Vals?

Here's a simple question for anyone who thinks they can answer it.

How is a val different from an ET-mapping? i.e. a list of the numbers
of steps approximating each prime in some ET.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> Here's a simple question for anyone who thinks they can answer it.
>
> How is a val different from an ET-mapping? i.e. a list of the
numbers
> of steps approximating each prime in some ET.

You can identify them; however a val, as I defined it, would define a
homomorphic mapping from every positive rational number to an
associated integer. Usually you would restrict this to a given prime
limit.

Dave:
> > Here's a simple question for anyone who thinks they can answer it.
> >
> > How is a val different from an ET-mapping? i.e. a list of the
> > numbers of steps approximating each prime in some ET.

Gene:
> You can identify them; however a val, as I defined it, would define a
> homomorphic mapping from every positive rational number to an
> associated integer. Usually you would restrict this to a given prime
> limit.

Couldn't I (in fact didn't I) just define an (unqualified) ET-mapping
in exactly the same way? And then a "p-limit ET-mapping" would be a
restricted one.

So they're exactly the same!!!!

So why have we been calling them "vals" all this time? A mathworld
search on the term finds nothing. Did you invent the term? Is it
merely an obscure synonym for "homomorphism", or "group homomorphism"?
The fact that it _is_ a group homomorphism is far from being its most
important characteristic as far as microtonality is concerned. The
fact that it maps ratios to steps of ETs is of far more interest.

Even if the term is in common use in some area of mathematics, I don't
understand why you have been using it all this time when the perfectly
transparent terms ET-mapping, or ET-prime-mapping would have served us
far better. I wouldn't be the only one who has failed to understand
many of your posts over the years through the lack of this simple
substitution.

And I don't think I'll be the only person reading this who will be
saying to themselves now, "Is that all a val is?.

So can that obscure definition of "val" on Monz's site now be replaced
with "ET-prime-mapping", which can in turn be defined as
"a list of the numbers of steps approximating each prime number in
some ET", and proceeding to show how to use one to calculate the
number of steps corresponding to some ratio in some ET?

Somehow I don't think the wedge-invariant or wedgie will be so easily
disposed of. I'll have to reread that Grassman Algebra book some time.
I remember a glimmer of understanding last time I did that, but it was
a long time ago.
http://www.ses.swin.edu.au/homes/browne/grassmannalgebra/book/

Dave Keenan wrote:

> So can that obscure definition of "val" on Monz's site now be replaced
> with "ET-prime-mapping", which can in turn be defined as
> "a list of the numbers of steps approximating each prime number in
> some ET", and proceeding to show how to use one to calculate the
> number of steps corresponding to some ratio in some ET?

AIUI, the only difference is that vals don't have to describe any temperament, so there one step of abstraction beyond an ET.

Graham

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

> Couldn't I (in fact didn't I) just define an (unqualified) ET-mapping
> in exactly the same way?

Go ahead and do so, however a val is not necessarily an et-mapping of
any kind.

And then a "p-limit ET-mapping" would be a
> restricted one.
>
> So they're exactly the same!!!!

I havn't seen your definition.

> So why have we been calling them "vals" all this time?

Because there wasn't a good word for "finitely generated homomorphic
mapping from Q+ to Z" or "Z-linear combination of padic valuations"
already in existence.

A mathworld
> search on the term finds nothing. Did you invent the term?

You bet. We needed a term for it, and there wasn't one.

Is it
> merely an obscure synonym for "homomorphism", or "group homomorphism"?
> The fact that it _is_ a group homomorphism is far from being its most
> important characteristic as far as microtonality is concerned. The
> fact that it maps ratios to steps of ETs is of far more interest.

A val *does not necessarily* map ratios to steps of an ET, but it *is*
always a homomorphism. If you insist, you could replace the term with
"finitely generated homomorphic mapping from Q+ to Z", I suppose, but
I imagine "a p-limit homomorphic mapping to the integers" or something
like that would suit you better.

> Even if the term is in common use in some area of mathematics, I don't
> understand why you have been using it all this time when the perfectly
> transparent terms ET-mapping, or ET-prime-mapping would have served us
> far better.

It most certainly would not have served me better. You do what you
like, but please don't expect me to follow your lead.

in fairness to gene, i believe he derived it from the
term "valuation" which is an entire field of mathematics.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
>
> > Couldn't I (in fact didn't I) just define an (unqualified) ET-
mapping
> > in exactly the same way?
>
> Go ahead and do so, however a val is not necessarily an et-mapping
of
> any kind.
>
> And then a "p-limit ET-mapping" would be a
> > restricted one.
> >
> > So they're exactly the same!!!!
>
> I havn't seen your definition.
>
> > So why have we been calling them "vals" all this time?
>
> Because there wasn't a good word for "finitely generated homomorphic
> mapping from Q+ to Z" or "Z-linear combination of padic valuations"
> already in existence.
>
> A mathworld
> > search on the term finds nothing. Did you invent the term?
>
> You bet. We needed a term for it, and there wasn't one.
>
> Is it
> > merely an obscure synonym for "homomorphism", or "group
homomorphism"?
> > The fact that it _is_ a group homomorphism is far from being its
most
> > important characteristic as far as microtonality is concerned. The
> > fact that it maps ratios to steps of ETs is of far more interest.
>
> A val *does not necessarily* map ratios to steps of an ET, but it
*is*
> always a homomorphism. If you insist, you could replace the term
with
> "finitely generated homomorphic mapping from Q+ to Z", I suppose,
but
> I imagine "a p-limit homomorphic mapping to the integers" or
something
> like that would suit you better.
>
> > Even if the term is in common use in some area of mathematics, I
don't
> > understand why you have been using it all this time when the
perfectly
> > transparent terms ET-mapping, or ET-prime-mapping would have
served us
> > far better.
>
> It most certainly would not have served me better. You do what you
> like, but please don't expect me to follow your lead.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > Couldn't I (in fact didn't I) just define an (unqualified) ET-mapping
> > in exactly the same way?
>
> Go ahead and do so, however a val is not necessarily an et-mapping of
> any kind.

In this message
/tuning-math/message/7528
you said the two could be identified (except for a question about
finiteness)?

Here is a possible Monz dictionary definition of an ET-prime-mapping
(improved from the one I already gave in the above message, that you
seem to have missed):
----------------------------------------------------------------------
ET-prime-mapping

A list of the (whole) numbers of steps of some equal temperament (ET)
(not necessarily octave based) used to approximate each prime number
(considered as a frequency ratio). An "n-limit" ET-prime-mapping
(where n is a whole number) only lists numbers of steps for primes no
greater than n.

The "standard" prime mapping for an ET is the one that gives the best
approximation for each prime, but note that this is not guaranteed to
give the best approximation for all ratios, and other mappings may be
more useful in some cases.

To find the number of steps approximating some ratio in some ET,
express the ratio as a prime-exponent-vector and multiply its elements
by the corresponding elements in the chosen ET-prime-mapping and sum
the products. This is called the dot product or inner product or
scalar product of the two vectors.

For example the standard 7-limit ET-prime-mapping for 12-EDO is [12 19
28 34]. These numbers can be calculated for any EDO as
Round(N*ln(p)/ln(2)) where N is the number of divisons per octave and
p is the prime number.

To find how many steps of 12-EDO approximate a 7/5 frequency ratio,
first express the ratio as a prime-exponent-vector.
7/5 = 2^0 * 3^0 * 5^-1 * 7^1 = [0 0 -1 1]
now find its dot product with the prime-mapping
[0 0 -1 1].[12 19 28 34] = 0*12 + 0*19 + -1*28 + 1*34 = 6
So 7/5 is approximated by 6 steps, a tritone.
----------------------------------------------------------------------

> And then a "p-limit ET-mapping" would be a
> > restricted one.
> >
> > So they're exactly the same!!!!
>
> I havn't seen your definition.

I suspect you just didn't recognise it as such, because it was in
plain English.

> > So why have we been calling them "vals" all this time?
>
> Because there wasn't a good word for "finitely generated homomorphic
> mapping from Q+ to Z" or "Z-linear combination of padic valuations"
> already in existence.
>
> A mathworld
> > search on the term finds nothing. Did you invent the term?
>
> You bet. We needed a term for it, and there wasn't one.

_You_ might have needed a term for "finitely generated homomorphic
mapping from Q+ to Z" or "Z-linear combination of padic valuations",
but I don't think anyone else on this list did.

> Is it
> > merely an obscure synonym for "homomorphism", or "group homomorphism"?
> > The fact that it _is_ a group homomorphism is far from being its most
> > important characteristic as far as microtonality is concerned. The
> > fact that it maps ratios to steps of ETs is of far more interest.
>
> A val *does not necessarily* map ratios to steps of an ET, but it *is*
> always a homomorphism.

So what would you estimate is the percentage of the vals posted to
tuning-math that could not be read as mapping ratios to steps of an
ET. Please give some examples of these and explain what they _do_ mean
in tuning terms.

> If you insist, you could replace the term with
> "finitely generated homomorphic mapping from Q+ to Z", I suppose, but
> I imagine "a p-limit homomorphic mapping to the integers" or something
> like that would suit you better.

Those certainly don't suit me. By all means use the word "val" to
stand for this abstract mathematical category. But this is the
_tuning_ math list. We want names that indicate their meaning as
applied to _tuning_.

For example, in the application area of electrical theory we use
vectors to represent the magnitude and phase of sinusoidal voltages
and currents. But we don't just call them all vectors. We want names
that tell us what they _mean_. We call them "voltage phasors" and
"current phasors".

I assume that the val, strictly speaking, is the operation of taking
the dot product with a vector of step numbers, not the actual vector
of step numbers itself. That's ok. The term "mapping" is used
similarly ambiguously. It doesn't usually cause any misunderstanding.

> It most certainly would not have served me better. You do what you
> like, but please don't expect me to follow your lead.

So are you saying that you don't really care if only two or three
people on this list understand how the things you write about apply to
tuning?

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

> In this message
> /tuning-math/message/7528
> you said the two could be identified (except for a question about
> finiteness)?

I understood you to mean any kind of prime mapping, whether it could
be called an et or not.

> Here is a possible Monz dictionary definition of an ET-prime-mapping
> (improved from the one I already gave in the above message, that you
> seem to have missed):
> ----------------------------------------------------------------------
> ET-prime-mapping
>
> A list of the (whole) numbers of steps of some equal temperament (ET)
> (not necessarily octave based) used to approximate each prime number
> (considered as a frequency ratio). An "n-limit" ET-prime-mapping
> (where n is a whole number) only lists numbers of steps for primes no
> greater than n.

OK, but this isn't equivalent to val.

> So what would you estimate is the percentage of the vals posted to
> tuning-math that could not be read as mapping ratios to steps of an
> ET.

Who knows? Maybe 50-50.

Please give some examples of these and explain what they _do_ mean
> in tuning terms.

<0 1 4 10| should be familiar from the meantone temperament.

> For example, in the application area of electrical theory we use
> vectors to represent the magnitude and phase of sinusoidal voltages
> and currents.

Unless you use complex numbers, of course.

> I assume that the val, strictly speaking, is the operation of taking
> the dot product with a vector of step numbers, not the actual vector
> of step numbers itself.

It's the mapping that generates, yes.

> So are you saying that you don't really care if only two or three
> people on this list understand how the things you write about apply to
> tuning?

I've explained what a val is numerous times. I can't insist you pay
attention to everything I say; these days you and George tend to lose
me, after all, which is fair enough.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > In this message
> > /tuning-math/message/7528
> > you said the two could be identified (except for a question about
> > finiteness)?
>
> I understood you to mean any kind of prime mapping, whether it could
> be called an et or not.

I wrote:
> > How is a val different from an ET-mapping? i.e. a list of the
> > numbers of steps approximating each prime in some ET.

So I'm rather surprised you didn't know I was talking about ETs?

But that's good, because now it looks like "val", as applied to
tuning, can be replaced by "prime-mapping", which is even simpler than
"ET-prime-mapping".

> <0 1 4 10| should be familiar from the meantone temperament.

This looks like one row of the 7-limit prime-mapping for the meantone
linear temperament using a fifth as the generator, in particular the
row giving the mapping to fifth generators. Isn't it somewhat
incomplete without the other row that gives the mapping to octave
generators (periods)?

Why do we want to give the same name to something which in one case is
the complete mapping for an ET (a 1D temperament), and in the other
case only a part of the mapping for an LT (a 2D temperament)?

But assuming that there's a good reason, I'd simply call them
"prime-mappings" or "1D-prime mappings".

But I'd prefer to be more specific and call one an ET-mapping and the
other an LT-generator-mapping. I'd call the missing row the
LT-period-mapping. Together the LT-generator-mapping and the
LT-period-mapping make up the LT-mapping. The word "prime" can be
inserted before the word "mapping" whenever this is not clear from the
context.

> > So are you saying that you don't really care if only two or three
> > people on this list understand how the things you write about
apply > > to tuning?
>
> I've explained what a val is numerous times. I can't insist you pay
> attention to everything I say; these days you and George tend to lose
> me, after all, which is fair enough.

If all your explanations were similar to this one
http://sonic-arts.org/dict/val.htm
I'm afraid it wouldn't have made any difference if I'd read them all.
But I can't hold it against anyone that they are not good at
explaining things.

I'm pretty sure I did read an early one, and said to myself, "I have
no idea what that means. I guess I need a bit more mathematical
background. I'll look into it later."

But it appears that little or no explanation would have been necessary
if you had simply called them prime mappings.

So is a val, as applied to tuning theory, simply a prime-mapping, or a
1D-prime-mapping?

>I've explained what a val is numerous times. I can't insist you pay
>attention to everything I say; these days you and George tend to lose
>me, after all, which is fair enough.

But you haven't explained how it works.

>If the standard 5-limit val
>for 12-equal is [12 19 28] or something, how does it come from
>round(n log2(p))? Oh n is 12, eh? So vals are uniquely identified
>by this n?
>
>So how does one find a standard val for an odd limit (the start of
>this thread, which perhaps George is still following)? Where do you
>get your n?

-C.

Dave Keenan wrote:

> But it appears that little or no explanation would have been necessary
> if you had simply called them prime mappings.

I think I call them "equal mappings" to mean mappings of equal temperaments that don't imply you have an actual equal temperament. That is, I think this concept is the same as Gene's "Val".

Graham

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > <0 1 4 10| should be familiar from the meantone temperament.
>
> This looks like one row of the 7-limit prime-mapping for the
meantone
> linear temperament using a fifth as the generator, in particular the
> row giving the mapping to fifth generators. Isn't it somewhat
> incomplete without the other row that gives the mapping to octave
> generators (periods)?

The octaves would be another val.

> Why do we want to give the same name to something which in one case
is
> the complete mapping for an ET (a 1D temperament), and in the other
> case only a part of the mapping for an LT (a 2D temperament)?

Is it just barely possible I might know something about mathematics?
To me this is a bit like asking why we would care about a single
comma, when we need more than one of them for a Fokker block, BTW.

> But assuming that there's a good reason, I'd simply call them
> "prime-mappings" or "1D-prime mappings".

Call them what you like, but clearly your names are clumsier.

> But it appears that little or no explanation would have been
necessary
> if you had simply called them prime mappings.

They aren't prime mappings per se; that's just a basis. I *did*
explain they were homomorphic mappings, and give examples with column
vectors, etc etc.

> So is a val, as applied to tuning theory, simply a prime-mapping,
or a
> 1D-prime-mapping?

More or less.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I've explained what a val is numerous times. I can't insist you pay
> >attention to everything I say; these days you and George tend to
lose
> >me, after all, which is fair enough.
>
> But you haven't explained how it works.

I have done just that many times.

>> >I've explained what a val is numerous times. I can't insist you pay
>> >attention to everything I say; these days you and George tend to
>> >lose me, after all, which is fair enough.
>>
>> But you haven't explained how it works.
>
>I have done just that many times.

Here's what you've given us so far...

>Consider the otonal chord of the n-odd-limit. This has (n+1)/2 octave
>reduced elements, 1 < q[i] <= 2, where the q[i], i from 1 to (n+1)/2,
>are arranged in increasing size. The n-odd-limit has pi(n) primes; we
>may solve the (n+1)/2 linear equations for the val which sends q[1]
>to 1, q[2] to 2, up to q[(n+1)/2]=2 to (n+1)/2. These linear
>equations have a unique solution in the 3, 5, 7, 9, and 13 odd limits.
>For 3 we get [2, 3], for 5 [3, 5, 7] and so forth--the standard vals
>in the respective prime limits 3, 5, 7, 7, 11 for 2, 3, 4, 5, and 7.

>To give a simple example, in the 5-limit, (5+1)/2 = 3, and we may
>start from the 3-chord [5/4, 3/2, 2]. If we solve for a val [a, b, c]
>such that 5/4, or [-2, 0, 1] is mapped to 1, 3/2 is mapped to 2, and
>2 is mapped to 3 we get the equations a5 - 2 a2 = 1, a3 - a2 = 2, and
>a2 = 3, the solution of which is a2 = 3, a3 = 5, and a5 = 7, so the
>val in question is uniquely determined to be [3, 5, 7], the standard
>3-val for the 5-limit.

standard val-
>The vector consisting of round(n log2(p)) for primes p in ascending
>order up to the chosen prime limit, considered as defining a val.

...It appears that in the case of the "standard 3-val for the 5-limit",
n=3. Is that why you called it a 3-val? Where did 3 come from?

Further, is the standard val supposed to be the val with the smallest
possible numbers that works? Or, I don't get your criterion for
deciding you want "the val which sends q[1] to 1, q[2] to 2, up to
q[(n+1)/2]=2 to (n+1)/2".

Further, why are you sending 5/4 to 1 and 3/2 to 2 and 2/1 to 3,
instead of the reverse? I thought "the q[i], i from 1 to (n+1)/2,
are arranged in increasing size".

-Carl

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

> Here's what you've given us so far...

I've given way, way way more than that. I can't force anyone to read
it.

> ...It appears that in the case of the "standard 3-val for the 5-
limit",
> n=3. Is that why you called it a 3-val?

> Where did 3 come from?

A division of the octave into three parts, or in other words, a
mapping of 2 to 3.

>
> Further, is the standard val supposed to be the val with the
smallest
> possible numbers that works?

It's simply what you get by rounding log2(p) to the nearest integer.

Or, I don't get your criterion for
> deciding you want "the val which sends q[1] to 1, q[2] to 2, up to
> q[(n+1)/2]=2 to (n+1)/2".
>
> Further, why are you sending 5/4 to 1 and 3/2 to 2 and 2/1 to 3,
> instead of the reverse? I thought "the q[i], i from 1 to (n+1)/2,
> are arranged in increasing size".

Eh? clearly 5/4 < 3/2 < 2.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > Here's what you've given us so far...
>
> I've given way, way way more than that. I can't force anyone to
read
> it.
>
> > ...It appears that in the case of the "standard 3-val for the 5-
> limit",
> > n=3. Is that why you called it a 3-val?
>
> > Where did 3 come from?
>
> A division of the octave into three parts, or in other words, a
> mapping of 2 to 3.

excuse me, but i think the answer to carl's question is "the complete
5-limit otonal chord has *3* notes". right?

>> Here's what you've given us so far...
>
>I've given way, way way more than that. I can't force anyone to
>read it.

I've read everything you've ever posted to this list, much of it
more than once, and much of it I've saved locally.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> The octaves would be another val.
>
> > Why do we want to give the same name to something which in one case
> is
> > the complete mapping for an ET (a 1D temperament), and in the other
> > case only a part of the mapping for an LT (a 2D temperament)?
>
> Is it just barely possible I might know something about mathematics?

There is no doubt about that. We're all very grateful for the new
tools you've given us. If only we could figure out what they actually
are in tuning terms, and how to use them.

But the fact is, that was not a pure-math question. It was a
tuning-math question. And it was not intended as a jibe, it was a
sincere question. I had hoped that would be clear from my following
sentence where I assumed that there _was_ a good reason.

> To me this is a bit like asking why we would care about a single
> comma, when we need more than one of them for a Fokker block, BTW.

I don't don't see why it is like asking that. You _could_ just try
answering the original question.

> > But assuming that there's a good reason, I'd simply call them
> > "prime-mappings" or "1D-prime mappings".
>
> Call them what you like, but clearly your names are clumsier.

I really don't think 3 syllables is excessive (nor 5 if we need to add
the "1D"). We can always make up a brand new shorter word for
something, but it isn't always a good idea. We'd be in a sorry state
now if we invented a new (initially meaningless) term every time a
descriptive term went over two syllables.

> > But it appears that little or no explanation would have been
> > necessary if you had simply called them prime mappings.
>
> They aren't prime mappings per se; that's just a basis.

I understand that. But isn't it the _only_ basis that we are using
them with for _tuning_ purposes?

> I *did*
> explain they were homomorphic mappings, and give examples with column
> vectors, etc etc.

OK. Sorry I missed them. But like I said, if you'd used an obvious
name that related to the application area, instead of a newly invented
abstract pure-math term, the explanations would barely have even been
necessary. Talk about secret decoder rings. :-)

> > So is a val, as applied to tuning theory, simply a prime-mapping,
> > or a 1D-prime-mapping?
>
> More or less.

OK. So now I feel like the boy who cried "the emperor has no clothes".
While we can't hold it against you that you think better in pure-math
terms and are not very good at explaining the relationships to tuning
in a way the rest of us can understand, I think we _can_ hold it
against you if you insist on continuing to use an obscure term when
you've been presented with perfectly transparent alternatives
_for_the_application_to_tuning_,
which is, after all, what this list is about.

I notice that even Paul, who along with Graham, appears to understand
your stuff better than any of us, was gently suggesting a similar
thing in another thread (although I thought he caved in rather too
easily).

Regards,
-- Dave Keenan

Actually, if you need a shorter term than "prime-mapping", it seems
like "mapping" would do. What other kinds of mappings do we use in
tuning-math?

>Actually, if you need a shorter term than "prime-mapping", it seems
>like "mapping" would do. What other kinds of mappings do we use in
>tuning-math?

A "mapping", as it has been used, is sufficient to define a
linear temperament. A val is not. But choo got me as to the
exact relationship/difference between the two.

-Carl

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

> But the fact is, that was not a pure-math question.

It sounded to me like you wanted to tell me I should use matricies,
and forget about all the other issues I have in mind, such as for
instance wedge products. In other words, very much telling me not only
how to name things, but how to do my math.

> I don't don't see why it is like asking that. You _could_ just try
> answering the original question.

I want to talk about homomorphims to Z, because that is dual to
intevals, which has various implications. Concretely we see the
usefulness of that, to give one example, in the whole multilinear
algebra approach.

> I understand that. But isn't it the _only_ basis that we are using
> them with for _tuning_ purposes?

It's obviously the gold standard.

> I think we _can_ hold it
> against you if you insist on continuing to use an obscure term when
> you've been presented with perfectly transparent alternatives
> _for_the_application_to_tuning_,
> which is, after all, what this list is about.

Well, I did for my own mathematical reasons define vals as a finite
Z-linear combination of padic valuations, and that isn't *precisely*
like any of your proposed alternatives. More or less, but not 100%, as
I said.

Vals are an important concept and deserve a name. Why is this so
painful? I admit my names are not always terrific (eg "standard val")
and some of them (eg "icon") I haven't even attempted to inflict on
people here, while others (eg "notation") have generated no support,
but I really am not interested in using an inferior name for an
inferior definition. Why insist that everything must be done your way?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Actually, if you need a shorter term than "prime-mapping", it seems
> >like "mapping" would do. What other kinds of mappings do we use in
> >tuning-math?
>
> A "mapping", as it has been used, is sufficient to define a
> linear temperament. A val is not.

Agreed.

> But choo got me as to the
> exact relationship/difference between the two.

Then a val is just a mapping-row, which is itself still a mapping. A
vector can also be considered as an nx1 matrix (as it apparently has
been in the ET case).

In the case of an ET the complete mapping has 1 row, for an LT it has
2 rows, for a planar temperament (PT) it has 3 rows, etc.

But we can still refer to a single row of an LT or PT mapping as "the
mapping for the <something> generator". Or in the LT case, the
generator-mapping and the period-mapping. If you call them vals,
you're still going to have to say "the val for the <something>
generator", or the period val and the generator val. I don't see how
calling them vals adds anything to this. In fact I think it just
obscures things.

>Then a val is just a mapping-row,

What confuses the hell out of me is that Gene keeps using
the word "column" re. vals, but they don't give successive
approximations to the same prime, they give a single
approx. to various primes.

>I don't see how
>calling them vals adds anything to this. In fact I think it
>just obscures things.

By now it should be no surprise that I'm utterly confused
by your obsession over this word. Considered a career in
postmodern critical theory?

At the very least, I'd hope understand what vals are good
for before trying to rename them. Or maybe you understand
why the 11-limit has no standard val, and can explain it
to the rest of us.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > But the fact is, that was not a pure-math question.
>
> It sounded to me like you wanted to tell me I should use matricies,
> and forget about all the other issues I have in mind, such as for
> instance wedge products. In other words, very much telling me not only
> how to name things, but how to do my math.

I assure you this is not the case.

I think we have quite complementary skills. You come up with the the
math tools and methods and I may _eventually_ be able to understand
them enough to put them into terms that others on this list can more
easily understand. But I don't think you should worry too much if my
explanations or recasting of terminology misses some of the more
subtle points as far as the pure mathematician is concerned, at least
on a first pass.

> > I don't don't see why it is like asking that. You _could_ just try
> > answering the original question.
>
> I want to talk about homomorphims to Z, because that is dual to
> intevals, which has various implications. Concretely we see the
> usefulness of that, to give one example, in the whole multilinear
> algebra approach.

OK. Well I hope we can put those implications in tuning terms
eventually, but I'd prefer to get the basics translated first. The
sort of stuff people can do themselves using nothing more
sophisticated than Excel, and without needing to know the meaning of
the terms homomorphism, Z, dual or multilinear algebra.

> Vals are an important concept and deserve a name.

I agree.

> Why is this so painful?

That isn't painful per se. But it's a term that belongs to the
pure-math side of things and isn't specific enough to our applications
of it.

> I admit my names are not always terrific (eg "standard val")
> and some of them (eg "icon") I haven't even attempted to inflict on
> people here, while others (eg "notation") have generated no support,
> but I really am not interested in using an inferior name for an
> inferior definition. Why insist that everything must be done your way?

Now you're exaggerating. I don't insist on that. How could I anyway?
I'm just expressing my opinion like anyone else.

All I'm saying is, if you want people on this list to understand what
you are on about, it's a good idea to invest in names that are
descriptive of their specific application to tuning, or even ones that
are more like terms i everyday use. By all means tell us "in
mathematics we call this a <whatever>", but when someone tells you,
"Oh I think I understand what you're talking about. That's a <whatever
tuning thing>.", and no-one seriously disagrees, then I think it would
be a good idea to try to construct a descriptive name from that for
use in future discourse on this list, even it's ony 99% the same
concept to start with.

I'm just disappointed we got this far with "val" without someone
figuring out a more tuning-specific or everyday cognate (or very near
cognate). But I don't blame you for that.

>I'm just disappointed we got this far with "val"

...with only 1 -- two if we're lucky -- persons who know
how to use them for what they're capable of.

Gene, since you won't say what's desirable about being a
standard val, and you haven't said what the lack of a
standard 11-limit val means about the 11-limit, I can only
guess that the definition of standard val is an error,
since the 11-limit is one of the more useful ways to get
hexads.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Then a val is just a mapping-row,
>
> What confuses the hell out of me is that Gene keeps using
> the word "column" re. vals, but they don't give successive
> approximations to the same prime, they give a single
> approx. to various primes.
>
> >I don't see how
> >calling them vals adds anything to this. In fact I think it
> >just obscures things.
>
> By now it should be no surprise that I'm utterly confused
> by your obsession over this word. Considered a career in
> postmodern critical theory?

So are you telling me that things didn't become a lot clearer for you
when you figured out that a val was in fact a prime mapping (for
purposes of tuning theory)? When, incidentally, did you figure that out?

> At the very least, I'd hope understand what vals are good
> for before trying to rename them.

I believe I do understand what they are good for
_in_relation_to_tuning_, which is surely what matters for this list?

> Or maybe you understand
> why the 11-limit has no standard val, and can explain it
> to the rest of us.

I don't thing anyone is saying the 11-limit has no standard prime
mapping. That doesn't make sense.

I believe the discussion you're referring to is about the 11-limit
complete otonality. And the claim is not limited to standard mappings,
but any mappings at all.

I believe the claim is that there is no prime mapping that will map
the pitches of the 11-limit complete otonality, in any voicing, to
consecutive degrees of 6-tET.

Why 6-ET? Because that's how many pitches are in the chord.

Why is this interesting? Because there _is_ a mapping that maps some
voicing of the 3-limit complete otonality to consecutive degrees of
2-ET, and there's one that maps the 5-limit otonality to 3-ET, 7-limit
otonality to 4-ET, 9-limit to 5-tET and 13-limit to 7-tET. And in each
case it happens to be the "standard" mapping that does it.

The "standard" mapping for a tET is the one that gives the best
approximation to each prime number (and its octave equivalents). It
doesn't guarantee the best approximations to other ratios with
_combinations_ of primes. e.g. At the 5-limit, if some tET is
inconsistent, the "standard" mapping will give the best approximation
to 5/4 and 3/2 but not 5/3.

You can calculate the coefficient for prime p in the "standard"
mapping for n-tET as round(n*ln(p)/ln(2)).

>So are you telling me that things didn't become a lot clearer
>for you when you figured out that a val was in fact a prime
>mapping (for purposes of tuning theory)? When, incidentally,
>did you figure that out?

Months ago, when Gene showed me how to use his Maple routines
to find linear temperaments from a pair of vals.

>I don't thing anyone is saying the 11-limit has no standard prime
>mapping. That doesn't make sense.
//
>I believe the discussion you're referring to is about the 11-limit
>complete otonality.

Right, "limit" means odd-limit unless it's "prime-limit", as
established by Partch and Erlich.

