epimorphic

Hi Gene
I have no means in which to decipher this
--

Then it is epimorphic with val
h if h(s[i]) = i, where s[i] is the ith scale element.

Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

On Monday 11 July 2005 12:31 am, Kraig Grady wrote:
> Today 12:31:15 am
>
>
> Hi Gene
>  I have no means in which to decipher this
> --
>
> Then it is epimorphic with val
> h if h(s[i]) = i, where s[i] is the ith scale element

Yes, this is *too* abstract. Not everyone here has a math PhD. Perhaps an
example of an epimorphic scale and a non-epimorphic scale would do well.

I think hardcore mathematicians lose touch with what makes something
understandable quite a bit, because they are used to talking to each other.
And part of the problem is you are using obscure code-words to define obscure
code-words ('val' for example). I know it all seems so simple to *you*, Gene,
but you do highly abstract math that the general public (the rest of us) have
to study for hours to even parse.

In short, you are a math-genius, but a lousy teacher (at least for the general
student who is not motivated to continue finding out more, and even for those
that are curious (myself) who find getting caught in a morass of abstract
algebra terms really frustrating and unhelpful)

For the musicians who are *not* mathematicians here, the above definition
would be something like trying to identify a cat by its chromosomes:
technically precise and correct, but least useful in daily life. What Kraig
wants is a definition more like what an average person needs to identify 'a'
cat: a warm-blooded mammal with whiskers, meowing, 4 legs, a tail, smal nose,
a photograph of a cat (being the example), adept at hunting and sleeping,
etc.

If I knew for sure what epimorphic meant (and why it was an important property
of a scale), I would say something like: This scale below is epimorphic
because all of its steps have this property; give a simple algorithm for
finding that property, step by step, using functions available on a
(scientific) calculator; then the same for a non-epimorphic scale. Then, if
they cared enough, they would go back to your original definition and a light
bulb would go off, wherein they would see the general abstract truth of the
above. But the abstract can't easily be grasped if the specific tangible
isn't presented first.

In short, assume you are talking to mathematical idiots, and what you present
just might start making sense to us! Most of us here know arithmetic and
algebra, which can cover most tuning-topics. And by the way, keep doing all
the work you do, I for one appreciate that you are doing it. Formal thinking
is good.

-A.

It does get a bit tedious following all the bland formulas after a while. How about tuning for dummies (such as myself)?

Tiger: A vicious feline weighing about 150kgs with saber fangs and razor paws wearing a beautiful designer's coat. Has a voracious apettite and cannot be tamed. It's giant gaping mouth can deliver a bite of 250kgs of pressure. To view pictures of the beast, see http://www.ozanyarman.com/mainpage/pictures.html.

----- Original Message -----
From: Aaron Krister Johnson
To: tuning@yahoogroups.com
Sent: 11 Temmuz 2005 Pazartesi 16:37
Subject: Re: [tuning] epimorphic

On Monday 11 July 2005 12:31 am, Kraig Grady wrote:
> Today 12:31:15 am
>
>
> Hi Gene
> I have no means in which to decipher this
> --
>
> Then it is epimorphic with val
> h if h(s[i]) = i, where s[i] is the ith scale element

Yes, this is *too* abstract. Not everyone here has a math PhD. Perhaps an
example of an epimorphic scale and a non-epimorphic scale would do well.

I think hardcore mathematicians lose touch with what makes something
understandable quite a bit, because they are used to talking to each other.
And part of the problem is you are using obscure code-words to define obscure
code-words ('val' for example). I know it all seems so simple to *you*, Gene,
but you do highly abstract math that the general public (the rest of us) have
to study for hours to even parse.

In short, you are a math-genius, but a lousy teacher (at least for the general
student who is not motivated to continue finding out more, and even for those
that are curious (myself) who find getting caught in a morass of abstract
algebra terms really frustrating and unhelpful)

For the musicians who are *not* mathematicians here, the above definition
would be something like trying to identify a cat by its chromosomes:
technically precise and correct, but least useful in daily life. What Kraig
wants is a definition more like what an average person needs to identify 'a'
cat: a warm-blooded mammal with whiskers, meowing, 4 legs, a tail, smal nose,
a photograph of a cat (being the example), adept at hunting and sleeping,
etc.

