What exactly is TOP?

Hi!

This TOP business sounds really exciting for me mainly because we are
at last tempering the octaves too! But I'm lost in the archives of
tuning and tuning-math and can't quite figure out what it exactly is.
Please help!

I understand that it is some kind of minimax optimization over
intervals divided by log(n*d) but what exactly are those intervals?

And how is the calculation done?

Kalle

>Hi!
>
>This TOP business sounds really exciting for me mainly because we are
>at last tempering the octaves too! But I'm lost in the archives of
>tuning and tuning-math and can't quite figure out what it exactly is.
>Please help!
>
>I understand that it is some kind of minimax optimization over
>intervals divided by log(n*d) but what exactly are those intervals?
>
>And how is the calculation done?

Hi Kalle! Good to hear from you.

Try this thread...

/tuning-math/message/8387

-Carl

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

> Try this thread...
>
> /tuning-math/message/8387

I'm hoping to have formulas going directly from the wedgie to the
tuning some time.

Hi, TOPpers!

I still don't quite get this TOP business. I would be very pleased if
someone would write some kind of summary for dummies like me!

I understand what Tenney harmonic distance is and that it is related
to harmonic entropy. BUT:

What is this talk about all intervals?
Can TOP be used in all cases: linear, planar and equal temperaments?
How?
What is it good for?
How and in what way does it minimize the errors in (all?) intervals?

Would it be good for me to tune decatonic scales or 22-equal
differently? How to calculate these? And by the way, doesn't Pajara
have a half-octave period?

I know, too many questions. :)

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> Hi, TOPpers!
>
> I still don't quite get this TOP business. I would be very pleased
if
> someone would write some kind of summary for dummies like me!

Eventually . . . there are still some things that might be
simplifyable . . . I'd love to explain more but I must run off
now . . .

> I understand what Tenney harmonic distance is and that it is
related
> to harmonic entropy. BUT:
>
> What is this talk about all intervals?

All intervals that occur in the infinite just lattice you're
tempering. In the 5-limit, it's 3-dimensional. If you're tempering
out a comma, it will only affect the primes that occur in the comma,
so if you know the TOP tuning for a given comma being tempered out,
it's trivial to extend the result to a just lattice of any prime
limit.

> Can TOP be used in all cases: linear, planar and equal
>temperaments?

Yes but so far I've been able to figure out easy methods for only the
equal case and the co-dimension 1 case (that is, where only one comma
is tempered out).

> How?

If you're tempering one comma out, then for each prime p that occurs
in the comma n/d,

if p occurs in the numerator, then it gets approximated as (in cents)

cents(p) - cents(n/d)*log(p)/log(n*d)

if p occurs in the denominator, then it gets approximated as (in
cents)

cents(p) + cents(n/d)*log(p)/log(n*d)

> What is it good for?

Everything :)

> How and in what way does it minimize the errors in (all?)
>intervals?

It minimizes the maximum *weighted* error, meaning the error divided
by log(n*d), over all ratios n/d in the original just lattice.

> Would it be good for me to tune decatonic scales or 22-equal
> differently?

Yes.

> How to calculate these?

Gene did so on tuning-math.

> And by the way, doesn't Pajara
> have a half-octave period?

Yes, it does. Whether it's TOP or not. Though in TOP, it's not
precisely 600 cents.

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

> I still don't quite get this TOP business. I would be very pleased
if
> someone would write some kind of summary for dummies like me!

I'm not good at that, but will try to say something useful.

> Can TOP be used in all cases: linear, planar and equal
temperaments?

Yes.

> How?

The heavy-duty mathematical explanation can be found here:

http://66.98.148.43/~xenharmo/top.htm

So far this hasn't been translated into English; the gist is that
just intonation can be identified with a point in a certain space,
and temperaments with a subpsace. TOP tuning is the point in the
subspace closest to the just intonation point (JIP.)

> What is it good for?