>And the claim is not limited to standard mappings,
>but any mappings at all.

According to Gene, Gram and other vals may get around this 'problem'.

>I believe the claim is that there is no prime mapping that will map
>the pitches of the 11-limit complete otonality, in any voicing, to
>consecutive degrees of 6-tET.
>
>Why 6-ET? Because that's how many pitches are in the chord.

Why consecutive?

>Why is this interesting? Because there _is_ a mapping that maps some
>voicing of the 3-limit complete otonality to consecutive degrees of
>2-ET,

It would have to be consecutive.

>and there's one that maps the 5-limit otonality to 3-ET,

Consecutive?

>The "standard" mapping for a tET is the one that gives the best
>approximation to each prime number (and its octave equivalents).

2 is a prime, so octaves are included. But this doesn't mention
anything about consecuity (or ordering of any kind). And it
doesn't include why we care that the number of notes in an octave
equals the number of notes in the chord. And it only defines vals
for prime limits, not for odd limits.

>It
>doesn't guarantee the best approximations to other ratios with
>_combinations_ of primes. e.g. At the 5-limit, if some tET is
>inconsistent, the "standard" mapping will give the best approximation
>to 5/4 and 3/2 but not 5/3.
>
>You can calculate the coefficient for prime p in the "standard"
>mapping for n-tET as round(n*ln(p)/ln(2)).

Gene's already given that.

-Carl

hi Dave and Gene,

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

> I notice that even Paul, who along with Graham, appears
> to understand your stuff better than any of us,

that's the impression i have too.

i think Paul Hjelmstad and Carl Lumma can grasp a bit
of Gene's stuff too. more than me, anyway.

> was gently suggesting a similar thing in another thread
> (although I thought he caved in rather too easily).

Gene, i hope you don't feel like i too am joining the
bandwagon to deride you. in any case, i always try hard
to temper what i write in an opposing debate with some
compassion for the person on the other end.

but i must agree that while i feel (intuitively) that your
contribution to tuning theory in the last two years has
been truly mighty, i actually understand very little of
what you're doing and describing, and i unfortunately also
have to agree that the reason for that lack of understanding
is primarily the *way* you describe your work.

i applaud your eagerness to post definitions of terms as
you need to coin them, and i lemming-like pop them into
the Tuning Dictionary ... but they really don't help me
(nor, apparently, many others among us).

i'm anticipating that working with the software i now
have under development, upon its first release, will
aid me in comprehending your (Gene's) ideas, and i look
forward to your criticisms and comments on it after
you try it.

-monz

hi Gene and Dave,

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

> Vals are an important concept and deserve a name. Why is
> this so painful? I admit my names are not always terrific
> (eg "standard val") and some of them (eg "icon") I haven't
> even attempted to inflict on people here, while others
> (eg "notation") have generated no support, but I really am
> not interested in using an inferior name for an inferior
> definition. Why insist that everything must be done your way?

Gene, Dave can't help it ... he's a systems designer.

Dave, i can go along with Gene to some extent. that's
why the Tuning Dictionary is there. i understand your
desire for standardization, and agree with that too.

but so what if there are two different terms which mean
*very nearly* the same thing, but not exactly. i'll put
them both in the Dictionary and you guys provide me with
proper definitions that show exactly what they have in
common and exactly where they differ.

i've been thru this same sort of thing myself with
my term "xenharmonic bridge" and its similarity to
Fokker's "unison vector".

but despite the extreme similarity between the two,
i perceive a distinction (and i grant the possibility
that i could be wrong about that too).

so, they both go into the Dictionary and their
definitions both grow as i learn more and more about
them.

what's the problem?

i think both of you guys need to chill out a little.

:)

-monz

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

> I believe the discussion you're referring to is about the
> 11-limit complete otonality. And the claim is not limited
> to standard mappings, but any mappings at all.
>
> I believe the claim is that there is no prime mapping
> that will map the pitches of the 11-limit complete otonality,
> in any voicing, to consecutive degrees of 6-tET.
>
> Why 6-ET? Because that's how many pitches are in the chord.
>
> Why is this interesting? Because there _is_ a mapping that
> maps some voicing of the 3-limit complete otonality to
> consecutive degrees of 2-ET, and there's one that maps
> the 5-limit otonality to 3-ET, 7-limit otonality to 4-ET,
> 9-limit to 5-tET and 13-limit to 7-tET. And in each
> case it happens to be the "standard" mapping that does it.
>
> The "standard" mapping for a tET is the one that gives the
> best approximation to each prime number (and its octave
> equivalents). It doesn't guarantee the best approximations
> to other ratios with _combinations_ of primes. e.g. At the
> 5-limit, if some tET is inconsistent, the "standard" mapping
> will give the best approximation to 5/4 and 3/2 but not 5/3.
>
> You can calculate the coefficient for prime p in the "standard"
> mapping for n-tET as round(n*ln(p)/ln(2)).

this is the first time this has been explained in a way
that made sense to me.

and this is good enough to put in the Tuning Dictionary
... but under what definition? should "standard mapping"
be a Dictionary entry? or is this an amendment to "val"?

-monz

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >So are you telling me that things didn't become a lot clearer
> >for you when you figured out that a val was in fact a prime
> >mapping (for purposes of tuning theory)? When, incidentally,
> >did you figure that out?
>
> Months ago, when Gene showed me how to use his Maple routines
> to find linear temperaments from a pair of vals.

So did it become a lot clearer what a val was, when you figured out
that it was a prime mapping?

Did you understand what they were mapping the primes _to_ at that time?

> >I don't thing anyone is saying the 11-limit has no standard prime
> >mapping. That doesn't make sense.
> //
> >I believe the discussion you're referring to is about the 11-limit
> >complete otonality.
>
> Right, "limit" means odd-limit unless it's "prime-limit", as
> established by Partch and Erlich.

Maybe so, but that's not why it doesn't make sense. It doesn't make
sense because limits don't have mappings. Temperaments have mappings.
These mapings are usually limited to some maximum prime. So mappings
have prime limits, not the other way 'round.

> >And the claim is not limited to standard mappings,
> >but any mappings at all.
>
> According to Gene, Gram and other vals may get around this 'problem'.

I missed the definition of a "Gram" mapping.

> >I believe the claim is that there is no prime mapping that will map
> >the pitches of the 11-limit complete otonality, in any voicing, to
> >consecutive degrees of 6-tET.
> >
> >Why 6-ET? Because that's how many pitches are in the chord.
>
> Why consecutive?

I think it may relate to the complete otonality being a constant
structure. But we'll have to wait until Gene tells me if I got it
right. Then he can tell you why consecutive.

> >Why is this interesting? Because there _is_ a mapping that maps some
> >voicing of the 3-limit complete otonality to consecutive degrees of
> >2-ET,
>
> It would have to be consecutive.

Why so? It should be easy to find a voicing of the 1:3 chord, or a
3-limit 2-tET mapping, that has an unused pitch between the two
pitches of the chord. A 1:3 voicing would be such a voicing, with the
standard mapping <2 3]. I didn't say anything about restricting
ourselves to one octave. However, if I did say we had to fit within
the octave then I wouldn't need to say consecutive for any of them. So
it may well have been the octave limitation that Gene had in mind,
rather than the consecutive steps, since they are equivalent restrictions.

> >and there's one that maps the 5-limit otonality to 3-ET,
>
> Consecutive?

Why not? The standard 5-limit mapping for 3-tET is <3 5 7]. i.e
round(n*ln(p)/ln(2)), for n = 3 and p = 2, 3, and 5.

Consider a 4:5:6 voicing of the complete 5-limit otonality. Convert
each pitch to a prime exponent vector. That gives us [2 0 0]:[0 0
1]:[1 1 0]. Now take the dot product of the mapping with each pitch in
turn.

<3 5 7].[2 0 0> = 3*2 + 5*0 + 7*0 = 6
<3 5 7].[0 0 1> = 3*0 + 5*0 + 7*1 = 7
<3 5 7].[1 1 0> = 3*1 + 5*1 + 7*0 = 8

So we see we have consecutive numbers of steps. No gaps. But as I said
we could instead just say we must use every pitch-class exactly once.
It's the same game.

> >The "standard" mapping for a tET is the one that gives the best
> >approximation to each prime number (and its octave equivalents).
>
> 2 is a prime, so octaves are included.

Sure. It's only because we've restricted ourselves to equal divisions
of the octave that octave-equivalents also get their best
approximations. I probably shouldn't have even mentioned it.

> But this doesn't mention
> anything about consecuity (or ordering of any kind). And it
> doesn't include why we care that the number of notes in an octave
> equals the number of notes in the chord.

You'll have to ask Gene about that. If I've got that right.

> And it only defines vals
> for prime limits, not for odd limits.

A mapping with an odd-limit of n has a number of steps (or generators)
for every prime <= n. So the standard 9-limit mapping for some ET is
identical to its standard 7-limit mapping.

Alternatively we could decide to say that there is no such thing as an
odd-limit mapping - that the limit of a mapping is always considered
to be its largest prime.

There are of course odd-limit temperaments because that's talking
about what ratios we want to optimise the approximations of.

>> >So are you telling me that things didn't become a lot clearer
>> >for you when you figured out that a val was in fact a prime
>> >mapping (for purposes of tuning theory)? When, incidentally,
>> >did you figure that out?
>>
>> Months ago, when Gene showed me how to use his Maple routines
>> to find linear temperaments from a pair of vals.
>
>So did it become a lot clearer what a val was, when you figured
>out that it was a prime mapping?

Everything I've ever figured out about vals made them clearer,
obviously. The words "prime mapping" wouldn't have helped a bit.

Terminology has absolutely nothing to do with why Gene's writing
is utterly incomprehensible to me. I think it has more to do
with:

() Assumes too much prior math skill. This doesn't mean jargon,
which I can look up. It's something else. I can often perform
text substitution on his writing and still be either utterly
lost or left with something that is utterly obvious (but useful
in some aspect I'm not seeing).
() Unable or unwilling to explain the same thing with multiple
tools (ie math, natural language, diagrams).
() Is unable or unwilling to explain the same thing in multiple
ways.
() Unable or unwilling to give reasoning/motivation behind
definitions.

>Did you understand what they were mapping the primes _to_ at that
>time?

I still don't think I do.

>> >I don't thing anyone is saying the 11-limit has no standard prime
>> >mapping. That doesn't make sense.
>> //
>> >I believe the discussion you're referring to is about the 11-limit
>> >complete otonality.
>>
>> Right, "limit" means odd-limit unless it's "prime-limit", as
>> established by Partch and Erlich.
>
>Maybe so, but that's not why it doesn't make sense. It doesn't make
>sense because limits don't have mappings. Temperaments have mappings.
>These mapings are usually limited to some maximum prime. So mappings
>have prime limits, not the other way 'round.

But Gene's talking about finding vals for limits!!!

>> According to Gene, Gram and other vals may get around this 'problem'.
>
>I missed the definition of a "Gram" mapping.

You're not the only one.

>> >Why is this interesting? Because there _is_ a mapping that maps some
>> >voicing of the 3-limit complete otonality to consecutive degrees of
>> >2-ET,
>>
>> It would have to be consecutive.
>
>Why so?

Because there are only 2 elements!

>I didn't say anything about restricting ourselves to one octave.

Then the standard 5-limit 3-val that Gene gave isn't consecutive.

>> >and there's one that maps the 5-limit otonality to 3-ET,
>>
>> Consecutive?
>
>Why not? The standard 5-limit mapping for 3-tET is <3 5 7]. i.e
>round(n*ln(p)/ln(2)), for n = 3 and p = 2, 3, and 5.

Note that I have no idea what the bra ket notation stuff is about.

>> But this doesn't mention
>> anything about consecuity (or ordering of any kind). And it
>> doesn't include why we care that the number of notes in an octave
>> equals the number of notes in the chord.
>
>You'll have to ask Gene about that. If I've got that right.
>
>> And it only defines vals
>> for prime limits, not for odd limits.
>
>A mapping with an odd-limit of n has a number of steps (or generators)
>for every prime <= n. So the standard 9-limit mapping for some ET is
>identical to its standard 7-limit mapping.

Then I don't know why the standard 11-limit mapping wouldn't be
identical to the standard 11-prime-limit mapping. Anyway, saying
mapping instead of val is already confusing me here.

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > > Where did 3 come from?
> >
> > A division of the octave into three parts, or in other words, a
> > mapping of 2 to 3.
>
> excuse me, but i think the answer to carl's question is "the
complete
> 5-limit otonal chord has *3* notes". right?

Maybe I misunderstood the question.

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

> What confuses the hell out of me is that Gene keeps using
> the word "column" re. vals, but they don't give successive
> approximations to the same prime, they give a single
> approx. to various primes.

I don't understand this. The column business is to tell you they are
dual to intervals. Calling vals "bra vectors" and monzos "ket
vectors" does this also, and comes complete with a notational
convention easier to deal with that row vs column vectors, so I
suggest shifting to that.

> At the very least, I'd hope understand what vals are good
> for before trying to rename them.

To start with, I wanted intervals and vals to work in precisely the
same way in terms of the wedge product, and hence my particular
definition. But, obviously, we are always talking about these things
implicitly, and so why not name them? And if we name them, why not
give a precise, mathematically correct defintion at some point, even
if we don't normally need to sweat the details?

Or maybe you understand
> why the 11-limit has no standard val, and can explain it
> to the rest of us.

Paul seems to have figured that out. 11-limit complete otonal and
utonal chords just don't work in a consistent way.

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

> I think we have quite complementary skills. You come up with the the
> math tools and methods and I may _eventually_ be able to understand
> them enough to put them into terms that others on this list can more
> easily understand. But I don't think you should worry too much if my
> explanations or recasting of terminology misses some of the more
> subtle points as far as the pure mathematician is concerned, at
least
> on a first pass.

Sounds reasonable, but I don't think you should worry to much if I
want to make precise mathematical definitions for things, or make the
definitions the way they are for reasons not immediately apparent to
you.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I'm just disappointed we got this far with "val"
>
> ...with only 1 -- two if we're lucky -- persons who know
> how to use them for what they're capable of.

This hardly is the case.

> Gene, since you won't say what's desirable about being a
> standard val...

Purely a matter of being easy to calculate.

and you haven't said what the lack of a
> standard 11-limit val means about the 11-limit...

It's not a *standard* 11-limit val, but one associated to 11-limit
complete harmony. I haven't given it a name. At this point, I don't
know if I should even consider doing such a thing.

What it means for the 11-limit is that a systematic way of looking at
harmony by reducing it to an approximation plus a chord description
does not work in the 11-limit, but does work in the 3, 5, 7, 9 and 13
limits.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >And the claim is not limited to standard mappings,
> >but any mappings at all.
>
> According to Gene, Gram and other vals may get around
this 'problem'.

No, you get inconsistency no matter what you try. The way around it
would be to not use complete 11-limit chords, or to follow up on your
suggestion and scramble the ordering, but that means abandoning the
problem as stated as impossible.

> >I believe the claim is that there is no prime mapping that will map
> >the pitches of the 11-limit complete otonality, in any voicing, to
> >consecutive degrees of 6-tET.
> >
> >Why 6-ET? Because that's how many pitches are in the chord.
>
> Why consecutive?

Your going to get a dog's breakfast otherwise, but we could try the
idea, I suppose.

> >and there's one that maps the 5-limit otonality to 3-ET,
>
> Consecutive?

5/4-3/2-2

> >You can calculate the coefficient for prime p in the "standard"
> >mapping for n-tET as round(n*ln(p)/ln(2)).
>
> Gene's already given that.

It's always helpful to repeat things anyway, considering the
communication problems I seem to engender. I thought Dave's article
was nicely clear.

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

> I think it may relate to the complete otonality being a constant
> structure. But we'll have to wait until Gene tells me if I got it
> right. Then he can tell you why consecutive.

Right; or that it is epimorphic. Put it into Scala and it will tell
you both.

> Alternatively we could decide to say that there is no such thing as
an
> odd-limit mapping - that the limit of a mapping is always considered
> to be its largest prime.

Which can map to 0.

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

> > > [Dave Keenan:]
> > > You can calculate the coefficient for prime p in
> > > the "standard" mapping for n-tET as round(n*ln(p)/ln(2)).
> >
> > [Carl Lumma:]
> > Gene's already given that.
>
> It's always helpful to repeat things anyway, considering
> the communication problems I seem to engender. I thought
> Dave's article was nicely clear.

amen! by all means, *please* repeat stuff in as many
different ways as possible! it sure helps me to understand.

-monz

hi Gene,

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

> > and you haven't said what the lack of a
> > standard 11-limit val means about the 11-limit...
>
> It's not a *standard* 11-limit val, but one associated
> to 11-limit complete harmony. I haven't given it a name.
> At this point, I don't know if I should even consider
> doing such a thing.
>
> What it means for the 11-limit is that a systematic way
> of looking at harmony by reducing it to an approximation
> plus a chord description does not work in the 11-limit,
> but does work in the 3, 5, 7, 9 and 13 limits.

i'm getting hopelessly confused about this, but it seems
like something i'd really like to understand.

can you *please* give a very detailed explanation of what
you're saying? ... with lots and lots of 11-limit examples
that don't work and 3-, 5-, 7-, 9-, 13-limit examples that do?

thanks.

i've renamed the subject header in anticipation that this
will become a big thread in its own right, but unfortunately
since you haven't named this phenomenon it's an ugly header.
feel free to rename the subject line if you define a good name.

-monz

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

> can you *please* give a very detailed explanation of what
> you're saying? ... with lots and lots of 11-limit examples
> that don't work and 3-, 5-, 7-, 9-, 13-limit examples that do?
>
> thanks.

Here are the 5, 7, 9, 11 and 13 limit complete otonal chords as Scala
scale files. If you run "data" on them, you will find that 5, 7, 9
and 13 give Constant Structure scales, and 11 does not. You will also
find stuff about "JI epimorphic", but I don't understand what Manuel
is up to; it isn't what I expected.

! fivelim.scl
!
Five-limit otonal chord
3
!
5/4
3/2
2

! sevenlim.scl
!
Seven-limit otonal chord
4
!
5/4
3/2
7/4
2

! ninelim.scl
!
Nine-limit otonal chord
5
!
9/8
5/4
3/2
7/4
2

! elevenlim.scl
!
Eleven-limit otonal chord
6
!
9/8
5/4
11/8
3/2
7/4
2

! thirteenlim.scl
!
Thirteen-limit otonal chord
7
!
9/8
5/4
11/8
3/2
13/8
7/4
2

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

> >The "standard" mapping for a tET is the one that gives the best
> >approximation to each prime number (and its octave equivalents).
>
> 2 is a prime, so octaves are included. But this doesn't mention
> anything about consecuity (or ordering of any kind).

it does imply it in most cases, since the primes are already in
order, so the best approximations to them will have to be a non-
decreasing sequence.

> And it
> doesn't include why we care that the number of notes in an octave
> equals the number of notes in the chord.

george apparently cares about this very much too -- his observation
about the 11-limit hexad not being CS implies gene's observation.

> And it only defines vals
> for prime limits, not for odd limits.

all you need is the previous point.

monz, this is a very specific, and perhaps even unusual, application
of the val or mapping concept. it may warrant mention in an
Encyclopedia but probably not in a dictionary.

p.s. would it be OK for me to attempt a modification of your page

http://www.sonic-arts.org/dict/edo-prime-error.htm

? i realized that you *do* have the signed errors of the primes in
the text, despite your use of absolute values in the graph. so if i
just added the signed errors for the odds that you omitted, i could
then quickly locate any inconsistency, since inconsistency occurs if
and only if the signed relative error of one odd differs by over 50%
from the signed relative error of another odd. for example, in 43-
equal, the error on 7 is, as you show, +28%; the error on 9 is double
that on 3, so about -30%; the difference between these two signed
percentages (and thus the implied error on 9:7) is 58%; so 43-equal
is inconsistent in the 9-limit. what do you think? i think the page
would be sorely misleading, and much less useful, without this
information.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > I believe the discussion you're referring to is about the
> > 11-limit complete otonality. And the claim is not limited
> > to standard mappings, but any mappings at all.
> >
> > I believe the claim is that there is no prime mapping
> > that will map the pitches of the 11-limit complete otonality,
> > in any voicing, to consecutive degrees of 6-tET.
> >
> > Why 6-ET? Because that's how many pitches are in the chord.
> >
> > Why is this interesting? Because there _is_ a mapping that
> > maps some voicing of the 3-limit complete otonality to
> > consecutive degrees of 2-ET, and there's one that maps
> > the 5-limit otonality to 3-ET, 7-limit otonality to 4-ET,
> > 9-limit to 5-tET and 13-limit to 7-tET. And in each
> > case it happens to be the "standard" mapping that does it.
> >
> > The "standard" mapping for a tET is the one that gives the
> > best approximation to each prime number (and its octave
> > equivalents). It doesn't guarantee the best approximations
> > to other ratios with _combinations_ of primes. e.g. At the
> > 5-limit, if some tET is inconsistent, the "standard" mapping
> > will give the best approximation to 5/4 and 3/2 but not 5/3.
> >
> > You can calculate the coefficient for prime p in the "standard"
> > mapping for n-tET as round(n*ln(p)/ln(2)).
>
>
>
> this is the first time this has been explained in a way
> that made sense to me.
>
> and this is good enough to put in the Tuning Dictionary
> ... but under what definition? should "standard mapping"
> be a Dictionary entry? or is this an amendment to "val"?
>
>
>
> -monz

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

> But Gene's talking about finding vals for limits!!!

Come on Carl, this is no more true than that my Hypothesis concerned
temperaments.

> >I didn't say anything about restricting ourselves to one octave.
>
> Then the standard 5-limit 3-val that Gene gave isn't consecutive.

whaaaa???

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > > Where did 3 come from?
> > >
> > > A division of the octave into three parts, or in other words, a
> > > mapping of 2 to 3.
> >
> > excuse me, but i think the answer to carl's question is "the
> complete
> > 5-limit otonal chord has *3* notes". right?
>
> Maybe I misunderstood the question.

the question was "why 3"? your answer was not an answer at all, it
just assumed the number three all over again, without saying where it
came from.

maybe george would be better equipped to explain why this is
musically so significant.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > can you *please* give a very detailed explanation of what
> > you're saying? ... with lots and lots of 11-limit examples
> > that don't work and 3-, 5-, 7-, 9-, 13-limit examples that do?
> >
> > thanks.
>
> Here are the 5, 7, 9, 11 and 13 limit complete otonal chords as
Scala
> scale files. If you run "data" on them, you will find that 5, 7, 9
> and 13 give Constant Structure scales, and 11 does not. You will
also
> find stuff about "JI epimorphic", but I don't understand what
Manuel
> is up to; it isn't what I expected.
>
> ! fivelim.scl
> !
> Five-limit otonal chord
> 3
> !
> 5/4
> 3/2
> 2
>
>
> ! sevenlim.scl
> !
> Seven-limit otonal chord
> 4
> !
> 5/4
> 3/2
> 7/4
> 2
>
>
> ! ninelim.scl
> !
> Nine-limit otonal chord
> 5
> !
> 9/8
> 5/4
> 3/2
> 7/4
> 2
>
>
> ! elevenlim.scl
> !
> Eleven-limit otonal chord
> 6
> !
> 9/8
> 5/4
> 11/8
> 3/2
> 7/4
> 2
>
>
> ! thirteenlim.scl
> !
> Thirteen-limit otonal chord
> 7
> !
> 9/8
> 5/4
> 11/8
> 3/2
> 13/8
> 7/4
> 2

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > I think we have quite complementary skills. You come up with the the
> > math tools and methods and I may _eventually_ be able to understand
> > them enough to put them into terms that others on this list can more
> > easily understand. But I don't think you should worry too much if my
> > explanations or recasting of terminology misses some of the more
> > subtle points as far as the pure mathematician is concerned, at
> least
> > on a first pass.
>
> Sounds reasonable, but I don't think you should worry to much if I
> want to make precise mathematical definitions for things, or make the
> definitions the way they are for reasons not immediately apparent to
> you.

It's a deal. :-)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > can you *please* give a very detailed explanation of what
> > you're saying? ... with lots and lots of 11-limit examples
> > that don't work and 3-, 5-, 7-, 9-, 13-limit examples that do?
> >
> > thanks.
>
> Here are the 5, 7, 9, 11 and 13 limit complete otonal chords as Scala
> scale files. If you run "data" on them, you will find that 5, 7, 9
> and 13 give Constant Structure scales, and 11 does not.
...

Bravo! Gene, this is an excellent minimal-math explanation!

It certainly is a curious fact. And I'm looking forward to hearing
from George, why he thinks it matters so much musically that he
wouldn't consider using an 11-limit tuning.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> maybe george would be better equipped to explain why this is
> musically so significant.

Could be, though I've made very extensive use of it as a
compositional tool. That goes all the way back to my 1980s paper.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> [Dave Keenan:]
> >So did it become a lot clearer what a val was, when you figured
> >out that it was a prime mapping?
>
> Everything I've ever figured out about vals made them clearer,
> obviously. The words "prime mapping" wouldn't have helped a bit.

This seems like a logical contradiction. You say that figuring out
that vals were prime mappings made them clearer. Wouldn't calling them
"prime mappings" have helped you figure out that they were prime
mappings? And wouldn't this have happened sooner if they had been
called prime mappings instead of vals?

I guess what you're saying is that this wasn't a very important factor
in your understanding of the concept.

> >Did you understand what they were mapping the primes _to_ at that
> >time?
>
> I still don't think I do.

A prime-mapping (or val with log-prime basis) simply maps each prime
number (or strictly-speaking the logarithm of each prime number) to an
integer multiple of some interval (log of frequency ratio) that we
call a generator.

If we are told that the mapping is for a tET then _which_ tET it is
for can be read straight out of the mapping, as the coefficient for
the prime 2 (the first coefficient). And the generator is simply one
step of that tET.

If we were told it was for an ET3 then we could read off what ET3 it
was for as the second coefficient, and the generator is that fraction
of the tritave.

If however we are told that the mapping is for an arbitrary equal
temperament, a cET, then we would have to solve for the generator that
minimises some error function such as max-absolute (minimax) or rms
(sum of squares). It's fairly simple to do this numerically in Excel
using the Solver add-in. Let me know if you want more details on that.

In an ET, the generator is the step. But not necessarily so in higher
D temperaments

If we were told the mapping was one row (Gene says we can forget that
"column" stuff) of a linear temperament mapping, then to solve for the
generator that this row maps to, we would either need to know what the
other generator was, or what its mapping was, e.g. maybe the other
generator is the period and we are told it is an exact octave.

> But Gene's talking about finding vals for limits!!!

He's just abbreviating excessively and assuming the meaning will be
clear from your readings of his previous postings in the same thread.
He's really talking about finding vals-with-log-prime-basis
(prime-mappings) that map the complete chord of each limit to a tET
with the same cardinality. It's all about how evenly-spaced the chords
are.

Try the 6 possible possible voicings of the 11-limit otonality, that
fit within an octave, and you'll see that none of them are very even.

Why such an apparently melodic property should be considered important
when applied to a vertical harmony, I don't know.

> Note that I have no idea what the bra ket notation stuff is about.

It's just a way of distinguishing prime-mappings (vals) from
prime-exponent-vectors (monzos) without having to say it in words
every time. It only makes sense to multiply mappings by
exponent-vectors, not any other combination and these brackets try to
make that clear because ] and [ fit together, but > and <, > and [, ]
and < do not.

> Then I don't know why the standard 11-limit mapping wouldn't be
> identical to the standard 11-prime-limit mapping.

It is.

> Anyway, saying mapping instead of val is already confusing me here.

Sorry. I've given both in several places above.

As far as tuning is concerned the only important difference I can see
is that in the case of temperaments with more than one generator, "the
mapping" (unqualified) refers to the whole matrix (all the rows, one
per generator). There's no such thing as "the val" for such a
temperament. In this case a val is apparently only one row. But even
there, "the val for generator x" is the same as "the mapping for
generator x". So the term val isn't actually needed.

>> Gene, since you won't say what's desirable about being a
>> standard val...
>
>Purely a matter of being easy to calculate.

Adding our birthdays together is easy to calculate. There
must be some other reason. Dave's 'the best approx. to each
element of a chord in n-tET' is better, but why n should equal
the number of notes in a chord is still a mystery.

-Carl

>Here are the 5, 7, 9, 11 and 13 limit complete otonal chords as Scala
>scale files. If you run "data" on them, you will find that 5, 7, 9
>and 13 give Constant Structure scales, and 11 does not. You will also
>find stuff about "JI epimorphic", but I don't understand what Manuel
>is up to; it isn't what I expected.

So there's no val that sends all 11-limit intervals to integers
without collisions?

-Carl

>A prime-mapping (or val with log-prime basis) simply maps each prime
>number (or strictly-speaking the logarithm of each prime number) to an
>integer multiple of some interval (log of frequency ratio) that we
>call a generator.
>
>If we are told that the mapping is for a tET then _which_ tET it is
>for can be read straight out of the mapping, as the coefficient for
>the prime 2 (the first coefficient). And the generator is simply one
>step of that tET.

Yes, I know this. But why integers? And why can't there be collisions?
And in what sense could the order in which the identities of a chord
are considered have any bearing on things?

>> But Gene's talking about finding vals for limits!!!
>
>He's just abbreviating excessively and assuming the meaning will be
>clear from your readings of his previous postings in the same thread.
>He's really talking about finding vals-with-log-prime-basis
>(prime-mappings) that map the complete chord of each limit to a tET
>with the same cardinality.

I finally got that. Why the same card.?

>Try the 6 possible possible voicings of the 11-limit otonality, that
>fit within an octave, and you'll see that none of them are very even.

It's proper and for that matter seems to fit to 6-tET reasonably
well.

>> Note that I have no idea what the bra ket notation stuff is about.
>
>It's just a way of distinguishing prime-mappings (vals) from
>prime-exponent-vectors (monzos) without having to say it in words
>every time. It only makes sense to multiply mappings by
>exponent-vectors, not any other combination and these brackets try to
>make that clear because ] and [ fit together, but > and <, > and [, ]
>and < do not.

