Epimorphic

Gene,

I'm implementing the epimorphic property in Scala, but
find the name a bit terse. Shall I call it prime-epimorphic
or do you have a better name?

Manuel

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:

> I'm implementing the epimorphic property in Scala, but
> find the name a bit terse. Shall I call it prime-epimorphic
> or do you have a better name?

Great! It seems to me it would be better to say "JI-epimorphic" or
"RI-epimorphic", leaving open the possibility of also implementing
"meantone-epimorphic" or "starling-epimorphic" some fine day.

Gene wrote:

>Great! It seems to me it would be better to say "JI-epimorphic" or
>"RI-epimorphic", leaving open the possibility of also implementing
>"meantone-epimorphic" or "starling-epimorphic" some fine day.

It turns out the question was moot since Pierre showed that it's
equivalent to CS. Anyway I don't need to throw the new code straight
away if I use it to print out the characterising val. I'll call that
epimorphic prime-degree mapping.

Isn't "meantone-epimorphic" covered by Myhill's property?

Manuel

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:

> It turns out the question was moot since Pierre showed that it's
> equivalent to CS.

Not so far as I can see.

Gene wrote:
>Not so far as I can see.

I haven't found a CS and non-epimorphic counterexample yet.

Manuel

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> Gene wrote:
> >Not so far as I can see.
>
> I haven't found a CS and non-epimorphic counterexample yet.

Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> > Gene wrote:
> > >Not so far as I can see.
> >
> > I haven't found a CS and non-epimorphic counterexample yet.
>
> Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1

By the way, if you decide to impliment the epimorphism feature, I'd suggest "Scale is epimorphic with val ---" or "Scale is epimorphic with mapping ---"

>Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1

Allright, but are there any monotonic examples?

Manuel

>By the way, if you decide to impliment the epimorphism feature, I'd
suggest "Scale is epimorphic with val >---" or "Scale is epimorphic with
mapping ---"

Yes, it will show that.

Manuel

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> >Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1
>
> Allright, but are there any monotonic examples?

I dunno--define "monotonic". To me it means monotonically increasing, which this scale does.

Gene wrote:
Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1
Do you consider that unordered list as a scale ? Need correction or reordering.

Pierre

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> > >Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1
> >
> > Allright, but are there any monotonic examples?
>
> I dunno--define "monotonic". To me it means monotonically
>increasing, which this scale does.

2401/2400 is not between 5/3 and 2/1, gene!

>>It turns out the question was moot since Pierre showed that it's
>>equivalent to CS.
>
>Not so far as I can see.

Pierre didn't say that. He said Epimorphic -> CS, which is
exactly what Gene said. Clearly CS /-> Epimorphic. See my
post in this thread.

-Carl

>>Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1
>
>Allright, but are there any monotonic examples?

Why does this fail?

The stronger argument against CS /-> Epimorphic is
that CS doesn't require JI, as Gene pointed out.

-C.

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:
> Gene wrote:
> Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1
> Do you consider that unordered list as a scale ? Need correction or reordering.

Is that what I wrote? It should be

1/1--2700/2401--5/4-4/3--3/2--5/3--2401/1280--2/1

Barely distinguishable from a well-known scale.

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> 2401/2400 is not between 5/3 and 2/1, gene!

OK, so be picky about it. :)

2401/1280 is, though. I intended to put up a slightly modified version of 1--9/8--5/4--4/3--3/2--5/3--15/8--2, with the 9/8 adjusted
down by 2400/2401 and the 15/8 adjusted up by the same amount. Since
h7(2401/2400)=2, this throws a spanner in the works.

Gene wrote:
>Is that what I wrote? It should be
>1/1--2700/2401--5/4-4/3--3/2--5/3--2401/1280--2/1

My routine claims this scale is epimorphic. Being
puzzled, I found out why. The val was [7, 11, 16, 19.5]
and not the [7, 11, 16, 20] which was printed out.
So, do the components need to be integer, if so, why?
You said that h7(2401/2400)=2, but 2401/2400 isn't in the
scale so this is irrelevant?

Manuel

Carl wrote:
>The stronger argument against CS /-> Epimorphic is
>that CS doesn't require JI, as Gene pointed out.

Right, I was thinking that all scales being both CS and RI
are epimorphic. Now we still need a watertight definition
of epimorphic. I must have misunderstood Pierre's definition
of it.

