speak of chromatic versus vanishing commas, >>just as we can speak of chromatic versus vanishing unison vectors? >>In that case, doesn't it sound silly to speak of "commatic commas".
> > > What do you think of "chroma" and "comma"?
> > I can see monz's point about the generic definition of "comma"
> resulting in confusion.

BTW, since monz doesn't seem to have answered your question about the chroma as pitch class: I don't have his source, but I know that in psychology they make a distinction between pitch=the nearing range and chroma=recurring qualities within that.

klaus

> > -Carl
> >

🔗Carl Lumma <ekin@lumma.org>

🔗klaus schmirler <KSchmir@z.zgs.de>

klaus schmirler schrieb:

> Carl Lumma schrieb:
> >>>>http://www.xs4all.nl/~huygensf/doc/fokkerpb.html
>>>
>>>
>>>The terms "unison vector" and "chromatic" are fine, but I believe >>>you are misusing the term "commatic" here. By normal conventions of >>>English "commatic" should mean merely "relating to a comma or >>>commas". You are using it to mean more specifically "vanishing" >>>or "tempered out" or "not notated". If you look at the very article >>>you quote above, you will find that Fokker uses the term "commatic" >>>in relation to commas which do _not_ vanish and which he in fact >>>_does_ notate.
>>
>>
>>But on these lists we've usually said that such-and-such ratio
>>was a "comma" of such-and-such temperament it if vanished in that
>>temperament.
> > > The way I see it, this doesn't matter too much. Temperaments > are tuning systems with vanishing commas, and it makes more > sense in the context of temperaments to talk about the > vanishing ones that define them. Anybody who has been able > to follow the temperament making on this list should also be > able to infer that. No special term needed.
> > >>
>>>Isn't "unison vector" almost a synonym for "comma"? At least >>>shouldn't we be able to speak of chromatic versus vanishing commas, >>>just as we can speak of chromatic versus vanishing unison vectors? >>>In that case, doesn't it sound silly to speak of "commatic commas".
>>
>>
>>What do you think of "chroma" and "comma"?
>>
>>I can see monz's point about the generic definition of "comma"
>>resulting in confusion.
> > > BTW, since monz doesn't seem to have answered your question > about the chroma as pitch class: I don't have his source, > but I know that in psychology they make a distinction > between pitch=the nearing range

hearing, of course

and chroma=recurring
> qualities within that.
> > klaus
> > > > >>-Carl
>>
>>
>

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> > But on these lists we've usually said that such-and-such ratio
> > was a "comma" of such-and-such temperament it if vanished in that
> > temperament.
>
> In the right context the meaning may have been clear enough, but
> I've always disagreed with overloading the word comma in this way.
> How hard is it to write "vanishing comma" if that's what you mean?

A word for "interval which vanishes in a temperament" is much more
useful than a word which simply means "small interval" for the uses of
tuning theory. How hard is it to say "small interval" if that is what
you mean?

> This will avoid possible confusion with commas that correspond to
> chromatic accidentals for the temperament.

Those should not be called commas; that invites confusion.

> > What do you think of "chroma" and "comma"?
>
> As Monz said, these have both already been used with several
> different meanings, other than the ones you want. "Chromatic comma"
> and "vanishing comma" work fine for me.

I won't use them myself. I think the second is verbose and the first
is simply very bad, since it is confusing to call something a comma
relative to a temperament for which it does not vanish.

The point of a chroma is not that it vanishes; it is sent to zero by
the val defining epimorphicity for some scale, but it won't vanish
unless you temper it out and make the scale an equal temperament. The
point is that it does *not* vanish, and that what its value is or can
be after tempering is what is most interesting. How large is it? Is it
something which leads to interesting results, such as 21/20, 25/24,
36/35, 45/44, 49/48 or even 40/39?

It should not be called a comma by anyone who values clear
communication and does not want to lead people into massive confusion.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> > In the right context the meaning may have been clear enough, but
> > I've always disagreed with overloading the word comma in this
> > way. How hard is it to write "vanishing comma" if that's what
> > you mean?
>
> A word for "interval which vanishes in a temperament" is much more
> useful than a word which simply means "small interval" for the
uses of
> tuning theory. How hard is it to say "small interval" if that is
what
> you mean?

The problem is that "small" is relative. And to many, an "interval"
is two notes of a scale. So I suspect that "small interval" would
often be taken to mean a second or an augmented unison, generally
much bigger than a comma.

The ancient Greeks coined the term "comma". Their usage is
consistent with a size range of around 12 to 35 cents and certainly
did not imply anything about "vanishing in a temperament" since they
had no concept of temperament. The comma of Pythagoras most
certainly did not vanish in a Pythagorean scale.

> > This will avoid possible confusion with commas that correspond
to
> > chromatic accidentals for the temperament.
>
> Those should not be called commas; that invites confusion.

It only invites confusion if one accepts your definition of a comma
as necessarily vanishing, which is not a valid generalisation from
past usage (starting with the ancient Greeks).

The most obvious chromatic alteration for the miracle temperament is
the septimal comma (63;64). Should we be required to call it
something else when it rudely refuses to vanish like this? I think
not.

It's certainly debatable whether we should use the term comma in a
broader sense, when we need a term that covers schismas, kleismas,
commas, dieses, and maybe even sometimes limmas and apotomes.
i.e. "an interval smaller than a scale step". But the possible
objection to this should only be that it includes intervals too
large and too small. I contend that question of whether it includes
intervals which are not necessarily vanishing in a temperament is
irrelevant.

Can you show me any documents prior to Paul Hahn's unfortunate
juxtaposition of "commatic" versus "chromatic", that clearly
indicate that "commatic" means vanishing. If you want something more
recent than the ancient Greeks as evidence of my contention, I've
already pointed out that Fokker (or at least his translators)
used "commatic" for things which did not vanish, but were in fact
notated. So to Fokker a "commatic unison vector" would simply be one
whose untempered size is in the appropriate size range for a comma.