We are still finding that out. One thing it is good for is giving us
a theoretically justifified canonical tuning for every temperament,
but that is just theory. My rendition of a Haydn string quartet in
TOP can be found here, so you can form your own judgment:

http://66.98.148.43/~xenharmo/meantop.htm

So far I haven't heard any comments, possibly because people are
afraid of getting caught in the crossfire if they say anything. I've
also finished Act I of Rigoletto, and used TOP where some high
sceechy notes I wanted to make screech even more, and then resolve to
something very smooth and quiet. Act I has various interesting tunings
(meantone top, Marvel, extended meantone, meaning more than 12 notes
to the octave of meantone) but it also has a lot of things to offend
the purists, even excluding the tuning. Given the atmosphere here, I
am hesitating about putting it up, but probably will. If I do, I
would ask (as I would ask about any music I put up, whether an
original composition or an arragement) that any critical comments not
be personal comments, but concern themselves solely with the music.

> How and in what way does it minimize the errors in (all?)
intervals?

It minimizes a certain kind of weighted error; the math which shows
it can do that is on my web page.

> Would it be good for me to tune decatonic scales or 22-equal
> differently?

Try it and see if you like it.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> > I understand what Tenney harmonic distance is and that it is
> related
> > to harmonic entropy. BUT:
> >
> > What is this talk about all intervals?
>
> All intervals that occur in the infinite just lattice you're
> tempering.

I knew it!

> In the 5-limit, it's 3-dimensional. If you're tempering
> out a comma, it will only affect the primes that occur in the
comma,

Aha!

> so if you know the TOP tuning for a given comma being tempered out,
> it's trivial to extend the result to a just lattice of any prime
> limit.

But what if you introduce more commas?

> > Can TOP be used in all cases: linear, planar and equal
> >temperaments?
>
> Yes but so far I've been able to figure out easy methods for only
the
> equal case and the co-dimension 1 case (that is, where only one
comma
> is tempered out).

Ok. Co-dimension 1 case is linear temperament, right? What's co-
dimension?

> > How?
>
> If you're tempering one comma out, then for each prime p that
occurs
> in the comma n/d,
>
> if p occurs in the numerator, then it gets approximated as (in
cents)
>
> cents(p) - cents(n/d)*log(p)/log(n*d)
>
> if p occurs in the denominator, then it gets approximated as (in
> cents)
>
> cents(p) + cents(n/d)*log(p)/log(n*d)

Thanks for this!

> > What is it good for?
>
> Everything :)

Great!

> > How and in what way does it minimize the errors in (all?)
> >intervals?
>
> It minimizes the maximum *weighted* error, meaning the error
divided
> by log(n*d), over all ratios n/d in the original just lattice.

Wow! That's pretty awesome!

> > Would it be good for me to tune decatonic scales or 22-equal
> > differently?
>
> Yes.

Cool! I'll try that.

> > How to calculate these?
>
> Gene did so on tuning-math.

Right. I found Pajara but where's 22-equal? Is that Diminished
Seventh-temperament the one I was talking about in my
dorian/mixolydian thread?

> > And by the way, doesn't Pajara
> > have a half-octave period?
>
> Yes, it does. Whether it's TOP or not. Though in TOP, it's not
> precisely 600 cents.

So I have to divide by 2 the value Gene gave for 2:1 to get the
period, right?

Kalle

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
>
> > I still don't quite get this TOP business. I would be very
pleased
> if
> > someone would write some kind of summary for dummies like me!
>
> I'm not good at that, but will try to say something useful.

:)

> > Can TOP be used in all cases: linear, planar and equal
> temperaments?
>
> Yes.
>
> > How?
>
> The heavy-duty mathematical explanation can be found here:
>
> http://66.98.148.43/~xenharmo/top.htm

Yep. I had a course in linear algebra but I've already forgotten
almost everything. :)

> So far this hasn't been translated into English; the gist is that
> just intonation can be identified with a point in a certain space,
> and temperaments with a subpsace. TOP tuning is the point in the
> subspace closest to the just intonation point (JIP.)

So every point in a subspace is a different tuning? What's the metric
of this space, Tenney harmonic distance?

> > What is it good for?
>
> We are still finding that out. One thing it is good for is giving
us
> a theoretically justifified canonical tuning for every temperament,

That's a good thing in my opinion.

