Yahoo Tuning Groups Ultimate Backup tuning webpage update: multimonzo

hi Gene,

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

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > try this latest version, complete with illustrative
> > examples, and its related links:
> >
> > http://tonalsoft.com/enc/multimonzo.htm
>
> You write "The enclosing brackets [[ >> are for
> 3-dimensional space and describe a bimonzo; [[[ >>> would
> describe a 4-dimensional trimonzo, etc." In fact, it's
> not a matter of the dimension of the space, but of how
> many monzos are being wedged togther. ||>> is two
> monzos, and so forth.

oops! ... did i leave that in the multimonzo definition?

right, i understand how it works now, and fixed it on
most (i think) of the related pages ... i just forgot
to fix it on this one. will do.

> What does Graham think of this "breed" business--Graham?
> I don't see why people want to change the name.

in the past Paul used a similar argument against "monzo",
insisting on calling it simply "vector". i went with
"val" as my primary choice until today, simply because
you're the one using it all the time, and that's the
word you always use.

i'm still flexible about this, and willing to change
the definitions back to "val" again ... but since i
obviously would like to keep using "monzo", i thought
it nice to use "breed" for the complement. having
met Graham in real life, i think we "complement"
each other well.

;-)

-monz

🔗monz <monz@tonalsoft.com>

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

> hi Gene,
>
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > > try this latest version, complete with illustrative
> > > examples, and its related links:
> > >
> > > http://tonalsoft.com/enc/multimonzo.htm
> >
> > You write "The enclosing brackets [[ >> are for
> > 3-dimensional space and describe a bimonzo; [[[ >>> would
> > describe a 4-dimensional trimonzo, etc." In fact, it's
> > not a matter of the dimension of the space, but of how
> > many monzos are being wedged togther. ||>> is two
> > monzos, and so forth.
>
>
>
> oops! ... did i leave that in the multimonzo definition?
>
> right, i understand how it works now, and fixed it on
> most (i think) of the related pages ... i just forgot
> to fix it on this one. will do.

OK, i've fixed that, and also cleaned up a few other
things and added some more stuff.

please "reload" and tell me if it's OK. i'm particularly
unsure about saying "the 1-dimensional set is equivalent to
the simple monzo" ... perhaps it should be "the 1-dimensional
multimonzo is simply the regular monzo" ...?

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

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

🔗monz <monz@tonalsoft.com>

hi Gene,

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

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > please "reload" and tell me if it's OK. i'm particularly
> > unsure about saying "the 1-dimensional set is equivalent to
> > the simple monzo" ... perhaps it should be "the 1-dimensional
> > multimonzo is simply the regular monzo" ...?
>
> It's got two problems--one is that it isn't a set, and the
> other is that it isn't 1-dimensional.

but it is! ... in the case where the interval described
by the monzo is tempered out.

in that case, each monzo represents not only the
interval indicated by the exponents, but also all
multiples and submultiples of it. i.e., [ -4 4, -1 >
represents not only the syntonic comma 81/80, but
rather makes all of these equivalent:

etc.
[-4 4, -1> * 3 = [-12 12, -3>
[-4 4, -1> * 2 = [-8 8, -2>
[-4 4, -1> * 1 = [-4 4, -1>
[-4 4, -1> * 0 = [0 0, 0>
[-4 4, -1> *-1 = [4 -4, 1>
[-4 4, -1> *-2 = [8 -8, 2>
[-4 4, -1> *-3 = [12 -12, -3>
etc.

so in the case of a vanishing comma, the monzo
describes not just a point in tone-space, but rather
a line.

of course, this does not apply if the monzo
describes an interval that does not vanish.
then it only represents that particular interval.
so my webpage should say this.

> I'd call it a "list" (keeping things
> nontechnical) and scrap the dimension stuff.

the dimensionality seems important, because that's
what got me confused. i thought that *monzo and *val,
where * is a numerical prefix, were always complements.
but i see that in 3 dimensions that's not the case.

-monz

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

> > It's got two problems--one is that it isn't a set, and the
> > other is that it isn't 1-dimensional.
>
>
> but it is! ... in the case where the interval described
> by the monzo is tempered out.
>
> in that case, each monzo represents not only the
> interval indicated by the exponents, but also all
> multiples and submultiples of it. i.e., [ -4 4, -1 >
> represents not only the syntonic comma 81/80, but
> rather makes all of these equivalent:

In math terms you seem to be saying it represents a point in the
projective plane, but that really makes sense if you are concerned
only with the temperament it gives, not with the interval itself. You
need as its primary meaning that it represents the exact interval
81/80. You could write eg |-4; 4; -1> to indicate it is supposed to be
homogenous coordinates for a projective point, so that only the ratios
-4:4, -4:-1, 4:-1 actually matter. In general wedgies for temperaments
can be thought of as projective.

In case you are wondering why I am calling this a line and not a
point, a line through the origin in n dimensional space is a point in
the projective space of dimension n-1.

> so in the case of a vanishing comma, the monzo
> describes not just a point in tone-space, but rather
> a line.

You can't have the same notation for both without sowing confusion.

> > I'd call it a "list" (keeping things
> > nontechnical) and scrap the dimension stuff.
>
> the dimensionality seems important, because that's
> what got me confused. i thought that *monzo and *val,
> where * is a numerical prefix, were always complements.
> but i see that in 3 dimensions that's not the case.

No, if you are in the p-limit, where p is the nth prime, then an a-val
and a b-monzo may be complementary if a+b=n. I am not sure you mean by
"dimensions" so I don't think the readers of your web page will know
either.

🔗monz <monz@tonalsoft.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Robert Walker <robertwalker@ntlworld.com>

Hi Monz,

Might vum be better? (or vuv).

I understand monzo as meaning a row vector written
in the [ ..> form and it seems helpful to have a word
for that, primarily referring to a form of notation
for a vector.

A unison monzo then should be the same as a unison
vector except for notation, and a vanishing unison
monzo then needs extra qualification, I'd have thought
- it seems confusing to distinguish a unison monzo
from a unison vector on the basis of whether the
vector vanishes.