🔗monz <monz@tonalsoft.com>

hi Dave, Gene,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> The problem is that "small" is relative. And to many, an "interval"
> is two notes of a scale. So I suspect that "small interval" would
> often be taken to mean a second or an augmented unison, generally
> much bigger than a comma.

that's a very good point to make.

> The ancient Greeks coined the term "comma". Their usage is
> consistent with a size range of around 12 to 35 cents and certainly
> did not imply anything about "vanishing in a temperament" since they
> had no concept of temperament. The comma of Pythagoras most
> certainly did not vanish in a Pythagorean scale.
>
> <snip>
>
> It's certainly debatable whether we should use the term comma in a
> broader sense, when we need a term that covers schismas, kleismas,
> commas, dieses, and maybe even sometimes limmas and apotomes.
> i.e. "an interval smaller than a scale step". But the possible
> objection to this should only be that it includes intervals too
> large and too small. I contend that question of whether it includes
> intervals which are not necessarily vanishing in a temperament is
> irrelevant.
>
> Can you show me any documents prior to Paul Hahn's unfortunate
> juxtaposition of "commatic" versus "chromatic", that clearly
> indicate that "commatic" means vanishing. If you want something more
> recent than the ancient Greeks as evidence of my contention, I've
> already pointed out that Fokker (or at least his translators)
> used "commatic" for things which did not vanish, but were in fact
> notated. So to Fokker a "commatic unison vector" would simply be one
> whose untempered size is in the appropriate size range for a comma.

this is a great argument, and illustrates exactly why
i think it's so silly for people to complain about
"jargon".

the only reason we are able to talk about these things
at all is because words have been invented to represent
the important concepts. enriching one's vocabulary only
makes discussion easier.

yes, of course i understand how hard it is for a newbie
... but that's just part of the learning curve. if we're
going to be here online discussing this stuff, a confused
reader can always turn to the Encyclopaedia for help,
and google other sources if appropriate.

so anyway, i personally don't want to have to keep
adding "vanishing" to "promo" (which, yes, i will
always use from now on, it's a very useful concept).

i want one short word that is defined as a
"vanishing promo".

-monz

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> > The ancient Greeks coined the term "comma". Their usage is
> > consistent with a size range of around 12 to 35 cents and
certainly
> > did not imply anything about "vanishing in a temperament" since
they
> > had no concept of temperament. The comma of Pythagoras most
> > certainly did not vanish in a Pythagorean scale.
...

> this is a great argument, and illustrates exactly why
> i think it's so silly for people to complain about
> "jargon".

I fail to see how it illustrates that at all. I'm trying to maintain
a connection to terminology that has been in consistent use for
thousands of years. You and Gene seem to be busy inventing new words
for old concepts (or minor modifications of old concepts) or
assigning new concepts to old words, apparently because you don't
like having to type so much. These activities break connections
rather than maintaining them.

> the only reason we are able to talk about these things
> at all is because words have been invented to represent
> the important concepts. enriching one's vocabulary only
> makes discussion easier.

It is extremely rare that a concept is so new, and so hard to
describe with a few existing words, that a completely new term must
be invented. What makes you guys so sure that we suddenly need
dozens of them, and that you are the chosen ones to bring
these "gifts".

By the way, your example of "logarithmic pitch height" and "cents"
is quite false. These are not synonyms, nor is "cents" a refinement
or modification of the concept of "logarithmic pitch
height". "Logarithmic pitch height" is the name of the quantity
and "cent" is the name of one unit used for measuring that quantity.
This is the same as the relationship between the concepts
of "length" and "meters" or "feet".

> yes, of course i understand how hard it is for a newbie
> ... but that's just part of the learning curve. if we're
> going to be here online discussing this stuff, a confused
> reader can always turn to the Encyclopaedia for help,
> and google other sources if appropriate.

The cost of new terminology is not so small as you make out. Your
total immersion in it makes you blind to how it appears to
a "newbie" or even to someone like me who stops reading the list for
several months.

Some Encyclopedia entries are starting too look ridiculous. Brand
new jargon defined in terms of other brand new jargon. One gets
dizzy following the links to try and figure out what it all means.
And then when it finally dawns I find myself saying "what was wrong
with such-and-such a term", and the only thing I can see is that it
has more syllables.

> i want one short word that is defined as a
> "vanishing promo".

I think pretty soon you're going to need a word for "vanishing
encyclopedia" as it becomes so inbred it disappears up its own
fundamental orifice.

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗monz <monz@tonalsoft.com>

hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> I'm trying to maintain a connection to terminology that
> has been in consistent use for thousands of years.

and with my interest in the history of music and
music-theory, i really do appreciate that a lot.

> You and Gene seem to be busy inventing new words
> for old concepts (or minor modifications of old concepts)
> or assigning new concepts to old words, apparently because
> you don't like having to type so much. These activities
> break connections rather than maintaining them.

i *never* advocate assigning new concepts to old words.
i will always advocate creating a new word for a new concept.

i speak only for myself, not for Gene.

now, inventing new words for old concepts ... if the
old concept can only be described by a phrase of several
words, then yes, i definitely prefer to have one short
word for it. handy as they may be, i really hate terms
like "distributional evenness" because they're so verbose.

as far as the modifications being minor ... well,
i do make an effort to judge that and try to decide
judiciously whether a concept is new enough, or
whether a modification of an old concept is important
enough to great enough, to merit a new term.

superficially, it may seem that verbosity only concerns
the amount of typing we have to do ... but since i
can type up to about 120 wpm that doesn't really
concern me much. i'm more concerned that long strings
of words become meaningless to readers where a short
word "packs a bigger punch".