So are monzos are now kets written [ ... > ?
and vals are bras written < ... ] ?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Gene, since you won't say what's desirable about being a
> >> standard val...
> >
> >Purely a matter of being easy to calculate.
>
> Adding our birthdays together is easy to calculate. There
> must be some other reason.

No. That's really it. Although I suppose we should say "and it is also
the best (i.e. most accurate) mapping when the ET is consistent. So
you'd only be interested in mappings other than this easy-to-calculate
one, if the ET is inconsistent at the given limit, and we are rarely
interested in those anyway, so you can get away with it most of the
time. It's a lot more complicated to find the best mapping for an
inconsistent ET.

> Dave's 'the best approx. to each
> element of a chord in n-tET' is better,

Err. I don't remember writing that. The "standard" mapping for a tET
gives the best approximation to intervals whose ratio contains a
single prime number other than 2 (with that odd-prime raised only to
the first power).

You've certainly gotten hold of some wrong idea about all this, and I
don't blame you. I just wish I knew what it is.

Standard vals (standard mappings), or vals (mappings) of any kind,
have absolutely nothing to do with chords. Except that you can apply
vals (mappings) to rational pitches to see where they end up in the
temperament that the val (mapping) corresponds to. And chords can
contain rational pitches, so vals (mappings) can be applied to chords
to see where they end up when so tempered.

A val (mapping) for some temperament simply tells you how to map a
rational pitch to iterations of a single generator of that temperament
(which is the _only_ generator in the case of an ET, and is its step).

> but why n should equal
> the number of notes in a chord is still a mystery.

It's pretty much of a mystery to me too. This is not a necessary
property of vals or even of standard vals. It has nothing to do with them.

It's just that Gene and George found it interesting to look at how
complete chords can be mapped to a single octave of the ET of the same
cardinality as the chord. It turns out that the 11-limit otonality
can't be. There is no mapping and no voicing of the chord that will do
this.

>> But Gene's talking about finding vals for limits!!!
>
>Come on Carl, this is no more true than that my Hypothesis concerned
>temperaments.

Gene did use those words, apparently abbreviating excessively. I'm
closer to what he meant now, but I have no idea what you're referring
to re. the Hypothesis. It clearly concerns temperaments, since it
states things about what happens when you temper uvs out of a PB.

>> >I didn't say anything about restricting ourselves to one octave.
>>
>> Then the standard 5-limit 3-val that Gene gave isn't consecutive.
>
>whaaaa???

Gene's going from smallest to largest interval, though as I just
Confessed, I have no idea what order has to do with anything.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Here are the 5, 7, 9, 11 and 13 limit complete otonal chords as Scala
> >scale files. If you run "data" on them, you will find that 5, 7, 9
> >and 13 give Constant Structure scales, and 11 does not. You will also
> >find stuff about "JI epimorphic", but I don't understand what Manuel
> >is up to; it isn't what I expected.
>
> So there's no val that sends all 11-limit intervals to integers
> without collisions?

Assuming you mean 11-odd-limit intervals then that's not true, and its
not what's being discussed.

The "integers" referred to are simply the degrees of some ET, and
clearly there are ETs and mappings of primes to them, that give a
unique degree for every interval in the diamond, for any odd limit.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Why the same card.?

Dunno. But it's too easy otherwise.

> >Try the 6 possible possible voicings of the 11-limit otonality, that
> >fit within an octave, and you'll see that none of them are very even.
>
> It's proper and for that matter seems to fit to 6-tET reasonably
> well.

OK. Well maybe I got something wrong about what's being claimed. I'd
better let the experts sort it out.

> So are monzos are now kets written [ ... > ?
> and vals are bras written < ... ] ?

That's Gene's proposal (that we should write them that way, not that
we should call them bras and kets) and it seems like a reasonable one
to me. Except I wonder how we should write a complete mapping matrix
for a more-than-1D temperament. I suppose
< ......
......
...... ]
would be best, although to put them on one line we could use
< ...... ; ...... ; ...... ]

>I suppose we should say "and it is also
>the best (i.e. most accurate) mapping when the ET is consistent.

Which fits with:
>> Dave's 'the best approx. to each
>> element of a chord in n-tET' is better,

>> Dave's 'the best approx. to each
>> element of a chord in n-tET' is better,
>
>Err. I don't remember writing that.

You wrote "primes" not 'elements of any chord'.

>Standard vals (standard mappings), or vals (mappings) of any kind,
>have absolutely nothing to do with chords. Except that you can apply
>vals (mappings) to rational pitches to see where they end up in the
>temperament that the val (mapping) corresponds to.

So far so good.

>> but why n should equal
>> the number of notes in a chord is still a mystery.
>
>It's pretty much of a mystery to me too. This is not a necessary
>property of vals or even of standard vals. It has nothing to do
>with them.

Ok....

>It's just that Gene and George found it interesting to look at how
>complete chords can be mapped to a single octave of the ET of the same
>cardinality as the chord. It turns out that the 11-limit otonality
>can't be. There is no mapping and no voicing of the chord that will do
>this.

Voicing shouldn't matter, since the voicing of the thing you're
mapping to (an ET) doesn't matter. If I set...

1= 9/8
2= 5/4
3= 11/8
4= 3/2
5= 7/4
6= 2/1

...can you show me the problem? Lessee, would the val would be
(the parens pending clarification on the ketbra situation)...

(val 0 10 14 17 21)

and reversing the process to get the above I only need to worry
about 9/8, which is (monzo -3 2 0 0 0). It looks like this gives
me a 20-18 = 2, which is supposed to be 5/4. Is *this* the
problem?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >If we are told that the mapping is for a tET then _which_ tET it is
> >for can be read straight out of the mapping, as the coefficient for
> >the prime 2 (the first coefficient). And the generator is simply one
> >step of that tET.
>
> Yes, I know this. But why integers?

Because that's what a temperament is. A mapping from rational pitches
to integer numbers of some (usually irrational) generator (or
generators). If they didn't have to be integers then there would be
nothing to it.

> And why can't there be collisions?

I don't know what you're referring to here. I don't think I said
anything about collisions. I suspect that's just part of the
particular investigation. It's certainly nothing to do with vals or
mappings per se. They can certainly result in collisions.

> And in what sense could the order in which the identities of a chord
> are considered have any bearing on things?

If you bring everything back to the first octave then probably no
bearing at all.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >cardinality as the chord. It turns out that the 11-limit otonality
> >can't be. There is no mapping and no voicing of the chord that will
> >do this.
>
> Voicing shouldn't matter, since the voicing of the thing you're
> mapping to (an ET) doesn't matter.

You're probably right, forget about voicing.

> If I set...
>
> 1= 9/8
> 2= 5/4
> 3= 11/8
> 4= 3/2
> 5= 7/4
> 6= 2/1
>
> ...can you show me the problem? Lessee, would the val would be
> (the parens pending clarification on the ketbra situation)...
>
> (val 0 10 14 17 21)

Except for the typo where you have a 0 instead of a 6. Yes
<6 10, 14 17 21] is the standard prime-mapping for 6-tET.

> and reversing the process to get the above I only need to worry
> about 9/8, which is (monzo -3 2 0 0 0). It looks like this gives
> me a 20-18 = 2,

Yes, we can write <6 10, 14 17 21].[-3 2, 0 0 0> = 2 (step generators).

> which is supposed to be 5/4. Is *this* the
> problem?

Yes! Congratulations!

So it's proper, but not a constant structure. I was under the
misapprehension that proper always implied constant structure, i.e
that propriety was a stronger condition. Hmm.

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> monz, this is a very specific, and perhaps even unusual,
> application of the val or mapping concept. it may warrant
> mention in an Encyclopedia but probably not in a dictionary.

well, OK ... i don't have anything about it in the Dictionary
yet, so i guess i'll leave it out, or just include it in
the upcoming Encyclopedia.

> p.s. would it be OK for me to attempt a modification of your page
>
> http://www.sonic-arts.org/dict/edo-prime-error.htm
>
> ? i realized that you *do* have the signed errors of the
> primes in the text, despite your use of absolute values in
> the graph. so if i just added the signed errors for the
> odds that you omitted, i could then quickly locate any
> inconsistency, since inconsistency occurs if and only if
> the signed relative error of one odd differs by over 50%
> from the signed relative error of another odd. for example,
> in 43-equal, the error on 7 is, as you show, +28%; the error
> on 9 is double that on 3, so about -30%; the difference
> between these two signed percentages (and thus the implied
> error on 9:7) is 58%; so 43-equal is inconsistent in the
> 9-limit. what do you think? i think the page would be
> sorely misleading, and much less useful, without this
> information.

paul, you already know that i think that the information
given on my "EDO prime error" page is useful as it is.

but if you envision a better version of the page, sure,
send me the modification. you know i trust your judgment
on tuning matters! :)

-monz

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> A prime-mapping (or val with log-prime basis) simply maps each prime
> number (or strictly-speaking the logarithm of each prime number) to
an
> integer multiple of some interval (log of frequency ratio) that we
> call a generator.

This is absolutely not what I mean by a val, which maps to integers.

> > Note that I have no idea what the bra ket notation stuff is about.
>
> It's just a way of distinguishing prime-mappings (vals) from
> prime-exponent-vectors (monzos) without having to say it in words
> every time.

It also shows the covariant vs contravariant aspect.

>Yes! Congratulations!

Yeah well, the choice of 6 here still hasn't been accounted for.

>So it's proper, but not a constant structure. I was under the
>misapprehension that proper always implied constant structure, i.e
>that propriety was a stronger condition. Hmm.

Nope. The set of all non-CS scales is equivalennt to the set of
all non-strictly-proper scales.

I remember surprise when John Chalmers first pointed this out
to me, by way of this example:

[private communication]
>However, the Enharmonic of Archytas is. Translate the scale 28/27 x
>36/35 x 5/4 x 9/8 x 28/27 x 36/35 x 5/4 into cents and generate the
>D-matrix.
>
>63 49 386 204 63 49 386
>112 435 590 267 112 435 449
>498 639 653 316 498 498 498
>702 702 702 702 561 547 884
>765 751 1088 765 610 933 1088
>814 1137 1151 814 996 1137 1151
>1200 .......

-Carl

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

> It's proper and for that matter seems to fit to 6-tET reasonably
> well.

It's not a Constant Structure.

> So are monzos are now kets written [ ... > ?
> and vals are bras written < ... ] ?

I think that's a good suggestion. It is a standard (especially in
physics), clever notation due to Dirac. We just don't worry about
complex numbers and certainly not about quantum mechanics.

>> So are monzos are now kets written [ ... > ?
>> and vals are bras written < ... ] ?
>
>I think that's a good suggestion. It is a standard (especially in
>physics), clever notation due to Dirac. We just don't worry about
>complex numbers and certainly not about quantum mechanics.

But Dave is right that you weren't suggesting using the names
"bra" and "ket", right?

Carl

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

> Voicing shouldn't matter, since the voicing of the thing you're
> mapping to (an ET) doesn't matter. If I set...
>
> 1= 9/8
> 2= 5/4
> 3= 11/8
> 4= 3/2
> 5= 7/4
> 6= 2/1
>
> ...can you show me the problem?

3/2 is 4 steps, so 9/4 is 8 steps, so 9/8 is 2 steps--except it is
also 1 step, contradiction.

>> Voicing shouldn't matter, since the voicing of the thing you're
>> mapping to (an ET) doesn't matter. If I set...
>>
>> 1= 9/8
>> 2= 5/4
>> 3= 11/8
>> 4= 3/2
>> 5= 7/4
>> 6= 2/1
>>
>> ...can you show me the problem?
>
>3/2 is 4 steps, so 9/4 is 8 steps, so 9/8 is 2 steps--except it is
>also 1 step, contradiction.

You may be wondering why I ask a question and then answer it in
the same message. If I'm wrong it saves a volley because the
reader knows what I was trying to do.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> So are monzos are now kets written [ ... > ?
> >> and vals are bras written < ... ] ?
> >
> >I think that's a good suggestion. It is a standard (especially in
> >physics), clever notation due to Dirac. We just don't worry about
> >complex numbers and certainly not about quantum mechanics.
>
> But Dave is right that you weren't suggesting using the names
> "bra" and "ket", right?

No--that would really lead to confusion anyway.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > A prime-mapping (or val with log-prime basis) simply maps each prime
> > number (or strictly-speaking the logarithm of each prime number) to
> an
> > integer multiple of some interval (log of frequency ratio) that we
> > call a generator.
>
> This is absolutely not what I mean by a val, which maps to integers.

I think we're picking nits here. What do the integers represent in
tuning terms? What are they counting?

>I think we're picking nits here. What do the integers represent in
>tuning terms? What are they counting?

Not necessarily the most musically-obvious generator, if you're
coming at vals in the context of linear temperaments. I don't
think integers is what confuses me.

I'm still wondering about 6. In 22, the 11-prime-limit val
consistently maps the 9/8, and the resulting hexad taken as a
scale is a Constant Structure. 22 is even generally 11-limit
consistent. Why not use 22?

-Carl

Gene wrote:
>You will also find stuff about "JI epimorphic", but I don't understand
>what Manuel is up to; it isn't what I expected.

Can you give an example of what you expected to be different?
I thought I implemented the epimorphism you discussed on this list.

Manuel

hey paul,

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

> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:

> > p.s. would it be OK for me to attempt a modification of your page
> >
> > http://www.sonic-arts.org/dict/edo-prime-error.htm
> >
> > ? i realized that you *do* have the signed errors of the
> > primes in the text, despite your use of absolute values in
> > the graph. so if i just added the signed errors for the
> > odds that you omitted, i could then quickly locate any
> > inconsistency, since inconsistency occurs if and only if
> > the signed relative error of one odd differs by over 50%
> > from the signed relative error of another odd. for example,
> > in 43-equal, the error on 7 is, as you show, +28%; the error
> > on 9 is double that on 3, so about -30%; the difference
> > between these two signed percentages (and thus the implied
> > error on 9:7) is 58%; so 43-equal is inconsistent in the
> > 9-limit. what do you think? i think the page would be
> > sorely misleading, and much less useful, without this
> > information.
>
>
>
> paul, you already know that i think that the information
> given on my "EDO prime error" page is useful as it is.
>
> but if you envision a better version of the page, sure,
> send me the modification. you know i trust your judgment
> on tuning matters! :)

take a look at this:

http://sonic-arts.org/dict/edo-11-odd-limit-error.htm

-monz

This is in reply to two messages:

/tuning-math/message/7634
--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> > --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > can you *please* give a very detailed explanation of what
> > > you're saying? ... with lots and lots of 11-limit examples
> > > that don't work and 3-, 5-, 7-, 9-, 13-limit examples that do?
> > >
> > > thanks.
> >
> > Here are the 5, 7, 9, 11 and 13 limit complete otonal chords as
Scala
> > scale files. If you run "data" on them, you will find that 5, 7,
9
> > and 13 give Constant Structure scales, and 11 does not.
> ...
>
> Bravo! Gene, this is an excellent minimal-math explanation!
>
> It certainly is a curious fact. And I'm looking forward to hearing
> from George, why he thinks it matters so much musically that he
> wouldn't consider using an 11-limit tuning.

I'm not sure that I said that I wouldn't consider it, only that I
have never been (and am still not) interested in it.

/tuning-math/message/7630
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> maybe george would be better equipped to explain why this is
> musically so significant.
> ...

Gene's observation about constant structure is not only relevant, but
it goes right to the heart of the matter. Having said that, I guess
that the question, then, is: Is it important that a musical scale be
a constant structure, and if so, why?

I've never really thought very much about this, because for me this
was something that seemed to be fairly obvious: that a musical scale
that is not a constant structure will tend to result in confusion or
disorientation by an inherent contradiction between the acoustical
properties of certain intervals and their identity (or ability to
function) as members (i.e., degrees or steps) of that scale.
(By "scale", I am referring to a set of tones that may be used to
write a simple melody. A scale may or may not be a subset of a
larger set of tones, which I would call a "tuning". Thus a major or
minor scale could be considered a subset of the pythagorean tuning,
meantone temperament, or 12-ET, for example.)

If I'm using a pentatonic scale made from a 9-limit otonal chord:
8 : 9 : 10 : 12 : 14 : 16
then I have two intervals each of 2:3 (both pentatonic "4ths") and
3:4 (both pentatonic "3rds").

Likewise, if I'm using a heptatonic scale made from a 13-limit otonal
chord:
8 : 9 : 10 : 11 : 12 : 13 : 14 : 16
then I have two intervals each of 2:3 (both heptatonic 5ths) and 3:4
(both heptatonic 4ths).

So far, so good.

But if I try to use hexatonic scale made from an 11-limit otonal
chord:
8 : 9 : 10 : 11 : 12 : 14 : 16
then one of my 2:3s is a hexatonic "5th" and the other is a
hexatonic "4th", and likewise one of my 3:4s is a hexatonic "4th" and
the other is a hexatonic "3rd". Most attempts to transfer a melodic
figure beginning on a certain scale degree to another scale degree
(such as is required in the musical device called a "sequence") will
tend to produce undesirable consequences (such as listener
disorientation) due to the fact that the 2:3 and/or 3:4 must switch
degree-roles in the process.

Now we could go on to ask why this scale-member identity or
functionality is so important, and this is the point at which I
really had to dig deep for an answer. I believe that, at least with
the examples given above, it has something to do with the role that
the simplest ratios of 3 play in establishing the roots of chords.
If a chord contains a *single* 2:3 or 3:4 (whether just or tempered),
then I can almost guarantee that the tone represented by the 3 will
*never* be heard as the root of the chord. (There are instances,
e.g., 8:10:15, that a tone not in the 2:3 interval will be perceived
as the root, but that's not critical to the point that I'm making.)
It is this property of the simple ratios of 3 that makes it possible
to *invert* many conventional triads and seventh chords *without*
changing our *perception* of which note of the chord functions as the
*root*.

So it would not have been possible for the methods of conventional (5-
limit) harmony to have reached such sophistication if the major and
minor scales were not constant structures, because our whole method
of building chords (by 3rds) has depended on the fact that the simple
ratios of 3 would always be heptatonic 4ths and 5ths and that the
simple ratios of 5 would always be 3rds and 6ths. (Don't make the
mistake of calling the augmented 2nd of a harmonic minor scale a
minor 3rd, because it isn't; the two intervals just happen to be the
the same size in 12-ET, but the meantone temperament will reveal that
they are different.)

One can similarly demonstrate that pentatonic melodies are perceived
as coherent, because pentatonic scales that contain multiple 3:4s and
2:3s are also constant structures.

But take away the property of constant structure while retaining
multiple 2:3s, and you invite confusion and disorientation.

If you want 11-limit otonal harmony in a conherent scale, then I
think it will have to be at least heptatonic and that you're going to
have to fill that extra position with something or other, such as:
8:9:10:11:12:27/2:14:16.
Hmmm, that's really not a bad choice, if you'll notice that 22:27:32
is an isoharmonic triad. I remember that this scale works very
nicely in 31-ET, since the 27/2:16:20 ends up as an ordinary minor
triad.

Likewise, you can have constant-structure 17 and 19-limit otonal
scales:
16:17:18:20:21:22:24:26:28:30:32 (with chords built in
decatonic "4ths")
and
16:17:18:19:20:21:22:24:25:26:28:30:32

--George

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

> > But Dave is right that you weren't suggesting using the names
> > "bra" and "ket", right?
>
> No--that would really lead to confusion anyway.

Speaking of which, I should probably have used < ... | for the val
and | ... > for the monzo, as being the actual bra-ket notation. The
idea is that then you just stick them together, without putting a dot
in the middle, for the product: < ... | ... > notating the product.

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

> I think we're picking nits here. What do the integers represent in
> tuning terms? What are they counting?

I am not picking nits. This is very important to the math. I don't
understand what your objection is anyway--we have a lot of defintions
of tuning and scale terms tossed around which are neither clear nor
elegant sounding. "Val" is short, and I gave it a precise definition,
thereby doing what I wish other people would more often do, judging
by the definitions in Joe's dictionary.

>Speaking of which, I should probably have used < ... | for the val
>and | ... > for the monzo, as being the actual bra-ket notation.

The only reason I put it the other way was so that bra-ket would
tell you the position of the square brackets. But you're using
a pipe instead of a square bracket, so this way the word still
tells you the location of the most bracket-like things, the angle
bracket.

-Carl

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

> I'm still wondering about 6. In 22, the 11-prime-limit val
> consistently maps the 9/8, and the resulting hexad taken as a
> scale is a Constant Structure. 22 is even generally 11-limit
> consistent. Why not use 22?

You started with 6 and ended up with 22. Where is your 22 note
scale/chord?

I wrote...
>>So it's proper, but not a constant structure. I was under the
>>misapprehension that proper always implied constant structure, i.e
>>that propriety was a stronger condition. Hmm.
>
>Nope. The set of all non-CS scales is equivalennt to the set of
>all non-strictly-proper scales.

Sorry, it was too late. There improper non-CS scales. There's
no convenient way to express CS with Rothenberg's language that
I know of.

-Carl

>I've never really thought very much about this, because for me this
>was something that seemed to be fairly obvious: that a musical scale
>that is not a constant structure will tend to result in confusion or
>disorientation by an inherent contradiction between the acoustical
>properties of certain intervals and their identity (or ability to
>function) as members (i.e., degrees or steps) of that scale.

Does that include the diatonic scale in 12-equal?

>Now we could go on to ask why this scale-member identity or
>functionality is so important, and this is the point at which I
>really had to dig deep for an answer. I believe that, at least with
>the examples given above, it has something to do with the role that
>the simplest ratios of 3 play in establishing the roots of chords.

Paul E. has suggested that we only care about collisions if they
occur to a consonant interval. That allows the diatonic scale
in 12-equal to pass.

But incidentally, I'd love a musical example of a hexatonic 11-limit
melody where the non-CS "collision" causes a problem with constructing
a musical sequence. With all the ink I've spilled on this subject,
I'm probably more guilty than anyone of not having come up with
musical examples to demonstrate propriety...

-Carl

>> I'm still wondering about 6. In 22, the 11-prime-limit val
>> consistently maps the 9/8, and the resulting hexad taken as a
>> scale is a Constant Structure. 22 is even generally 11-limit
>> consistent. Why not use 22?
>
>You started with 6 and ended up with 22. Where is your 22 note
>scale/chord?

Ok, now we're on the right track, but I'm still not grokking
you. I started with six rationals and ended up with 6 integers.
What's the problem?

-Carl

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

> If we are told that the mapping is for a tET then _which_ tET it is
> for can be read straight out of the mapping, as the coefficient for
> the prime 2 (the first coefficient). And the generator is simply one
> step of that tET.

just wondering why you keep saying "tET" -- 'If we are told that the
mapping is for a tone equal temperament then . . .' ??

> Why such an apparently melodic property should be considered
important
> when applied to a vertical harmony, I don't know.

well, my 22 paper does give a bit of indication as to why that might
be. by those considerations too, 11-limit hexads would be rather
difficult to find a reasonable tonal system for. but my
considerations had to do with getting the utonal and otonal complete
consonances from the same pattern of scale steps, while george is
probably uninterested in the utonalities . . .

> > Note that I have no idea what the bra ket notation stuff is about.
>
> It's just a way of distinguishing prime-mappings (vals) from
> prime-exponent-vectors (monzos) without having to say it in words
> every time. It only makes sense to multiply mappings by
> exponent-vectors, not any other combination and these brackets try
to
> make that clear because ] and [ fit together, but > and <, > and
[, ]
> and < do not.

actually, > and < fit together and create a X (as in times) !

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >> Gene, since you won't say what's desirable about being a
> > >> standard val...
> > >
> > >Purely a matter of being easy to calculate.
> >
> > Adding our birthdays together is easy to calculate. There
> > must be some other reason.
>
> No. That's really it. Although I suppose we should say "and it is
also
> the best (i.e. most accurate) mapping when the ET is consistent. So
> you'd only be interested in mappings other than this easy-to-
calculate
> one, if the ET is inconsistent at the given limit, and we are rarely
> interested in those anyway, so you can get away with it most of the
> time. It's a lot more complicated to find the best mapping for an
> inconsistent ET.

right, but gene just did that (two days ago?), and hopefully will do
more.

> It's just that Gene and George found it interesting to look at how
> complete chords can be mapped to a single octave of the ET of the
same
> cardinality as the chord. It turns out that the 11-limit otonality
> can't be. There is no mapping and no voicing of the chord that will
do
> this.

let me just repeat dave and say that this has *nothing* to do with
the definition of vals -- it's a separate question that you can
safely ignore if you want to understand vals.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > Why the same card.?
>
> Dunno. But it's too easy otherwise.
>
> > >Try the 6 possible possible voicings of the 11-limit otonality,
that
> > >fit within an octave, and you'll see that none of them are very
even.
> >
> > It's proper and for that matter seems to fit to 6-tET reasonably
> > well.
>
> OK. Well maybe I got something wrong about what's being claimed. I'd
> better let the experts sort it out.

it's not strictly proper.

> > So are monzos are now kets written [ ... > ?
> > and vals are bras written < ... ] ?
>
> That's Gene's proposal (that we should write them that way, not that
> we should call them bras and kets) and it seems like a reasonable
one
> to me.

it'll help me, since i'm used to them.

> Except I wonder how we should write a complete mapping matrix for a
more-than-1D temperament.

a matrix is a matrix, not a bra or a ket. i never understood
covariant vs. contravariant, though . . .

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Yes! Congratulations!
>
> Yeah well, the choice of 6 here still hasn't been accounted for.

i explained why it was 3 in the case of 5-limit, and here it's the
same -- 6 is the number of notes in the 11-limit complete otonality!
this has nothing to do with the definition of vals, it's just one
particular problem that gene and george happen to be interested in.

> >So it's proper, but not a constant structure. I was under the
> >misapprehension that proper always implied constant structure, i.e
> >that propriety was a stronger condition. Hmm.
>
> Nope. The set of all non-CS scales is equivalennt to the set of
> all non-strictly-proper scales.
>
> I remember surprise when John Chalmers first pointed this out
> to me, by way of this example:
>
> [private communication]
> >However, the Enharmonic of Archytas is. Translate the scale 28/27 x
> >36/35 x 5/4 x 9/8 x 28/27 x 36/35 x 5/4 into cents and generate the
> >D-matrix.
> >
> >63 49 386 204 63 49 386
> >112 435 590 267 112 435 449
> >498 639 653 316 498 498 498
> >702 702 702 702 561 547 884
> >765 751 1088 765 610 933 1088
> >814 1137 1151 814 996 1137 1151
> >1200 .......
>
> -Carl

this is an example of . . . ?

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

> >You started with 6 and ended up with 22. Where is your 22 note
> >scale/chord?
>
> Ok, now we're on the right track, but I'm still not grokking
> you. I started with six rationals and ended up with 6 integers.
> What's the problem?

Are your integers consecutive?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >> So are monzos are now kets written [ ... > ?
> > >> and vals are bras written < ... ] ?
> > >
> > >I think that's a good suggestion. It is a standard (especially
in
> > >physics), clever notation due to Dirac. We just don't worry
about
> > >complex numbers and certainly not about quantum mechanics.
> >
> > But Dave is right that you weren't suggesting using the names
> > "bra" and "ket", right?
>
> No--that would really lead to confusion anyway.

but those are the names of the mathematical objects.

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

> I'm still wondering about 6. In 22, the 11-prime-limit val
> consistently maps the 9/8, and the resulting hexad taken as a
> scale is a Constant Structure.

no, it's still not a constant structure, since one 3:2 subtends more
notes than the other ('9:6').

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hey paul,
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
>
> > > p.s. would it be OK for me to attempt a modification of your
page
> > >
> > > http://www.sonic-arts.org/dict/edo-prime-error.htm
> > >
> > > ? i realized that you *do* have the signed errors of the
> > > primes in the text, despite your use of absolute values in
> > > the graph. so if i just added the signed errors for the
> > > odds that you omitted, i could then quickly locate any
> > > inconsistency, since inconsistency occurs if and only if
> > > the signed relative error of one odd differs by over 50%
> > > from the signed relative error of another odd. for example,
> > > in 43-equal, the error on 7 is, as you show, +28%; the error
> > > on 9 is double that on 3, so about -30%; the difference
> > > between these two signed percentages (and thus the implied
> > > error on 9:7) is 58%; so 43-equal is inconsistent in the
> > > 9-limit. what do you think? i think the page would be
> > > sorely misleading, and much less useful, without this
> > > information.
> >
> >
> >
> > paul, you already know that i think that the information
> > given on my "EDO prime error" page is useful as it is.
> >
> > but if you envision a better version of the page, sure,
> > send me the modification. you know i trust your judgment
> > on tuning matters! :)
>
>
>
> take a look at this:
>
> http://sonic-arts.org/dict/edo-11-odd-limit-error.htm
>
>
>
>
> -monz

nice! replied on main tuning list.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> > That's Gene's proposal (that we should write them that way, not
that
> > we should call them bras and kets) and it seems like a reasonable
> one
> > to me.
>
> it'll help me, since i'm used to them.

Sounds like we are achieving consensus on something!

> > Except I wonder how we should write a complete mapping matrix for
a
> more-than-1D temperament.
>
> a matrix is a matrix, not a bra or a ket. i never understood
> covariant vs. contravariant, though . . .

Let's leave matricies alone. As for covariant vs contravariant, if
you change the basis for monzos to something other than primes, you
have to make a complimentary change in basis for the val basis. The
standard basis is that monzos have a basis e_2, e_3 etc.
corresponding to primes, and vals v_2, v_3 corresponding to (whether
we want to call them that or not) padic valuations.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> but those are the names of the mathematical objects.

Not really. Monzos are a particular type of ket vector, with a
particular interpretation, and vals a particular type (in both cases,
composed of integers) of bra vector, with a particular interpretation.
The "mathematical object" involves more than merely being a vector.
Any integer bra vector can be interpreted as a val, and any integer
ket vector as a monzo (and hence, a positive rational number) but we
need the interpretation.