On using names for terms, its common in maths
to do that but usually as an adjective as
in Reimann space, or Gaussian, or Cantor's dust
or abelian group, Clifford algebra, Euler's constant...
etc. I can't think of an example of a name
in maths used as a noun right now. Can
anyone think of an example?

If that's so, really it should be a Monzian vector and
a Breedian prime mapping or primes evaluation
I suppose, or Monzian or Breedian for short
if one wanted to fit in with what seems to be
established mathematical practice - not
saying one needs to do that.

Using a name as a noun makes it sounds
a bit like astronomy to me :-).
The multimonzo asteroid or planet or something.
Multimonzian sounds more mathematical to me.
Just an observation. I suppose it makes
it seem more like a science or applied
subject and less like a pure theory
subject, which one may want. I'm not
trying to change the notation system.

Robert

🔗monz <monz@tonalsoft.com>

hi Carl and Gene,

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

> --- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
>
> > >also, readers should notice yet another new term:
> > >"um". we were saying "vanishing comma" or
> > >"vanishing vector" or "vanishing monzo" so much
> > >that we decided that a new short term was called for.
> >
> > Oh dear. What's wrong with "uv"? We've been through
> > all this before. Mathematicians don't own the word
> > "vector". Fokker used it, so can we.
>
> Mathematicians know what they mean when they use the
> word, which is very helpful in many cases. If you don't
> have precise definitions in mind, your meaning will
> necessarily be more context depended and likely be
> downright murky.
>
> Anyway, "unison vector" is awkward and verbose.
> I like "comma".

first: "comma" used to refer more specifically to
an interval about 1/8-tone in pitch-size, ~25 cents.
then it started being used to designate all sorts of
intervals <~120 cents, which i really objected to.

"anomaly" was something i already had in the Dictionary
to designate the more general sense of a small interval
which may be ignored. i had hoped that others would use it.

second: "um" and its multi-relatives refers specifically
to "commas" which **vanish**.

Fokker's use of the term "unison-vector" certainly
found its practical application in his adoption of
31edo, but AFAIK his explanations of it were always
in terms of JI, where the unison-vector is assumed
to be imperceptible, and thus may be ignored ... but
is still *present*. in temperaments, certain ones
do vanish.

as i explained, we got so tired of having to keep typing
"vanishing" that we just wanted one short word to
represent the concept. Gene, you should appreciate
this, with your preference for "precise definitions".

anyway, as far as i'm concerned the work i've done
on these two webpages makes it firmly established
now. ;-)

http://tonalsoft.com/enc/um.htm

http://tonalsoft.com/enc/bium.htm

and besides, "um" is even less verbose than "comma".

:^P

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...> wrote:
> Hi Monz,
>
> Might vum be better? (or vuv).
>
> I understand monzo as meaning a row vector written
> in the [ ..> form and it seems helpful to have a word
> for that, primarily referring to a form of notation
> for a vector.

I came up with the word, at first merely for my own private use, and I
did not intend for it to simply mean a row vector. The coefficients
are supposed to be exponents of successive prime numbers.

> A unison monzo then should be the same as a unison
> vector except for notation, and a vanishing unison
> monzo then needs extra qualification, I'd have thought
> - it seems confusing to distinguish a unison monzo
> from a unison vector on the basis of whether the
> vector vanishes.

A monzo of the form |0 ... 0> represents 1, and this is the only monzo
which is a unison monzo so far as I can see. The point of commas is
not that they are unisons, but that some set of vals maps them to unisons.

> On using names for terms, its common in maths
> to do that but usually as an adjective as
> in Reimann space, or Gaussian, or Cantor's dust
> or abelian group, Clifford algebra, Euler's constant...
> etc. I can't think of an example of a name
> in maths used as a noun right now. Can
> anyone think of an example?

So we are breaking new ground here--look at how much easier it would
be to say "a cube is a Plato" rather than "a cube is a Platonic solid."

> If that's so, really it should be a Monzian vector and
> a Breedian prime mapping or primes evaluation
> I suppose, or Monzian or Breedian for short
> if one wanted to fit in with what seems to be
> established mathematical practice - not
> saying one needs to do that.

Monzian vector is ugly and verbose and assumes you want to talk about
vectors rather than abelian groups.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> first: "comma" used to refer more specifically to
> an interval about 1/8-tone in pitch-size, ~25 cents.
> then it started being used to designate all sorts of
> intervals <~120 cents, which i really objected to.

You need a word which means "interval tempered out by a temperament",
and "comma" has been serving as that.

> second: "um" and its multi-relatives refers specifically
> to "commas" which **vanish**.

Which is what the word "comma" is mostly used for in theory
discussions. Are you proposing "um" as a word to mean "interval
tempered out by a temperament"?

> as i explained, we got so tired of having to keep typing
> "vanishing" that we just wanted one short word to
> represent the concept. Gene, you should appreciate
> this, with your preference for "precise definitions".

I always mean that it vanishes when I say "comma" unless I am
referring to a specific interval of 81/80, 3^12/2^19, or 64/63, or am
referring to a size or using a size measure ("1/4 comma"). I don't
neccessarily mean that it is small, or that it is larger than 1, or
anything else, but I do mean it vanishes. If people are not assuming
that meaning then maybe we need to umm.

🔗monz <monz@tonalsoft.com>

hi Gene,

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

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > first: "comma" used to refer more specifically to
> > an interval about 1/8-tone in pitch-size, ~25 cents.
> > then it started being used to designate all sorts of
> > intervals <~120 cents, which i really objected to.
>
> You need a word which means "interval tempered out by
> a temperament", and "comma" has been serving as that.
>
> > second: "um" and its multi-relatives refers specifically
> > to "commas" which **vanish**.
>
> Which is what the word "comma" is mostly used for in theory
> discussions. Are you proposing "um" as a word to mean "interval
> tempered out by a temperament"?

exactly. have you read the "um" and "bium"
Encyclopaedia webpages?