> > the only reason we are able to talk about these things
> > at all is because words have been invented to represent
> > the important concepts. enriching one's vocabulary only
> > makes discussion easier.
>
> It is extremely rare that a concept is so new, and so
> hard to describe with a few existing words, that a
> completely new term must be invented. What makes you
> guys so sure that we suddenly need dozens of them,

the post i wrote, to which you responded here, clearly
laid out my argument that discourse on a technical
subject is facilitated by concise -- and *well-defined*
-- vocabulary.

> and that you are the chosen ones to bring
> these "gifts".

again, i can speak only for myself. i created what
used to be the Tuning Dictionary, and what is now the
Encyclopaedia of Tuning, precisely because i felt such
a strong need for an easily accessible reference defining
tuning terminology.

so the only person who chose me was ... me.

if someone else doesn't like that, fine ... they can make
up their own words, or use long strings of already-existing
words, or just make diagrams or do whatever you're going
to do. me, i'll keep writing about tuning, and if i
think there's an important concept for which i don't
know any currently-accepted *concise* term, i'll make
one up.

in any case, i continue (and will continue) to strive
to include any tuning terminology i find or invent,
in the Encyclopaedia. there's no limit to its growth.

> By the way, your example of "logarithmic pitch height" and "cents"
> is quite false. These are not synonyms, nor is "cents" a refinement
> or modification of the concept of "logarithmic pitch
> height". "Logarithmic pitch height" is the name of the quantity
> and "cent" is the name of one unit used for measuring that quantity.
> This is the same as the relationship between the concepts
> of "length" and "meters" or "feet".

right ... which is exactly why i gave the side-by-side
examples of "meride" and "savart". but how many people
use those?

so, then i should have said that "logarithmic pitch height
measured in 1/100ths of a 12-et semitone" was replaced by
"cents".

good -- that makes my point even better.

my real point here is that *frequent usage* makes
a short word familiar, and makes discourse corresponding
much easier. the idea is that if i get the new word
into the Encyclopaedia quickly enough, and with a
good enough definition, then people will use it a
lot and only newbies will have to be concerned about it.
(and yes, i do keep that in mind too.)

> > yes, of course i understand how hard it is for a newbie
> > ... but that's just part of the learning curve. if we're
> > going to be here online discussing this stuff, a confused
> > reader can always turn to the Encyclopaedia for help,
> > and google other sources if appropriate.
>
> The cost of new terminology is not so small as you make out. Your
> total immersion in it makes you blind to how it appears to
> a "newbie" or even to someone like me who stops reading the list for
> several months.
>
> Some Encyclopedia entries are starting too look ridiculous. Brand
> new jargon defined in terms of other brand new jargon. One gets
> dizzy following the links to try and figure out what it all means.
> And then when it finally dawns I find myself saying "what was wrong
> with such-and-such a term", and the only thing I can see is that it
> has more syllables.

i've tried to emphasize how hard it was for me to learn
about tuning-theory in the beginning because of the
roadblock of terminology.

in those days, i went crazy searching out hefty volumes
on dusty library shelves, usually to find that i didn't
learn much after all that effort.

my goal is to help other newbies who have that problem.

if the Encyclopaedia is really getting as bad as you
say it is, it's only because i'm the *only* person
working on it (aside from Paul Erlich spending a *lot*
of his time chatting with me to help me weed out the
most egregious errors).

and many of the terms i've added lately describe
concepts that i've had a hard time understanding
myself.

in any case, my whole point in creating the tuning-jargon
list -- which not many people around here seem interested
in particpating in -- was to keep the jargon discussion
off this list as much as possible.

> > i want one short word that is defined as a
> > "vanishing promo".
>
> I think pretty soon you're going to need a word for "vanishing
> encyclopedia" as it becomes so inbred it disappears up its own
> fundamental orifice.

hey, everything that's always been in there that so many
have found valuable over the years, is still there. as
it becomes integrated with the software it will probably
become moreso.

i'm firmly convinced that it will continue to become
more and more valuable as i keep working on it. if you
think otherwise, you can choose to stop using it ... but
i hope you don't.

part of the problem, especially with the latest batch
of new terms, is that my definitions are sorely
inadequate. i beg anyone out there who can improve
any of my definitions -- with text, graphs, musical
illustrations, or audio examples -- to please send
material to me, for inclusion in the Encyclopaedia.

it was begun in a spirit of cooperation, and i continue
to manage it that way. even tho i've done probably
more than 99% of the actual work on it, it is very
much a collaborative effort.

-monz

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> > You and Gene seem to be busy inventing new words
> > for old concepts (or minor modifications of old concepts)
> > or assigning new concepts to old words, apparently because
> > you don't like having to type so much. These activities
> > break connections rather than maintaining them.

> i *never* advocate assigning new concepts to old words.

If I understand correctly, Dave claims that's what I did with "comma",
but it ain't so. It's a classic lose-lose situation for me; I called
these things "kernel elements" a lot at first, and people thought I
was being too mathematical. "Comma" was in use with the meaning, more
or less, of kernel element. So was "unison vector", but I find that
phrase simply unacceptable, since it suggests two things, both of
which are false--one, that we are talking about a unison, and the
other, that we are talking about a vector.

> i will always advocate creating a new word for a new concept.

I create new words when it seems either at worst harmless (such as
thinking up names for temperaments so we can keep track of them in our
heads) or something which really ought to be done, such as my
introduction of words for genuinely important concepts, such as vals
or wedgies, or for concepts I at least make a lot of use of and which
greatly expedite discourse, such as poptimal generator. Paul does the
same--TOP needed a word, and he supplied it. I don't think "schismina"
is in the same category, but I have no objection to it. I do think it
absurd that someone who is willing to use such a word complains of
introducing and then using words for concepts which are absolutely
central to mathematical tuning theory.