If I knew for sure what epimorphic meant (and why it was an important property
of a scale), I would say something like: This scale below is epimorphic
because all of its steps have this property; give a simple algorithm for
finding that property, step by step, using functions available on a
(scientific) calculator; then the same for a non-epimorphic scale. Then, if
they cared enough, they would go back to your original definition and a light
bulb would go off, wherein they would see the general abstract truth of the
above. But the abstract can't easily be grasped if the specific tangible
isn't presented first.

In short, assume you are talking to mathematical idiots, and what you present
just might start making sense to us! Most of us here know arithmetic and
algebra, which can cover most tuning-topics. And by the way, keep doing all
the work you do, I for one appreciate that you are doing it. Formal thinking
is good.

-A.

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> Hi Gene
> I have no means in which to decipher this
> --
>
> Then it is epimorphic with val
> h if h(s[i]) = i, where s[i] is the ith scale element.

Ah Gene,

What a sense of humour.

Here it is in English. ;-)

Introduction

"Epimorphic" is yet another attempt, along with "proper", and
"constant structure", to capture in a yes/no formula the kinds of
combinations of evenness and interval-consistency that we humans find
useful in a musical scale or tuning. "Epimorphic" only applies to
rational scales while the other two apply to any scale.

Definition

A rational scale is epimorphic if and only if it is a consistent
order-preserving detempering of the ET with the same number of notes.

Or putting it another way:
A rational scale is epimorphic if and only if the ET with the same
number of notes is a consistent order-preserving tempering of it.

By "consistent order-preserving tempering" here we mean that there is
some mapping from prime numbers to steps of the ET, so that when this
mapping is applied to consecutive ratios of the rational tuning (in
pitch order), it generates consecutive degrees of the ET (i.e. no
crossovers in pitch).

Testing

So if you want to test if a tuning is epimorphic you'd probably first
find an example of each prime interval, 1:2, 1:3, 1:5, etc up to its
prime limit, and count the number of steps of the tuning each one
spans. This gives you a prime-mapping to try. You then apply that
prime-mapping to every ratio of the scale (relative to 1/1) and see
that you get consective numbers of steps.

Here's some more tutorial for those unfamiliar with what it means to
"apply a prime mapping".

To apply a prime mapping to a ratio you first find the ratio's prime
factors and list the exponent of each prime (the number of times the
prime occurs in the prime factorisation, with those on the bottom of
the ratio shown as negative). This is the prime-exponent-vector for
the ratio. Then multiply each prime-exponent by the corresponding
number of steps in the prime-mapping and add up all those products to
get the number of steps for the ratio.

For example consider the prime-mapping

<12 19, 28] (Notice the angle-bracket points to the left)

This happens to be the standard 5-limit prime-mapping for 12-ET)
It means that the octave 1:2 is approximated by 12 generators (in this
case steps of the ET), and 1:3 (an octave and fifth) is approximated
by 19 steps (12 + 7), and 1:5 (two octaves and a major third) by 28
steps (12+12+4).

Now consider the ratio 25/24. Its prime factorisation is

5 * 5
-------------
2 * 2 * 2 * 3

So can be written as 2^-3 * 3^-1 * 5^2 or more simply as the
prime-exponent-vector

[-3 -1, 2> (Notice the angle-bracket points to the right)

One way of remembering that angle-brackets are "mappings left,
exponents right", is to note that when we write say 5 to the power of
2, we put the exponent on the right, 5^2. Another way is to notice
that applying a mapping is like applying a function (such as sine,
cosine or square-root) and when we write those we usually write the
function to the left.

Now to apply the above mapping to this ratio we can use the "dot
product" notation and write:

<12 19, 28].[-3 -1, 2>

which is just the sum of the products of the corresponding entries

12*-3 + 19*-1 + 28*2

= 1

I note that since we're usually dealing with octave-equivalent scales
we often ignore the exponents of 2 and just perform our arithmetic
modulo the number of steps per octave. In which case the above mapping
can be written in octave-equivalent form as

<7, 4]

i.e. 7 steps per 2:3 fifth and 4 steps per 4:5 major third

And in octave-equivalent form the ratio 25/24 is

[-1, 2>

Note that the comma is always after the 3 exponent (and then every
third place after that) so we know whether we are looking at a full
exponent vector or an octave-equivalent one.

Then we have <7, 4].[-1, 2> = 7*-1 + 4*2 = 1

This is the same answer as before. If the answer had been outside the
range 0 to 11 we would have added or subtracted octaves (12's in this
case) to bring it back to that range.

Further discussion

The definition of "constant structure" in Monz's encyclopedia is that
a scale or tuning has (or is a) constant structure if every instance
of the same size interval (in cents or as a ratio) spans the same
number of scale steps. e.g. all 2:3 fifths might span 7 steps etc.