George,

I was already convinced that Constant Structure is a valuable melodic
property of a scale. But what's wrong with using complete 11-limit
hexads as vertical harmony within a larger CS scale? Why should we
care that the hexads _themselves_ are not CS?

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> Is it important that a musical scale be
> a constant structure, and if so, why?

but we're talking about a chord, not a whole scale . . .

> (By "scale", I am referring to a set of tones that may be used to
> write a simple melody.

exactly . . . the two champions would have to be the diatonic
pentatonic and heptatonic scales . . .

> If I'm using a pentatonic scale made from a 9-limit otonal chord:
> 8 : 9 : 10 : 12 : 14 : 16
> then I have two intervals each of 2:3 (both pentatonic "4ths") and
> 3:4 (both pentatonic "3rds").

personally, i'm not fond of this as a scale or melodic entity at all -
- when i improvise over a dominant ninth chord, simply using its
notes is about the worst way to come up with a melody . . .

> So far, so good.
>
> But if I try to use hexatonic scale made from an 11-limit otonal
> chord:
> 8 : 9 : 10 : 11 : 12 : 14 : 16
> then one of my 2:3s is a hexatonic "5th" and the other is a
> hexatonic "4th", and likewise one of my 3:4s is a hexatonic "4th"
and
> the other is a hexatonic "3rd". Most attempts to transfer a
melodic
> figure beginning on a certain scale degree to another scale degree
> (such as is required in the musical device called a "sequence")
will
> tend to produce undesirable consequences (such as listener
> disorientation) due to the fact that the 2:3 and/or 3:4 must switch
> degree-roles in the process.

yet even an algorithmic composition program, such as those written by
prent rodgers, can produce lovely music by simply using a single such
hexad at a time, for both harmony and melody (or for polyphony). i'm
not disputing your constant structure argument too vehemently,
especially when it concerns such an important interval as 3:2, but
note especially that the diatonic scale in 12-equal is not CS, and
yet doesn't cause any more listener disorientation than the diatonic
scale in, say, 19-equal or 17-equal, where it it CS.

> Now we could go on to ask why this scale-member identity or
> functionality is so important, and this is the point at which I
> really had to dig deep for an answer. I believe that, at least
with
> the examples given above, it has something to do with the role that
> the simplest ratios of 3 play in establishing the roots of chords.
> If a chord contains a *single* 2:3 or 3:4 (whether just or
tempered),
> then I can almost guarantee that the tone represented by the 3 will
> *never* be heard as the root of the chord.

another commonality with my 22 paper.

> (There are instances,
> e.g., 8:10:15, that a tone not in the 2:3 interval will be
perceived
> as the root, but that's not critical to the point that I'm
making.)
> It is this property of the simple ratios of 3 that makes it
possible
> to *invert* many conventional triads and seventh chords *without*
> changing our *perception* of which note of the chord functions as
the
> *root*.
>
> So it would not have been possible for the methods of conventional
(5-
> limit) harmony to have reached such sophistication if the major and
> minor scales were not constant structures, because our whole method
> of building chords (by 3rds) has depended on the fact that the
simple
> ratios of 3 would always be heptatonic 4ths and 5ths and that the
> simple ratios of 5 would always be 3rds and 6ths.

ah, but you're depending on the heptatonic scale here! if we used
some sort of heptatonic or other scalar framework to understand 11-
limit harmony, the same property might hold, despite the fact that
the hexad itself is not CS. so the latter fact seems irrelevant.

> One can similarly demonstrate that pentatonic melodies are
perceived
> as coherent, because pentatonic scales that contain multiple 3:4s
and
> 2:3s are also constant structures.
>
> But take away the property of constant structure while retaining
> multiple 2:3s, and you invite confusion and disorientation.
>
> If you want 11-limit otonal harmony in a conherent scale, then I
> think it will have to be at least heptatonic and that you're going
to
> have to fill that extra position with something or other, such as:
> 8:9:10:11:12:27/2:14:16.
> Hmmm, that's really not a bad choice, if you'll notice that
22:27:32
> is an isoharmonic triad. I remember that this scale works very
> nicely in 31-ET, since the 27/2:16:20 ends up as an ordinary minor
> triad.
>
> Likewise, you can have constant-structure 17 and 19-limit otonal
> scales:
> 16:17:18:20:21:22:24:26:28:30:32 (with chords built in
> decatonic "4ths")
> and
> 16:17:18:19:20:21:22:24:25:26:28:30:32

ok, but not much harmonic movement possible here.

p.s. i enjoy using 8:9:10:11:12 as a consonant chord, and not only
isn't it CS, it's not even proper! voice-leading can be tricky,
though, when the consonant chord isn't spread roughly evenly over the
octave . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I am not picking nits. This is very important to the math. I don't
> understand what your objection is anyway--we have a lot of defintions
> of tuning and scale terms tossed around which are neither clear nor
> elegant sounding. "Val" is short, and I gave it a precise definition,
> thereby doing what I wish other people would more often do, judging
> by the definitions in Joe's dictionary.

I thought we just agreed that you wouldn't worry too much if my
explanations didn't capture the precise pure-math meaning, and in
return I wouldn't worry that you give definitions that are
incomprehensible to most tuning-math readers

But if I've made a serious mistake I really need to know:
What do the integers [the val's coefficients] represent in tuning
terms? What are they counting?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > That's Gene's proposal (that we should write them that way, not
> that
> > > we should call them bras and kets) and it seems like a
reasonable
> > one
> > > to me.
> >
> > it'll help me, since i'm used to them.
>
> Sounds like we are achieving consensus on something!
>
> > > Except I wonder how we should write a complete mapping matrix
for
> a
> > more-than-1D temperament.
> >
> > a matrix is a matrix, not a bra or a ket. i never understood
> > covariant vs. contravariant, though . . .
>
> Let's leave matricies alone. As for covariant vs contravariant, if
> you change the basis for monzos to something other than primes, you
> have to make a complimentary change in basis for the val basis. The
> standard basis is that monzos have a basis e_2, e_3 etc.
> corresponding to primes, and vals v_2, v_3 corresponding to
(whether
> we want to call them that or not) padic valuations.

so how can i tell which one is covariant and which one is
contravariant?

>> Ok, now we're on the right track, but I'm still not grokking
>> you. I started with six rationals and ended up with 6 integers.
>> What's the problem?
>
>Are your integers consecutive?

No, and that's part of the def. of standard val, but what
motivates it?

-Carl

>> Yeah well, the choice of 6 here still hasn't been accounted for.
>
>i explained why it was 3 in the case of 5-limit, and here it's the
>same -- 6 is the number of notes in the 11-limit complete otonality!
>this has nothing to do with the definition of vals, it's just one
>particular problem that gene and george happen to be interested in.

Right, got that, just don't see why it's a "problem".

>> >So it's proper, but not a constant structure. I was under the
>> >misapprehension that proper always implied constant structure, i.e
>> >that propriety was a stronger condition. Hmm.
//
>> >However, the Enharmonic of Archytas is. Translate the scale 28/27 x
>> >36/35 x 5/4 x 9/8 x 28/27 x 36/35 x 5/4 into cents and generate the
>> >D-matrix.
>> >
>> >63 49 386 204 63 49 386
>> >112 435 590 267 112 435 449
>> >498 639 653 316 498 498 498
>> >702 702 702 702 561 547 884
>> >765 751 1088 765 610 933 1088
>> >814 1137 1151 814 996 1137 1151
>> >1200 .......
>>
>> -Carl
>
>this is an example of . . . ?

A (wildly) improper constant structure.

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Ok, now we're on the right track, but I'm still not grokking
> >> you. I started with six rationals and ended up with 6 integers.
> >> What's the problem?
> >
> >Are your integers consecutive?
>
> No, and that's part of the def. of standard val, but what
> motivates it?
>
> -Carl

i can't make heads or tails of this question. the standard val puts
the primes in order because it's easy to remember them that way. you
could put them in a different order but you would have to remember
which entry refers to which prime. so i don't see what there is to
motivate.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Yeah well, the choice of 6 here still hasn't been accounted for.
> >
> >i explained why it was 3 in the case of 5-limit, and here it's the
> >same -- 6 is the number of notes in the 11-limit complete
otonality!
> >this has nothing to do with the definition of vals, it's just one
> >particular problem that gene and george happen to be interested in.
>
> Right, got that, just don't see why it's a "problem".

well, george tried to explain it, and i had point of agreement and
disagreement with him . .

> >> >So it's proper, but not a constant structure. I was under the
> >> >misapprehension that proper always implied constant structure,
i.e
> >> >that propriety was a stronger condition. Hmm.
> //
> >> >However, the Enharmonic of Archytas is. Translate the scale
28/27 x
> >> >36/35 x 5/4 x 9/8 x 28/27 x 36/35 x 5/4 into cents and generate
the
> >> >D-matrix.
> >> >
> >> >63 49 386 204 63 49 386
> >> >112 435 590 267 112 435 449
> >> >498 639 653 316 498 498 498
> >> >702 702 702 702 561 547 884
> >> >765 751 1088 765 610 933 1088
> >> >814 1137 1151 814 996 1137 1151
> >> >1200 .......
> >>
> >> -Carl
> >
> >this is an example of . . . ?
>
> A (wildly) improper constant structure.

a random scale has essentially a 100% chance of being one.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > If we are told that the mapping is for a tET then _which_ tET it is
> > for can be read straight out of the mapping, as the coefficient for
> > the prime 2 (the first coefficient). And the generator is simply one
> > step of that tET.
>
> just wondering why you keep saying "tET" -- 'If we are told that the
> mapping is for a tone equal temperament then . . .' ??

I agree it's awkward. Carl objected so vehemently to EDO and I wanted
to reserve ET for the most general term (including EDOs ED3s cETs).
Perhaps this would be a misuse of ET. Do we have some other term for
the most general category of 1D temperaments, i.e. any single
generator temperament whether or not it is an integer fraction of any
ratio? I guess "1D-temperament" will do.

> actually, > and < fit together and create a X (as in times) !

Oops. Well we could interpret that as the matrix-product as opposed to
the scalar-product (dot-product), but I don't know of any meaning for
that in tuning.

>> I'm still wondering about 6. In 22, the 11-prime-limit val
>> consistently maps the 9/8, and the resulting hexad taken as a
>> scale is a Constant Structure.
>
>no, it's still not a constant structure, since one 3:2 subtends more
>notes than the other ('9:6').

Oh bloody hell; I had checked this by eye only, and missed that.
It's still sitting in my scheme terminal from last night.

-Carl

>> >> Ok, now we're on the right track, but I'm still not grokking
>> >> you. I started with six rationals and ended up with 6 integers.
>> >> What's the problem?
>> >
>> >Are your integers consecutive?
>>
>> No, and that's part of the def. of standard val, but what
>> motivates it?
>>
>> -Carl
>
>i can't make heads or tails of this question. the standard val puts
>the primes in order because it's easy to remember them that way. you
>could put them in a different order but you would have to remember
>which entry refers to which prime. so i don't see what there is to
>motivate.

You lost me. My 22-val example doesn't reorder the primes!

-C.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> >
> > > If we are told that the mapping is for a tET then _which_ tET
it is
> > > for can be read straight out of the mapping, as the coefficient
for
> > > the prime 2 (the first coefficient). And the generator is
simply one
> > > step of that tET.
> >
> > just wondering why you keep saying "tET" -- 'If we are told that
the
> > mapping is for a tone equal temperament then . . .' ??
>
> I agree it's awkward. Carl objected so vehemently to EDO and I
wanted
> to reserve ET for the most general term (including EDOs ED3s cETs).
> Perhaps this would be a misuse of ET. Do we have some other term for
> the most general category of 1D temperaments, i.e. any single
> generator temperament whether or not it is an integer fraction of
any
> ratio? I guess "1D-temperament" will do.
>
> > actually, > and < fit together and create a X (as in times) !
>
> Oops. Well we could interpret that as the matrix-product as opposed
to
> the scalar-product (dot-product), but I don't know of any meaning
for
> that in tuning.

the symbol normally indicates the cross-product, which is extremely
useful in tuning: for example, if i take the monzo for the diaschisma

[-4 4 -1>

and cross it with the (transpose of the?) monzo for the syntonic comma

<-11 4 2]

i get the val for the et where they both vanish:

[12 19 28]

not sure how gene would do this notationally, probably i did
something terrible, but without it i could not have made those
charts . . .

>> Right, got that, just don't see why it's a "problem".
>
>well, george tried to explain it, and i had point of agreement and
>disagreement with him . .

Without an musical example (referenced or constructed), I'm skeptical.

-Carl

>Let's leave matricies alone. As for covariant vs contravariant, if
>you change the basis for monzos to something other than primes, you
>have to make a complimentary change in basis for the val basis. The
>standard basis is that monzos have a basis e_2, e_3 etc.
>corresponding to primes, and vals v_2, v_3 corresponding to (whether
>we want to call them that or not) padic valuations.

What happens if we change the bases to odd numbers. There's no
longer a unique monzo for any given interval, which seems bad.
What other bases did you have in mind?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> Ok, now we're on the right track, but I'm still not grokking
> >> >> you. I started with six rationals and ended up with 6
integers.
> >> >> What's the problem?
> >> >
> >> >Are your integers consecutive?
> >>
> >> No, and that's part of the def. of standard val, but what
> >> motivates it?
> >>
> >> -Carl
> >
> >i can't make heads or tails of this question. the standard val
puts
> >the primes in order because it's easy to remember them that way.
you
> >could put them in a different order but you would have to remember
> >which entry refers to which prime. so i don't see what there is to
> >motivate.
>
> You lost me. My 22-val example doesn't reorder the primes!
>
> -C.

what does that have to do with the definition of standard val? the
definition, as dave keenan explained it, seems perfectly well
motivated, if not always optimal. in a special case it is turned
toward the particular problem of an odd-limit complete otonality, but
that should be a separate concern . . .

>the symbol normally indicates the cross-product, which is extremely
>useful in tuning: for example, if i take the monzo for the diaschisma
>
>[-4 4 -1>
>
>and cross it with the (transpose of the?) monzo for the syntonic comma
>
><-11 4 2]

Huh; I thought all monzos were supposed to be written | ... > from
here on out.

>i get the val for the et where they both vanish:
>
>[12 19 28]
>
>not sure how gene would do this notationally, probably i did
>something terrible, but without it i could not have made those
>charts . . .

And I thought all vals were supposed to be written < ... |.

Did I miss something?

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > time. It's a lot more complicated to find the best mapping for an
> > inconsistent ET.
>
> right, but gene just did that (two days ago?), and hopefully will do
> more.

I was quite aware of that.

> > It's just that Gene and George found it interesting to look at how
> > complete chords can be mapped to a single octave of the ET of the
> same
> > cardinality as the chord. It turns out that the 11-limit otonality
> > can't be. There is no mapping and no voicing of the chord that will
> do
> > this.
>
> let me just repeat dave and say that this has *nothing* to do with
> the definition of vals -- it's a separate question that you can
> safely ignore if you want to understand vals.

I was quite aware of that.

I was merely trying to answer Carl's questions.

I understand vals now (as of this week), and would have understood
them several _years_ ago, if the term prime-mapping had been used. Or
since I have undertood prime-mappings for years (pre-Gene) you could
say I understood vals the whole time (as applied to tuning). I just
didn't have a clue that's what he was talking about. The new term and
the pure-math definitions obscured this simple fact. For years!

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >the symbol normally indicates the cross-product, which is
extremely
> >useful in tuning: for example, if i take the monzo for the
diaschisma
> >
> >[-4 4 -1>
> >
> >and cross it with the (transpose of the?) monzo for the syntonic
comma
> >
> ><-11 4 2]
>
> Huh; I thought all monzos were supposed to be written | ... > from
> here on out.

if so, then the second is the transpose of a monzo.

> >i get the val for the et where they both vanish:
> >
> >[12 19 28]
> >
> >not sure how gene would do this notationally, probably i did
> >something terrible, but without it i could not have made those
> >charts . . .
>
> And I thought all vals were supposed to be written < ... |.
>
> Did I miss something?

that's why i said i probably did something terrible notationally!

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Let's leave matricies alone. As for covariant vs contravariant, if
> >you change the basis for monzos to something other than primes,
you
> >have to make a complimentary change in basis for the val basis.
The
> >standard basis is that monzos have a basis e_2, e_3 etc.
> >corresponding to primes, and vals v_2, v_3 corresponding to
(whether
> >we want to call them that or not) padic valuations.
>
> What happens if we change the bases to odd numbers. There's no
> longer a unique monzo for any given interval, which seems bad.
> What other bases did you have in mind?

overtones of inharmonic spectra is one kind graham has used.

similarly, if i take the (transpose of the?) val for 12-equal:

|12 19 28>

and take the cross product with the val for 22-equal:

<22 35 51|

i get the monzo for the diaschisma, the interval that vanishes in
both tunings:

[-11 4 2]

again, not sure what's going on notationally, but the numbers
work . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > --- In tuning-math@yahoogroups.com, "Dave Keenan"
<d.keenan@b...>
> > > wrote:
> > >
> > > > If we are told that the mapping is for a tET then _which_ tET
> it is
> > > > for can be read straight out of the mapping, as the
coefficient
> for
> > > > the prime 2 (the first coefficient). And the generator is
> simply one
> > > > step of that tET.
> > >
> > > just wondering why you keep saying "tET" -- 'If we are told
that
> the
> > > mapping is for a tone equal temperament then . . .' ??
> >
> > I agree it's awkward. Carl objected so vehemently to EDO and I
> wanted
> > to reserve ET for the most general term (including EDOs ED3s
cETs).
> > Perhaps this would be a misuse of ET. Do we have some other term
for
> > the most general category of 1D temperaments, i.e. any single
> > generator temperament whether or not it is an integer fraction of
> any
> > ratio? I guess "1D-temperament" will do.
> >
> > > actually, > and < fit together and create a X (as in times) !
> >
> > Oops. Well we could interpret that as the matrix-product as
opposed
> to
> > the scalar-product (dot-product), but I don't know of any meaning
> for
> > that in tuning.
>
> the symbol normally indicates the cross-product, which is extremely
> useful in tuning: for example, if i take the monzo for the
diaschisma
>
> [-4 4 -1>
>
> and cross it with the (transpose of the?) monzo for the syntonic
comma
>
> <-11 4 2]
>
> i get the val for the et where they both vanish:
>
> [12 19 28]
>
> not sure how gene would do this notationally, probably i did
> something terrible, but without it i could not have made those
> charts . . .

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

> > let me just repeat dave and say that this has *nothing* to do
with
> > the definition of vals -- it's a separate question that you can
> > safely ignore if you want to understand vals.
>
> I was quite aware of that.

you should be, because as i said, i was just repeating you!

> I was merely trying to answer Carl's questions.

me too!

>>>>>>I started with six rationals and ended up with 6
>>>>>>integers. What's the problem?
>>>>>
>>>>>Are your integers consecutive?
>>>>
>>>> No, and that's part of the def. of standard val, but what
>>>> motivates it?
>
>what does that have to do with the definition of standard val?

Sorry, it doesn't. I forgot the definition doesn't mention
consecutive. It's just this particular case.

Wait... is this true: 'For a scale with card k, if there is
no standard val with n=k that consistently maps the scale, the
scale is not a Constant Structure.'

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > Except I wonder how we should write a complete mapping matrix for a
> more-than-1D temperament.
>
> a matrix is a matrix, not a bra or a ket.

True, but the dot product of two vectors generalises naturally to the
product of a matrix with a column vector. When we go beyond 1D
temperaments we have prime-mappings which are matrices (one row per
generator) and we multiply that by the transpose of a ratio's
prime-exponent-vector (monzo) to get a vector giving the count of each
generator.

> i never understood
> covariant vs. contravariant, though . . .

Me neither. Paul, we've gotta keep on Gene to tell us what things mean
AS APPLIED TO TUNING. And for him to keep trying different
explanations until we get it.

I propose a new term for that, since we'll probably be writing it so
often. BWDIMAATT. Pronounced "BWOOD-ee-mart" As in "Gene, BWDIMAATT"
Meaning "Gene, But What Does It Mean As Applied To Tuning". ;-)

And when one of us gets it we've gotta try to explain it to the rest,
- if it actually has any relevance at all.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > > Except I wonder how we should write a complete mapping matrix
for a
> > more-than-1D temperament.
> >
> > a matrix is a matrix, not a bra or a ket.
>
> True, but the dot product of two vectors generalises naturally to
the
> product of a matrix with a column vector.

> When we go beyond 1D
> temperaments we have prime-mappings which are matrices (one row per
> generator) and we multiply that by the transpose of a ratio's
> prime-exponent-vector (monzo) to get a vector giving the count of
each
> generator.

can you show an example? obviously i'm plenty confused as to how to
correctly notate these things . . .

> > i never understood
> > covariant vs. contravariant, though . . .
>
> Me neither. Paul, we've gotta keep on Gene to tell us what things
mean
> AS APPLIED TO TUNING. And for him to keep trying different
> explanations until we get it.

right, but i still want to understand it, since it was in my
relativity textbooks . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > That's Gene's proposal (that we should write them that way, not
> that
> > > we should call them bras and kets) and it seems like a reasonable
> > one
> > > to me.
> >
> > it'll help me, since i'm used to them.
>
> Sounds like we are achieving consensus on something!
>
> > > Except I wonder how we should write a complete mapping matrix for
> a
> > more-than-1D temperament.
> >
> > a matrix is a matrix, not a bra or a ket. i never understood
> > covariant vs. contravariant, though . . .
>
> Let's leave matricies alone.

But mappings for more-than-1D temperaments are naturally expressed as
matrices that map prime-exponent vectors to generator-count vectors.

In general a temperament has a prime-mapping matrix "M" that has a
column for each prime and a row for each generator. And a ratio has a
prime-exponent-vector (monzo) "a" of the same width. If we want to
know how many of each generator correspond to that ratio in that
temperament we simply calculate the matrix product
transpose(M*transpose(a)), or simply M*a if the monzo is already a
column vector and the result can be a column vector.

> As for covariant vs contravariant, if
> you change the basis for monzos to something other than primes, you
> have to make a complimentary change in basis for the val basis. The
> standard basis is that monzos have a basis e_2, e_3 etc.
> corresponding to primes, and vals v_2, v_3 corresponding to (whether
> we want to call them that or not) padic valuations.

Sure. BWDIMAATT? :-) Never mind. I don't think we need to worry about
it. We just need to remember that mappings and monzos are different
kinds of things. They have different "units" as it were.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>>>>>I started with six rationals and ended up with 6
> >>>>>>integers. What's the problem?
> >>>>>
> >>>>>Are your integers consecutive?
> >>>>
> >>>> No, and that's part of the def. of standard val, but what
> >>>> motivates it?
> >
> >what does that have to do with the definition of standard val?
>
> Sorry, it doesn't. I forgot the definition doesn't mention
> consecutive. It's just this particular case.
>
> Wait... is this true: 'For a scale with card k, if there is
> no standard val with n=k that consistently maps the scale, the
> scale is not a Constant Structure.'

Carl,

Drop the word "standard". There's absolutely no relationship between
"standard" vals and Constant Structure. Sorry if anything I said
mislead you in that direction.

Perhaps I should have said: There's absolutely no relationship between
the "standardness" of the vals and Constant Structure.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > > wrote:
> > >
> > > > If we are told that the mapping is for a tET then _which_ tET
> it is
> > > > for can be read straight out of the mapping, as the coefficient
> for
> > > > the prime 2 (the first coefficient). And the generator is
> simply one
> > > > step of that tET.
> > >
> > > just wondering why you keep saying "tET" -- 'If we are told that
> the
> > > mapping is for a tone equal temperament then . . .' ??
> >
> > I agree it's awkward. Carl objected so vehemently to EDO and I
> wanted
> > to reserve ET for the most general term (including EDOs ED3s cETs).
> > Perhaps this would be a misuse of ET. Do we have some other term for
> > the most general category of 1D temperaments, i.e. any single
> > generator temperament whether or not it is an integer fraction of
> any
> > ratio? I guess "1D-temperament" will do.
> >
> > > actually, > and < fit together and create a X (as in times) !
> >
> > Oops. Well we could interpret that as the matrix-product as opposed
> to
> > the scalar-product (dot-product), but I don't know of any meaning
> for
> > that in tuning.
>
> the symbol normally indicates the cross-product, which is extremely
> useful in tuning: for example, if i take the monzo for the diaschisma
>
> [-4 4 -1>
>
> and cross it with the (transpose of the?) monzo for the syntonic comma
>
> <-11 4 2]

Should have been [-11 4 2>

> i get the val for the et where they both vanish:
>
> [12 19 28]

Now you could write <12 19 28]

That's magic! I never knew that! But of course if someone ever said it
before I wouldn't have understood it since I didn't have a clue what a
val was.

So [-4 4 -1> (x) [-11 4 2> = <12 19 28]

Where (x) is a rather poor ASCII version of the cross-product
operator. Is there a standard ASCII version of that. And while we're
at it how about an ASCII version of the matrix transpose operator.
"^T" ? It's obviously bad having letters in operator symbols since
they invite confusion with variables.

For non-math types:

The cross-product of vectors
<a1 a2 a3} and <b1 b2 b3]
is [a2b3-a3u2 a3b1-a1b3 a1b2-a2b1>

I should probably have made it clearer by writing:

The cross-product of vectors

<a1 a2 a3}

and

<b1 b2 b3]

is

[a2*b3-a3*b2 a3*b1-a1*b3 a1*b2-a2*b1>

How does that generalise to other than the 5-limit? i.e. vectors with
other than 3 components?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> similarly, if i take the (transpose of the?) val for 12-equal:
>
> |12 19 28>
>
> and take the cross product with the val for 22-equal:
>
> <22 35 51|
>
> i get the monzo for the diaschisma, the interval that vanishes in
> both tunings:
>
> [-11 4 2]

I never knew this either!

Although your use of notation sucks, as you suggested it might.

I'd write

<12 19 28] (x) <22 35 51] = [-11 4 2>

I don't think the bra and ket notation was particularly designed to
help us with what can be crossed with what and what the result is.
Although we can see that ]< and >[ are both cross products while ][
(which can be relaced by | is the dot product and >< is the matrix
product.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > > let me just repeat dave and say that this has *nothing* to do
> with
> > > the definition of vals -- it's a separate question that you can
> > > safely ignore if you want to understand vals.
> >
> > I was quite aware of that.
>
> you should be, because as i said, i was just repeating you!
>
> > I was merely trying to answer Carl's questions.
>
> me too!

Sorry. I read it as "just let me repeat, dave". i.e. I thought you
were talking to me. Duh.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > When we go beyond 1D
> > temperaments we have prime-mappings which are matrices (one row per
> > generator) and we multiply that by the transpose of a ratio's
> > prime-exponent-vector (monzo) to get a vector giving the count of
> each
> > generator.
>
> can you show an example? obviously i'm plenty confused as to how to
> correctly notate these things . . .

Me too, since I want it to be generalisable to matrices, and it seems
Gene doesn't care about that.

Here's a 5-limit mapping matrix for meantone (call it "M") in one
possible notation.

<1 2 4]
<0 -1 -4]

or on one line <1 2 2; 0 -1 -4]

The first row related the primes to the octave generator, the second
row relates them to the fourth generator. In Gene's terminology, each
row is a val.

And let "a" be a prime-exponent-vector for some ratio, say 5/3
[0 -1 1>

By treating M as a single matrix instead of a pair of vectors (vals)
we can just use software that has matrix operations (even Excel) and write
M*a (in that would be Excel {=MMULT(M,a)}).

However, the fine details are that "a" has to be a column vector for
this to work, and the result will be a column vector. If we want them
to be rows we have to write (M*aT)T where T is the transpose operator.
In Excel {=TRANSPOSE(MMULT(M,TRANSPOSE(a)))}

The result is <2, -3> meaning 2 octaves up and 3 fourth-generators down.

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

> But if I've made a serious mistake I really need to know:
> What do the integers [the val's coefficients] represent in tuning
> terms? What are they counting?

They can be counting different things; steps of an equal temperament,
number of octaves, number of generators, exponent of a particular
prime number being the obvious examples, but not the only ones.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> so how can i tell which one is covariant and which one is
> contravariant?

Which one do you regard as the vectors you start from (contravariant
vector) and which as linear functions on the space of such vectors
(covariant vector?) Obviously, in our case the monzos are the
objects, and the vals are the mappings, and not the other way around.
However, we *can* consider linear mappings of vals, which can be
identifified via unique isomorphim with monzos.

Anyway, we have this:

monzo = ket = contravariant

val = bra = covariant

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Ok, now we're on the right track, but I'm still not grokking
> >> you. I started with six rationals and ended up with 6 integers.
> >> What's the problem?
> >
> >Are your integers consecutive?
>
> No, and that's part of the def. of standard val, but what
> motivates it?

Forget standard vals. The point of it is that if you leave gaps, you
have no interpretation for what to put in the gap, which fits into
your chord.

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

> I agree it's awkward. Carl objected so vehemently to EDO and I
wanted
> to reserve ET for the most general term (including EDOs ED3s cETs).
> Perhaps this would be a misuse of ET. Do we have some other term for
> the most general category of 1D temperaments, i.e. any single
> generator temperament whether or not it is an integer fraction of
any
> ratio? I guess "1D-temperament" will do.

Not 1D. These are 0-dimensional temperaments, I'm afraid.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> not sure how gene would do this notationally, probably i did
> something terrible, but without it i could not have made those
> charts . . .

It's a special case of the wedge product, and I'd notate it that way.

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

> Without an musical example (referenced or constructed), I'm
skeptical.

What do you mean by an example?

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

> What happens if we change the bases to odd numbers.

You can't. It's not a basis.

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

> Wait... is this true: 'For a scale with card k, if there is
> no standard val with n=k that consistently maps the scale, the
> scale is not a Constant Structure.'