Do we really need words for projective versions of monzos, vals, and
their wedge products? I dunno. I don't know if it will help, but the
added insight is hardly likely to hurt. I probably would not have
dared such a suggestion myself--I recall thinking about the fact that
really, we were talking projectively when I was discussing
standardizing the form of a wedgie--but I don't think I brought it up.
I did mention later on that there are projective varieties involved,
but I haven't tried to jam that down anyone. Now that Paul and Monz
have put forward the idea of projectivizing our nomenclaure, the heat
is off me, however. I can blame them. :)

I'm ready to go back to using "kernel element" if this is really a
huge deal, but I don't think I was the one who started the comma ball
rolling. Meanwhile I'm pretty happy with "promo", since it seems to
have a definite meaning, at least in my mind.

> superficially, it may seem that verbosity only concerns
> the amount of typing we have to do ... but since i
> can type up to about 120 wpm that doesn't really
> concern me much.

It's pretty obvious that giving a name to a concept does much more
than save breath.

i'm more concerned that long strings
> of words become meaningless to readers where a short
> word "packs a bigger punch".

A nice short word, made up for the occasion, tells you someone thinks
the matter is important enough to deserve one, and actually makes it
easier to think about the subjects making use of the word.

> > It is extremely rare that a concept is so new, and so
> > hard to describe with a few existing words, that a
> > completely new term must be invented.

In math it is really, really common. It would be truly horrible to
have to write pages of incomprehensible prose rather than just say
"scheme", after all. Other technical subjects introduce terminology
for similar reasons.

And yes, tuning theory is a technical subject.

> right ... which is exactly why i gave the side-by-side
> examples of "meride" and "savart". but how many people
> use those?

How many people would use cents if they had to call them "logarithms
base the 1200th root of 2", which is the long way to say it?

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

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

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > i will always advocate creating a new word for a new concept.
>
> I create new words when it seems either at worst harmless
> (such as thinking up names for temperaments so we can keep
> track of them in our heads)

... and i've been meaning to specifically say something
about that too, but kept forgetting.

by all means, keep the creation of names of temperaments
flowing lavishly! there have been so many new tunings
invented in the last couple of years that we really
need lots of colorful names. i really love the recent
ideas on families of tunings with anthropomorphic
family-type names for all the members.

i got feedback on some tunings about a month ago
for which i devised a format which Gene edited into
a template. *please* keep more of those coming!
i want to have the Encyclopaedia chock-full of
weird and wonderful names of tunings! then we'll
also have all the big family-tree diagrams.

that will be such an aid to newbies trying to
make sense of it all, not to mention a great
resource for those of us who actually want to
learn about or compose with any particular tuning.

> or something which really ought to be done, such as my
> introduction of words for genuinely important concepts,
> such as vals or wedgies, or for concepts I at least make
> a lot of use of and which greatly expedite discourse,
> such as poptimal generator. Paul does the
> same--TOP needed a word, and he supplied it.

and that's pretty much been my argument.

and for those who appreciate what miracle tuning is ...
think of how hard it would have been to discuss all of
its wonderful properties if it didn't have such a
short and meaningful name... ingeniously, a name
which is an acronym laying out all of those properties!

> Do we really need words for projective versions of monzos,
> vals, and their wedge products? I dunno. I don't know if
> it will help, but the added insight is hardly likely to hurt.
> I probably would not have dared such a suggestion myself--
> I recall thinking about the fact that really, we were talking
> projectively when I was discussing standardizing the form of
> a wedgie--but I don't think I brought it up. I did mention
> later on that there are projective varieties involved,
> but I haven't tried to jam that down anyone. Now that
> Paul and Monz have put forward the idea of projectivizing
> our nomenclaure, the heat is off me, however. I can blame
> them. :)

very funny. :P

here i am, right here. standing tall and proudly,

(... "The Good, The Bad, and The Ugly" background music ...)

in defense of "promo"-ting the new projective terminology !!

:PPP

> I'm ready to go back to using "kernel element" if this
> is really a huge deal, but I don't think I was the one
> who started the comma ball rolling.

let's just try to get a shorter, prettier term for
"kernel element". there are more than a few good
creative minds reading this, who could help out.

(and please feel free to join the tuning-jargon list
and do it there, for the sake of those here who
really want this train to stop before they jump off.)

> Meanwhile I'm pretty happy with "promo", since it seems to
> have a definite meaning, at least in my mind.

i really like it. i'm pretty sure Paul came up with it.
(or maybe me, i don't remember.)

i like "vapro" (for "VAnishing PROmo") even more !!

> > superficially, it may seem that verbosity only concerns
> > the amount of typing we have to do ... but since i
> > can type up to about 120 wpm that doesn't really
> > concern me much.
>
> It's pretty obvious that giving a name to a concept does
> much more than save breath.

thank you, thank you, thank you.

> > i'm more concerned that long strings
> > of words become meaningless to readers where a short
> > word "packs a bigger punch".
>
> A nice short word, made up for the occasion, tells you
> someone thinks the matter is important enough to deserve
> one, and actually makes it easier to think about the
> subjects making use of the word.

not only that ... but as concepts continue to build
upon, and combine with, other concepts, the terms themselves
are there, as "kernels" ;-) which can form larger terms,
which in turn can generate new short terms.

this is exactly what happened with:

vanishing prime-factor-exponent unison-vector and all its multiples
==> vanishing monzo and its multiples
==> vanishing projective monzo
==> vanishing promo
==> vapro

now we have one nice little word, whose definition
is a whole mouthful of technical jargon.

if people accept promo and vapro, and use them a lot,
they'll become as ....... "ess-'cents'-ial" ...
as some other terms we use a lot these days. ;->

> > > It is extremely rare that a concept is so new, and so
> > > hard to describe with a few existing words, that a
> > > completely new term must be invented.
>
> In math it is really, really common. It would be truly
> horrible to have to write pages of incomprehensible
> prose rather than just say "scheme", after all. Other
> technical subjects introduce terminology for similar
> reasons.
>
> And yes, tuning theory is a technical subject.

i say, just keep inventing lots and lots of new terms,
and history will sort out what stays and what goes.