It seems to me that this might well be equivalent to "epimorphic", in
the case of rational tunings. Is that right Gene? Maybe I missed the
counterexample.

-- Dave Keenan

Hi Dave,

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

> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> > Hi Gene
> > I have no means in which to decipher this
> > --
> >
> > Then it is epimorphic with val
> > h if h(s[i]) = i, where s[i] is the ith scale element.
>
>
> Ah Gene,
>
> What a sense of humour.
>
> Here it is in English. ;-)

Thanks *SO MUCH* for this nice tutorial!
It certainly helps me a lot.

With your permission, i'd like to add it to
the Encyclopedia in some form.

> I note that since we're usually dealing with
> octave-equivalent scales we often ignore the exponents
> of 2 and just perform our arithmetic modulo the number
> of steps per octave. In which case the above mapping
> can be written in octave-equivalent form as
>
> <7, 4]
>
> i.e. 7 steps per 2:3 fifth and 4 steps per 4:5 major third
>
> And in octave-equivalent form the ratio 25/24 is
>
> [-1, 2>
>
> Note that the comma is always after the 3 exponent
> (and then every third place after that) so we know
> whether we are looking at a full exponent vector or
> an octave-equivalent one.

I am *so glad* to see you following the proposed
standard for notation of monzos. I wish Gene would.

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> Thanks *SO MUCH* for this nice tutorial!
> It certainly helps me a lot.
>
> With your permission, i'd like to add it to
> the Encyclopedia in some form.

You may certainly use it verbatim, or with minor changes, but if "in
some form" means you want to replace every (or any) ocurrence of
"prime mapping" with "val" or "breed" and every (or any) ocurrence of
"prime exponent vector" with "monzo", forget it.

In my opinion that would be wilful obfuscation. Such terms are quite
unnecessary.

Sorry.

-- Dave Keenan

Dave Keenan, after an explanation that looked fearsomely mathematical but as it turned out could be followed without, or should I say even with a simple, pocket calculator, wrote:

> You may certainly use it verbatim, or with minor changes, but if "in
> some form" means you want to replace every (or any) ocurrence of
> "prime mapping" with "val" or "breed" and every (or any) ocurrence of
> "prime exponent vector" with "monzo", forget it.
> > In my opinion that would be wilful obfuscation. Such terms are quite
> unnecessary.

db
(thumbs up)

klaus

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
"Epimorphic" only applies to
> rational scales while the other two apply to any scale.

"Epimorphic" actually applies to any scale which is regularly
generated in terms of a certain set of generators with fixed tunings,
but it's only really been used for JI scales. Manuel calls that "JI
epimorphic" because of this, but Scala does not attempt to find if a
scale is epimorphic in any broader sense than JI but in a
non-ascending order.

> So if you want to test if a tuning is epimorphic you'd probably first
> find an example of each prime interval, 1:2, 1:3, 1:5, etc up to its
> prime limit, and count the number of steps of the tuning each one
> spans.

I don't know how Manuel does it, but the way I do it is simply to find
if the set of linear equations you get if the epimorphic assumption is
true has a solution.

> I note that since we're usually dealing with octave-equivalent scales
> we often ignore the exponents of 2 and just perform our arithmetic
> modulo the number of steps per octave.

This is probably not a good idea if you are working with the
epimorphic property, since unless the number of steps n to the octave
is a prime number (eg, n=19) you don't have a field on reduction
modulo n, which messes up solving linear equations.

> It seems to me that this might well be equivalent to "epimorphic", in
> the case of rational tunings. Is that right Gene? Maybe I missed the
> counterexample.

It's clear epimorphic ==> constant structure. For the reverse
implication, we need that CS is a strong enough condition to force
a solution to {h(s[i])-i} for some h, not necessarily uniquely. For a
counterexample, consider the "scale" [1,4375/4374,2401/2400]. If we
solve for what the val would need to be, we get

<3 a3 (3/2)a3 - 1/18 a3+38/9|.

Since a3 is an integer, (3/2)a3-1/18 and a3+38/9 are not integers.
However, any 4375/4374 is one step, and any 2401/2400 is two steps,
anywhere, so it is CS. Scala says of it that "scale is JI epimorphic
without unique prime-degree mapping", a statement I have not seen
before, but this isn't true--it doesn't *have* such a mapping.