Manuel

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> Gene wrote:
> >Is that what I wrote? It should be
> >1/1--2700/2401--5/4-4/3--3/2--5/3--2401/1280--2/1
>
> My routine claims this scale is epimorphic. Being
> puzzled, I found out why. The val was [7, 11, 16, 19.5]
> and not the [7, 11, 16, 20] which was printed out.
> So, do the components need to be integer, if so, why?

It needs to define an epimorphic mapping to the integers; that means the components must be integers, and the gcd must be 1. The second part your routine presumably takes care of automatically, but it means that you can't multiply the above mapping by 2 and get
[14, 22, 32, 40].

> You said that h7(2401/2400)=2, but 2401/2400 isn't in the
> scale so this is irrelevant?

It's not an accident that your mapping tempers out 2401/2400, since this is what I mundged by. However, a more convincing example is one where there isn't a mapping, for instance

! nonepi.scl
!
Non epimorphic scale
7
!
2701/2400
5/4
4/3
98415/65536
5/3
2401/1280
2/1

> Manuel

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> Carl wrote:
> >The stronger argument against CS /-> Epimorphic is
> >that CS doesn't require JI, as Gene pointed out.
>
> Right, I was thinking that all scales being both CS and RI
> are epimorphic. Now we still need a watertight definition
> of epimorphic. I must have misunderstood Pierre's definition
> of it.

Joe's dictionary gives the definition I introduced:

epimorphic

A scale has the epimorphic property, or is epimorphic, if there is a val h such that if qn is the nth scale degree, then h(qn)=n. The val h is the characterizing val of the scale.

[from Gene Ward Smith, Yahoo tuning-math message 2569 (Thu Jan 10, 2002 11:11 pm)

Yes that scale is indeed not epimorphic. So I'll
add the additional integer test to the code.
So doesn't that need to be added to the definition
in Joe's dictionary too, since all components being
integer doesn't follow automatically from
h(qn)=n for n = 1 .. (number of notes - 1)?

Manuel

So I may conclude that the simplest example of a JI,
CS and non-epimorphic scale is this one: 1/1--4/1

Manuel

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:

> So doesn't that need to be added to the definition
> in Joe's dictionary too, since all components being
> integer doesn't follow automatically from
> h(qn)=n for n = 1 .. (number of notes - 1)?

It follows automatically from the definition of a val, which is also in Joe's dictionary:

/tuning-math/files/dict/val.htm

I've also called it a finite Z-linear combination of p-adic valuations defined additively, but I doubt that helps anybody.

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> So I may conclude that the simplest example of a JI,
> CS and non-epimorphic scale is this one: 1/1--4/1

If you want to call that a scale.

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> I've also called it a finite Z-linear combination of p-adic
> valuations defined additively, but I doubt that helps anybody.

If there is a bartender that pours martinis as dry as your wit, I wanna know!

Cheers,
Jon

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> > So I may conclude that the simplest example of a JI,
> > CS and non-epimorphic scale is this one: 1/1--4/1
>
> If you want to call that a scale.

i'm confused. are we assuming a 2/1 interval of equivalence? then
wouldn't be a scale at all, would it?

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
>
> > --- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> > > So I may conclude that the simplest example of a JI,
> > > CS and non-epimorphic scale is this one: 1/1--4/1
> >
> > If you want to call that a scale.
>
> i'm confused. are we assuming a 2/1 interval of equivalence? then
> wouldn't be a scale at all, would it?

I made no assumptions about intervals of equivalence, but I did assume any mapping would map every positive rational number to an integer, which v2(q)/2, where v2 is the 2-adic valuation, does not.
Other definitions are possible.

I've uploaded the new Scala version so the epimorphic code
can be tried now by others (show data).

Also the Edit->Sound settings dialog has been improved a bit.

Perhaps I should once mention another new feature which
is the automatic keyboard mapping available in the MIDI
relay dialog. You need to set "Automatic note name mapping"
to use it, and select an appropriate key for the music.
It doesn't respond to key changes, but it's not a trivial
feature either. So if you have a midi loopback or keyboard,
you can use bigger scales like 31-tET and the results will
still make sense. For ETs it's easy to make the notation
system follow the tuning if you set the "Set corresponding
notation system" tick box in File->New->Equal temperament.

Manuel