> > right ... which is exactly why i gave the side-by-side
> > examples of "meride" and "savart". but how many people
> > use those?
>
> How many people would use cents if they had to call them
> "logarithms base the 1200th root of 2", which is the long
> way to say it?

thanks, Gene ... that was what i was groping towards. ;-)

imagine if Einstein had not had the foresight to
choose such a concise term as "relativity" to
describe the mind-warping theories he proposed !

... which, i hasten to add, have been
confirmed pretty far beyond doubt.

-monz

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >
> > > So was "unison vector", but I find that phrase
> > > simply unacceptable, since it suggests two things,
> > > both of which are false--one, that we are talking
> > > about a unison, and the other, that we are talking
> > > about a vector.
> >
> > It is a unison (by definition) and a vector in both
> > the n-tuple and vector-space senses.
>
> No, it is not a unison. It is mapped to 1 by some
> temperament, but being mapped to 1 and not equal to 1
> is, obviously, a form of not being equal to 1.

a unison-vector in the "pure" sense is an interval which
is *not* an actual unison. it is an interval with a real
pitch difference by however small or large an amount.
at least as often as not, it is a rational interval.

now for some reason i'll presume to speak with the
Voice of Authority ... (... booming canyon-type echo ...)

... with the disclaimer preamble:
"as i understand it, and particularly what i think
Fokker meant by it (but i could be partly or totally
wrong)" ...

the essential concepts embodied in the term
"unison-vector", are these:

- the sonic foundation from which a tuning system is derived,
is a theoretically infinite lattice of pitches, arranged
as coordinates along axes according to the prime-factorization
of the frequency-ratios of the pitches, where axes represent
primes, and steps along those axes represent increments
of the exponent of those prime-factors, postive or negative;

- a particular tuning system is a subset of that infinite
source lattice, and may be modeled as a geometric shape,
typcially a parallogram/parallelpiped;

- that shape is a periodicity-block, which is a cell
enclosing a certain area of the lattice and filled
with a certain number of pitches, and whose shape
may be repeated ad infinitum to fill the lattice;

- there is some interval between a note within the
periodicity-block and a note similar in pitch outside
the block, and this interval is smaller than any of
the intervals between the degrees of the tuning system
-- this small interval is the actual unison-vector.

- the unison-vector is responsible for the finity
of the tuning system, by linking the notes inside the
block to those related notes outside of it, acting
as a sort of "bridge";

- the prime-space vector (monzo) of this interval
also describes the shape of two parallel edges of
the periodicity-block on the lattice -- and the
angles of these vectors usually do *not* follow
the taxicab metric along the prime-axes, thus
the sloping angles of the parallelogram.

now venturing into my own terminology ...

there is also a specific type of unison-vector
called a "xenharmonic-bridge", which links a note
within the periodicity-block of a certain prime-space,
to one which lies not only in another "tiling" of
the periodicity-block, but also in a different prime-space.

i coined this term, but Fokker often invoked the
septimal-kleisma in precisely the way i describe here
-- to make singers become familiar with the 5-limit JI
"augmented-6th", then have them use it as a "target pitch"
when he notates a harmonic-7th in his compositions.
this was at least as early as his 1949 English book on
singing JI.

........ interval .......... ratio .. 2,3,5,7-monzo .. ~cents

5-limit "augmented-6th" ... 225:128 . [-7 2, 2 0> . 976.5374295
- "harmonic-7th" ............ 7:4 ... [-2 0, 0 1> . 968.8259065
----------------------- ... ------- . ----------- . ------------
septimal-kleisma .......... 225:224 = [-5 2, 2 -1> .. 7.711522991

the unison-vector has, within the past couple of years,
also been referred to as a "comma", a usage which i
personally do not encourage, because "comma" has already
had a millennia-long established meaning, as an interval
of about 12 to 35 cents.

a unison-vector certainly may be a "real" comma ...
for example, the syntonic-comma (ratio 81:80 =
2,3,5-monzo [-4 4, -1> = ~ 21.5 cents) definitely
does exist in theory in 5-limit JI.

in theory and in practice, a unison-vector may be
tempered-out ("vanish"), ignored, or deliberately
used as a valid scale degree (or "step"), either
to play around with a listener's expectations or
to secure exact JI with commatic drift.

a unison-vector's multiples, both positive and negative,
are all equivalent to it. thus they may be modeled
geometrically as a line cutting across the lattice.

the monzo describing the one, of the pair of unison-vectors
closest to the origin-point of the lattice, which has
a positive pitch-height (the other will always be
negative), is chosen to represent the whole line, and
is a "promo" (projective monzo).

if a promo vanishes in a temperament,
i propose to call that a "vapro".

> As an algebraist I reject the idea that an element of
> an abelian group is a vector "by definition". It is no
> such thing by *definition*. It only becomes a member
> of a vector space if you define a vector space and
> an inclusion mapping from the group into the vector space,
> and it can become a member of many different vector spaces
> in this way. For example, I can claim we must be talking
> about the vector space of 2-adic 3-tuples as the space in
> which 5-limit intervals live; it is *not* uniquely defined.
> By definition, a vector *is* a member of an abelian group,
> but that's the reverse of your claim.

Gene, you seem to feel rather strongly against using
"unison-vector", and i suppose this is a cogent argument
to support your case.