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

> You may certainly use it verbatim, or with minor changes, but if "in
> some form" means you want to replace every (or any) ocurrence of
> "prime mapping" with "val" or "breed" and every (or any) ocurrence of
> "prime exponent vector" with "monzo", forget it.
>
> In my opinion that would be wilful obfuscation. Such terms are quite
> unnecessary.

"Prime exponent vector" suggests that the notes of JI constitute a
vector space, not an abelian group, so I think the phrase is both
clumbersome and not altogether accurate. "Prime mapping" is not
usable, since we use that phrase for mappings which map to things
other than the integers, consisting of more than one val; the obvious
example being the map to period and generator. Moreover, to say that
it is obfuscation overlooks the fact that it is a fundamental
theoretical concept, which having a specialist term helps to drive
home. If you are going to interest yourself in tuning theory you need
the concept in some form, and neither "homomorphism from a free
subgroup of finite rank of the positive rational numbers under
multiplication to the integers" nor "finite Z-linear combination of
additive padic valuations" strike me as likely to elicit either
understanding or enthusiasm. Recall, this was the sort of language I
used when I first arrived on these lists, to bad reviews. Using "val"
and "monzo" and the bra-ket notation seems to have resulted in far
greater understanding that what we started out with.

The bottom line is, it works.

Hi Dave,

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

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > Thanks *SO MUCH* for this nice tutorial!
> > It certainly helps me a lot.
> >
> > With your permission, i'd like to add it to
> > the Encyclopedia in some form.
>
> You may certainly use it verbatim, or with minor changes,
> but if "in some form" means you want to replace every
> (or any) ocurrence of "prime mapping" with "val" or "breed"
> and every (or any) ocurrence of "prime exponent vector"
> with "monzo", forget it.
>
> In my opinion that would be wilful obfuscation. Such terms
> are quite unnecessary.
>
> Sorry.

Well, i think you know by now that i (respectfully)
disagree with you about new terminology.

But anyway, what i meant by "in some form" was
that it might be better (for the purposes of the
Encyclopedia) to chop your post up and put
different parts of it into the relevant Encylopedia
definitions.

In any case, i'm so busy with the software that
nothing new is being added to the Encyclopedia right now
anyway ... so hopefully at some future time my definition
of "epimorphic" will be enhanced by incorporation of
your post.

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> "Epimorphic" actually applies to any scale which is regularly
> generated in terms of a certain set of generators with fixed tunings,
> but it's only really been used for JI scales.

That makes sense, but neither the definition you gave most recently
nor the one in the Tonalsoft Encyclopedia include these cases.

> I don't know how Manuel does it, but the way I do it is simply to find
> if the set of linear equations you get if the epimorphic assumption is
> true has a solution.

That makes sense.

> > I note that since we're usually dealing with octave-equivalent scales
> > we often ignore the exponents of 2 and just perform our arithmetic
> > modulo the number of steps per octave.
>
> This is probably not a good idea if you are working with the
> epimorphic property, since unless the number of steps n to the octave
> is a prime number (eg, n=19) you don't have a field on reduction
> modulo n, which messes up solving linear equations.

We were asked for a simple algorithm that required at most a simple
pocket calculator. So algebra was out of the question. But sure, if
you're doing it automatically you'd keep the 2's.

> It's clear epimorphic ==> constant structure. For the reverse
> implication, we need that CS is a strong enough condition to force
> a solution to {h(s[i])-i} for some h, not necessarily uniquely. For a
> counterexample, consider the "scale" [1,4375/4374,2401/2400].
...

It would be good if you could give a counterexample with much smaller
numbers and more realistic intervals so it's easy for the algebraicly
challenged to see _why_ it is a constant structure but not epimorphic.

-- Dave

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> "Prime exponent vector" suggests that the notes of JI constitute a
> vector space, not an abelian group, so I think the phrase is both
> clumbersome and not altogether accurate. "Prime mapping" is not
> usable, since we use that phrase for mappings which map to things
> other than the integers, consisting of more than one val; the obvious
> example being the map to period and generator. Moreover, to say that
> it is obfuscation overlooks the fact that it is a fundamental
> theoretical concept, which having a specialist term helps to drive
> home. If you are going to interest yourself in tuning theory you need
> the concept in some form, and neither "homomorphism from a free
> subgroup of finite rank of the positive rational numbers under
> multiplication to the integers" nor "finite Z-linear combination of
> additive padic valuations" strike me as likely to elicit either
> understanding or enthusiasm. Recall, this was the sort of language I
> used when I first arrived on these lists, to bad reviews. Using "val"
> and "monzo" and the bra-ket notation seems to have resulted in far
> greater understanding that what we started out with.
>
> The bottom line is, it works.