> > as i explained, we got so tired of having to keep typing
> > "vanishing" that we just wanted one short word to
> > represent the concept. Gene, you should appreciate
> > this, with your preference for "precise definitions".
>
> I always mean that it vanishes when I say "comma" unless
> I am referring to a specific interval of 81/80, 3^12/2^19,
> or 64/63, or am referring to a size or using a size measure
> ("1/4 comma"). I don't neccessarily mean that it is small,
> or that it is larger than 1, or anything else, but I do mean
> it vanishes. If people are not assuming that meaning then
> maybe we need to umm.

i think that lately most people have been assuming
"vanishing" to be part of the definition of "comma",
but that was never made explicit.

as the self-appointed :-) lexicographer of the tuning
community, i prefer to acknowledge that "comma" has a
very long history of referring to a small interval
that is *not* tempered-out ... the word was common
currency in tuning-theory centuries before anyone
wrote anything about temperament.

i'm already unhappy that "comma" has already (and only
fairly recently) been generalized more than it ever
was before to refer to many more and much wider intervals.

it seems much better to me to have a new word to
refer to the cases where small intervals actually
do vanish. and in fact, it's not just one word, but
a whole series of words (um, bium, trium, etc.)
which are logically arranged to reflect certain
mathematical properties.

-monz

🔗Robert Walker <robertwalker@ntlworld.com>

Hi Gene,

> I came up with the word, at first merely for my own private use, and I
> did not intend for it to simply mean a row vector. The coefficients
> are supposed to be exponents of successive prime numbers.

Rightio, fair enough. Actually I've got it defined like that in
the dictionary entry that I posted here now that I look at it.

But it is a notational thing partly too isn't it
- a row vector written as {4,4,-1} wouldn't be a monzo would
it - even if used to define a vector in a Tenney lattice?

Surely not - as it has been called just a periodicity
block vector before hasn't it... and it isn't a new concept, rather
quite an old one - seems that what is new is the bra-ket notation
and the way it is used with the vals notation and all that
context to it.

> A monzo of the form |0 ... 0> represents 1, and this is the only monzo
> which is a unison monzo so far as I can see. The point of commas is
> not that they are unisons, but that some set of vals maps them to unisons.

Surely that's an identity monzo, not a unison monzo?

So far, I have understand a unison interval to mean an interval that is
used to induce an equivalence relation. So it doesn't have to map intervals
to identical intervals, just to ones that the user has chosen
to treat as equivalent under the periodicity block mapping.

In terms of your val notation, a unison interval (vector, monzo)
is a generating element of the kernel of the map defined by the val - or a set
of generating elements of the kernel if it is multi-dimensional.

Non techy aside - the kernel of a map is the set of all
the things that get mapped to the identity (1 or 0 whichever it is)
So here, it is the set of all monzos that are evaluated as 0 scale degrees
by the val. Then the generators of it would be a set of elements
chosen so that all elements of the kernel can be obtaned
from them.

So the kernel of the val for 12-et would be generated
by e.g. the monzos for 81/80 and 128/125 - but not
by their squares or cubes.

In terms of periodicity blocks, the kernel of the val
just means the set of all the vectors generated by
the unison intervals closing them under addition.
so as vectors the kernel is: a*<-4 4, 1> + b*[7 0, 5>
AS intervals: (81/80)^a + (128/125) ^ b

> So we are breaking new ground here--look at how much easier it would
> be to say "a cube is a Plato" rather than "a cube is a Platonic solid."

Well that's a matter of taste really.
Here it seems already to have become established usage
so that's okay and this isn't pure maths though
uses a lot of pure maths so we can make up our
own rules. You do use proper names like that in science.

I'd object to it in a pure mathematical context though,
if no-one can come up wit ha precedent particularly - here
for Platonic solids, because Plato was a philosopher
and not a cube, and Platonic solid is a familiar word
- it sounds strange, and there is some value in
keeping to convention even if slightly more verbose,
if it avoids confusion amongst those who are familiar
with the convention and if it doesn't propogate
wrong ideas or sweep necessary distinctions under
the carpet as it were. I.e. you can break the established
convention in a field, but there needs to be a good reason
for doing it, rather than just personal inclination or whim
- at least if one wants to be understood easily.
If one wants to spread confusion of course one can do
whatever one likes :-).

> > If that's so, really it should be a Monzian vector and
> > a Breedian prime mapping or primes evaluation
> > I suppose, or Monzian or Breedian for short
> > if one wanted to fit in with what seems to be
> > established mathematical practice - not
> > saying one needs to do that.

> Monzian vector is ugly and verbose and assumes you want to talk about
> vectors rather than abelian groups.

Well, don't know about ugly, sounds fine to me but
that is a matter of taste and familiarity.
To discuss it more would be like
discussing whether one should like
Rhubarb crumble. Maybe some might think
the opposite, that monzo on its own as
a mathematical term doesn't sound
so good as monzian. But everyone is entitled
to their personal opinion on that surely.

Anyway, on what it means,
I've been understanding it as a vector
certainly, and a bimonzo as a bivector
(wedge product of two vectors).
What else is it? It's not an abelian
group, maybe you regard it as essentially
a member of an abelian group - but then
vectors form an abelian group under addition
with some additional structure to it,
(multiplication by a scalar, also
vector multiplication if you add that)
and it is fine to talk about them
as vectors rather than abelian group
elements - so I'm not sure I understand
your point here.

Robert

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...> wrote:
> Hi Gene,
>
> > I came up with the word, at first merely for my own private use, and I
> > did not intend for it to simply mean a row vector. The coefficients
> > are supposed to be exponents of successive prime numbers.
>
> Rightio, fair enough. Actually I've got it defined like that in
> the dictionary entry that I posted here now that I look at it.
>
> But it is a notational thing partly too isn't it
> - a row vector written as {4,4,-1} wouldn't be a monzo would
> it - even if used to define a vector in a Tenney lattice?

It is a notational thing but I would call that a monzo if it was
supposed to mean 1296/5; however I would not call 1296/5 a monzo.