Sorry, but no. You are far too hung up on standard vals!

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> right, but i still want to understand it, since it was in my
> relativity textbooks . . .

It's more complicated in relativity. There you have tangent spaces
and cotangent spaces *at every point*, which have to connect
together, plus you have a non-positive inner product which changes
from point to point. We've got it easy and should enjoy ourseves.

>> Without an musical example (referenced or constructed), I'm
>skeptical.
>
>What do you mean by an example?

A short melody which doesn't sounds a certain way when tuned
in a CS scale, and then, by changing as little else as possible,
change things for the worse by warping the scale until its
non-CS. Would be ideal.

But even an example of music where the failure of "sequence"
(even from a theoretical point of view, even without hearing it)
would help to communicate what "sequence" is.

-Carl

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

> Drop the word "standard". There's absolutely no relationship between
> "standard" vals and Constant Structure. Sorry if anything I said
> mislead you in that direction.

I may have started it by remarking that all the vals we get from
complete p-limit chords turn out to be standard. Which is true, but
of small importance; it simply serves as a way to keep easily in mind
what they are.

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

> So [-4 4 -1> (x) [-11 4 2> = <12 19 28]

Or if you like, |-4 4 -1> ^ |-11 4 2> = <12 19 28|. Welcome to the
wonderful world of wedge products.

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

> How does that generalise to other than the 5-limit? i.e. vectors
with
> other than 3 components?

In the 7-limit, the wedge product of two monzos is a 6D wedge product
vector, (which is the two intervals are commas gives us on reduction
a wedgie for a temperament) wedging it with a monzo again gives us a
val. The wedge product of two vals (I'm assuming things are set up
the way I define them) gives us, once again, a 6D wedge product
vector, (which if the two vals are et vals gives us on reduction a
wedgie for a temperament) wedging it with a val again gives us a
monzo. This has to be done carefully in terms of basis elements to
make the equivalencies work.

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

> Me too, since I want it to be generalisable to matrices, and it
seems
> Gene doesn't care about that.

It's not a standard notation; however we might find it useful if it
doesn't confuse people.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > But if I've made a serious mistake I really need to know:
> > What do the integers [the val's coefficients] represent in tuning
> > terms? What are they counting?
>
> They can be counting different things; steps of an equal temperament,
> number of octaves, number of generators,

But these are all temperament "generators" in its most general sense.
As I said.

> exponent of a particular
> prime number

Now I'm really confused. I thought these were called monzos, not vals.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > I agree it's awkward. Carl objected so vehemently to EDO and I
> wanted
> > to reserve ET for the most general term (including EDOs ED3s cETs).
> > Perhaps this would be a misuse of ET. Do we have some other term for
> > the most general category of 1D temperaments, i.e. any single
> > generator temperament whether or not it is an integer fraction of
> any
> > ratio? I guess "1D-temperament" will do.
>
> Not 1D. These are 0-dimensional temperaments, I'm afraid.

Yes. You're right. Of course I was coming from the mathematical
standpoint.

Either way, we have descending dimensions planar temperaments, linear
temperaments, <then what?> temperaments?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > not sure how gene would do this notationally, probably i did
> > something terrible, but without it i could not have made those
> > charts . . .
>
> It's a special case of the wedge product, and I'd notate it that way.

So the wedge product is a generalisation of the 3D Cartesian product
or cross product. Awesome! There are really some light-bulbs coming on
in my head today. :-) Thanks Gene.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > So [-4 4 -1> (x) [-11 4 2> = <12 19 28]
>
> Or if you like, |-4 4 -1> ^ |-11 4 2> = <12 19 28|. Welcome to the
> wonderful world of wedge products.

Yes indeed. Hoorah!

Although I think I'd rather use the square brackets and only use the
pipe if you put them together so
< ... ].[ ... > is equivalent to
< ... | ... >

The square brackets alook more enclosing, and can't be mistaken for a one.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > How does that generalise to other than the 5-limit? i.e. vectors
> with
> > other than 3 components?
>
> In the 7-limit, the wedge product of two monzos is a 6D wedge product
> vector, (which is the two intervals are commas gives us on reduction
> a wedgie for a temperament)

How do you reduce it?

Is there a direct interpretation of the coefficients of the 6D
wedge-product in tuning terms, either before or after the reduction?
As there is in the 3D case?

> wedging it with a monzo again gives us a
> val. The wedge product of two vals (I'm assuming things are set up
> the way I define them) gives us, once again, a 6D wedge product
> vector, (which if the two vals are et vals gives us on reduction a
> wedgie for a temperament) wedging it with a val again gives us a
> monzo. This has to be done carefully in terms of basis elements to
> make the equivalencies work.

I think this is coming a bit too fast for me yet.

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> In the 7-limit, the wedge product of two monzos is a
> 6D wedge product vector, (which is the two intervals are
> commas gives us on reduction a wedgie for a temperament)
> wedging it with a monzo again gives us a val. The wedge
> product of two vals (I'm assuming things are set up
> the way I define them) gives us, once again, a 6D wedge
> product vector, (which if the two vals are et vals gives
> us on reduction a wedgie for a temperament) wedging it
> with a val again gives us a monzo. This has to be done
> carefully in terms of basis elements to make the
> equivalencies work.

i sure wish i knew what the hell this was all about.
especially since my name is being used as a term all thru it.

you guys (Gene, paul, Dave) lost me on this long ago.
but it sure seems interesting.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> i sure wish i knew what the hell this was all about.
> especially since my name is being used as a term all thru it.
>
> you guys (Gene, paul, Dave) lost me on this long ago.
> but it sure seems interesting.

I think I'll eventually be able to explain it in a way you can
understand it. But it wouldn't do to try until I'm sure I've actually
got it all sorted out myself. Paul could probably do the job too.

Dave Keenan wrote:

> So the wedge product is a generalisation of the 3D Cartesian product
> or cross product. Awesome! There are really some light-bulbs coming on
> in my head today. :-) Thanks Gene.

No, it's the other way around. The 3-D dot and cross products are special cases of Grassman algebra, which came first.

Graham

Paul Erlich wrote:
> similarly, if i take the (transpose of the?) val for 12-equal:
> > |12 19 28>
> > and take the cross product with the val for 22-equal:
> > <22 35 51|
> > i get the monzo for the diaschisma, the interval that vanishes in > both tunings:
> > [-11 4 2]
> > again, not sure what's going on notationally, but the numbers > work . . .

As Gene's said, this should be written

<12 19 28] ^ <22 35 51] = <-11 4 2]

meaning the wedge product of the two vals is equivalent to that monzo. Here's how you write it using my Python module:

>>> from temper import Wedgie as Val
>>> (Val((12,19,28))^Val((22,35,51))).complement().flatten()
(-11, 4, 2)

That the interval vanishes in both tunings can be expressed by the brakets (or whatever the products are called without complex numbers) equalling zero:

<12 19 28 | -11 4 2> = 0

<22 35 51 | -11 4 2> = 0

to check:

>>> Monzo=Val
>>> int(Val((22,35,51))^~Monzo((-11,4,2)))
0
>>> int(Val((12,19,28))^~Monzo((-11,4,2)))
0

(If the module knew the difference between covariant an contravariant vectors (as one version did) you wouldn't need that ~ .)

The complement (~ or .complement()) is not the same as a matrix transpose. Which way round you do the wedge product only affects the sign of the result.

>>the symbol normally indicates the cross-product, which is extremely >>useful in tuning: for example, if i take the monzo for the > > diaschisma
> >>[-4 4 -1>
>>
>>and cross it with the (transpose of the?) monzo for the syntonic > > comma
> >><-11 4 2]
>>
>>i get the val for the et where they both vanish:
>>
>>[12 19 28]
>>
>>not sure how gene would do this notationally, probably i did >>something terrible, but without it i could not have made those >>charts . . .

>>> (Monzo((-4,4,-1))^Monzo((-11,4,2))).complement().flatten()
(12, 19, 28)

Graham

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

> > exponent of a particular
> > prime number
>
> Now I'm really confused. I thought these were called monzos, not
vals.

The mapping which for any rational number gives the p-exponent of
that rational number is called a padic valuation, and is the basis
for the vals in the same way that prime numbers are the basis for
monzos.

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

> So the wedge product is a generalisation of the 3D Cartesian product
> or cross product. Awesome! There are really some light-bulbs coming
on
> in my head today. :-) Thanks Gene.

You've got it. It also defines the determinant, come to that.

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

> The square brackets alook more enclosing, and can't be mistaken for
a one.

I'm happy either way. Paul? What does a physics major think?

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

> > In the 7-limit, the wedge product of two monzos is a 6D wedge
product
> > vector, (which is the two intervals are commas gives us on
reduction
> > a wedgie for a temperament)
>
> How do you reduce it?

Take out any common factor, and make the first nonzero entry a one.
The exception is for commas, which are standardized by making the
comma greater than one. The point is to standardize so that there is
one and only one wedgie for a given temperament, of any dimension.

> Is there a direct interpretation of the coefficients of the 6D
> wedge-product in tuning terms, either before or after the reduction?

The first three are simply the mapping to generators, times a common
factor of the number of periods per octave. The last three can be
interpreted also, but less interestingly.

> As there is in the 3D case?

The 3D case is simply 5-limit monzos and vals; I don't get your
question.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> As Gene's said, this should be written
>
> <12 19 28] ^ <22 35 51] = <-11 4 2]

I think you mean

<12 19 28] ^ <22 35 51] = [-11 4 2>

monz wrote:

> i sure wish i knew what the hell this was all about.
> especially since my name is being used as a term all thru it.
> > you guys (Gene, paul, Dave) lost me on this long ago.
> but it sure seems interesting.

We've established that the wedge product of two monzos corresponds to the temperament in which they vanish. So, with Gene's notation, a comma and diaschisma give 12-equal.

|-4 4 -1> ^ |-11 4 2> = <12 19 28|

Well, on top of that, you can temper out 36:35, or |2 2 -1 -1>

|-4 4 -1> ^ |-11 4 2> ^ |2 2 -1 -1> = <12 19 28 34|

Which, to check with my Python module:

>>> from temper import Wedgie as Monzo
>>> syntonic = Monzo((-4,4,-1))
>>> diaschisma = Monzo((-11,4,2))
>>> septimal = Monzo((2,2,-1,-1))
>>> (syntonic^diaschisma^septimal).complement().flatten()
(12, 19, 28, 34)

If you only temper out two commas, you get a linear temperament.

|-4 4 -1> ^ |2 2 -1 -1> = 7-limit meantone.

I don't know how to write linear temperaments as bras, but there are some things you can show. For example, an octave equivalent mapping is like tempering out the octave.

>>> octave = Monzo((1,0,0))
>>> (syntonic^septimal^octave).complement().flatten()
(0, 1, 4, -2)

which means

|-4 4 -1> ^ |2 2 -1 -1> ^ |1 0 0> = <0 1 4 -2|

and (1 4 -2) is the octave-equivalent mapping for this particular version of meantone, where C-Bb approximates 4:7, rather than C-A#. For the more accurate one, you can do

|-4 4 -1> ^ |1 2 -3 1> ^ |1 0 0> = <0 1 4 10|

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
>> If you only temper out two [7-limit] commas, you get a linear
temperament.
>
> |-4 4 -1> ^ |2 2 -1 -1> = 7-limit meantone.

I'd have added the zero to the end to make it [-1 4 -1 0> just to keep
things clear.

> I don't know how to write linear temperaments as bras,
...

Here's an idea. Use << ... ]] for a linear temperament (bivector), <<<
... ]]] for a planar temperament (trivector) etc.

So we have a Pascal's triangle of covariant-multivector (multival)
types like this (where the number indicates how many coefficients
there are).

,vals
1 , ' ,bivals 1-limit
1 <1] , ' ,trivals 2-limit
1 <2] <<1]] , ' 3-limit
1 <3] <<3]] <<<1]]] 5-limit
1 <4] <<6]] <<<4]]] <<<<1]]]] 7-limit
1 <5] <<10]] <<<10]]] <<<<5]]]]<<<<<1]]]]] 11-limit
etc.

And I would have thought we'd have this for the contravariant
multivectors (multimonzos).
,monzos
1 , ' ,bimonzos 1-limit
1 [1> , ' ,trimonzos 2-limit
1 [2> [[1>> , ' 3-limit
1 [3> [[3>> [[[1>>> 5-limit
1 [4> [[6>> [[[4>>> [[[[1>>>> 7-limit
1 [5> [[10>> [[[10>>> [[[[5>>>>[[[[[1>>>>> 11-limit
etc.

However it seems that, at least in our application, we have

1 1-limit
1 1 2-limit
1 <2] 1 3-limit
1 <3] [3> 1 5-limit
1 <4] <<6]] [4> 1 7-limit
1 <5] <<10]] [[10>> [5> 1 11-limit

where those on the midline could go either way.

Why is that? I thought they had incomensurable bases. One lot has
units of "log-prime" and the other "per-log-prime".

e.g. Why is a 5-limit bi-val a monzo?

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Dave Keenan wrote:
>
> > So the wedge product is a generalisation of the 3D Cartesian product
> > or cross product. Awesome! There are really some light-bulbs coming on
> > in my head today. :-) Thanks Gene.
>
> No, it's the other way around. The 3-D dot and cross products are
> special cases of Grassman algebra, which came first.

I wasn't implying any history. If B is a special case of A then A is a
generalisation of B. But thanks for the history.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > > exponent of a particular
> > > prime number
> >
> > Now I'm really confused. I thought these were called monzos, not
> vals.
>
> The mapping which for any rational number gives the p-exponent of
> that rational number is called a padic valuation, and is the basis
> for the vals in the same way that prime numbers are the basis for
> monzos.

So what I said was correct. Putting it another way, more in
engineering terminology that pure math, the "units" for the
coefficients of a monzo are the logs of the respective primes, and the
"units" for the coefficients of the vals (in our application) or
prime-mappings are "generators per log prime" so that the dot product
of a val and a monzo has units of simply "generators".

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > So the wedge product is a generalisation of the 3D Cartesian product
> > or cross product. Awesome! There are really some light-bulbs coming
> on
> > in my head today. :-) Thanks Gene.
>
> You've got it. It also defines the determinant, come to that.

Right. I reread John Browne's Introduction to Grassman Algebra and
it's making a lot more sense this time.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > The square brackets alook more enclosing, and can't be mistaken for
> a one.
>
> I'm happy either way. Paul? What does a physics major think?

Hey I majored in Physics too (as well as Comp Sci). :-)

I learned the Dirac notation in second year Quantum Physics, from the
Feynman Lectures Volume III. But that _was_ a long time ago.

When I just looked at that book again now. You hardly ever see < .. |
or |.. > alone, but usually < ... | ... >. Also, you hardly ever see
actual numbers inside, usually variables, so the issue of confusion of
| with 1 or it's lack of visual enclosingness (is that a word?) don't
arise.

But yeah. What do others think? Square brackets or vertical bars (pipes)?

>But yeah. What do others think? Square brackets or vertical
>bars (pipes)?

I don't care, but I think we should standardize.

-Carl

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

> Here's an idea. Use << ... ]] for a linear temperament (bivector),
<<<
> ... ]]] for a planar temperament (trivector) etc.

That's a thought. I haven't seen this notation ever used, but it is
logical.

> And I would have thought we'd have this for the contravariant
> multivectors (multimonzos).

If we fix a prime limit, we have duality between multivals and
multimonzos; in the 7-limit, a bival and a bimonzo can be identified.
In the 11-limit, bivals and 4-monzos, and bimonzos and 4-vals, can be
identified, as can trivals with trimonzos. This involves changing the
basis of the n-monzos to make them numerically correspond to the
(pi(p)-n)-vals. My approach to all this has been to swap the
n-monzo for the corresponding (pi(p)-n)-val, and use that, but this
does require we fix a prime limit.

> e.g. Why is a 5-limit bi-val a monzo?

The above duality. It isn't, except if you make the identification.

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

> >But yeah. What do others think? Square brackets or vertical
> >bars (pipes)?
>
> I don't care, but I think we should standardize.

I'd also suggest that, unless we specify otherwise, a 3D val or monzo
is 5-limit, a 4D val or monzo 7-limit, etc. and that the basis for
the n-multimonzos is ordered so that they correspond to the (n-pi(p))-
multivals. That, at least, is how I wrote my software.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> monz wrote:
>
> > i sure wish i knew what the hell this was all about.
> > especially since my name is being used as a term all thru it.
> >
> > you guys (Gene, paul, Dave) lost me on this long ago.
> > but it sure seems interesting.
>
> We've established that the wedge product of two monzos
> corresponds to the temperament in which they vanish. So,
> with Gene's notation, a comma
> and diaschisma give 12-equal.
>
> |-4 4 -1> ^ |-11 4 2> = <12 19 28|
>
> Well, on top of that, you can temper out 36:35, or |2 2 -1 -1>
>
> |-4 4 -1> ^ |-11 4 2> ^ |2 2 -1 -1> = <12 19 28 34|
>
> Which, to check with my Python module:
>
> >>> from temper import Wedgie as Monzo
> >>> syntonic = Monzo((-4,4,-1))
> >>> diaschisma = Monzo((-11,4,2))
> >>> septimal = Monzo((2,2,-1,-1))
> >>> (syntonic^diaschisma^septimal).complement().flatten()
> (12, 19, 28, 34)
>
> If you only temper out two commas, you get a linear temperament.
>
> |-4 4 -1> ^ |2 2 -1 -1> = 7-limit meantone.
>
> I don't know how to write linear temperaments as bras, but
> there are some things you can show. For example, an
> octave equivalent mapping is like tempering out the octave.
>
> >>> octave = Monzo((1,0,0))
> >>> (syntonic^septimal^octave).complement().flatten()
> (0, 1, 4, -2)
>
> which means
>
> |-4 4 -1> ^ |2 2 -1 -1> ^ |1 0 0> = <0 1 4 -2|
>
> and (1 4 -2) is the octave-equivalent mapping for this
> particular version of meantone, where C-Bb approximates 4:7,
> rather than C-A#. For the more accurate one, you can do
>
> |-4 4 -1> ^ |1 2 -3 1> ^ |1 0 0> = <0 1 4 10|

thanks, Graham ... this helps a lot!

-monz

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

> > [Dave Keenan:]
> > But yeah. What do others think? Square brackets or vertical
> > bars (pipes)?
>
> I don't care, but I think we should standardize.
>
> -Carl

i think the best solution is to use square brackets for the
bra and ket when written separately, and to use the pipe when
they're put together.

bra <...]
ket [...>
braket <...|...>

didn't someone just suggest that yesterday?

-monz

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > Here's an idea. Use << ... ]] for a linear temperament
> > (bivector), <<< ... ]]] for a planar temperament (trivector)
> > etc.
>
> That's a thought. I haven't seen this notation ever used,
> but it is logical.

i'll second the notational idea, because i can see the logic
in it.

unfortunately, i have no clue what "bivectors" and "trivectors"
are.

> > And I would have thought we'd have this for the contravariant
> > multivectors (multimonzos).
>
> If we fix a prime limit, we have duality between multivals
> and multimonzos; in the 7-limit, a bival and a bimonzo can
> be identified.
> In the 11-limit, bivals and 4-monzos, and bimonzos and 4-vals,
> can be identified, as can trivals with trimonzos. This
> involves changing the basis of the n-monzos to make them
> numerically correspond to the (pi(p)-n)-vals. My approach
> to all this has been to swap the n-monzo for the corresponding
> (pi(p)-n)-val, and use that, but this does require we fix
> a prime limit.
>
> > e.g. Why is a 5-limit bi-val a monzo?
>
> The above duality. It isn't, except if you make the
> identification.

oh dear ... Graham, i was beginning to understand this stuff
after your last post, but now i'm hopelessly lost again. help!

-monz

hi Gene,

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

> I'd also suggest that, unless we specify otherwise, a 3D val
> or monzo is 5-limit, a 4D val or monzo 7-limit, etc. and that
> the basis for the n-multimonzos is ordered so that they
> correspond to the (n-pi(p))-multivals. That, at least, is
> how I wrote my software.

i'll second these suggestions.

but even tho i understand your last sentence,
what's (n-pi(p)) ?

-monz

monz wrote:

> unfortunately, i have no clue what "bivectors" and "trivectors"
> are.

A bivector will be the wedge product of two vectors, and a trivector the wedge product of three vectors.

>>>e.g. Why is a 5-limit bi-val a monzo?
>>
>>The above duality. It isn't, except if you make the
>>identification.
> > oh dear ... Graham, i was beginning to understand this stuff
> after your last post, but now i'm hopelessly lost again. help!

A bi-val must be the wedge product of two vals. If I can get it the right way round this time

<12 19 28] ^ <22 35 51] = [-11 4 2>

you can see that this particular bi-vector is the same as a monzo. But it only works in the 5-limit. A more familiar equation should be

<12 19 28] ^ <19 30 44] = [-4 4 -1>

so the intersection of 12-equal and 19-equal is a linear temperament in which the syntonic comma vanishes. In general terms, you can write that

<h12] ^ <h19] = <<meantone]]

where <<meantone]] is a bi-val defining some kind of meantone. I've previously shown the usual 7-limit case

[-4 4 -1 0> ^ [1 2 -3 1> = <<meantone7]]

so combining the two equations gives

<12 19 28 34] ^ <19 30 44 53] = [-4 4 -1 0> ^ [1 2 -3 1>

where a bi-val happens to equal a bi-monzo.

Here's the Python to check with:

>>> import temper
>>> h12 = temper.BestET(12, temper.limit7)
>>> h19 = temper.BestET(19, temper.limit7)
>>> syntonic = temper.Wedgie((-4, 4, -1))
>>> septimal = temper.Wedgie((1, 2, -3, 1))
>>> h12^h19 == ~(syntonic^septimal)
1
>>> h12
[12, 19, 28, 34]
>>> h19
[19, 30, 44, 53]

understanding the bi-val <<meantone]] itself is difficult, but you should always be able to get the octave-equivalent mapping by taking the wedge product with the octave

<<meantone]] ^ [1,0...> = +/- <0, 1, 4, ...]

Here's how you calculate it:

>>> octave = temper.Wedgie((1,))
>>> (octave^(h12^h19).complement()).complement().flatten()
(0, -1, -4, -10)

There are two ways of doing the left hand side. Firstly, you could take the complement of the octave, which will be a tri-val.

<<meantone]] ^ <<<octave_complement]]] = ????

in fact, this is illegal, because you can't have a 5-val in 4-space. So instead, we have to take the complement of <<meantone]] which is a bi-monzo

[[meantone_complement>> ^ [1,> = [[[mapping_complement>>>

In this space, the complement of a tri-monzo is a val, so we can say that

[[[mapping_complement>>> = <mapping] = -<0 1 4 10]

As the dimensions don't always make it unique, it may be worth specifying whether to take the complement of the left or right hand side in an operation involving mixed bras and kets. And probably coercing to the right hand side would make more sense, because I think the brakets usually give the right results then when the results are scalars. That is

<et|interval> = steps to interval

or

<primes|interval> = size of interval

always give positive numbers for an ascending interval. The calculation is as follows:

Take the complement of [interval> to get a multi-val

Multiply this by <primes] to get an even multi-er val

Take the complement of this to get a scalar.

so the general equation for linear temperaments can be written

<<lintemp]] ^ [octave> = +/- <mapping]

And, if you're really daring, you can even get rid of the ^

<<lintemp]][octave> = +/- <mapping]

I don't know if anybody's following this, but I think the multi-bra notation makes it clearer than anything we've had before.

Graham

hi Graham,

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> monz wrote:
>
> > unfortunately, i have no clue what "bivectors" and
> > "trivectors" are.
>
> A bivector will be the wedge product of two vectors,
> and a trivector the wedge product of three vectors.

OK, thanks. i knew that they referred to two and three
of something.

now if only i understood what a "wedge product" is ...
i understand it intuitively, but not mathematically.

> A bi-val must be the wedge product of two vals.
> If I can get it the right way round this time
>
> <12 19 28] ^ <22 35 51] = [-11 4 2>
>
> you can see that this particular bi-vector is the same
> as a monzo. But it only works in the 5-limit. A more
> familiar equation should be
>
> <12 19 28] ^ <19 30 44] = [-4 4 -1>
>
> so the intersection of 12-equal and 19-equal is a
> linear temperament in which the syntonic comma vanishes.
> In general terms, you can write that
>
> <h12] ^ <h19] = <<meantone]]
>
> where <<meantone]] is a bi-val defining some kind of meantone.

thanks so much for this! it's *much* clearer now!

>
> <snip>
>
> so the general equation for linear temperaments can be written
>
> <<lintemp]] ^ [octave> = +/- <mapping]
>
> And, if you're really daring, you can even get rid of the ^
>
> <<lintemp]][octave> = +/- <mapping]
>
>
> I don't know if anybody's following this, but I think
> the multi-bra notation makes it clearer than anything
> we've had before.

yes, i totally agree. i couldn't follow the part of your post
which i snipped, or that last bit quoted above, but the
new notation *is* helping a lot. using plain terms like
"meantone" and "meantone7" also helps.

-monz

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> As the dimensions don't always make it unique, it may be worth
> specifying whether to take the complement of the left or right hand
side
> in an operation involving mixed bras and kets.

And probably coercing to
> the right hand side would make more sense, because I think the
brakets
> usually give the right results then when the results are scalars.

My approach has been coercion; I routinely convert multimonzos into
multivals. In my practice multivals are only complimented over to
multimonzos when the result is simply a monzo, ie a 1-monzo. Of
course if you have a multival and want to wedge it with a monzo, it
has to be converted first to the complimentary multimonzo, and then
(if you are playing the coercion game) converted back again. This can
also be looked at as a matter of having two kinds of product. My own
software just calls it "up" vs "down", a matter of the dimension of
the corresponding temperament increasing (wedge with a val) or
decreasing (wedge with a monzo.)

> I don't know if anybody's following this, but I think the multi-bra
> notation makes it clearer than anything we've had before.

I think you and I are probably the only ones without headaches, but I
do agree. We seem to have passed rapidly from disappointing
incomprehension to far more understanding of this stuff than I've
learned to expect, and I think the notation thing is a big help. I'm
going to revise my web pages, and I think revised dictionary entries
are in order.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > >But yeah. What do others think? Square brackets or vertical
> > >bars (pipes)?
> >
> > I don't care, but I think we should standardize.
>
> I'd also suggest that, unless we specify otherwise, a 3D val or monzo
> is 5-limit, a 4D val or monzo 7-limit, etc. and that the basis for
> the n-multimonzos is ordered so that they correspond to the (n-pi(p))-
> multivals. That, at least, is how I wrote my software.

Yes. I totally agree. It is unfortunate that John Browne didn't do it
that way. See chapter 2 page 18
http://www.ses.swin.edu.au/homes/browne/grassmannalgebra/book/
Although he does mention complementary bases.

I assume your "pi()" function is the prime index function where pi(2)
= 1, pi(3) = 2, pi(5) = 3, etc.

So does this mean that since a 5-limit monzo has coefficients [e2 e3
e5> then a 5-limit bimonzo has coefficients ordered [[e35 e52 e23>>
where the subscripts indicate the primes referred-to in the basis? And
the same with maps and bimaps (vals and bivals).

And since a 7-limit monzo has coefficients [e2 e3 e5 e7> then a
7-limit trimonzo will have coefficients ordered [[[e357 e572 e723 e235>>>.

Is this how your software does it too Graham?

But how do you order the coefficents of a 7-limit bimonzo or bimap
(bival) so it's its own complement???

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> If we fix a prime limit, we have duality between multivals and
> multimonzos; in the 7-limit, a bival and a bimonzo can be identified.
> In the 11-limit, bivals and 4-monzos, and bimonzos and 4-vals, can be
> identified, as can trivals with trimonzos. This involves changing the
> basis of the n-monzos to make them numerically correspond to the
> (pi(p)-n)-vals. My approach to all this has been to swap the
> n-monzo for the corresponding (pi(p)-n)-val, and use that, but this
> does require we fix a prime limit.

Don't we always fix the prime limit anyway? Why might this be a problem?

> > e.g. Why is a 5-limit bi-val a monzo?
>
> The above duality. It isn't, except if you make the identification.

But why is that identification even possible, when, as I understand
it, the bases are incommensurable? There's something I'm not getting here.

Gene, can you please post your code for calculating the wedge-product
for arbitrary dimensions and arbitrary combinations of grades.

And Graham, if yours uses the same ordering of coefficients, please do
the same. (I know it's in the Python code on your website somewhere,
but I'm lazy).

Any accompanying comments will be gratefully received.

For those who haven't followed John Browne's intro, the "grade" of the
multivector is what we're now indicating by the number of nested
brackets and the "bi-" "tri-" etc.

In the Pascal's-triangle of multivector types, dimension increases
downward and grade increases from left to right.

hi Gene and Graham (and probably paul too)

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

> --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>
wrote:
>

> > I don't know if anybody's following this, but I think
> > the multi-bra notation makes it clearer than anything
> > we've had before.
>
> I think you and I are probably the only ones without
> headaches, but I do agree. We seem to have passed rapidly
> from disappointing incomprehension to far more
> understanding of this stuff than I've learned to expect,
> and I think the notation thing is a big help. I'm going
> to revise my web pages, and I think revised dictionary
> entries are in order.

yes, absolutely. i totally agree. would you guys
*please* rewrite them for me? thanks.

-monz

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
>
> > Is it important that a musical scale be
> > a constant structure, and if so, why?
>
> but we're talking about a chord, not a whole scale . . .

Aha! That's something that wasn't clear to me. If someone starts
talking about 5 to 7 tones in some sort of combination, I start
thinking of them in terms of a "scale" rather than a "chord". Sure,
I would say that it's not necessary or even important for a chord to
be a constant structure.