> but that is just theory. My rendition of a Haydn string quartet in
> TOP can be found here, so you can form your own judgment:
>
> http://66.98.148.43/~xenharmo/meantop.htm

I'm listening to the third movement right now. That one sounds quite
good to me though I'm hearing some weird noises in some of the
attacks of the samples. I have to listen to the other ones too.

> So far I haven't heard any comments, possibly because people are
> afraid of getting caught in the crossfire if they say anything.

Really? I'm not afraid of that.

I've
> also finished Act I of Rigoletto, and used TOP where some high
> sceechy notes I wanted to make screech even more, and then resolve
to
> something very smooth and quiet. Act I has various interesting
tunings
> (meantone top, Marvel, extended meantone, meaning more than 12
notes
> to the octave of meantone) but it also has a lot of things to
offend
> the purists, even excluding the tuning. Given the atmosphere here,
I
> am hesitating about putting it up, but probably will. If I do, I
> would ask (as I would ask about any music I put up, whether an
> original composition or an arragement) that any critical comments
not
> be personal comments, but concern themselves solely with the music.

Fair enough.

> > How and in what way does it minimize the errors in (all?)
> intervals?
>
> It minimizes a certain kind of weighted error; the math which shows
> it can do that is on my web page.

> > Would it be good for me to tune decatonic scales or 22-equal
> > differently?
>
> Try it and see if you like it.

I'll do that.

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

> So every point in a subspace is a different tuning?

Past a certain point it becomes absurd to call it a tuning, but if
you ignore that little bit of reality, yes.

What's the metric
> of this space, Tenney harmonic distance?

It's the dual of Tenney harmonic distance.

|| <v2 v3 v5 ... vp| || = Max(|v2/log2(2)|, |v3/log2(3)|, ...,
|vp/log2(p)|)

> > http://66.98.148.43/~xenharmo/meantop.htm
>
> I'm listening to the third movement right now. That one sounds
quite
> good to me though I'm hearing some weird noises in some of the
> attacks of the samples. I have to listen to the other ones too.

Soundfonts always seem a little flakey one way or another. I thought
of using the Cadenza Strings font on this movement and perhaps I
should have.

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
> > --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> > > I understand what Tenney harmonic distance is and that it is
> > related
> > > to harmonic entropy. BUT:
> > >
> > > What is this talk about all intervals?
> >
> > All intervals that occur in the infinite just lattice you're
> > tempering.
>
> I knew it!
>
> > In the 5-limit, it's 3-dimensional. If you're tempering
> > out a comma, it will only affect the primes that occur in the
> comma,
>
> Aha!
>
> > so if you know the TOP tuning for a given comma being tempered
out,
> > it's trivial to extend the result to a just lattice of any prime
> > limit.
>
> But what if you introduce more commas?

Then things get a bit messier . . . but there's still a solution
(sometimes there'll be a range of possible values for one of the
primes, if that prime never appears in any of the intervals that hit
the maximum).

> > > Can TOP be used in all cases: linear, planar and equal
> > >temperaments?
> >
> > Yes but so far I've been able to figure out easy methods for only
> the
> > equal case and the co-dimension 1 case (that is, where only one
> comma
> > is tempered out).
>
> Ok. Co-dimension 1 case is linear temperament, right?

Only in 5-limit. In 7-limit, it's planar temperament, etc.

> What's co-
> dimension?

The dimension of the lattice of tempered-out commas.

> > > Would it be good for me to tune decatonic scales or 22-equal
> > > differently?
> >
> > Yes.
>
> Cool! I'll try that.
>
> > > How to calculate these?
> >
> > Gene did so on tuning-math.
>
> Right. I found Pajara but where's 22-equal?

Let's see if Carl or someone can beat me to this.

> Is that Diminished
> Seventh-temperament the one I was talking about in my
> dorian/mixolydian thread?

Yes.

> > > And by the way, doesn't Pajara
> > > have a half-octave period?
> >
> > Yes, it does. Whether it's TOP or not. Though in TOP, it's not
> > precisely 600 cents.
>
> So I have to divide by 2 the value Gene gave for 2:1 to get the
> period, right?

Yes.