Fokker is the person who, AFAIK, invented the term.
from everything i know about the man, he was an erudite
physicist who certainly knew a good deal about math.
can you please tell us, after reading his papers, why
you think he used the term? when i see how he illustrates
it on a lattice, the name makes sense to me.

> > > Do we really need words for projective versions
> > > of monzos, vals, and their wedge products? I dunno.
> > > I don't know if it will help, but the added insight
> > > is hardly likely to hurt.
> >
> > It's already hurt. You and monz are using it like
> > secret knowledge.
>
> Eh? It's in his dictionary.

http://tonalsoft.com/enc/index2.htm?promo.htm

really, i don't see how you can say that, Carl.

there was a little confusion for a while because i
had proposed "um", then changed that to "vum", then
paul and i worked out "promo" but without the vanishing
aspect, and now i've made up "vapro" to replace "um/vum".

but i've had a detailed webpage explaining it all
thru this change-over. things will start to settle
down a bit now.

-monz

🔗Carl Lumma <ekin@lumma.org>

>So was "unison vector", but I find that
>> >phrase simply unacceptable, since it suggests two things, both of
>> >which are false--one, that we are talking about a unison, and the
>> >other, that we are talking about a vector.
>>
>> It is a unison (by definition) and a vector in both the n-tuple
>> and vector-space senses.
>
>No, it is not a unison. It is mapped to 1 by some temperament, but
>being mapped to 1 and not equal to 1 is, obviously, a form of not
>being equal to 1.

It is clearly meant to reflect the situation after mapping.

>As an algebraist I reject the idea that an element of an abelian group
>is a vector "by definition".

You're quoting me out of context! "By definition" applied to "unison".

>It is no such thing by *definition*. It
>only becomes a member of a vector space if you define a vector space
>and an inclusion mapping from the group into the vector space,

It seems there's at least one way to do this, as discussed in the
previous thread.

>> >Do we really need words for projective versions of monzos, vals, and
>> >their wedge products? I dunno. I don't know if it will help, but the
>> >added insight is hardly likely to hurt.
>>
>> It's already hurt. You and monz are using it like secret knowledge.
>
>Eh? It's in his dictionary.

And, as I already asked: What in blazes is a "projective monzo"?

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> Fokker is the person who, AFAIK, invented the term.
> from everything i know about the man, he was an erudite
> physicist who certainly knew a good deal about math.
> can you please tell us, after reading his papers, why
> you think he used the term? when i see how he illustrates
> it on a lattice, the name makes sense to me.

If he assumes the lattice point of view to start out with, then they
*are* vectors. Anyway, as you point out, he is a physicist, and my
problem is in part that as an algebraist I am trained to a different
point of view. I've seen algebraists become upset when other
mathematicians assume an algebraic integer is by definition a complex
number, because in algebraic number theory it is important to know
that it might not be.

If I consider the p-limit monzos to be a group, I can ask what their
homomorphisms to the integers are, and I end up with vals. From there,
I find I can talk about equal temperaments, linear temperaments, etc
in a natural way. If I merely take them to be elements of a real
vector space and ask for homomorphisms to the reals, I get tuning
maps. These tuning maps are not ordinarily vals, and the kernel of the
mapping gives us "commas" which are not ordinarily rational numbers. I
no longer have a natural mathematical language in which to talk about
temperaments. Hence for some purposes we clearly want to consider the
p-limit to be a group, not a vector space. It certainly makes sense to
embed the monzos in a vector space and look at the resulting lattice,
but from a algebraic point of view it is a bad idea to start off by
assuming that is all they are; it is important to look at them as an
abelian group in the category of abelian groups and abelian group
homomorphims, and not attempt to frame everything in the language of
the category of real vector spaces and real vector space homomorphisms.

And I don't know how much of that makes any sense. However, I think
the math is simply slicker and more accurate if you start out by
saying the p-limit is a *group*, but is NOT in a vector space unless
and until you construct an embedding into a vector space. That's the
technically correct way to do it, and it makes things make more sense
in certain important respects.

🔗Kurt Bigler <kkb@breathsense.com>

on 8/14/04 12:34 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
>> Fokker is the person who, AFAIK, invented the term.
>> from everything i know about the man, he was an erudite
>> physicist who certainly knew a good deal about math.
>> can you please tell us, after reading his papers, why
>> you think he used the term? when i see how he illustrates
>> it on a lattice, the name makes sense to me.
>
> If he assumes the lattice point of view to start out with, then they
> *are* vectors. Anyway, as you point out, he is a physicist, and my
> problem is in part that as an algebraist I am trained to a different
> point of view. I've seen algebraists become upset when other
> mathematicians assume an algebraic integer is by definition a complex
> number, because in algebraic number theory it is important to know
> that it might not be.
>
> If I consider the p-limit monzos to be a group, I can ask what their
> homomorphisms to the integers are, and I end up with vals. From there,
> I find I can talk about equal temperaments, linear temperaments, etc
> in a natural way. If I merely take them to be elements of a real
> vector space

Is that the reason for *your* objection to the user of the term "vector"?
Because a vector space is taken to be a real vector space rather than an
integral one?

> and ask for homomorphisms to the reals, I get tuning
> maps. These tuning maps are not ordinarily vals, and the kernel of the
> mapping gives us "commas" which are not ordinarily rational numbers. I
> no longer have a natural mathematical language in which to talk about
> temperaments. Hence for some purposes we clearly want to consider the
> p-limit to be a group, not a vector space. It certainly makes sense to
> embed the monzos in a vector space and look at the resulting lattice,
> but from a algebraic point of view it is a bad idea to start off by
> assuming that is all they are; it is important to look at them as an
> abelian group in the category of abelian groups and abelian group
> homomorphims, and not attempt to frame everything in the language of
> the category of real vector spaces and real vector space homomorphisms.
>
> And I don't know how much of that makes any sense. However, I think
> the math is simply slicker and more accurate if you start out by
> saying the p-limit is a *group*, but is NOT in a vector space unless
> and until you construct an embedding into a vector space.