> Surely not - as it has been called just a periodicity
> block vector before hasn't it... and it isn't a new concept, rather
> quite an old one - seems that what is new is the bra-ket notation
> and the way it is used with the vals notation and all that
> context to it.
>
> > A monzo of the form |0 ... 0> represents 1, and this is the only monzo
> > which is a unison monzo so far as I can see. The point of commas is
> > not that they are unisons, but that some set of vals maps them to
unisons.
>
> Surely that's an identity monzo, not a unison monzo?

It's the monzo for 1, which is the unison, so why isn't it the unison
monzo?

> So far, I have understand a unison interval to mean an interval that is
> used to induce an equivalence relation. So it doesn't have to map
intervals
> to identical intervals, just to ones that the user has chosen
> to treat as equivalent under the periodicity block mapping.

That's what I have been calling a comma, though usually not when
talking about periodicity blocks, but when discussing temperaments.

> In terms of your val notation, a unison interval (vector, monzo)
> is a generating element of the kernel of the map defined by the val
- or a set
> of generating elements of the kernel if it is multi-dimensional.

A member of a generating set is what Monz means by "um". A member of
the kernel is what I've been using "comma" to mean, which of course
means they may be huge.

> Non techy aside - the kernel of a map is the set of all
> the things that get mapped to the identity (1 or 0 whichever it is)
> So here, it is the set of all monzos that are evaluated as 0 scale
degrees
> by the val. Then the generators of it would be a set of elements
> chosen so that all elements of the kernel can be obtaned
> from them.

Or mapped to 0 by a temperament, which would mean by every val in some
set of vals, or which could be defined in terms of wedge products.

> Anyway, on what it means,
> I've been understanding it as a vector
> certainly, and a bimonzo as a bivector
> (wedge product of two vectors).
> What else is it?

It's an element of a group which is free and finitely generated is
what else it is; the generators being the primes in some p-limit. The
coefficients of a monzo, in other words, should be integers. We can
extend this to a vector space over Q but I don't think that should be
the primary meaning. This way the vals are homomorphisms to Z, not
linear functionals to Q, which means they are composed of Q-valuations
and not valuations on a field of algebaic numbers. This is, after all,
what we want if we are going to interpret temperaments as
approximations to just intervals.

🔗Robert Walker <robertwalker@ntlworld.com>

Hi Gene,

> It's the monzo for 1, which is the unison, so why isn't it the unison
> monzo?

Just established convention that's all. The periodicity block vectors
have been called unison intervals, so unison in this context
seems to mean equivalence under the mapping induced
by the vectors. Either that or redefine unison vector, or make
it so unison vector and unison monzo use different ideas of
unison. But it is a bit late to redefine unison vector and
seems okay to me understood as an equivalence relation.

> It's an element of a group which is free and finitely generated is
> what else it is; the generators being the primes in some p-limit. The
> coefficients of a monzo, in other words, should be integers. We can
> extend this to a vector space over Q but I don't think that should be
> the primary meaning. This way the vals are homomorphisms to Z, not
> linear functionals to Q, which means they are composed of Q-valuations
> and not valuations on a field of algebaic numbers. This is, after all,
> what we want if we are going to interpret temperaments as
> approximations to just intervals.

Yes - but it is still a vector. The group is a free finitely generated
group indeed (non techy note - free here just means that there
are no additional equivalence type relationships between the elements
added to reduce the number of elements on the group),
but its elements are vectors. Just as the elements of
a symmetry group are symmetries.

Seems strange on first sight to extend it to rational powers of a prime.
But I suppose any number that can be written in the form
2^a*3^b*5^c*... for a b c rational has a unique
representation in that form (if two were identical
then you would have some expression of that form
equal to 1 which is easily seen to be impossible
by multiplying out and unique factorisation),

So it is consistent to do that and well formed.
Obviously relevant for ets as e.g. 2^1/12 etc.

Well it makes sense mathematically anyway.
Though having a dense field instead of a
discrete lattice makes it rather different
from what one is used to. I wonder what
it would be useful for.

I see what you mean though - calling them vectors
implies to a mathematician that they
need to be elements of a vector space
which is fair enough
- and the lattice isn't a vector space
because the scalar multiplication
is by integers and they don't form a field.

Putting that in plain english, for them
to be vectors, it must make sense to talk
about a vector of half the length of a lattice
unit vector such as 2^1, 3^1 or whatever as well
as double the length, and a third of the length.
of any vector in the lattice and so on.

However, I think that does make sense.

One way to think of it is that
the lattice is embedded in
the pitch space of not just rational
multiples but even of real multiples
of the generating vectors.

2^pi * 3^e *5^rho makes
sense as a way of defining an interva;
- one could define an interval that
way if one wanted to - doesn't have to be a unique
representation of an interval.
Just means that the map from
the intervals to the lattice is
now one to many - and from the
lattice back to intervals remains
many to one. That's okay.

It seems to be the underlying
implied vector space that the
lattice is embedded in and the
reason why it is mathematically
correct to refer to them as vectors.

Though the rational exponents type
vectors will also do as a
vector space to embed it in as well.

So I see no problem calling them
vectors and making that part of the
definition.

Robert

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> Yes - but it is still a vector.

Only if you embed it in the vector space over the field of fractions;
admittedly, with a ring of integers that's pretty much a given, since
the construction is natural.

> Well it makes sense mathematically anyway.
> Though having a dense field instead of a
> discrete lattice makes it rather different
> from what one is used to.

Now you see why I don't want to call them vectors unless they in fact
really *are* vectors.

I wonder what
> it would be useful for.

Monz finds them useful. It could be useful to know that the tetrad has
a centroid of 105^(1/4) perhaps, or to draw a line connecting 1/4
comma to 2/7 comma meantone, and stick meantones along it, though for
that you will probably be thinking real numbers.

> I see what you mean though - calling them vectors
> implies to a mathematician that they
> need to be elements of a vector space

Right, and immediately we find we don't know for certain what vector
space--the reals and the rationals seem like the two strong
contenders, and lead to different mathmatical descriptions of what is
involved. Moriever, we don't know if it is a normed vector space, and
if it is, what the norm is.