But we could take something that I said a step further: just as a
major or minor scale is a subset of a pythagorean, meantone, or 12-ET
tuning, major and minor triads are subsets of major and minor
scales. Then we could ask the question: Is it important that new
musical scales be organized in such a manner that they can be
described as a set of tones that is the union of several subsets of
tones (chords) that are alike in structure, differing only in that
they built on different tones (i.e., have different roots), just as a
major scale could be described as the union of the tones in a tonic,
dominant, and subdominant triads? Certainly this would be a very
desirable characteristic in a scale -- hmmm, it seems to me that you
wrote a very brilliant paper on the possibilities of doing this in a
22-tone octave that would serve as a prime example. ;-). I also
found a couple of scale subsets of a 17-tone octave that have this
property, and I have a composition in progress in 17-WT that will
illustrate one of these using non-5 harmony. (The process of
composing is very slow going; I try lots of things, and some of them
work, but some don't.)

But I would say that this property is not essential in a new scale --
just that it's a very nice thing to have working in your favor. (But
I digress.)

> > (By "scale", I am referring to a set of tones that may be used to
> > write a simple melody.
>
> exactly . . . the two champions would have to be the diatonic
> pentatonic and heptatonic scales . . .
>
> > If I'm using a pentatonic scale made from a 9-limit otonal chord:
> > 8 : 9 : 10 : 12 : 14 : 16
> > then I have two intervals each of 2:3 (both pentatonic "4ths")
and
> > 3:4 (both pentatonic "3rds").
>
> personally, i'm not fond of this as a scale or melodic entity at
all -
> - when i improvise over a dominant ninth chord, simply using its
> notes is about the worst way to come up with a melody . . .

I understand, and I wouldn't have much to say about the harmonic
possibilities either. But I think that we've been spoiled by the
harmonic sophistication of the major-minor system to such an extent
that it's difficult to appreciate the resources of a simple scale.
We would have to immerse ourselves in gamelan music (particularly
slendro) to get in the proper frame of mind to be able to even begin
to create something decent with such limited tonal resources.
(Again, we're off on another topic.)

> > So far, so good.
> >
> > But if I try to use hexatonic scale made from an 11-limit otonal
> > chord:
> > 8 : 9 : 10 : 11 : 12 : 14 : 16
> > then one of my 2:3s is a hexatonic "5th" and the other is a
> > hexatonic "4th", and likewise one of my 3:4s is a hexatonic "4th"
and
> > the other is a hexatonic "3rd". Most attempts to transfer a
melodic
> > figure beginning on a certain scale degree to another scale
degree
> > (such as is required in the musical device called a "sequence")
will
> > tend to produce undesirable consequences (such as listener
> > disorientation) due to the fact that the 2:3 and/or 3:4 must
switch
> > degree-roles in the process.
>
> yet even an algorithmic composition program, such as those written
by
> prent rodgers, can produce lovely music by simply using a single
such
> hexad at a time, for both harmony and melody (or for polyphony).
i'm
> not disputing your constant structure argument too vehemently,
> especially when it concerns such an important interval as 3:2, but
> note especially that the diatonic scale in 12-equal is not CS, and
> yet doesn't cause any more listener disorientation than the
diatonic
> scale in, say, 19-equal or 17-equal, where it it CS.

I agree. I thought about this over the weekend, and I've figured out
how this differs from the 11-limit otonal scale. Suppose that we
were to modify this 11-limit otonal hexatonic scale:
8 : 9 : 10 : 11 : 12 : 14 : 16
so that the 8:12 and 9:12 are each tempered wide by one cent, so that:
8:12 = 703c, 4 scale degrees
12:18 = 701c, 3 deg
9:12 = 499c, 3 deg
12:16 = 497c, 2deg
thereby giving us a constant structure.

Would this eliminate the problem of the potential for
disorientation? Of course not, because it still remains that tones
having the same functional relationship are separated by different
numbers of degrees in the scale. We still perceive slightly tempered
fifths as fifths (i.e., members of the same interval-class), and
tempering does not change their harmonic identity or function within
a scale one bit.

Carl mentioned (in msg. #7670) that:
> Paul E. has suggested that we only care about collisions if they
> occur to a consonant interval. That allows the diatonic scale
> in 12-equal to pass.

That's a good hypothesis, but I think that there's more to it than
that. Observe that there is a collision in a harmonic minor scale
between the augmented 2nd (a dissonance between the 6th and 7th scale
degrees) and the minor 3rds in the scale (which are *consonant*), but
I wouldn't say that this results in any disorientation.

Also consider this: If we were to make each of the 12-ET fifths in a
major scale narrower by 0.1 cents (so as to make the augmented 4th
slightly different in size from the diminished 5th), then the scale
would be CS, even though we would be hard pressed to hear any
difference from 12-ET. So constant structure (taken alone) is not
the whole issue.

I believe that the potential for _functional scale disorientation_
(if I may attempt to coin a term) is caused by a particular
combination of circumstances:

1) If there are two *functionally* different intervals (i.e., aug4 &
dim5, or aug2 & min3) in a scale that only *happen* to be the exact
same size (because their ratios, which may be defined either as
rational or not, happen to be conflated in the tuning in which the
scale is being used) then there is *no* potential for functional
scale disorientation.

2) But if there are two intervals in a scale that are *not
functionally different* (such as the two 2:3s or 3:4s in our 11-limit
hexatonic otonality), but which span different numbers of steps in
the scale, then the possibility for functional scale disorientation
exists. Since it would be very difficult to perceive any interval
anywhere in the ballpark of 2:3 or 3:4 as having some other identity
or functional role, even tempering to make the scale CS would not
address the problem of functional scale disorientation. So we see
that consonance and harmonic entropy are involved here, but it is
*functional equality* in combination with a *non-CS* condition for
two intervals that are the conditions for disorientation.

To summarize: Two intervals with *different* functionality that are
the same size will not cause disorientation so long as each one spans
the proper number of steps in the scale. It is only when two
intervals with the *same* functionality span differing numbers of
steps that the problem arises (regardless of whether they are the
same size or not).

I'm not familiar with any of the 11-limit music that Prent Rodgers
has produced, so I can only make a guess about why it works. Suppose
that I happen to write a composition using a major scale with the 6th
degree omitted. Is it hexatonic or heptatonic? I think that we
would hear it as heptatonic (due in no small part to the fact that we
are so heptatonic-oriented, but on the other hand, I don't know how
someone coming from a culture that uses only a pentatonic scale would
interpret it). So I think that it's possible to avoid functional
scale disorientation with an 11-limit otonal scale by interpreting it
as an incomplete heptatonic scale.

> > ...
> > So it would not have been possible for the methods of
conventional (5-
> > limit) harmony to have reached such sophistication if the major
and
> > minor scales were not constant structures, because our whole
method
> > of building chords (by 3rds) has depended on the fact that the
simple
> > ratios of 3 would always be heptatonic 4ths and 5ths and that the
> > simple ratios of 5 would always be 3rds and 6ths.
>
> ah, but you're depending on the heptatonic scale here! if we used
> some sort of heptatonic or other scalar framework to understand 11-
> limit harmony, the same property might hold, despite the fact that
> the hexad itself is not CS. so the latter fact seems irrelevant.

So I guess I'm agreeing with you by what I said above.

> > ...
> > If you want 11-limit otonal harmony in a conherent scale, then I
> > think it will have to be at least heptatonic and that you're
going to
> > have to fill that extra position with something or other, such as:
> > 8:9:10:11:12:27/2:14:16.
> > Hmmm, that's really not a bad choice, if you'll notice that
22:27:32
> > is an isoharmonic triad. I remember that this scale works very
> > nicely in 31-ET, since the 27/2:16:20 ends up as an ordinary
minor
> > triad.
> >
> > Likewise, you can have constant-structure 17 and 19-limit otonal
scales:
> > 16:17:18:20:21:22:24:26:28:30:32 (with chords built in
decatonic "4ths")
> > and
> > 16:17:18:19:20:21:22:24:25:26:28:30:32
>
> ok, but not much harmonic movement possible here.

I also find that the dodecatonic scale has more tones than I would
like to see in anything that I would consider a scale. While it's
not quite the same challenge as trying to writing a successful melody
using all the tones of 12-ET (e.g., problems with identifying a tonal
center), it's still a bit much for a listener to take in.

I would guess that your statement about limitations on harmonic
movement comes from looking at this as a single scale, but I find
that if I look at it as a just tuning, then it possesses considerable
harmonic richness by virtue of the great variety of heptatonic scales
that can be found in it -- a direct result of the large number of
intervals present in the tuning (especially compared to 12-ET).
There's almost nothing that you can transpose (in the strict sense of
the term), but one thing that you *can* do is transpose in a looser
sense: just as a theme may be transposed from a major to its relative
minor key, so may a pentatonic or heptatonic theme be transposed in
this 19-limit tuning into another "key", and *always* with a change
of mode. And if the term "modulation" doesn't apply, then where does
it?

--George

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> George,
>
> I was already convinced that Constant Structure is a valuable
melodic
> property of a scale. But what's wrong with using complete 11-limit
> hexads as vertical harmony within a larger CS scale? Why should we
> care that the hexads _themselves_ are not CS?

Nothing wrong at all with that, just as long as your larger scale
doesn't get so large that there are too many tones for it to be
comprehended as a scale. See my reply to Paul.

--George

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I've never really thought very much about this, because for me
this
> >was something that seemed to be fairly obvious: that a musical
scale
> >that is not a constant structure will tend to result in confusion
or
> >disorientation by an inherent contradiction between the acoustical
> >properties of certain intervals and their identity (or ability to
> >function) as members (i.e., degrees or steps) of that scale.
>
> Does that include the diatonic scale in 12-equal?

No. See my reply to Paul.

> ...
> But incidentally, I'd love a musical example of a hexatonic 11-limit
> melody where the non-CS "collision" causes a problem with
constructing
> a musical sequence. With all the ink I've spilled on this subject,
> I'm probably more guilty than anyone of not having come up with
> musical examples to demonstrate propriety...

I'd have to compose something to illustrate that and produce a midi
or mp3 file that we could listen to. Then we would have to decide
that, if we both agreed that it didn't work, that it was due to the
non-CS property and not because I had a bad composing day. Or worse
yet, maybe I might have an extraordinarily good composing day and end
up turning out something that did work, in spite of the non-CS scale
property.

Unfortunately, I haven't found much time lately to compose things
that exemplify ideas that I believe *will* work. :-(

--George

>I also
>found a couple of scale subsets of a 17-tone octave that have this
>property, and I have a composition in progress in 17-WT that will
>illustrate one of these using non-5 harmony.

Sweet! Can't wait to hear this...

>Carl mentioned (in msg. #7670) that:
>> Paul E. has suggested that we only care about collisions if they
>> occur to a consonant interval. That allows the diatonic scale
>> in 12-equal to pass.
>
>That's a good hypothesis, but I think that there's more to it than
>that. Observe that there is a collision in a harmonic minor scale
>between the augmented 2nd (a dissonance between the 6th and 7th scale
>degrees) and the minor 3rds in the scale (which are *consonant*), but
>I wouldn't say that this results in any disorientation.

Hmm...

>Also consider this: If we were to make each of the 12-ET fifths in a
>major scale narrower by 0.1 cents (so as to make the augmented 4th
>slightly different in size from the diminished 5th), then the scale
>would be CS, even though we would be hard pressed to hear any
>difference from 12-ET. So constant structure (taken alone) is not
>the whole issue.

It's a bit of a red herring, to use tolerance in this way. Things
like CS can be assumed to operate through a blur filter. It's always
better to explicitly spec the filter, as I did for some of Rothenberg's
measures, but anyway....

>2) But if there are two intervals in a scale that are *not
>functionally different* (such as the two 2:3s or 3:4s in our 11-limit
>hexatonic otonality), but which span different numbers of steps in
>the scale, then the possibility for functional scale disorientation
>exists. Since it would be very difficult to perceive any interval
>anywhere in the ballpark of 2:3 or 3:4 as having some other identity
>or functional role, even tempering to make the scale CS would not
>address the problem of functional scale disorientation. So we see
>that consonance and harmonic entropy are involved here, but it is
>*functional equality* in combination with a *non-CS* condition for
>two intervals that are the conditions for disorientation.

Can you write a melody in the 11-limit 'scale' that sounds wrong
because of functional scale disorientation?

>To summarize: Two intervals with *different* functionality that are
>the same size will not cause disorientation so long as each one spans
>the proper number of steps in the scale. It is only when two
>intervals with the *same* functionality span differing numbers of
>steps that the problem arises (regardless of whether they are the
>same size or not).

[I'm quoting this here so I only have to save this message.]

>I'm not familiar with any of the 11-limit music that Prent Rodgers
>has produced, so I can only make a guess about why it works. Suppose
>that I happen to write a composition using a major scale with the 6th
>degree omitted. Is it hexatonic or heptatonic? I think that we
>would hear it as heptatonic (due in no small part to the fact that we
>are so heptatonic-oriented, but on the other hand, I don't know how
>someone coming from a culture that uses only a pentatonic scale would
>interpret it). So I think that it's possible to avoid functional
>scale disorientation with an 11-limit otonal scale by interpreting it
>as an incomplete heptatonic scale.

Of all the scales that might sound like a diatonic scale, I think
the 11-limit otonal scale is probably least. Anyway, you should
definitely download some of Prent's music, even though I don't think
it's a very good example of melodic writing with harmonic series
segments (Prent's music isn't very melodic) -- Denny Genovese or
Jules Siegel have provided much better examples (though not
downloadable at this point).

>I also find that the dodecatonic scale has more tones than I would
>like to see in anything that I would consider a scale.

We agree on that.

-Carl

>> ...
>> But incidentally, I'd love a musical example of a hexatonic 11-limit
>> melody where the non-CS "collision" causes a problem with
>> constructing a musical sequence.
>
>I'd have to compose something to illustrate that and produce a midi
>or mp3 file that we could listen to. Then we would have to decide
>that, if we both agreed that it didn't work, that it was due to the
>non-CS property and not because I had a bad composing day.

And would it be so much harder than writing all these messages,
full of so much speculation?

-Carl

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > wrote:
> > > > --- In tuning-math@yahoogroups.com, "Dave Keenan"
<d.keenan@b...>
> > > > wrote:
> > > >
> > > > > If we are told that the mapping is for a tET then _which_
tET
> > it is
> > > > > for can be read straight out of the mapping, as the
coefficient
> > for
> > > > > the prime 2 (the first coefficient). And the generator is
> > simply one
> > > > > step of that tET.
> > > >
> > > > just wondering why you keep saying "tET" -- 'If we are told
that
> > the
> > > > mapping is for a tone equal temperament then . . .' ??
> > >
> > > I agree it's awkward. Carl objected so vehemently to EDO and I
> > wanted
> > > to reserve ET for the most general term (including EDOs ED3s
cETs).
> > > Perhaps this would be a misuse of ET. Do we have some other
term for
> > > the most general category of 1D temperaments, i.e. any single
> > > generator temperament whether or not it is an integer fraction
of
> > any
> > > ratio? I guess "1D-temperament" will do.
> > >
> > > > actually, > and < fit together and create a X (as in times) !
> > >
> > > Oops. Well we could interpret that as the matrix-product as
opposed
> > to
> > > the scalar-product (dot-product), but I don't know of any
meaning
> > for
> > > that in tuning.
> >
> > the symbol normally indicates the cross-product, which is
extremely
> > useful in tuning: for example, if i take the monzo for the
diaschisma
> >
> > [-4 4 -1>
> >
> > and cross it with the (transpose of the?) monzo for the syntonic
comma
> >
> > <-11 4 2]
>
> Should have been [-11 4 2>

no, the point is that you transpose it so that the angle bracket is
at the beginning.

> > i get the val for the et where they both vanish:
> >
> > [12 19 28]
>
> Now you could write <12 19 28]

maybe.

> That's magic! I never knew that! But of course if someone ever said
it
> before I wouldn't have understood it since I didn't have a clue
what a
> val was.
>
> So [-4 4 -1> (x) [-11 4 2> = <12 19 28]

the idea, though, is that the if the second vector has the angle
bracket at the beginning, you end up with the symbology ><, which
already looks like a "x".

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > so how can i tell which one is covariant and which one is
> > contravariant?
>
> Which one do you regard as the vectors you start from
(contravariant
> vector) and which as linear functions on the space of such vectors
> (covariant vector?) Obviously, in our case the monzos are the
> objects, and the vals are the mappings, and not the other way
around.
> However, we *can* consider linear mappings of vals, which can be
> identifified via unique isomorphim with monzos.
>
> Anyway, we have this:
>
> monzo = ket = contravariant
>
> val = bra = covariant

so it's pretty much a matter of convention which ones you consider
covariant and which ones you consider contravariant, but you're ok as
long as you keep the two categories straight? a little math wouldn't
hurt :)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > I agree it's awkward. Carl objected so vehemently to EDO and I
> wanted
> > to reserve ET for the most general term (including EDOs ED3s
cETs).
> > Perhaps this would be a misuse of ET. Do we have some other term
for
> > the most general category of 1D temperaments, i.e. any single
> > generator temperament whether or not it is an integer fraction of
> any
> > ratio? I guess "1D-temperament" will do.
>
> Not 1D. These are 0-dimensional temperaments, I'm afraid.

if linear temperaments are 2-dimesional as you always stress, why
would these be 0-dimensional and not 1-dimensional? for example,
88cET has a single generator of 88 cents . . . seems 1 dimensional to
me!

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > so how can i tell which one is covariant and which one is
> > contravariant?
>
> Which one do you regard as the vectors you start from
(contravariant
> vector) and which as linear functions on the space of such vectors
> (covariant vector?) Obviously, in our case the monzos are the
> objects, and the vals are the mappings, and not the other way
around.
> However, we *can* consider linear mappings of vals, which can be
> identifified via unique isomorphim with monzos.
>
> Anyway, we have this:
>
> monzo = ket = contravariant
>
> val = bra = covariant

in quantum mechanics though, a state can be represented as either a
bra or a ket, depending on the mathematical operation being
performed . . .

??

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > right, but i still want to understand it, since it was in my
> > relativity textbooks . . .
>
> It's more complicated in relativity. There you have tangent spaces
> and cotangent spaces *at every point*, which have to connect
> together, plus you have a non-positive inner product which changes
> from point to point. We've got it easy and should enjoy ourseves.

aren't you talking about general relativity? i was only taking
special relativity . . .

Dave Keenan wrote:

> And since a 7-limit monzo has coefficients [e2 e3 e5 e7> then a
> 7-limit trimonzo will have coefficients ordered [[[e357 e572 e723 e235>>>.
> > Is this how your software does it too Graham?

The wedgies are stored in a dictionary, indexed by the bases. So the order only becomes important for some display functions. I order them by increasing index. And everything uses increasing numbers left to right. So it'd be [[[e235 e237 e257 e357>>>.

> But how do you order the coefficents of a 7-limit bimonzo or bimap
> (bival) so it's its own complement???

Gene does it so you reverse the order to do the complement. But he's never given the general case, and I haven't worked it out. If I could, I might be able to go on to write an efficient implementation in C.

Graham

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > If we fix a prime limit, we have duality between multivals and
> > multimonzos; in the 7-limit, a bival and a bimonzo can be
identified.
> > In the 11-limit, bivals and 4-monzos, and bimonzos and 4-vals,
can be
> > identified, as can trivals with trimonzos. This involves changing
the
> > basis of the n-monzos to make them numerically correspond to the
> > (pi(p)-n)-vals. My approach to all this has been to swap the
> > n-monzo for the corresponding (pi(p)-n)-val, and use that, but
this
> > does require we fix a prime limit.
>
> Don't we always fix the prime limit anyway? Why might this be a
problem?

sometimes you want to use a set of nonconsecutive primes, as you've
mentioned yourself, dave.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> the idea, though, is that the if the second vector has the angle
> bracket at the beginning, you end up with the symbology ><, which
> already looks like a "x".

I'm afraid I don't like that at all. It would only work for two
arguments, not 3 or more, and in any case we don't need to use the
cross-product operator since we're using the more general exterior
product (wedge product) ^. And the < ... ] is meant to tell us we're
looking at a map not a monzo. And it is beneficial to know that you
can't wedge maps with monzos. You have to convert them to the same
kind first.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > Anyway, we have this:
> >
> > monzo = ket = contravariant
> >
> > val = bra = covariant
>
> in quantum mechanics though, a state can be represented as either a
> bra or a ket, depending on the mathematical operation being
> performed . . .
>
> ??

I have to admit I'm not too concerned if analogies between quantum
mechanics and tuning theory don't pan out. :-)

I'm only concerned with whether (our extension of) the notation makes
the Grassman Algebra clearer for us. And it certainly seems to be
doing so.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> > exactly . . . the two champions would have to be the diatonic
> > pentatonic and heptatonic scales . . .
> >
> > > If I'm using a pentatonic scale made from a 9-limit otonal
chord:
> > > 8 : 9 : 10 : 12 : 14 : 16
> > > then I have two intervals each of 2:3 (both pentatonic "4ths")
> and
> > > 3:4 (both pentatonic "3rds").
> >
> > personally, i'm not fond of this as a scale or melodic entity at
> all -
> > - when i improvise over a dominant ninth chord, simply using its
> > notes is about the worst way to come up with a melody . . .
>
> I understand, and I wouldn't have much to say about the harmonic
> possibilities either. But I think that we've been spoiled by the
> harmonic sophistication of the major-minor system to such an extent
> that it's difficult to appreciate the resources of a simple scale.
> We would have to immerse ourselves in gamelan music (particularly
> slendro) to get in the proper frame of mind to be able to even
begin
> to create something decent with such limited tonal resources.
> (Again, we're off on another topic.)

i don't know . . . i mentioned the diatonic pentatonic scale above.
that's an equally simple scale, isn't it, and yet i could probably
live a happy life with no other melodic resources. so it seems you
missed my point entirely.

i realized, since i made my original post, that the "dominant
pentatonic" is not CS in 12-equal. perhaps that's one source of my
difficulty?

> Carl mentioned (in msg. #7670) that:
> > Paul E. has suggested that we only care about collisions if they
> > occur to a consonant interval. That allows the diatonic scale
> > in 12-equal to pass.
>
> That's a good hypothesis, but I think that there's more to it than
> that. Observe that there is a collision in a harmonic minor scale
> between the augmented 2nd (a dissonance between the 6th and 7th
scale
> degrees) and the minor 3rds in the scale (which are *consonant*),
but
> I wouldn't say that this results in any disorientation.

the suggestion of mine that carl was referring to . . . did i ever
make it quite clear? i don't remember :(

> Also consider this: If we were to make each of the 12-ET fifths in
a
> major scale narrower by 0.1 cents (so as to make the augmented 4th
> slightly different in size from the diminished 5th), then the scale
> would be CS, even though we would be hard pressed to hear any
> difference from 12-ET. So constant structure (taken alone) is not
> the whole issue.

well yes, that's exactly the point i was trying to make in my
original post.

> I believe that the potential for _functional scale disorientation_
> (if I may attempt to coin a term) is caused by a particular
> combination of circumstances:
>
> 1) If there are two *functionally* different intervals (i.e., aug4
&
> dim5, or aug2 & min3) in a scale that only *happen* to be the exact
> same size (because their ratios, which may be defined either as
> rational or not, happen to be conflated in the tuning in which the
> scale is being used) then there is *no* potential for functional
> scale disorientation.
>
> 2) But if there are two intervals in a scale that are *not
> functionally different* (such as the two 2:3s or 3:4s in our 11-
limit
> hexatonic otonality),

why aren't they functionally different? because we don't have a well-
defined sense of hexatonic musical function, while we know all too
much about the history and theory of the diatonic scale? i don't
think that the "happen to" above can be defined in any precise or
perceptually relevant sense -- though it would be nice . . .

> To summarize: Two intervals with *different* functionality that
are
> the same size will not cause disorientation so long as each one
spans
> the proper number of steps in the scale. It is only when two
> intervals with the *same* functionality span differing numbers of
> steps

i look forward to a definition of "functionality" . . .

> I'm not familiar with any of the 11-limit music that Prent Rodgers
> has produced,

please do listen to it as soon as possible . . .

http://home.comcast.net/~prodgers13/

> so I can only make a guess about why it works. Suppose
> that I happen to write a composition using a major scale with the
6th
> degree omitted. Is it hexatonic or heptatonic? I think that we
> would hear it as heptatonic (due in no small part to the fact that
we
> are so heptatonic-oriented, but on the other hand, I don't know how
> someone coming from a culture that uses only a pentatonic scale
would
> interpret it). So I think that it's possible to avoid functional
> scale disorientation with an 11-limit otonal scale by interpreting
it
> as an incomplete heptatonic scale.

does the composer or algorithm have to so interpret it in some way
for this to work?

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Dave Keenan wrote:
>
> > And since a 7-limit monzo has coefficients [e2 e3 e5 e7> then a
> > 7-limit trimonzo will have coefficients ordered [[[e357 e572 e723
e235>>>.
> >
> > Is this how your software does it too Graham?
>
> The wedgies are stored in a dictionary, indexed by the bases. So the
> order only becomes important for some display functions. I order them
> by increasing index. And everything uses increasing numbers left to
> right. So it'd be [[[e235 e237 e257 e357>>>.

OK. That's the "alphabetical" ordering that John Browne uses. I
suppose it's a path-of-least-resistance when writing software using a
"dictionary", but it's definitely not the most useful ordering for
human consumption, and nor is the one I gave.

> > But how do you order the coefficents of a 7-limit bimonzo or bimap
> > (bival) so it's its own complement???
>
> Gene does it so you reverse the order to do the complement.

Aha! Since posting my previous message on this, I had figured out that
was the best way to do it too.

> But he's
> never given the general case, and I haven't worked it out. If I could,
> I might be able to go on to write an efficient implementation in C.

OK. Well I think we have to work this out, and standardise on it,
since it seems to work so well with the new notation.

For example, from the Pascal's triangle of types I posted earlier it
seems that in the 3-limit (2 dimensions) if you want to know what
comma vanishes in an ET you just should just flip the ET's map left
for right, brackets and all.

The 3-limit map for 12-tET is <12 19], which we read as saying there
are 12 generators (steps, in this case) per octave and 19 per tritave.

The comma that vanishes is of course the Pythagorean comma whose monzo
(prime-exponent-vector) is [-19 12>, which we read as the ratio 2^-19
* 3^12.

So where did the minus sign come from, on the 19?

In 41-ET the map is <41 65], and the comma is [65 -41>. The minus
sign's on the other side here?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > Don't we always fix the prime limit anyway? Why might this be a
> problem?
>
> sometimes you want to use a set of nonconsecutive primes, as you've
> mentioned yourself, dave.

Good point.

There must be a convenient way of dealing with these. Does it actually
matter if you use non-consecutive primes, as long as you do it
consistently throughout the calculations. Isn't it really just the
_dimension_ of the multi-vectors that must be fixed for any given set
of calculations?

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Dave Keenan wrote:
>
> > And since a 7-limit monzo has coefficients [e2 e3 e5 e7> then a
> > 7-limit trimonzo will have coefficients ordered [[[e357 e572 e723
e235>>>.
> >
> > Is this how your software does it too Graham?
>
> The wedgies are stored in a dictionary, indexed by the bases. So the
> order only becomes important for some display functions. I order them
> by increasing index. And everything uses increasing numbers left to
> right. So it'd be [[[e235 e237 e257 e357>>>.
>
> > But how do you order the coefficents of a 7-limit bimonzo or bimap
> > (bival) so it's its own complement???
>
> Gene does it so you reverse the order to do the complement. But he's
> never given the general case, and I haven't worked it out. If I could,
> I might be able to go on to write an efficient implementation in C.

And I might be able to write an Excel Add-in, inefficiently in VBA :-)
(Visual Basic for Applications).

So the 3D wedge product is not quite the same as the cross-product.
The cross product is actually the wedge-product followed by a
complementation. What should we use for the complement operator? Tilde?

E.g. With Pauls's example of the syntonic comma and diaschisma and
12-ET, wedging the two comma monzos gives

[-4 4 -1> ^ [-11 4 2> = [[28 19 12>> a bimonzo

whereas their cross product gives

[-4 4 -1> (x) [-11 4 2> = <12 19 28] a map

and one is the complement of the other

~[[28 19 12>> = <12 19 28]

So why no problems with minus signs in 3D?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > Don't we always fix the prime limit anyway? Why might this be a
> problem?
>
> sometimes you want to use a set of nonconsecutive primes, as you've
> mentioned yourself, dave.

I suspect if you just put in "don't cares" for some of the
coefficients, they will propagate sensibly. e.g. Use NaNs
(Not-a-number) in IEEE Floating point, and display them as "X"s.

Further to my question of whether the cross-product is the 3D wedge
product followed by complementation, is it the case that the
dot-product is complementation of the second argument followed by the
wedge-product followed by complementation?

i.e. for a map M and a monzo E (for exponents), of the same
prime-limit p (dimension d),

M.E = ~(M ^ ~E)

i.e. <m2 m3 m5 ... mp] . [e2 e3 e5 ... ep>

= ~( <m2 m3 m5 ... mp] ^ ~[e2 e3 e5 ... ep> )

= ~( <m2 m3 m5 ... mp] ^ <d-1< ep ... e5 e3 e2 ]d-1] )

(with some minus sign on some of the e's?)

The notation <g< ... ]g] is meant to indicate g nested brackets, a
g-vector, where g is the grade.

= ~<d< a2*b2 + a3*b3 + a5*b5 + .... ap*mp ]d]

= a2*b2 + a3*b3 + a5*b5 + .... ap*mp

Is that correct? And what _is_ the general complement operation in
terms of scalar multiply, add, and negate operations?

For that matter, what is the general wedge-product in terms of scalar
multiply, add, and negate operations? Is there a convenient recursive
definition?