So not knowing much of this theory (but having been exposed to a *little* of
it) I'm making some guesses here. Vector implies vector space. Vector
space implies attributes that groups do not imply?

Yet monzo's are clearly n-tuples. My impression was that a vector space
does not imply the ability to have n-tuples, not until you have a basis set
or some other ability to make unambiguous n-dimensional measurements.
However I did think that an n-tuple implied the existence of a space.
Depending on the restrictions applied to the elements of the n-tuple you
have a different kind of space, e.g. real or integral. Monzo n-tuples imply
the existence of what I intuitively call a space, simply the space occupied
by the conceivable values of the n-tuples. In the case of a monzo in the
usual sense, it is a space that can be mapped (?) by a set of n integers.

The ability to have a vector is then a separate property from the existence
of a space? Does a vector imply the existence of *relative* measure within
a space, i.e. requires that another space exists which relates to the first
space as offsets relates to positions? Or can you have a vector system
based on an assumed single origin and avoid the issue of relative position?

-Kurt

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Is that the reason for *your* objection to the user of the term
"vector"?
> Because a vector space is taken to be a real vector space rather than an
> integral one?

The word is "module". Yes, my problem with it is that we want to began
by talking about Z-modules, also known as abelian groups, before we
talk about vector spaces over either the reals or the rationals. By
definition, a vector space is over a field (or sometimes a skew field)
not just a ring.

> So not knowing much of this theory (but having been exposed to a
*little* of
> it) I'm making some guesses here. Vector implies vector space. Vector
> space implies attributes that groups do not imply?

Exactly. It adds some extra structure, and for some purposes this is
not a good thing.

> Yet monzo's are clearly n-tuples. My impression was that a vector space
> does not imply the ability to have n-tuples, not until you have a
basis set
> or some other ability to make unambiguous n-dimensional measurements.

Any finite-dimensional vector space is isomorphic to one you can
notate using n-tuples.

> However I did think that an n-tuple implied the existence of a space.

Why? And which space?

> Depending on the restrictions applied to the elements of the n-tuple you
> have a different kind of space, e.g. real or integral. Monzo
n-tuples imply
> the existence of what I intuitively call a space, simply the space
occupied
> by the conceivable values of the n-tuples. In the case of a monzo
in the
> usual sense, it is a space that can be mapped (?) by a set of n
integers.

Why do n-tuples of integers make up a space?

> The ability to have a vector is then a separate property from the
existence
> of a space? Does a vector imply the existence of *relative* measure
within
> a space, i.e. requires that another space exists which relates to
the first
> space as offsets relates to positions? Or can you have a vector system
> based on an assumed single origin and avoid the issue of relative
position?

Vector spaces by definition have a zero element; so do abelian groups.

🔗Kurt Bigler <kkb@breathsense.com>

on 8/14/04 3:29 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> Is that the reason for *your* objection to the user of the term
> "vector"?
>> Because a vector space is taken to be a real vector space rather than an
>> integral one?
>
> The word is "module".

So you wouldn't call a module a "modular space" I guess, because this would
contradict some requirement that has been established for the term "space"?
Wikipedia didn't really define space very restrictively but left it to
qualified types of spaces to have significant defintions.

> Yes, my problem with it is that we want to began
> by talking about Z-modules, also known as abelian groups, before we
> talk about vector spaces over either the reals or the rationals. By
> definition, a vector space is over a field (or sometimes a skew field)
> not just a ring.

Ah, this *feels* unfortunate, but perhaps only because my intuitive sense of
a vector does not seem to be inapplicable to a ring. This is probably
because I think of a vector as if it is in a field anyway with the ring
being a convenient subset.

Is there an x such that x is to ring as vector is to field?

>> So not knowing much of this theory (but having been exposed to a
> *little* of
>> it) I'm making some guesses here. Vector implies vector space. Vector
>> space implies attributes that groups do not imply?
>
> Exactly. It adds some extra structure, and for some purposes this is
> not a good thing.

You indicated the distinction has to do with mathematical clarity. Do you
have examples in mind of where it would actually be wrong rather than just
less clear? Or is it because new math is being created here attached to an
application and that work will not be directly available to the mathematical
community because it becamse polluted with unnecessary assumptions?

>> Yet monzo's are clearly n-tuples. My impression was that a vector space
>> does not imply the ability to have n-tuples, not until you have a
> basis set
>> or some other ability to make unambiguous n-dimensional measurements.
>
> Any finite-dimensional vector space is isomorphic to one you can
> notate using n-tuples.

Yes I think my point was that you have to define the isomorphism, it is not
a given.

>> However I did think that an n-tuple implied the existence of a space.
>
> Why? And which space?

I was using the word space intuitively. You corrected me. I actually meant
a ring which I visualize spatially. But we are an applied group here and as
such it seems appropriate to allow a language which relates to the
intuitions and distinctions that are substantive here. The fact that we
don't have a mathematical space doesn't mean we shouldn't use the word space
to refer to that which is spatial for us.

> Why do n-tuples of integers make up a space?

Now you get it I think. In my mind.

> Vector spaces by definition have a zero element; so do abelian groups.

Yes I guess I was thinking of a set for which subtraction was defined even
though the "difference" would not be in the same set and thus not require
zero to be defined for that set. Maybe you wouldn't call that subraction.
But this is far off-topic now.

-Kurt

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 8/14/04 3:29 PM, Gene Ward Smith <gwsmith@s...> wrote:

> > The word is "module".
>
> So you wouldn't call a module a "modular space" I guess, because
this would
> contradict some requirement that has been established for the term
"space"?