I would prefer to introduce vector spaces in the context of an
explcitly given embedding into a normed vector space, turning the
monzos into a lattice, such as what I've called Tenney space. In the
usual way this gets done around here, people say "lattice" but they
don't know a lattice in what, so they don't know exactly what they are
talking about--much of the time they could really be talking graph
theory. Then Paul and I start to throw things at each other, but
that's another story.

> One way to think of it is that
> the lattice is embedded in
> the pitch space of not just rational
> multiples but even of real multiples
> of the generating vectors.

The most obvious sense in which something you might call pitch space
could be involved--expressing pitch in terms of hertz or cents--gives
a dense embedding with no lattices in sight. "Pitch space" sounds like
an excellent phrase to avoid like the plague.

> 2^pi * 3^e *5^rho makes
> sense as a way of defining an interva;
> - one could define an interval that
> way if one wanted to - doesn't have to be a unique
> representation of an interval.

Well, as a matter of fact we've got Lucy Tuning, which has a meantone
fifth of 2^(1/2 + 1/(4*pi)), or 600+300/pi cents. Of course with
non-unique representatives you've opened up a can of worms, and you
shouldn't do that unless you have a use for the worms, such as lattices.

> Just means that the map from
> the intervals to the lattice is
> now one to many - and from the
> lattice back to intervals remains
> many to one. That's okay.

It's just a linear functional, ie a real valued val.

> It seems to be the underlying
> implied vector space that the
> lattice is embedded in and the
> reason why it is mathematically
> correct to refer to them as vectors.

What lattice? You do not have an implied *normed* vector space, nor do
you have a lattice, until you've defined them. You have a canonical
embedding into a real vector space without norm, but this does not
give you a canonical norm. Many different possibilities are possible
and have been suggested!

> So I see no problem calling them
> vectors and making that part of the
> definition.

It leads to confusion, a good example being this business of implied
vector spaces. What, exactly, does that mean?

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_55183.html#55215

> --- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
>
> > >also, readers should notice yet another new term:
> > >"um". we were saying "vanishing comma" or
> > >"vanishing vector" or "vanishing monzo" so much
> > >that we decided that a new short term was called for.
> >
> > Oh dear. What's wrong with "uv"? We've been through
> > all this before. Mathematicians don't own the word
> > "vector". Fokker used it, so can we.
>
> Mathematicians know what they mean when they use the word, which is
> very helpful in many cases. If you don't have precise definitions in
> mind, your meaning will necessarily be more context depended and
> likely be downright murky.
>
> Anyway, "unison vector" is awkward and verbose. I like "comma".

****Hmmm... I guess it's true there really was no need to create a
new term for this...

J. Pehrson

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_55183.html#55225

> hi Gene,
>
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > > first: "comma" used to refer more specifically to
> > > an interval about 1/8-tone in pitch-size, ~25 cents.
> > > then it started being used to designate all sorts of
> > > intervals <~120 cents, which i really objected to.
> >
> > You need a word which means "interval tempered out by
> > a temperament", and "comma" has been serving as that.
> >
> > > second: "um" and its multi-relatives refers specifically
> > > to "commas" which **vanish**.
> >
> > Which is what the word "comma" is mostly used for in theory
> > discussions. Are you proposing "um" as a word to mean "interval
> > tempered out by a temperament"?
>
>
>
> exactly. have you read the "um" and "bium"
> Encyclopaedia webpages?
>
>
>
> > > as i explained, we got so tired of having to keep typing
> > > "vanishing" that we just wanted one short word to
> > > represent the concept. Gene, you should appreciate
> > > this, with your preference for "precise definitions".
> >
> > I always mean that it vanishes when I say "comma" unless
> > I am referring to a specific interval of 81/80, 3^12/2^19,
> > or 64/63, or am referring to a size or using a size measure
> > ("1/4 comma"). I don't neccessarily mean that it is small,
> > or that it is larger than 1, or anything else, but I do mean
> > it vanishes. If people are not assuming that meaning then
> > maybe we need to umm.
>
>
>
> i think that lately most people have been assuming
> "vanishing" to be part of the definition of "comma",
> but that was never made explicit.
>
> as the self-appointed :-) lexicographer of the tuning
> community, i prefer to acknowledge that "comma" has a
> very long history of referring to a small interval
> that is *not* tempered-out ... the word was common
> currency in tuning-theory centuries before anyone
> wrote anything about temperament.
>
> i'm already unhappy that "comma" has already (and only
> fairly recently) been generalized more than it ever
> was before to refer to many more and much wider intervals.
>
> it seems much better to me to have a new word to
> refer to the cases where small intervals actually
> do vanish. and in fact, it's not just one word, but
> a whole series of words (um, bium, trium, etc.)
> which are logically arranged to reflect certain
> mathematical properties.
>
>
>
> -monz

***If "comma" has traditionally been associated with a small
interval, why not call a large comma a "large comma."

Also, if the comma vanishes, why not call it a "vanishing comma."

Why create new terms for everything? I'm pretty sure nobody is ever
going to use these terms...

J. Pehrson

🔗monz <monz@tonalsoft.com>

hi Joe,

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> ***If "comma" has traditionally been associated with a small
> interval, why not call a large comma a "large comma."
>
> Also, if the comma vanishes, why not call it a "vanishing comma."
>
> Why create new terms for everything? I'm pretty sure nobody is ever
> going to use these terms...
>
> J. Pehrson

sorry, guess again.

the reason why Paul and i decided to come up with a
new term for this (i think we've finally settled on
"vum" for "vanishing unison monzo") is because we got
tired of typing the long 3-word version so much in
our discussions. essentially it's an acronym, but
i prefer to just make it a simple word.

when something is similar to something else but the
difference is an important one, it's worthwhile to
distinguish the two terms with clearly separable
terminology.

it's an important fact that in a temperament, not
only does the "typical" unison-vector vanish, but
so do all of its multiples and all of the reciprocals.

for example, all meantones temper out not only the
syntonic-comma 81:80 but also the double-syntonic-comma,
the "reverse" syntonic-comma, etc.

and because these intervals do *not* vanish in JI,
it's clearly good to have a term which distinguishes
them from their JI counterparts (which we're still
calling "unison-vectors": a vum is a specific type
of unison-vector).

there are thus two important concepts embedded in "vum":

1)
the fact that a whole set of intervals which is
linear in prime-space, is defined uniquely by a
simple set of numbers (appearing as a "monzo").