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

Oops I changed my variables in the first part and forgot to fix them
up in the second part. That should have been:

> i.e. for a map M and a monzo E (for exponents), of the same
> prime-limit p (dimension d),
>
> M.E = ~(M ^ ~E)
>
> i.e. <m2 m3 m5 ... mp] . [e2 e3 e5 ... ep>
>
> = ~( <m2 m3 m5 ... mp] ^ ~[e2 e3 e5 ... ep> )
>
> = ~( <m2 m3 m5 ... mp] ^ <d-1< ep ... e5 e3 e2 ]d-1] )
>
> (with some minus sign on some of the e's?)
>
> The notation <g< ... ]g] is meant to indicate g nested brackets, a
> g-vector, where g is the grade.
>
> = ~<d< m2*e2 + m3*e3 + m5*e5 + .... mp*ep ]d]
>
> = m2*e2 + m3*e3 + m5*e5 + .... mp*ep
>
> Is that correct? And what _is_ the general complement operation in
> terms of scalar multiply, add, and negate operations?
>
> For that matter, what is the general wedge-product in terms of scalar
> multiply, add, and negate operations? Is there a convenient recursive
> definition?

I guess it would be better to use the index of the prime as the
subscript, rather than the prime itself, in describing the general
complement and wedge-product operations in terms of scalar operations.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I also
> >found a couple of scale subsets of a 17-tone octave that have this
> >property, and I have a composition in progress in 17-WT that will
> >illustrate one of these using non-5 harmony.
>
> Sweet! Can't wait to hear this...

Me too. I conceived the idea for it over 25 years ago and started
working on it in Cakewalk a little over a year ago. It's in a 9-tone
MOS scale structure -- very different from anything else I've ever
tried. It's been very slow going to figure out how to make
everything work to my satisfaction. I have only about 20 bars of
music done, but I'm absolutely delighted with it so far. At this
rate, maybe I'll have it done by the end of this decade. ;-)

> ...
> Can you write a melody in the 11-limit 'scale' that sounds wrong
> because of functional scale disorientation?

I actually tried to run a couple of ideas through my head last night
about how to do it, and it's not as easy as I thought.
Paradoxically, it takes a certain lack of talent or lack or
familiarity with microtonal resources to write something
intentionally bad enough that it sounds wrong. -- it's only when I'm
not trying to do it that I seem to be able to succeed. Hmmm, perhaps
I've stumbled on a new method of composition -- guaranteed to give
good results. ;-)

--George

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> ...
> >> But incidentally, I'd love a musical example of a hexatonic 11-
limit
> >> melody where the non-CS "collision" causes a problem with
> >> constructing a musical sequence.
> >
> >I'd have to compose something to illustrate that and produce a
midi
> >or mp3 file that we could listen to. Then we would have to decide
> >that, if we both agreed that it didn't work, that it was due to
the
> >non-CS property and not because I had a bad composing day.
>
> And would it be so much harder than writing all these messages,
> full of so much speculation?
>
> -Carl

Yes, I know from experience that it would be both harder and more
time-consuming, because I would need to make a lot of decisions on
how I would go about it. It's much quicker and easier just to rattle
thoughts out of my head into a keyboard. Speculation is cheap, but
creating music (even bad music) requires a bigger investment -- it
takes a bit of work.

That said, I think I've convinced myself that I need to take a break
from the tuning lists for a while and use the time to do some
composing. That's the way to discover first-hand which techniques
and ideas will work (at least for me) and which ones won't. (And
then perhaps I'll also save excerpts of some of the bad ideas I try,
just for illustration when discussions like these come up.)

--George

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
>
> > > exactly . . . the two champions would have to be the diatonic
> > > pentatonic and heptatonic scales . . .
> > >
> > > > If I'm using a pentatonic scale made from a 9-limit otonal
> chord:
> > > > 8 : 9 : 10 : 12 : 14 : 16
> > > > then I have two intervals each of 2:3 (both
pentatonic "4ths")
> > and
> > > > 3:4 (both pentatonic "3rds").
> > >
> > > personally, i'm not fond of this as a scale or melodic entity
at all -
> > > - when i improvise over a dominant ninth chord, simply using
its
> > > notes is about the worst way to come up with a melody . . .
> >
> > I understand, and I wouldn't have much to say about the harmonic
> > possibilities either. But I think that we've been spoiled by the
> > harmonic sophistication of the major-minor system to such an
extent
> > that it's difficult to appreciate the resources of a simple
scale.
> > We would have to immerse ourselves in gamelan music (particularly
> > slendro) to get in the proper frame of mind to be able to even
begin
> > to create something decent with such limited tonal resources.
> > (Again, we're off on another topic.)
>
> i don't know . . . i mentioned the diatonic pentatonic scale above.
> that's an equally simple scale, isn't it, and yet i could probably
> live a happy life with no other melodic resources. so it seems you
> missed my point entirely.

No, I don't think I did. I recognize that a scale that is
essentially a just dominant 9th chord makes it a little more
difficult to imply any changes in the harmonic element other than
alternation between 4:5:6 to 6:7:9 triads in an accompaniment. An
unaccompanied melody using a diatonic pentatonic scale, on the other
hand (such as _Auld Lang Syne_), can easily evoke a heptatonic
chordal accompaniment in our imaginations -- it takes only the
presence of the 5/3 in the scale to imply that there should be a 4/3
(subdominant) somewhere in context.

My motive in suggesting "slendro therapy" was to experience that
there is a feeling of satisfaction that can be achieved with music
that has little or no harmonic motion.

> i realized, since i made my original post, that the "dominant
> pentatonic" is not CS in 12-equal. perhaps that's one source of my
> difficulty?

I wouldn't think so.

> > ...
> > 2) But if there are two intervals in a scale that are *not
> > functionally different* (such as the two 2:3s or 3:4s in our 11-
limit
> > hexatonic otonality),
>
> why aren't they functionally different? because we don't have a
well-
> defined sense of hexatonic musical function, while we know all too
> much about the history and theory of the diatonic scale? i don't
> think that the "happen to" above can be defined in any precise or
> perceptually relevant sense -- though it would be nice . . .

As I see it, interval function is independent of the number of tones
in the scale, but instead has to do with the (just) *ratio* that is
either directly expressed (in JI) or implied (in a temperament) by
that interval. So two tempered intervals that (in a given context)
are implying the same just interval are functionally the same, even
if they are not exactly the same size (such as in a well-
temperament). But two tempered intervals that (by context) imply
different just intervals are functionally different, even if they are
exactly the same size in a particular tuning.

In the context of a diatonic scale the tones are all assumed to be in
a chain of fifths. If one member of that chain is taken to represent
1/1, then each of the other members can be assigned at least one
(rational) ratio that is unique to that member. An augmented 4th and
diminished 5th (or a minor 3rd and augmented 2nd, etc.) will
therefore be considered to be serving different harmonic functions,
since they represent different ratios.

In an 8:9:10:11:12:14:16 scale there is no question that the two 2:3s
(or the two 3:4s) are for all intents and purposes identical (since
this is JI), so on a *harmonic* level they are functionally
equivalent. But since these pairs of intervals subtend different
steps in the scale, the potential for _functional scale
disorientation_ (if you don't like the term, then please suggest
something else) exists.

>
> > To summarize: Two intervals with *different* functionality that
are
> > the same size will not cause disorientation so long as each one
spans
> > the proper number of steps in the scale. It is only when two
> > intervals with the *same* functionality span differing numbers of
> > steps
>
> i look forward to a definition of "functionality" . . .

I think that I've given enough information above to arrive at one.
Perhaps I haven't chosen the right term -- would something containing
the words "harmonic identity" or "interval identity" be better?

> > I'm not familiar with any of the 11-limit music that Prent
Rodgers
> > has produced,
>
> please do listen to it as soon as possible . . .
>
> http://home.comcast.net/~prodgers13/
>
> > so I can only make a guess about why it works. Suppose
> > that I happen to write a composition using a major scale with the
6th
> > degree omitted. Is it hexatonic or heptatonic? I think that we
> > would hear it as heptatonic (due in no small part to the fact
that we
> > are so heptatonic-oriented, but on the other hand, I don't know
how
> > someone coming from a culture that uses only a pentatonic scale
would
> > interpret it). So I think that it's possible to avoid functional
> > scale disorientation with an 11-limit otonal scale by
interpreting it
> > as an incomplete heptatonic scale.
>
> does the composer or algorithm have to so interpret it in some way
> for this to work?

It's hard to say. It's possible for a composition to be so free-form
that we (as listeners) just take it in without attempting to perceive
a particular scale structure in our minds.

--George

>Me too. I conceived the idea for it over 25 years ago and started
>working on it in Cakewalk a little over a year ago. It's in a 9-tone
>MOS scale structure -- very different from anything else I've ever
>tried. It's been very slow going to figure out how to make
>everything work to my satisfaction. I have only about 20 bars of
>music done, but I'm absolutely delighted with it so far. At this
>rate, maybe I'll have it done by the end of this decade. ;-)

Are you writing it in Sagittal? ;-)

>> Can you write a melody in the 11-limit 'scale' that sounds wrong
>> because of functional scale disorientation?
>
>I actually tried to run a couple of ideas through my head last night
>about how to do it, and it's not as easy as I thought.
>Paradoxically, it takes a certain lack of talent or lack or
>familiarity with microtonal resources to write something
>intentionally bad enough that it sounds wrong. -- it's only when I'm
>not trying to do it that I seem to be able to succeed. Hmmm, perhaps
>I've stumbled on a new method of composition -- guaranteed to give
>good results. ;-)

Remember, we're trying to ignore composition as a factor. We don't
care if it's bad, just if it has functional scale disorientation.
If we try and try but can't ever hear functional scale disorientation,
it's important -- it means it probably doesn't exist.

-Carl

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> ... I think I've convinced myself that I need to take a break
> from the tuning lists for a while and use the time to do some
> composing. That's the way to discover first-hand which
> techniques and ideas will work (at least for me) and which
> ones won't. (And then perhaps I'll also save excerpts of
> some of the bad ideas I try, just for illustration when
> discussions like these come up.)

this is interesting to me.

i have one piece which i wrote in 12edo, and later converted
to JI, but i was never really happy with the JI version.

_In A Minute_:
http://sonic-arts.org/monzo/inminute/inminute.htm

this page is supposed to open with the mp3, but in case
there's a problem, here it is:
http://sonic-arts.org/monzo/inminute/inminute.mp3

i'm expecting that once my software has music composition
capability (which will be within a month), i'll be able
to do a better job of this.

but anyway, even tho i'm not happy with it, i thought it
was interesting to document my "justification" of this piece.

-monz

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Me too. I conceived the idea for it over 25 years ago and started
> >working on it in Cakewalk a little over a year ago. It's in a 9-
tone
> >MOS scale structure -- very different from anything else I've ever
> >tried. It's been very slow going to figure out how to make
> >everything work to my satisfaction. I have only about 20 bars of
> >music done, but I'm absolutely delighted with it so far. At this
> >rate, maybe I'll have it done by the end of this decade. ;-)
>
> Are you writing it in Sagittal? ;-)

No, because it's not actually "written", and if I were to print out
anything from Cakewalk, it would only show the nearest 12-ET
pitches. I'll have to wait for Monz's software to come out before I
can do any better than that.

> >> Can you write a melody in the 11-limit 'scale' that sounds wrong
> >> because of functional scale disorientation?
> >
> >I actually tried to run a couple of ideas through my head last
night
> >about how to do it, and it's not as easy as I thought.
> >Paradoxically, it takes a certain lack of talent or lack or
> >familiarity with microtonal resources to write something
> >intentionally bad enough that it sounds wrong. -- it's only when
I'm
> >not trying to do it that I seem to be able to succeed. Hmmm,
perhaps
> >I've stumbled on a new method of composition -- guaranteed to give
> >good results. ;-)
>
> Remember, we're trying to ignore composition as a factor. We don't
> care if it's bad, just if it has functional scale disorientation.
> If we try and try but can't ever hear functional scale
disorientation,
> it's important -- it means it probably doesn't exist.
>
> -Carl

What I was concerned about was that disorientation might occur, but
the result wouldn't be perceived as a problem, but rather as
something that, though unexpected, ends up sounding new and exciting
(rather like a deceptive cadence). (Oh boy, I think I'm just digging
myself into a hole. Maybe I'll get a chance to try this out
sometime, but don't hold your breath.)

--George

>No, because it's not actually "written",

Are you entering notes from a keyboard?

-C.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> > > Don't we always fix the prime limit anyway? Why might this be a
> > problem?
> >
> > sometimes you want to use a set of nonconsecutive primes, as
you've
> > mentioned yourself, dave.
>
> Good point.
>
> There must be a convenient way of dealing with these. Does it
actually
> matter if you use non-consecutive primes, as long as you do it
> consistently throughout the calculations. Isn't it really just the
> _dimension_ of the multi-vectors that must be fixed for any given
set
> of calculations?

of course. it's just that you might not be dealing with a "prime
limit"!

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "George D. Secor"
> <gdsecor@y...>
> > wrote:
> >
> > > > exactly . . . the two champions would have to be the diatonic
> > > > pentatonic and heptatonic scales . . .
> > > >
> > > > > If I'm using a pentatonic scale made from a 9-limit otonal
> > chord:
> > > > > 8 : 9 : 10 : 12 : 14 : 16
> > > > > then I have two intervals each of 2:3 (both
> pentatonic "4ths")
> > > and
> > > > > 3:4 (both pentatonic "3rds").
> > > >
> > > > personally, i'm not fond of this as a scale or melodic entity
> at all -
> > > > - when i improvise over a dominant ninth chord, simply using
> its
> > > > notes is about the worst way to come up with a melody . . .
> > >
> > > I understand, and I wouldn't have much to say about the
harmonic
> > > possibilities either. But I think that we've been spoiled by
the
> > > harmonic sophistication of the major-minor system to such an
> extent
> > > that it's difficult to appreciate the resources of a simple
> scale.
> > > We would have to immerse ourselves in gamelan music
(particularly
> > > slendro) to get in the proper frame of mind to be able to even
> begin
> > > to create something decent with such limited tonal resources.
> > > (Again, we're off on another topic.)
> >
> > i don't know . . . i mentioned the diatonic pentatonic scale
above.
> > that's an equally simple scale, isn't it, and yet i could
probably
> > live a happy life with no other melodic resources. so it seems
you
> > missed my point entirely.
>
> No, I don't think I did.
> I recognize that a scale that is
> essentially a just dominant 9th chord makes it a little more
> difficult to imply any changes in the harmonic element other than
> alternation between 4:5:6 to 6:7:9 triads in an accompaniment. An
> unaccompanied melody using a diatonic pentatonic scale, on the
other
> hand (such as _Auld Lang Syne_), can easily evoke a heptatonic
> chordal accompaniment in our imaginations -- it takes only the
> presence of the 5/3 in the scale to imply that there should be a
4/3
> (subdominant) somewhere in context.

I didn't mean to imply any harmonic dimension whatsoever, or at least
not any harmonic changes. My original comment, above, concerned
improvising over *one* chord. So I think you did misundestand me.

> My motive in suggesting "slendro therapy" was to experience that
> there is a feeling of satisfaction that can be achieved with music
> that has little or no harmonic motion.

I'm the last person that needs to be convinced of this -- one of my
microtonal examples on mp3.com used to be about 9 minutes over an
unchanging "harmony" or drone! Also, one of my main musical
activities, sometimes quite lucrative actually, is improvising
dextrously on acoustic guitar with open strings tuned to a 1/1-3/2
drone.

> > i realized, since i made my original post, that the "dominant
> > pentatonic" is not CS in 12-equal. perhaps that's one source of
my
> > difficulty?
>
> I wouldn't think so.

Well, I'm interested in investigating further . . .

> > > 2) But if there are two intervals in a scale that are *not
> > > functionally different* (such as the two 2:3s or 3:4s in our 11-
> limit
> > > hexatonic otonality),
> >
> > why aren't they functionally different? because we don't have a
> well-
> > defined sense of hexatonic musical function, while we know all
too
> > much about the history and theory of the diatonic scale? i don't
> > think that the "happen to" above can be defined in any precise or
> > perceptually relevant sense -- though it would be nice . . .
>
> As I see it, interval function is independent of the number of
tones
> in the scale, but instead has to do with the (just) *ratio* that is
> either directly expressed (in JI) or implied (in a temperament) by
> that interval. So two tempered intervals that (in a given context)
> are implying the same just interval are functionally the same, even
> if they are not exactly the same size (such as in a well-
> temperament). But two tempered intervals that (by context) imply
> different just intervals are functionally different, even if they
are
> exactly the same size in a particular tuning.

What just interval does the 12-equal augmented second imply? And how
is this implication effected, exactly?

> In the context of a diatonic scale the tones are all assumed to be
in
> a chain of fifths. If one member of that chain is taken to
represent
> 1/1, then each of the other members can be assigned at least one
> (rational) ratio that is unique to that member. An augmented 4th
and
> diminished 5th (or a minor 3rd and augmented 2nd, etc.) will
> therefore be considered to be serving different harmonic functions,
> since they represent different ratios.

I'd like to see this made more explicit.

> In an 8:9:10:11:12:14:16 scale there is no question that the two
2:3s
> (or the two 3:4s) are for all intents and purposes identical (since
> this is JI),

What if I tuned a harmonic minor scale in JI with a 6:5 augmented
second?

> so on a *harmonic* level they are functionally
> equivalent. But since these pairs of intervals subtend different
> steps in the scale, the potential for _functional scale
> disorientation_ (if you don't like the term, then please suggest
> something else) exists.

I'm hoping we can make this precise. Right now it seems fuzzy, with
meaning adapted differently to fit this fact and that. Please help me
remove the ambiguity.

Perhaps we are talking about epimorphic vs. non-epimorphic scales? If
so, realizing this could be a breakthrough. At least we could have a
precise (and very relevant to the material on this list) mathematical
characterization of what makes a scale have or not have "functional
scale disorientation" to you. That could be very helpful. Gene, would
you chime in?

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

> E.g. With Pauls's example of the syntonic comma and diaschisma and
> 12-ET, wedging the two comma monzos gives
>
> [-4 4 -1> ^ [-11 4 2> = [[28 19 12>> a bimonzo
>
> whereas their cross product gives
>
> [-4 4 -1> (x) [-11 4 2> = <12 19 28] a map
>
> and one is the complement of the other
>
> ~[[28 19 12>> = <12 19 28]
>
> So why no problems with minus signs in 3D?

maybe this is why:

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

"While the distinction between covariant and contravariant indices
must be made for general tensors, the two are equivalent for tensors
in three-dimensional Euclidean space, and such tensors are known as
Cartesian tensors."

i don't think my answer was right.

rather, i think it has something to do with the fact that the
complement of a vector (a,b) in the plane is a line parallel to the
vector (-b,a) or (b,-a), while the complement of a vector (a,b,c) in
3D space is a plane whose *normal* is simply (a,b,c) or -(a,b,c) . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > E.g. With Pauls's example of the syntonic comma and diaschisma and
> > 12-ET, wedging the two comma monzos gives
> >
> > [-4 4 -1> ^ [-11 4 2> = [[28 19 12>> a bimonzo
> >
> > whereas their cross product gives
> >
> > [-4 4 -1> (x) [-11 4 2> = <12 19 28] a map
> >
> > and one is the complement of the other
> >
> > ~[[28 19 12>> = <12 19 28]
> >
> > So why no problems with minus signs in 3D?
>
> maybe this is why:
>
> http://mathworld.wolfram.com/Tensor.html
>
> "While the distinction between covariant and contravariant indices
> must be made for general tensors, the two are equivalent for
tensors
> in three-dimensional Euclidean space, and such tensors are known as
> Cartesian tensors."

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> i don't think my answer was right.
>
> rather, i think it has something to do with the fact that the
> complement of a vector (a,b) in the plane is a line parallel to the
> vector (-b,a) or (b,-a), while the complement of a vector (a,b,c) in
> 3D space is a plane whose *normal* is simply (a,b,c) or -(a,b,c) . .

But did I actually get it right when I wrote

[-4 4 -1> ^ [-11 4 2> = [[28 19 12>> a bimonzo

Could a minus sign or two be required in the bimonzo, which then
disappear when you take the complement?

The basis for the monzos is
lg(2) lg(3) lg(5)
where lg() is a logarithm function of arbitrary base.

According to John Browne, the above bimonzo is correct if its basis is

lg(2)^lg(3) lg(5)^lg(2) lg(3)^lg(5) (indices 12 31 23)

where ^ is the wedge-product operator, not exponentiation.

= [[ (-4*4)-(4*-11) (-1*-11)-(-4*2) (4*2)-(-1*4) >>
= [[28 19 12>>

But if the basis is instead

lg(2)^lg(3) lg(2)^lg(5) lg(3)^lg(5) (indices 12 13 23)

(just swapped the order of lg(2) and lg(5) in the middle one)
then the bimonzo is
[[28 -19 12>>

And it starts to look like the general complement (for any grade and
dimension) should not only reverse the order of coefficients, but
negate every second one. But I'm really not sure. Gene, help!

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > i don't think my answer was right.
> >
> > rather, i think it has something to do with the fact that the
> > complement of a vector (a,b) in the plane is a line parallel to
the
> > vector (-b,a) or (b,-a), while the complement of a vector (a,b,c)
in
> > 3D space is a plane whose *normal* is simply (a,b,c) or -
(a,b,c) . .
>
> But did I actually get it right when I wrote
>
> [-4 4 -1> ^ [-11 4 2> = [[28 19 12>> a bimonzo

GABLE gives 28*e2^e3 + 12*e3^e5 + 19*e5^e2, where "e" is the unit
vector.

> Could a minus sign or two be required in the bimonzo, which then
> disappear when you take the complement?
>
> The basis for the monzos is
> lg(2) lg(3) lg(5)
> where lg() is a logarithm function of arbitrary base.
>
> According to John Browne, the above bimonzo is correct if its basis
is
>
> lg(2)^lg(3) lg(5)^lg(2) lg(3)^lg(5) (indices 12 31 23)
>
> where ^ is the wedge-product operator, not exponentiation.
>
> = [[ (-4*4)-(4*-11) (-1*-11)-(-4*2) (4*2)-(-1*4) >>
> = [[28 19 12>>
>
> But if the basis is instead
>
> lg(2)^lg(3) lg(2)^lg(5) lg(3)^lg(5) (indices 12 13 23)
>
> (just swapped the order of lg(2) and lg(5) in the middle one)
> then the bimonzo is
> [[28 -19 12>>

right, but if you keep the (directed) angle between the two vectors
in each basis bivector the same, you don't get this behavior in 3D
(since you use e5^e2 and not e2^e5 in your basis) -- but you *do* get
it in 2D.

> And it starts to look like the general complement (for any grade and
> dimension) should not only reverse the order of coefficients, but
> negate every second one.

what could be special about every second one? think about this purely
geometrically, so the order of the primes loses its significance . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > According to John Browne, the above bimonzo is correct if its basis
> is
> >
> > lg(2)^lg(3) lg(5)^lg(2) lg(3)^lg(5) (indices 12 31 23)
> >
> > where ^ is the wedge-product operator, not exponentiation.
> >
> > = [[ (-4*4)-(4*-11) (-1*-11)-(-4*2) (4*2)-(-1*4) >>
> > = [[28 19 12>>
> >
> > But if the basis is instead
> >
> > lg(2)^lg(3) lg(2)^lg(5) lg(3)^lg(5) (indices 12 13 23)
> >
> > (just swapped the order of lg(2) and lg(5) in the middle one)
> > then the bimonzo is
> > [[28 -19 12>>
>
> right, but if you keep the (directed) angle between the two vectors
> in each basis bivector the same, you don't get this behavior in 3D
> (since you use e5^e2 and not e2^e5 in your basis) -- but you *do* get
> it in 2D.
>
> > And it starts to look like the general complement (for any grade and
> > dimension) should not only reverse the order of coefficients, but
> > negate every second one.
>
> what could be special about every second one? think about this purely
> geometrically, so the order of the primes loses its significance . . .

True, but we have to agree on _some_ standard ordering of the
coefficients in a multivector of any grade and dimension, and in
addition to that, we have to agree on the ordering of the grade-1
basis components making up higher-grade basis components.

It should be something we can easily remember for any grade and
dimension.

Lexicographic ("alphabetical") ordering (in both of the above cases),
is something that's easy to remember. It's what Browne uses in his
Mathematica package. And it seems like it might give rise to a uniform
complementation rule of "negate every second one and reverse the order".

Clearly this rule is unavoidable in the 3-limit (2D) case. There's no
choice in the matter there.

I think I was wrong before when I said it sounded like Gene's ordering
was different to John Browne's. It also sounds like they might both be
the same as Graham's. But I'm not sure.

You needn't worry about this giving you unfamiliar results in 3D.
Remember that while the cross-product of two vectors gives you another
vector at right angles to both of them, the wedge-product of two
vectors does not. Instead it gives you a bivector representing the
plane (or "planar direction") containing those two vectors. If you
want the vector normal to that plane you have to take the complement
of that bivector.

So it doesn't matter how we shuffle the indices in the basis of the
bivector representing the plane, because the definition of the
complement operation will change accordingly, so the normal always
comes out the way you would expect.

A (x) B = ~(A ^ B)

At least I think that's right. Sigh.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

Sure,
> I would say that it's not necessary or even important for a chord
to
> be a constant structure.

It'sn certainly very important to me; I've used the idea extensively.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> so it's pretty much a matter of convention which ones you consider
> covariant and which ones you consider contravariant, but you're ok
as
> long as you keep the two categories straight? a little math
wouldn't
> hurt :)

The idea is that contravariant vectors are ordinary, garden variety
vectors, and linear mappings of vectors are covariant vectors. What
confuses the issue is that mappings of the mappings are canonically
isomorphic to the vectors you started with.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> if linear temperaments are 2-dimesional as you always stress, why
> would these be 0-dimensional and not 1-dimensional?

Don't blame me--you are the one who insisted linear temperaments were
to be called linear, not planar. If they are linear--ie 1D, then what
are really 1D temperaments (ets) now have to be called 0D.

for example,
> 88cET has a single generator of 88 cents . . . seems 1 dimensional
to
> me!

Of course, but I was mugged for saying this sort of thing in the
first place. If you have octave equivalence, you can reduce mod
octaves, and get cyclic groups, which is about as 0D a thing as this
business will afford you.

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

> I'm only concerned with whether (our extension of) the notation
makes
> the Grassman Algebra clearer for us. And it certainly seems to be
> doing so.

I would also suggest extending the inner product notation to
multivals with corresponding multimonzos

<<u|v]]
<<<u|v]]]

and so forth. This is one step towards defining conjugacy.

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

> There must be a convenient way of dealing with these. Does it
actually
> matter if you use non-consecutive primes, as long as you do it
> consistently throughout the calculations. Isn't it really just the
> _dimension_ of the multi-vectors that must be fixed for any given
set
> of calculations?

The dimensions only need to be fixed if you want to define the
compliment.

OK. With lexicographic ordering of the indices, it isn't as simple as
negating every second coefficient. There's sometimes a hiccup in the
middle. It's explained in Section 5.4 of

http://www.ses.swin.edu.au/homes/browne/grassmannalgebra/book/bookpdf/TheComplement.pdf

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

> So the 3D wedge product is not quite the same as the cross-product.

Nor is the 2D wedge product exactly multiplication of complex
numbers. You need to take a compliment, since a 3D wedge is a 2-
vector, not a vector.

> The cross product is actually the wedge-product followed by a
> complementation. What should we use for the complement operator?
Tilde?

I've been using asterisk, but if tilde is clearer I will change what
I've written.

> E.g. With Pauls's example of the syntonic comma and diaschisma and
> 12-ET, wedging the two comma monzos gives
>
> [-4 4 -1> ^ [-11 4 2> = [[28 19 12>> a bimonzo

Going strictly by alphabetical ordering, this would be

[-4 4 -1> ^ [-11 4 2> = [[28 -19 12>>

after which

[[28 -19 12>>* = [12 19 28>

and we have the cross product.

> So why no problems with minus signs in 3D?

You used a basis without them, because you didn't alphabetically
order the basis wedges.

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

> I suspect if you just put in "don't cares" for some of the
> coefficients, they will propagate sensibly. e.g. Use NaNs
> (Not-a-number) in IEEE Floating point, and display them as "X"s.

What does a don't care do to dimensionality? If we are not worried
about that, we can use 0.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> Further to my question of whether the cross-product is the 3D wedge
> product followed by complementation, is it the case that the
> dot-product is complementation of the second argument followed by
the
> wedge-product followed by complementation?

This sounds a little backwards--it's the inner product for n-vals
with n-monzos that lets you define the compliment.

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

> Could a minus sign or two be required in the bimonzo, which then
> disappear when you take the complement?

The most straightforward way to do it would be that.

5-limit

[a1,a2,a3>* = <<a3,-a2,a1]]
[[a1,a2,a3>>* = <a3,-a2,a1]

7-limit

[[a1,a2,a3,a4,a5,a6>>* = <<a6,-a5,a4,a3,-a2,a1]]

11-limit

[[a1,a2,a3,a4,a5,a6,a7,a8,a9,a10>>* =
<<<a10,-a9,a8,a7,-a6,a5,-a4,a3,-a2,a1]]]

[[[a1,a2,a3,a4,a5,a6,a7,a8,a9,a10>>>* =
<<-a10,a9,-a8,a7,-a6,a5,-a4,a3,a2,-a1]]

13-limit

[[a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15>>* =

<<<<a15,-a14,a13,-a12,a11,a10,-a9,a8,-a7,a6,-a5,a4,a3,-a2,a1]]]]

[[[a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,
a20>>>* =

<<<-a20,a19,-a18,a17,-a16,a15,-a14,-a13,a12,-a11,a10,-a9,a8,a7,-
a6,a5,-a4,a3,-a2,a1]]]

[[[[a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15>>>>* =
<<a15,-a14,a13,-a12,a11,a10,-a9,a8,-a7,a6,-a5,a4,a3,-a2,a1]]

Unless I've made a mistake; I'll check this.

> The basis for the monzos is
> lg(2) lg(3) lg(5)
> where lg() is a logarithm function of arbitrary base.