If something was called a space you'd expect it to have some kind of
added structure giving it a geometric, space-like aspect in some
general sense. You could, for example, use the cofinite topology, in
which the closed sets are either the finite sets or all of the points,
to add a topology and then call it a space. But why would you?

> Wikipedia didn't really define space very restrictively but left it to
> qualified types of spaces to have significant defintions.

I see that what it says is "In mathematics, a space is a set, usually
with some additional structure. For examples, see Euclidean space,
vector space, normed vector space, affine space, projective space,
Banach space, inner product space, Hilbert space, topological space,
uniform space, metric space, probability space." This is pretty much
what I said, but note that we are actually kind of fussy about what
sort of extra structure it has before we call it a "space".

> Is there an x such that x is to ring as vector is to field?

"Module element".

> You indicated the distinction has to do with mathematical clarity.
Do you
> have examples in mind of where it would actually be wrong rather
than just
> less clear?

As I pointed out, the category (the objects of a theory together with
the collection of mappings between them) in which we discuss the
p-limit should be the category of abelian groups, and needs to be
defined in terms of abelian groups and abelian group homomorphisms;
otherwise we end up with the wrong category and inapplicable concepts.

> > Any finite-dimensional vector space is isomorphic to one you can
> > notate using n-tuples.
>
> Yes I think my point was that you have to define the isomorphism, it
is not
> a given.

True enough. The fundamental theorem of arithmetic however makes the
primes seem like an obvious choice. In musical terms we are then
operating using only overtones to define things, and it seems to lead
to musically natural results. If I represent the 7-limit in terms of
6/5, 7/5, 3/2 and 5/3, I can describe subgroups in terms of 6/5, 7/5,
3/2, etc, but results do not seem as natural as using primes.

🔗Kurt Bigler <kkb@breathsense.com>

on 8/14/04 4:59 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> If something was called a space you'd expect it to have some kind of
> added structure giving it a geometric, space-like aspect in some
> general sense. You could, for example, use the cofinite topology, in
> which the closed sets are either the finite sets or all of the points,
> to add a topology and then call it a space. But why would you?

Well personally I wouldn't need topology to make what is intuitively spatial
(even though integral) to me to have a "space-like aspect". But in math,
space-likeness seems to depend on the availability of some kind of internal
in-between-ness (e.g. involving division) which what I would have called an
"integral space" does not have.

>> You indicated the distinction has to do with mathematical clarity.
> Do you
>> have examples in mind of where it would actually be wrong rather
> than just
>> less clear?
>
> As I pointed out, the category (the objects of a theory together with
> the collection of mappings between them) in which we discuss the
> p-limit should be the category of abelian groups, and needs to be
> defined in terms of abelian groups and abelian group homomorphisms;
> otherwise we end up with the wrong category and inapplicable concepts.

Ok, going back to your original statement:

on 8/14/04 12:34 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:
> However, I think the math is simply slicker and more accurate if you start out
> by saying the p-limit is a *group*, but is NOT in a vector space unless and
> until you construct an embedding into a vector space.

Are you saying that there is not "generlized" kind of embedding that would
always apply to tuning theory and could be considered to be "always there"
and allow some fluidity of language within this applied field? It seems to
me the embedding does not deny the abelian group properties and in some
sense the objects we work with cross the categories regularly. Is there no
way to deal with that? That's the clearest way I can now state what I was
originally trying to ask given the explanation you've provided.

>>> Any finite-dimensional vector space is isomorphic to one you can
>>> notate using n-tuples.
>>
>> Yes I think my point was that you have to define the isomorphism, it
> is not
>> a given.
>
> True enough. The fundamental theorem of arithmetic however makes the
> primes seem like an obvious choice. In musical terms we are then
> operating using only overtones to define things, and it seems to lead
> to musically natural results. If I represent the 7-limit in terms of
> 6/5, 7/5, 3/2 and 5/3, I can describe subgroups in terms of 6/5, 7/5,
> 3/2, etc, but results do not seem as natural as using primes.

On the other hand the ratios that form a given lattice also seem like an
obvious choice, e.g. possibly 3/2 and 5/4 for the 5-limit.

-Kurt

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> on 8/14/04 12:34 PM, Gene Ward Smith <gwsmith@s...> wrote:
> > However, I think the math is simply slicker and more accurate if
you start out
> > by saying the p-limit is a *group*, but is NOT in a vector space
unless and
> > until you construct an embedding into a vector space.
>
> Are you saying that there is not "generlized" kind of embedding that
would
> always apply to tuning theory and could be considered to be "always
there"
> and allow some fluidity of language within this applied field?

There's a natural embedding into the corresponding vector space over
the rational numbers which is always there. The rational numbers
themselves, and their vector spaces, have different types of
embeddings, but the most familiar and useful to us is into the real
numbers and real vector spaces. These embeddings are natural and
easily defined--we obviously have no problem thinking about integers
as a kind of real number. However, my point is that automatically
thinking in terms of the embedding does not always seem like a good idea.

It seems to
> me the embedding does not deny the abelian group properties and in some
> sense the objects we work with cross the categories regularly.

It's completely standard to do that sort of thing; my objection is
that it is also standard to simply be able to forget about any
embeddings. It's far from clear to me that people are doing this, or
are ready to do it, if they insist on always thinking in terms of vectors.

> On the other hand the ratios that form a given lattice also seem like an
> obvious choice, e.g. possibly 3/2 and 5/4 for the 5-limit.

There is no such thing as the ratios that form a given lattice in any
unique sense; there is such a thing as a lattice basis, but it is not
unique. So, indeed, you can define a lattice (assuming you have a norm
or at least a topology) using 3/2 and 5/4, but you could also use 3/2
and 6/5, or 6/5 and 5/4, etc.