2)
the fact that all of these intervals do physically
and literally vanish.

Gene started using "monzo" because it was much more
convenient than "prime-factor exponent vector".
ultimately, "monzo" is still just another name for
"ratio" ... but a "monzo" immediately conveys data
which allows the mind to visualize the ratio's location
in prime-space, something which is difficult with the
plain rational number.

can you see from this example how much easier it is
to discuss "monzo" than "prime-factor exponent vector
of a ratio"?

think about how convenient it is to use "cents"
in tuning discussions.

now, think about where we would be, if back in 1875
someone said to Ellis, "we don't need a term for
"cents" ... we can just call it 'logarithmic pitch-size
of an interval' (... and besides, we already have
merides and savarts)".

;-)

it always seems like jargon when it's brand-new
... over time people will either use the term or
change it to something else, but in any case,
once the concept has been embodied in a word,
it's not likely to go away.

-monz

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_55183.html#55291

> Gene started using "monzo" because it was much more
> convenient than "prime-factor exponent vector".
> ultimately, "monzo" is still just another name for
> "ratio" ... but a "monzo" immediately conveys data
> which allows the mind to visualize the ratio's location
> in prime-space, something which is difficult with the
> plain rational number.
>
> can you see from this example how much easier it is
> to discuss "monzo" than "prime-factor exponent vector
> of a ratio"?
>

***Hi Monz... Yes, I do see this. It's when you get to the vums, and
uums, voom, that I start to question things...

>
>
> think about how convenient it is to use "cents"
> in tuning discussions.
>
> now, think about where we would be, if back in 1875
> someone said to Ellis, "we don't need a term for
> "cents" ... we can just call it 'logarithmic pitch-size
> of an interval' (... and besides, we already have
> merides and savarts)".
>

***Quite frankly, Monz, I don't believe it's a situation of the same
magnitude, although I could be wrong.

Why not just accept one term, "monzo," and then in a *given paper*
say with an asterisk: "For the purposes of this paper, when a unison
monzo vanishes we will call it a *vum*" Uum?

(Rather than another defined generalized term...)

But, I should probably stay out of all of this anyway...

va, va voom...

🔗Carl Lumma <ekin@lumma.org>

🔗klaus schmirler <KSchmir@z.zgs.de>

monz schrieb:
> hi Joe,
> > > --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> > >>***If "comma" has traditionally been associated with a small >>interval, why not call a large comma a "large comma."
>>
>>Also, if the comma vanishes, why not call it a "vanishing comma."
>>
>>Why create new terms for everything? I'm pretty sure nobody is ever >>going to use these terms...
>>
>>J. Pehrson
> > > > > sorry, guess again.
> > the reason why Paul and i decided to come up with a
> new term for this (i think we've finally settled on
> "vum" for "vanishing unison monzo") is because we got
> tired of typing the long 3-word version so much in
> our discussions. essentially it's an acronym, but
> i prefer to just make it a simple word.

please be sincere and leave it as an acronym, preferably a capitalzed one that will never never be mistaken for a word.

> > > when something is similar to something else but the
> difference is an important one, it's worthwhile to
> distinguish the two terms with clearly separable > terminology.

and by using adjectives and nouns (plain english) you can draw attention to the difference directly using the adjective only.

> > > it's an important fact that in a temperament, not
> only does the "typical" unison-vector vanish, but
> so do all of its multiples and all of the reciprocals.

isn't that trivial? nought of nought is...

> > for example, all meantones temper out not only the
> syntonic-comma 81:80 but also the double-syntonic-comma,
> the "reverse" syntonic-comma, etc.
> > and because these intervals do *not* vanish in JI,

isn't that trivial?

> it's clearly good to have a term which distinguishes > them from their JI counterparts (which we're still
> calling "unison-vectors": a vum is a specific type
> of unison-vector).

wouldn't it be good policy in communication to establish first whether you are talking about ji or temperaments?

> > > there are thus two important concepts embedded in "vum":
> > 1)
> the fact that a whole set of intervals which is
> linear in prime-space, is defined uniquely by a
> simple set of numbers (appearing as a "monzo").
> > 2)
> the fact that all of these intervals do physically
> and literally vanish.
> > > > Gene started using "monzo" because it was much more
> convenient than "prime-factor exponent vector".
> ultimately, "monzo" is still just another name for
> "ratio" ... but a "monzo" immediately conveys data
> which allows the mind to visualize the ratio's location
> in prime-space, something which is difficult with the
> plain rational number.
> > can you see from this example how much easier it is
> to discuss "monzo" than "prime-factor exponent vector
> of a ratio"?
> > > > think about how convenient it is to use "cents"
> in tuning discussions.
> and fortunately they aren't called ellises...
cent was and is a widely used prefix for divisions into 100 parts.

> now, think about where we would be, if back in 1875 > someone said to Ellis, "we don't need a term for > "cents" ... we can just call it 'logarithmic pitch-size
> of an interval' (... and besides, we already have
> merides and savarts)".
> > ;-)
> > > > it always seems like jargon when it's brand-new
> ... over time people will either use the term or > change it to something else, but in any case, > once the concept has been embodied in a word,
> it's not likely to go away.

my (our? joe p?) fear is that it will go away as soon as the hard core of the list members here disband for some reason. there is a world outside yahoo, and i doubt very much that posterity will go the trouble of translating the findings of this rather arcane community from geek talk into music.

[i sometimes think that gene really has a different perspective and really wants to describe the behaviour of scales in mathematical terms. nothing wrong with that, but, let's face it, that - and the jargon explosion - is tuning for the sake of math and not math for the sake of tuning.]