This strikes me as a weird notion. It is true that
<1 log2(3) log2(5)] (not a val, but a perfectly acceptable bra
vector) is important in the 5-limit, as mapping monzos to log2 of the
corrsponding rational number--or we could use
<1200 1200*log2(3) 1200*log2(5)] instead.

> According to John Browne, the above bimonzo is correct if its basis
is
>
> lg(2)^lg(3) lg(5)^lg(2) lg(3)^lg(5) (indices 12 31 23)

Browne doesn't use alphabetical ordering of the basis wedges? I
thought alphabetical was what you had settled on.

> And it starts to look like the general complement (for any grade and
> dimension) should not only reverse the order of coefficients, but
> negate every second one. But I'm really not sure. Gene, help!

It isn't even that easy; the coefficients get reversed, but the sign
has to be figured out from whether the permutation you get is even or
odd.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> GABLE gives 28*e2^e3 + 12*e3^e5 + 19*e5^e2, where "e" is the unit
> vector.

Sounds great. What's GABLE?

Now you know why I tried to sweep all of this under the rug.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> GABLE gives 28*e2^e3 + 12*e3^e5 + 19*e5^e2, where "e" is the unit
> vector.

Sounds great. What's GABLE?

Now you know why I tried to sweep all of this under the rug.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> Lexicographic ("alphabetical") ordering (in both of the above
cases),
> is something that's easy to remember. It's what Browne uses in his
> Mathematica package. And it seems like it might give rise to a
uniform
> complementation rule of "negate every second one and reverse the
order".

It doesn't. It is, however, what we've been using ever since Graham
got me to switch by basis choice to conform to his.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Going strictly by alphabetical ordering, this would be
>
> [-4 4 -1> ^ [-11 4 2> = [[28 -19 12>>
>
> after which
>
> [[28 -19 12>>* = [12 19 28>
>
> and we have the cross product.

I'm still confused here.

So the complement operation keeps the braket pointing in the same
direction?

So <12 19 28] is not the complement of [[28 -19 12>> but is simply
_equal_ to it (because it has a complementary basis)?

Likewise in 3-limit, <12 19] is equal to [19 -12>?

I like the prefix tilde for complement since it supports
De-Morgan-like intuitions from Boolean algebra.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> I'm still confused here.
>
> So the complement operation keeps the braket pointing in the same
> direction?
>
> So <12 19 28] is not the complement of [[28 -19 12>> but is simply
> _equal_ to it (because it has a complementary basis)?

Sorry. That should have been "(because it has a _reciprocal_ basis)?".

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

> > [[28 -19 12>>* = [12 19 28>
> >
> > and we have the cross product.
>
> I'm still confused here.
>
> So the complement operation keeps the braket pointing in the same
> direction?

Sorry, that should have been

[[28 -19 12>>* = <12 19 28]

> I like the prefix tilde for complement since it supports
> De-Morgan-like intuitions from Boolean algebra.

Postfix seems more natural to me; that's normally how these things
are done.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > I like the prefix tilde for complement since it supports
> > De-Morgan-like intuitions from Boolean algebra.
>
> Postfix seems more natural to me; that's normally how these things
> are done.

Well, in ASCII * is of course most commonly infix for multiplication.
As postfix I'm used to it being the complex-conjugate operator which
doesn't seem as analogous as prefix ~ for the logical complement.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > > According to John Browne, the above bimonzo is correct if its
basis
> > is
> > >
> > > lg(2)^lg(3) lg(5)^lg(2) lg(3)^lg(5) (indices 12 31 23)
> > >
> > > where ^ is the wedge-product operator, not exponentiation.
> > >
> > > = [[ (-4*4)-(4*-11) (-1*-11)-(-4*2) (4*2)-(-1*4) >>
> > > = [[28 19 12>>
> > >
> > > But if the basis is instead
> > >
> > > lg(2)^lg(3) lg(2)^lg(5) lg(3)^lg(5) (indices 12 13 23)
> > >
> > > (just swapped the order of lg(2) and lg(5) in the middle one)
> > > then the bimonzo is
> > > [[28 -19 12>>
> >
> > right, but if you keep the (directed) angle between the two
vectors
> > in each basis bivector the same, you don't get this behavior in
3D
> > (since you use e5^e2 and not e2^e5 in your basis) -- but you *do*
get
> > it in 2D.
> >
> > > And it starts to look like the general complement (for any
grade and
> > > dimension) should not only reverse the order of coefficients,
but
> > > negate every second one.
> >
> > what could be special about every second one? think about this
purely
> > geometrically, so the order of the primes loses its
significance . . .
>
> True, but we have to agree on _some_ standard ordering of the
> coefficients in a multivector of any grade and dimension, and in
> addition to that, we have to agree on the ordering of the grade-1
> basis components making up higher-grade basis components.
>
> It should be something we can easily remember for any grade and
> dimension.
>
> Lexicographic ("alphabetical") ordering (in both of the above
cases),
> is something that's easy to remember. It's what Browne uses in his
> Mathematica package. And it seems like it might give rise to a
uniform
> complementation rule of "negate every second one and reverse the
order".

It sure doesn't seem that way to me, for the reasons I tried to
convey to you.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > if linear temperaments are 2-dimesional as you always stress, why
> > would these be 0-dimensional and not 1-dimensional?
>
> Don't blame me--you are the one who insisted linear temperaments
were
> to be called linear, not planar. If they are linear--ie 1D, then
what
> are really 1D temperaments (ets) now have to be called 0D.
>
> for example,
> > 88cET has a single generator of 88 cents . . . seems 1
dimensional
> to
> > me!
>
> Of course, but I was mugged for saying this sort of thing in the
> first place. If you have octave equivalence, you can reduce mod
> octaves, and get cyclic groups, which is about as 0D a thing as
this
> business will afford you.

yes, as you know i (and especially graham) like that idea very much --
BUT 88cET has no octaves!

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> OK. With lexicographic ordering of the indices, it isn't as simple
as
> negating every second coefficient. There's sometimes a hiccup in the
> middle. It's explained in Section 5.4 of
>
>
http://www.ses.swin.edu.au/homes/browne/grassmannalgebra/book/bookpdf/
TheComplement.pdf

The page cannot be displayed

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > GABLE gives 28*e2^e3 + 12*e3^e5 + 19*e5^e2, where "e" is the unit
> > vector.
>
> Sounds great. What's GABLE?

it's that matlab program you turned me on to -- Geometric AlgeBra
Learning Environment.

> Now you know why I tried to sweep all of this under the rug.

he he

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > Lexicographic ("alphabetical") ordering (in both of the above
> cases),
> > is something that's easy to remember. It's what Browne uses in his
> > Mathematica package. And it seems like it might give rise to a
> uniform
> > complementation rule of "negate every second one and reverse the
> order".
>
> It doesn't. It is, however, what we've been using ever since Graham
> got me to switch by basis choice to conform to his.

what was your original basis choice, and what do the patterns of
signs for duals look like under it?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >No, because it's not actually "written",
>
> Are you entering notes from a keyboard?
>
> -C.

No, I have to mouse back and forth around the screen, clicking on a
note duration in one place (if it needs to be changed from what was
set for the previous note) and then clicking on the staff in the
appropriate place to draw the note. This part would go much faster
if Cakewalk allowed me to use the keyboard to change the note
durations (with the left hand; only about a half-dozen different keys
would be needed) while I inserted the notes with the mouse (with the
right hand) -- MONZ, THIS IS AN IDEA YOU SHOULD USE IN YOUR NEW
SOFTWARE!!! (It would also be nice to be able to set sharps/flats
and microtonal accidentals to be applied to the next note with the
keyboard.)

Then, if the note isn't something that occurs in the key signature
(or if I chose not to have anything in the key signature), then I
have to right-click on the note and set a chromatic alteration with
the mouse. And if it isn't spelled the way I wanted it (e.g., if I
wanted G# instead of Ab), then I have to have to fix that with a
keyboard entry. In case you're wondering why I'm finicky about the
spelling (since I'm not intending to print out any music): this
enables me to use more than 12 pitch names in an octave, so when it
comes time to insert the pitch bends, I can easily see that G# and Ab
are different tones in my tuning and that I should bend them by
different amounts.

Once I've done all of the notes in a section, then I have to insert
the pitch bends manually in event view (which is mostly done with the
keyboard), also making sure that each note has the intended midi
channel selected.

As you can see, this is all very time-consuming, so I'm looking
forward to Monz's software to make all of this a whole lot easier.

--George

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > > --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> > > wrote:
> > > ...
> > > i realized, since i made my original post, that the "dominant
> > > pentatonic" is not CS in 12-equal. perhaps that's one source of
my
> > > difficulty?
> >
> > I wouldn't think so.
>
> Well, I'm interested in investigating further . . .

Okay. Once you're satisfied that we've adequately pinned down the
concept of functional scale disorientation (or whatever it is we end
up calling it -- the word "disorientation" suggests subjectivity),
then we can take another look at this.

> > > > 2) But if there are two intervals in a scale that are *not
> > > > functionally different* (such as the two 2:3s or 3:4s in our
11-limit
> > > > hexatonic otonality),
> > >
> > > why aren't they functionally different? because we don't have a
well-
> > > defined sense of hexatonic musical function, while we know all
too
> > > much about the history and theory of the diatonic scale? i
don't
> > > think that the "happen to" above can be defined in any precise
or
> > > perceptually relevant sense -- though it would be nice . . .
> >
> > As I see it, interval function is independent of the number of
tones
> > in the scale, but instead has to do with the (just) *ratio* that
is
> > either directly expressed (in JI) or implied (in a temperament)
by
> > that interval. So two tempered intervals that (in a given
context)
> > are implying the same just interval are functionally the same,
even
> > if they are not exactly the same size (such as in a well-
> > temperament). But two tempered intervals that (by context) imply
> > different just intervals are functionally different, even if they
are
> > exactly the same size in a particular tuning.
>
> What just interval does the 12-equal augmented second imply? And
how
> is this implication effected, exactly?

The diatonic system and traditional harmony do not exist only in 12-
ET, so I would first want to answer this question for the whole range
of tunings in which diatonic music could be performed.

The 6th and 7th members of a harmonic minor scale could be given
ratios of 8/5 and 15/8 (as 3rds of the subdominant and dominant
triads), so the interval between these tones would be 64:75. (In the
case of an extended meantone temperament, if you wanted to go to a 7-
prime-limit and call this 6:7, I would see no problem with that. The
only requirement is that, since the two tones are related by a
different number of positions in a chain of fifths than is the minor
3rd, the augmented 2nd must be identified by a different ratio than
5:6.)

To answer the question for 12-ET: that's a special case in which 5:6
and 64:75 are conflated, but even in 12-ET an augmented 2nd in the
context of traditional (diatonic) harmony still functions as a 64:75
(a dissonance), not a (consonant) 5:6.

> > In the context of a diatonic scale the tones are all assumed to
be in
> > a chain of fifths. If one member of that chain is taken to
represent
> > 1/1, then each of the other members can be assigned at least one
> > (rational) ratio that is unique to that member. An augmented 4th
and
> > diminished 5th (or a minor 3rd and augmented 2nd, etc.) will
> > therefore be considered to be serving different harmonic
functions,
> > since they represent different ratios.
>
> I'd like to see this made more explicit.

In a C major scale the diminished 5th B-F has the upper note of the
interval -6 positions in the chain of 5ths, relative to the lower
note. F to B (an augmented 4th) is +6 positions. If F is taken as
4/3 and B as 15/8, then a diminished 5th represents the ratio 45:64
and the augmented 4th represents 32:45. Thus their harmonic
functions differ.

> > In an 8:9:10:11:12:14:16 scale there is no question that the two
2:3s
> > (or the two 3:4s) are for all intents and purposes identical
(since
> > this is JI),
>
> What if I tuned a harmonic minor scale in JI with a 6:5 augmented
> second?

You wouldn't have consonant (5-limit) thirds on both the subdominant
and dominant triads, so I think that your scale would depart from
conventional thinking about what constitutes a harmonic minor scale.
Let's develop this further.

Suppose that you kept the 8/5 and raised the 15/8 to 48/25 so that
you have a dissonant dominant triad (a very interesting possibility,
since a higher leading tone would be desirable from a melodic
perspective). You would also have a consonant minor triad Ab-B-Eb
that makes a very nice resolution to G-C-E. Problem is, the spelling
of this minor triad should be Ab-Cb-Eb, so something's wrong here:
the tones don't *function* properly (i.e., consistently) in a
heptatonic scale, because there are two tones competing to be some
sort of "C".

Now suppose that you keep the 15/8 but lower the 8/5 to 25/16 so that
you have a dissonant subdominant triad. Now let's make the tonic
triad major, so the 3rd degree of the scale is 5/4. You would then
have a consonant major triad E-Ab-B (that would also resolve nicely
to E-G-C). Again there's a problem with the spelling, since Ab
should be G#, so again the tones don't *function* properly in a
heptatonic scale, since now you have two tones that are some sort
of "G".

I'm not saying that you can't compose music using intervals this way,
only that these tonal relationships are not consistent with a
heptatonic scale.

> > so on a *harmonic* level they are functionally
> > equivalent. But since these pairs of intervals subtend different
> > steps in the scale, the potential for _functional scale
> > disorientation_ (if you don't like the term, then please suggest
> > something else) exists.
>
> I'm hoping we can make this precise. Right now it seems fuzzy, with
> meaning adapted differently to fit this fact and that. Please help
me
> remove the ambiguity.

Are I making any progress with my examples?

> Perhaps we are talking about epimorphic vs. non-epimorphic scales?
If
> so, realizing this could be a breakthrough. At least we could have
a
> precise (and very relevant to the material on this list)
mathematical
> characterization of what makes a scale have or not have "functional
> scale disorientation" to you. That could be very helpful. Gene,
would
> you chime in?

I looked up this term in Monz's dictionary but gave up trying to
figure it out when I saw "val" in the definition. I'll wait for
Gene's comments before I try again.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> To answer the question for 12-ET: that's a special case in which
5:6
> and 64:75 are conflated, but even in 12-ET an augmented 2nd in the
> context of traditional (diatonic) harmony still functions as a
64:75
> (a dissonance), not a (consonant) 5:6.

I'm not sure i buy the idea that it 'functions as a 64:75' -- at
least not if you're excluding simultaneous 'functioning' as 108:125
and 1024:1215 . . .

> > Perhaps we are talking about epimorphic vs. non-epimorphic
scales?
> If
> > so, realizing this could be a breakthrough. At least we could
have
> a
> > precise (and very relevant to the material on this list)
> mathematical
> > characterization of what makes a scale have or not
have "functional
> > scale disorientation" to you. That could be very helpful. Gene,
> would
> > you chime in?
>
> I looked up this term in Monz's dictionary but gave up trying to
> figure it out when I saw "val" in the definition.

Uh-oh -- luckily Dave Keenan, at least, has recently cleared his
hurdle. You may want to look at his most recent posts here, where he
was trying to come up with a friendlier term for what 'val' means.
Although his attempts weren't entirely satisfactory, they should get
the relevant meaning of the term across to you.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
>
> > > Perhaps we are talking about epimorphic vs.
> > > non-epimorphic scales? If so, realizing this
> > > could be a breakthrough. At least we could have
> > > a precise (and very relevant to the material on
> > > this list) mathematical characterization of what
> > > makes a scale have or not have "functional scale
> > > disorientation" to you. That could be very helpful.
> > > Gene, would you chime in?
> >
> > I looked up this term in Monz's dictionary

which? "epimorphic"?

> > but gave up trying to figure it out when I saw "val"
> > in the definition.
>
> Uh-oh -- luckily Dave Keenan, at least, has recently
> cleared his hurdle. You may want to look at his most
> recent posts here, where he was trying to come up with
> a friendlier term for what 'val' means. Although his
> attempts weren't entirely satisfactory, they should get
> the relevant meaning of the term across to you.

i'd really like to amend, alter, or replace the definition
i have of "val". i suppose Gene's definition should stand,
but a hefty amendment of the kind of stuff Dave wrote
would help others to understand, and me too.

anyone willing to provide something that i can just paste in?

-monz

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> what was your original basis choice, and what do the patterns of
> signs for duals look like under it?

I'd suggest we forget about that. Alphabetical is the most usual
approach, and we are already using it. Moreover, it does allow us to
use the formula Sum indicies + m(m+1)/2 to determine the sign of the
compliment.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> As you can see, this is all very time-consuming, so I'm looking
> forward to Monz's software to make all of this a whole lot easier.

There's always the possibility of simply creating a Scala seq file
directly.

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

> i'd really like to amend, alter, or replace the definition
> i have of "val". i suppose Gene's definition should stand,
> but a hefty amendment of the kind of stuff Dave wrote
> would help others to understand, and me too.

No, that needs to be rewritten now that we've agreed on a notation.

> anyone willing to provide something that i can just paste in?

We could try something like this:

A monzo is a ket vector of exponents of a positive rational number in
a certain prime limit p; if q = 2^e2 3^e3 ... p^ep, then the
corresponding monzo is [e2 e3 ... ep>. A val, in the same prime limit,
is a bra vector of integers. For prime limit p, both the monzo and
the val have dimension pi(p), meaning the number of primes up to p.
The inner product of a val and monzo therefore defines a mapping from
p-limit positive rational numbers to integers; if v = <v2 v3 ... vp]
is a p-limit val and e = [e2 e3 ... ep> is a p-limit monzo, then

<v|e> = <v2 v3 ... vp |e2 e3 ... ep> = v2e2 + v3e3 + ... + vpep

is the homomorphic mapping v(e) defined by v.

The following may be too mathematical for your dictionary, but is
from my web site:

Intervals and Vals

For p an odd prime, the intervals of the p-limit Np may be taken as
the set of all frequency ratios which are positive rational numbers
whose factorization involves only primes less than or equal to p. If
q is such a ratio, it may be written in factored form as

q = 2^e2 3^e3 ... p^ep

where e2, e3, ... ep are integer exponents. We may write this in
factored form as a ket vector of the exponents, or monzo:

[e2 e3 ... ep>

The p-limit rational numbers Np form an abelian group, or Z-module,
under multiplication, so that it acts on itself as a transformation
group of a musical space; this becomes an additive group using vector
addition when written additively as a monzo.

Np is a free abelian group of rank pi(p), where pi(p) is the number
of primes less than or equal to p. The rank is the dimension of the
vector space in which Np written additively can be embedded as a
lattice; saying it is free means this embedding can be done, since
there are no torsion elements, meaning there are no positive rational
numbers q (called roots of unity) other than 1 itself, with the
property that for some positive power n, q^n = 1.

Given the p-limit group Np of intervals, there is a non-canonically
isomorphic dual group Vp of vals. A val is a homomorphism of Np to
the integers Z. Just as an interval may be regarded as a Z-linear
combination of basis elements representing the prime numbers, a val
may be regarded as a Z-linear combination of a dual basis, consisting
of the p-adic valuations. For a given prime p, the corresponding
p-adic valuation vp gives the p-exponent of an interval q, so for
instance v2(5/4) = -2, v3(5/4) = 0, v5(5/4) = 1. If intervals are
written as ket vectors, or monzos, vals are denoted by the
corresponding bra vector. The 5-limit 12-et val, for instance, would
be written <12 19 28].

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > what was your original basis choice, and what do the patterns of
> > signs for duals look like under it?
>
> I'd suggest we forget about that. Alphabetical is the most usual
> approach, and we are already using it.

OK -- but it's interesting to note that the cross product immediately
gives you the quantity of interest in 3D, regardless of indexing
conventions. The GABLE tutorial claims that cross products are
useless and should be dispensed with since geometric algebra has
better ways of solving all the problems that the cross product is
used for. I don't know . . .

> Moreover, it does allow us to
> use the formula Sum indicies + m(m+1)/2 to determine the sign of
the
> compliment.

my highest compliments, but it's spelled complement.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i'd really like to amend, alter, or replace the definition
> > i have of "val". i suppose Gene's definition should stand,
> > but a hefty amendment of the kind of stuff Dave wrote
> > would help others to understand, and me too.
>
> No, that needs to be rewritten now that we've agreed on a notation.
>
> > anyone willing to provide something that i can just paste in?
>
> We could try something like this:
>
> A monzo is a ket vector of exponents of a positive rational number
in
> a certain prime limit p; if q = 2^e2 3^e3 ... p^ep, then the
> corresponding monzo is [e2 e3 ... ep>. A val, in the same prime
limit,
> is a bra vector of integers. For prime limit p, both the monzo and
> the val have dimension pi(p), meaning the number of primes up to p.
> The inner product of a val and monzo therefore defines a mapping
from
> p-limit positive rational numbers to integers; if v = <v2 v3 ...
vp]
> is a p-limit val and e = [e2 e3 ... ep> is a p-limit monzo, then
>
> <v|e> = <v2 v3 ... vp |e2 e3 ... ep> = v2e2 + v3e3 + ... + vpep
>
> is the homomorphic mapping v(e) defined by v.

this is the definition for what? no offense, but regardless, it needs
to be *greatly* expanded upon to be useful to 99.9% of its likely
audience. i don't have time right now as i'm chatting with monz's
business partner and then have to leave . . .

whew! thanks, Gene!

... i appreciate this, but ... um ... really also need a
musician-friendly version along the lines of what Dave
was doing. i'm hoping that he, paul, or Graham can help
with that.

but i will update the definitions with yours.

-monz

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i'd really like to amend, alter, or replace the definition
> > i have of "val". i suppose Gene's definition should stand,
> > but a hefty amendment of the kind of stuff Dave wrote
> > would help others to understand, and me too.
>
> No, that needs to be rewritten now that we've agreed on a notation.
>
> > anyone willing to provide something that i can just paste in?
>
> We could try something like this:
>
> A monzo is a ket vector of exponents of a positive rational number
in
> a certain prime limit p; if q = 2^e2 3^e3 ... p^ep, then the
> corresponding monzo is [e2 e3 ... ep>. A val, in the same prime
limit,
> is a bra vector of integers. For prime limit p, both the monzo and
> the val have dimension pi(p), meaning the number of primes up to p.
> The inner product of a val and monzo therefore defines a mapping
from
> p-limit positive rational numbers to integers; if v = <v2 v3 ...
vp]
> is a p-limit val and e = [e2 e3 ... ep> is a p-limit monzo, then
>
> <v|e> = <v2 v3 ... vp |e2 e3 ... ep> = v2e2 + v3e3 + ... + vpep
>
> is the homomorphic mapping v(e) defined by v.
>
>
> The following may be too mathematical for your dictionary, but is
> from my web site:
>
> Intervals and Vals
>
> For p an odd prime, the intervals of the p-limit Np may be taken as
> the set of all frequency ratios which are positive rational numbers
> whose factorization involves only primes less than or equal to p.
If
> q is such a ratio, it may be written in factored form as
>
> q = 2^e2 3^e3 ... p^ep
>
> where e2, e3, ... ep are integer exponents. We may write this in
> factored form as a ket vector of the exponents, or monzo:
>
> [e2 e3 ... ep>
>
> The p-limit rational numbers Np form an abelian group, or Z-module,
> under multiplication, so that it acts on itself as a transformation
> group of a musical space; this becomes an additive group using
vector
> addition when written additively as a monzo.
>
> Np is a free abelian group of rank pi(p), where pi(p) is the number
> of primes less than or equal to p. The rank is the dimension of the
> vector space in which Np written additively can be embedded as a
> lattice; saying it is free means this embedding can be done, since
> there are no torsion elements, meaning there are no positive
rational
> numbers q (called roots of unity) other than 1 itself, with the
> property that for some positive power n, q^n = 1.
>
> Given the p-limit group Np of intervals, there is a non-canonically
> isomorphic dual group Vp of vals. A val is a homomorphism of Np to
> the integers Z. Just as an interval may be regarded as a Z-linear
> combination of basis elements representing the prime numbers, a val
> may be regarded as a Z-linear combination of a dual basis,
consisting
> of the p-adic valuations. For a given prime p, the corresponding
> p-adic valuation vp gives the p-exponent of an interval q, so for
> instance v2(5/4) = -2, v3(5/4) = 0, v5(5/4) = 1. If intervals are
> written as ket vectors, or monzos, vals are denoted by the
> corresponding bra vector. The 5-limit 12-et val, for instance,
would
> be written <12 19 28].

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
>
> > As you can see, this is all very time-consuming, so I'm looking
> > forward to Monz's software to make all of this a whole lot easier.
>
> There's always the possibility of simply creating a Scala seq file
> directly.

I haven't figured out how to do that yet. (Yes, I did see the recent
postings about that on the main list.)

Anyway, I haven't had much incentive to figure it out, because I
don't use midi channels (in Cakewalk) the same way Scala does -- I
prefer to keep a single melodic line in a single track and channel so
I can copy and paste something from one track to another, including
the pitch-bend events. I then change the channel for each note in
the new track to another number (which goes fairly quickly in
Cakewalk, once I figured out how to do it). Another reason for
assigning channels this way is that I experienced that having two
different patches assigned to the same channel results tends to
corrupt the quality of the sound.

--George

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
>
> > To answer the question for 12-ET: that's a special case in which
5:6
> > and 64:75 are conflated, but even in 12-ET an augmented 2nd in
the
> > context of traditional (diatonic) harmony still functions as a
64:75
> > (a dissonance), not a (consonant) 5:6.
>
> I'm not sure i buy the idea that it 'functions as a 64:75' -- at
> least not if you're excluding simultaneous 'functioning' as 108:125
> and 1024:1215 . . .

But yes, I would also include *both* 108:125 and 1024:1215 as ratios
(or *roles*, as Dave and I call them when a Sagittal symbol is used
to represent multiple ratios) that would be covered by the interval
function of an augmented 2nd in traditional diatonic harmony, just as
I would include both 8:9 and 9:10 as roles in which the interval of a
major 2nd functions. My only requirement is that a single ratio
*not* be represented by multiple vectors (or in this case multiple
scalars) in the scale construct.

I mentioned vectors in the previous paragraph, because in the case of
the 11-limit hexatonic otonal scale that we've been using as our
other example, the tones occur in a 4-dimensional structure. And
there will be only one ratio (or role) for each tone, since the scale
is JI.

> > > Perhaps we are talking about epimorphic vs. non-epimorphic
scales? If
> > > so, realizing this could be a breakthrough. At least we could
have a
> > > precise (and very relevant to the material on this list)
mathematical
> > > characterization of what makes a scale have or not
have "functional
> > > scale disorientation" to you. That could be very helpful. Gene,
would
> > > you chime in?
> >
> > I looked up this term in Monz's dictionary but gave up trying to
> > figure it out when I saw "val" in the definition.
>
> Uh-oh -- luckily Dave Keenan, at least, has recently cleared his
> hurdle. You may want to look at his most recent posts here, where
he
> was trying to come up with a friendlier term for what 'val' means.
> Although his attempts weren't entirely satisfactory, they should
get
> the relevant meaning of the term across to you.

I was just reacting to this like Gene did when he saw Dave's "komma".

Okay, I studied the definitions of "epimorphic" and "val" in Monz's
dictionary for a couple of minutes, and I think I got somewhere with
this.

Gene, did you intend the term "epimorphic" to apply to temperaments
as well as rational scales? If so, then is it possible that "qn"
*could* be an irrational number?

If the answer to the second question is yes, then I don't think that
Gene and I are dealing with the same thing, because I require that
the "qn" for an irrational scale (or tempered) interval be a rational
ratio (or set of ratios) which that interval is intended to
approximate or represent. We could test this out by seeing whether
Gene considers the various scale examples given here:
/tuning-math/message/7827
to be epimorphic:
1) If each scale is in 1/4-comma meantone temperament, and
2) If each scale is in 12-ET.

Different answers for 1) and 2) would indicate that we're *not*
dealing with the same thing.

--George

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > OK. With lexicographic ordering of the indices, it isn't as simple
> as
> > negating every second coefficient. There's sometimes a hiccup in the
> > middle. It's explained in Section 5.4 of
> >
> >
> http://www.ses.swin.edu.au/homes/browne/grassmannalgebra/book/bookpdf/
> TheComplement.pdf
>
> The page cannot be displayed

It does seem to be offline at the moment. I'll keep trying
occasionally and let you know if I get thru again.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> >
> > > what was your original basis choice, and what do the patterns of
> > > signs for duals look like under it?
> >
> > I'd suggest we forget about that. Alphabetical is the most usual
> > approach, and we are already using it.
>
> OK -- but it's interesting to note that the cross product immediately
> gives you the quantity of interest in 3D, regardless of indexing
> conventions.

Paul. You must have missed where I explained that the cross-product
stays the same no matter what the indexing conventions, because the
wedge-product and the complement change in "complementary" ways when
you change the indexing and A(x)B = ~(A^B).

Gene:
> > Moreover, it does allow us to
> > use the formula Sum indicies + m(m+1)/2 to determine the sign of
> > the compliment.

"m" here is the grade of the object, i.e. the number of nested
brakets. A more intuitive (for me) alternative to m(m+1)/2 is
Ceiling(m/2). If the sum of the indices plus this quantity is even
then you negate it when complementing.

Dave Keenan wrote:

> "m" here is the grade of the object, i.e. the number of nested
> brakets. A more intuitive (for me) alternative to m(m+1)/2 is
> Ceiling(m/2). If the sum of the indices plus this quantity is even
> then you negate it when complementing.

That's an interesting short cut. The way I've been doing it is:

Join the old basis on the left and the new basis on the right to get a list of all primitive bases (or whatever they are). If this is an odd permuatation, negate the coefficient.

This assumes all bases are being stored in numerical order of their components (what's all this talk about alphabetical order?)

You can test for an odd permutation by swapping adjacent pairs of primitive bases until their in the right order, and it's an odd permutation if you did an odd number of swaps. This follows from the antisymmetry of the wedge product, which is the main thing you need to know about it. You can also compare every pair of numbers in the basis, and it's an odd permutation iff an odd number of them are the wrong way round.

Graham

Paul Erlich wrote:

> yes, as you know i (and especially graham) like that idea very much --
> BUT 88cET has no octaves!

You can make 88cET equivalent with respect to some other interval -- like 88 cents for example.

Graham