All of this may be valid if you're talking to other mathematicians,
for example on the tuning-math list, but on this list it's all
irrelevant. Most readers of this list will not be confused by the
difference between a vector space and an abelian group since they have
no idea what either of them is, and don't need to. And I think it has
been explained to you before now that it is the engineering or
geometric meaning of "vector" that is alluded to here, not the algebraic.

And I didn't claim that "prime mapping" was a synonym for your "val"
in any case other than the one under discussion. Surely, even in your
terms, a prime mapping _can_ consist of a single "val", i.e. in the
case of ETs, which is what was under discussion.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> Scala says of it that "scale is JI epimorphic
> without unique prime-degree mapping", a statement I have not seen
> before, but this isn't true--it doesn't *have* such a mapping.

I admit it's a pathetic one, but isn't this a solution?
2 => 2
3 => 0
5 => 0
7 => 3

What were the 3 and 7 scales again you made and which I didn't put in
the archive yet?

Manuel

On Wednesday 13 July 2005 8:46 am, Manuel Op de Coul wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> > Scala says of it that "scale is JI epimorphic
> > without unique prime-degree mapping", a statement I have not seen
> > before, but this isn't true--it doesn't *have* such a mapping.
>
> I admit it's a pathetic one, but isn't this a solution?
> 2 => 2
> 3 => 0
> 5 => 0
> 7 => 3
>
> What were the 3 and 7 scales again you made and which I didn't put in
> the archive yet?

The original messages, where gene posted .scl files, are:

/tuning/topicId_53040.html#53059
and
/tuning/topicId_53061.html#53062

-Aaron.

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

> It would be good if you could give a counterexample with much smaller
> numbers and more realistic intervals so it's easy for the algebraicly
> challenged to see _why_ it is a constant structure but not epimorphic.

A much better counterexample is [1, 5/4, 9/5], which when you solve
for a val gives <3 9/5 7|.

--- In tuning@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:

> I admit it's a pathetic one, but isn't this a solution?
> 2 => 2
> 3 => 0
> 5 => 0
> 7 => 3

It doesn't work, since we need 2 => 3, as there are three notes to the
octave.

> What were the 3 and 7 scales again you made and which I didn't put in
> the archive yet?

I made some scales for Aaron; I looked quickly and didn't find any and
the best plan might be for Aaron to repost them here.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > It would be good if you could give a counterexample with much smaller
> > numbers and more realistic intervals so it's easy for the algebraicly
> > challenged to see _why_ it is a constant structure but not epimorphic.
>
> A much better counterexample is [1, 5/4, 9/5], which when you solve
> for a val gives <3 9/5 7|.

<3 9/2 7| sorry.

On Wednesday 13 July 2005 9:33 am, Aaron Krister Johnson wrote:
> On Wednesday 13 July 2005 8:46 am, Manuel Op de Coul wrote:
> > What were the 3 and 7 scales again you made and which I didn't put in
> > the archive yet?
>
> The original messages, where gene posted .scl files, are:
>
> /tuning/topicId_53040.html#53059
> and
> /tuning/topicId_53061.html#53062
>
> -Aaron.

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

> It doesn't work, since we need 2 => 3, as there are three notes to the
> octave.

Well yes if you add 2/1 as the third degree to the "scale", not if you
don't. In that case you're right that Scala is incorrect.

> A much better counterexample is [1, 5/4, 9/5], which when you solve
> for a val gives <3 9/5 7|.

<3 6 8| is an out of order solution, so in that sense it is epimorphic.

> I made some scales for Aaron; I looked quickly and didn't find any and
> the best plan might be for Aaron to repost them here.

I saved them from Aaron's post.

Manuel

Beginning today, if you missed a Concertzender broadcast, then most of
them can be heard during one month afterwards.
Go to http://www.concertzender.nl and click on "radio on demand".
Then choose the genre on the left. The occasional microtonal music is
mostly under "Nieuwe muziek".
Format is 64 Kb/s real-audio.

There's a 7-part Wolfgang Rihm series going now; he also wrote some
microtonal music if I'm not mistaken.

I don't use the horrible RealPlayer by the way, but the old Microsoft
Media Player Classic version 6.4.8.3 with an extra codec for
real-audio and it works fine.

Manuel