[what if something like sagittal notation could not have been developped without jargon? tell me so, and i'll quietly accept whatever terms you choose to cook up in the future...]

klaus

🔗monz <monz@tonalsoft.com>

hi Carl,

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

> > the reason why Paul and i decided to come up with a
> > new term for this
>
> Paul's the worst offender. His terminology revisionism is
> well-meaning but counterproductive and incredibly annoying.
>
> -Carl

i can see why you would say this, and have to agree
that there is some truth in it.

but Paul's endeavor, as is mine, is to base the terminology
on a firm logical foundation.

within just the last few years, the combination of

. internet discussion among theorists,

. software which is computationally very powerful, and

. growing realization of how more and more aspects
and methods of algebra and geometry can be applied to
tuning-theory ...

has resulted in a veritable explosion of new ideas.

creation of a compact -- and logical -- terminology with
which to discuss these ideas, is an important tool for
their further development.

i personally have never been too concerned about a
"jargon explosion", because i've taken on the job of
maintaining the tuning lexicon. as long as the
Encyclopaedia is freely available online -- as i expect
it to be permanently -- any reader of tuning literature
(including these lists) who has internet access, can
look up the jargon.

i realize that even some of the stuff in the Encyclopaedia
is jibberish to most people, but i strive to make it
all correct and understandable to a newbie.

i still remember vividly how i had to struggle to
get a grasp of tuning concepts when i kept reading
words like "proslambanomenos" and "diezeugmenon".
that's why i work so hard on the Encyclopaedia.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗klaus schmirler <KSchmir@z.zgs.de>

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

klaus schmirler wrote:

> wouldn't it be good policy in communication to establish > first whether you are talking about ji or temperaments?

Well, Fokker didn't. The 1969 paper in English that people take the term "unison vector" from doesn't establish that the tuning is JI. Much of it applies regardless of tuning. As does most written music.

It isn't that clear cut anyway. If you're talking about linear, planar, etc., temperaments there may be unison vectors that don't vanish along with those that do. In that case, you need a way of saying which ones vanish. And the word "vanishing" looks good for that.

A chromatic unison vector is a subtly different beastie. Not only does it not vanish, but it's explicitly notated. This doesn't look like what Fokker intended. When he talked about notation, it was always that notes differing by a unison vector could be considered equivalent, not that a sign should be invented for the unison vector.

Still, we could do with a word for the interval between two notes that differ by an accidental. Surprisingly, I don't think there is a common one.

Graham

🔗klaus schmirler <KSchmir@z.zgs.de>

Graham Breed schrieb:

> klaus schmirler wrote:
> > >>wouldn't it be good policy in communication to establish >>first whether you are talking about ji or temperaments?
> > > Well, Fokker didn't. The 1969 paper in English that people take the > term "unison vector" from doesn't establish that the tuning is JI. Much > of it applies regardless of tuning. As does most written music.
> > It isn't that clear cut anyway. If you're talking about linear, planar, > etc., temperaments there may be unison vectors that don't vanish along > with those that do. In that case, you need a way of saying which ones > vanish. And the word "vanishing" looks good for that.

thank you. nothing better than using the word "vanish" when talking about vanishingness.
(my take on the "not that clear cut" is that JI tries to realize all the commas whereas [please don't have me prove it] any temperament will temper out a rather low number of the infinite number of commas -- whether this matters or not boils down to whether you use temperaments for the things they were intended for or not.)

> > A chromatic unison vector is a subtly different beastie. Not only does > it not vanish, but it's explicitly notated. This doesn't look like what > Fokker intended. When he talked about notation, it was always that > notes differing by a unison vector could be considered equivalent, not > that a sign should be invented for the unison vector.
> > Still, we could do with a word for the interval between two notes that > differ by an accidental. Surprisingly, I don't think there is a common one.
> the musical community used to call it an "augmented prime". i would prefer this term to invade math territory rather than the other round.

klaus

> > Graham

🔗Carl Lumma <ekin@lumma.org>

>A chromatic unison vector is a subtly different beastie. Not only does
>it not vanish, but it's explicitly notated. This doesn't look like what
>Fokker intended. When he talked about notation, it was always that
>notes differing by a unison vector could be considered equivalent, not
>that a sign should be invented for the unison vector.

He does say:

"Care must be taken to avoid an ambiguity, in that not only 0,1,0, one
major third above D, bears the name f-sharp, but the fifth 4,0,0 too,
one place beyond b (3,0,0) is commonly called f-sharp. The latter,
owever, is sharper than the former, by one syntonic comma, the fraction
81/80. Therefore it is appropriate for 4,0,0 to use an additional
commatic sign, a stroke sloping upward, /, and to name it
f-sharp-comma-up, /f. Likewise there is a difference of a syntonic
comma between b-flat for 0,-1,0 and b-flat-comma-down, -4,0,0, or \b."

But already Paul's paper is far more important than any that have
come before.

>Still, we could do with a word for the interval between two notes that
>differ by an accidental. Surprisingly, I don't think there is a common
>one.

Since nobody seems to like "chromatic unison vector" and "commatic
unison vector", I'm voting for Gene's "chroma" and "comma", resp.

-Carl

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

Carl Lumma wrote:

> He does say:
> > "Care must be taken to avoid an ambiguity, in that not only 0,1,0, one
> major third above D, bears the name f-sharp, but the fifth 4,0,0 too,
> one place beyond b (3,0,0) is commonly called f-sharp. The latter,
> owever, is sharper than the former, by one syntonic comma, the fraction
> 81/80. Therefore it is appropriate for 4,0,0 to use an additional
> commatic sign, a stroke sloping upward, /, and to name it
> f-sharp-comma-up, /f. Likewise there is a difference of a syntonic
> comma between b-flat for 0,-1,0 and b-flat-comma-down, -4,0,0, or \b."

Fokker, that is. Yes, but he doesn't call it a unison vector at this point. He does use the notation later on: "As a rule no attention is paid to the difference of c(-2,0,0) and /c(2,-1,0)." But it's only because no attention is paid that it becomes a unison vector. And if no attention is paid, there's no reason to notate it.

> But already Paul's paper is far more important than any that have
> come before.

That's for history to decide.

> Since nobody seems to like "chromatic unison vector" and "commatic
> unison vector", I'm voting for Gene's "chroma" and "comma", resp.

What nice words! I wonder why I didn't think of them.

Graham

🔗Carl Lumma <ekin@lumma.org>

>> "Care must be taken to avoid an ambiguity, in that not only 0,1,0, one
>> major third above D, bears the name f-sharp, but the fifth 4,0,0 too,
>> one place beyond b (3,0,0) is commonly called f-sharp. The latter,
>> owever, is sharper than the former, by one syntonic comma, the fraction
>> 81/80. Therefore it is appropriate for 4,0,0 to use an additional
>> commatic sign, a stroke sloping upward, /, and to name it
>> f-sharp-comma-up, /f. Likewise there is a difference of a syntonic
>> comma between b-flat for 0,-1,0 and b-flat-comma-down, -4,0,0, or \b."
>
>Fokker, that is. Yes, but he doesn't call it a unison vector at this
>point. He does use the notation later on: "As a rule no attention is
>paid to the difference of c(-2,0,0) and /c(2,-1,0)." But it's only
>because no attention is paid that it becomes a unison vector. And if no
>attention is paid, there's no reason to notate it.

The point is you need 25:24 or similar to close a block around the
diatonic scale. One might not call it a unison vector, but it's a
vector that closes a block, and we notate it with accidentals.

>> Since nobody seems to like "chromatic unison vector" and "commatic
>> unison vector", I'm voting for Gene's "chroma" and "comma", resp.
>
>What nice words! I wonder why I didn't think of them.

And since Gene has been using them for a few years, they have the
advantage of having been in use for a few years.

-Carl

🔗monz <monz@tonalsoft.com>

🔗klaus schmirler <KSchmir@z.zgs.de>

monz schrieb:

> --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:
> > >>>Still, we could do with a word for the interval
>>>between two notes that differ by an accidental.
>>>Surprisingly, I don't think there is a common one.
>>>
>>
>>the musical community used to call it an "augmented prime". >>i would prefer this term to invade math territory rather >>than the other round.
> > > > > klaus is right ... an augmented prime is the interval
> between a plain nominal and the same nominal with
> a sharp, and a diminished prime is the interval between
> a plain nominal and the same nominal with a flat.
> > however, it's rare to find an example of the latter.
> well, not if you include the chromatic semitone. that occurs in keys of less than two sharps any time there's a switch from major to minor...

this drives home the fact that there are many kinds of augmented/diminished primes. but rather than giving them all names, i'd prefer explicit indications whether about the kind of "space" we're in: scale, temperament, open/closed. normally people will not talk about all conextual possibilities at once.

klaus

> > > -monz
>

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:

> monz schrieb:
>
> > klaus is right ... an augmented prime is the interval
> > between a plain nominal and the same nominal with
> > a sharp, and a diminished prime is the interval between
> > a plain nominal and the same nominal with a flat.
> >
> > however, it's rare to find an example of the latter.
> >
>
> well, not if you include the chromatic semitone. that occurs
> in keys of less than two sharps any time there's a switch
> from major to minor...

i'm not clear on what you mean there. examples?

> this drives home the fact that there are many kinds of
> augmented/diminished primes.

how so? in both pythagorean and meantone, an
augmented-prime always means the same thing: the
addition of a sharp or subtraction of a flat without
changing the nominal, and a widening in pitch by
the chromatic semitone (2,3-monzo [-11 7,> in
pythagorean tuning, and whatever tuning it may
have in the various meantones).

-monz

🔗klaus schmirler <KSchmir@z.zgs.de>

monz schrieb:
> --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:
> > >>monz schrieb:
>>
>>
>>>klaus is right ... an augmented prime is the interval
>>>between a plain nominal and the same nominal with
>>>a sharp, and a diminished prime is the interval between
>>>a plain nominal and the same nominal with a flat.
>>>
>>>however, it's rare to find an example of the latter.
>>>
>>
>>well, not if you include the chromatic semitone. that occurs >>in keys of less than two sharps any time there's a switch >>from major to minor...
> > > > i'm not clear on what you mean there. examples?
> >

the example at the outset was c as 1/1 and c# as the major third of its sixth degree, 135/128. this occurs in a double (well, triple) dominant and is much more common than its inverse (in a key of c, you're more likely to see ab-b than ab-cb).

for 25/24, examples of diminished primes aren't that hard to come by.

> > >>this drives home the fact that there are many kinds of >>augmented/diminished primes. > > > > how so? in both pythagorean and meantone, an
> augmented-prime always means the same thing: the
> addition of a sharp or subtraction of a flat without
> changing the nominal, and a widening in pitch by
> the chromatic semitone (2,3-monzo [-11 7,> in > pythagorean tuning, and whatever tuning it may
> have in the various meantones).
> > > > -monz

i don't want to be nosy, but what time is it in your place?

klaus

🔗monz <monz@tonalsoft.com>

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

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > in both pythagorean and meantone, an
> > augmented-prime always means the same thing: the
> > addition of a sharp or subtraction of a flat without
> > changing the nominal, and a widening in pitch by
> > the chromatic semitone (2,3-monzo [-11 7,> in
> > pythagorean tuning, and whatever tuning it may
> > have in the various meantones).
> >
> >
> >
> > -monz
>
>
>
> which prompted me to add a nice table and graphic
> of the sizes of various historical meantone
> augmented-primes (or chromatic semitones, whichever
> you prefer).
>
> http://tonalsoft.com/enc/index2.htm?chromatic.htm#semitone
>
>
>
> -monz

it's all the way down at the bottom of the page.

many thanks to Robert Walker for writing up the new
Encyclopaedia main page which allows links to open
entries within the frames.

Robert, in examples like the above the part after
".htm" ("#semitone") is not working -- it should
direct the browser to the named section of that page.
can you fix that?

-monz