Like TOP, but with pure octaves

Paul took exception to my description of NOT tuning in this manner on
MMM; however, it seems to me it has the best claim. If you wander into
Bohlen-Pierce land, and consider only intervals which are the product
of odd primes, then applying the TOP idea to such intervals gives NOT
tuning.

On 1/1/06, Gene Ward Smith <gwsmith@svpal.org> wrote:
> Paul took exception to my description of NOT tuning in this manner on
> MMM; however, it seems to me it has the best claim. If you wander into
> Bohlen-Pierce land, and consider only intervals which are the product
> of odd primes, then applying the TOP idea to such intervals gives NOT
> tuning.

What's NOT then? Does the error obey the relation
(max(w)-min(w))/(max(w)+min(w)) where w are the weighted prime
intervals?

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> On 1/1/06, Gene Ward Smith <gwsmith@s...> wrote:
> > Paul took exception to my description of NOT tuning in this manner on
> > MMM; however, it seems to me it has the best claim. If you wander into
> > Bohlen-Pierce land, and consider only intervals which are the product
> > of odd primes, then applying the TOP idea to such intervals gives NOT
> > tuning.
>
> What's NOT then? Does the error obey the relation
> (max(w)-min(w))/(max(w)+min(w)) where w are the weighted prime
> intervals?

It works exactly like TOP, but with no 2s. In other words, you look
only at intervals which are ratios of odd integers to determine the
tuning.

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

> Paul took exception to my description of NOT tuning in this manner on
> MMM; however, it seems to me it has the best claim.

I most certainly disagree. One of the important features of TOP you
have thrown out is that TOP is still TOP even if you only look at the
intervals narrower than X (where X can be an octave or whatever). I
hope Graham chimes in too.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > Paul took exception to my description of NOT tuning in this manner on
> > MMM; however, it seems to me it has the best claim.
>
> I most certainly disagree. One of the important features of TOP you
> have thrown out is that TOP is still TOP even if you only look at the
> intervals narrower than X (where X can be an octave or whatever). I
> hope Graham chimes in too.

The same is true of NOT. You merely ignore anything other than ratios
of odd integers, and you are in business *exactly* like TOP.

Paul:
>>I most certainly disagree. One of the important features of TOP you >>have thrown out is that TOP is still TOP even if you only look at the >>intervals narrower than X (where X can be an octave or whatever). I >>hope Graham chimes in too.

Gene:
> The same is true of NOT. You merely ignore anything other than ratios
> of odd integers, and you are in business *exactly* like TOP.

In that case it's "like TOP but without octaves" not "like TOP but with pure octaves" as the subject line says. You can hide 3, or any other prime, behind the sofa and TOP will still work with what you have left. Why have a special name for throwing away the most important one?

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> In that case it's "like TOP but without octaves" not "like TOP but with
> pure octaves" as the subject line says.

That would be completely wrong, since the tuning *does* include octaves.
Youy make it sound like some kind of Bohlen-Pierce deal; it's an
entirely practical alternative.

You can hide 3, or any other
> prime, behind the sofa and TOP will still work with what you have left.
> Why have a special name for throwing away the most important one?

Because lots and lots and lots of people insist on pure octaves, and
it does make life simpler in some theoretical respects.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >
> > > Paul took exception to my description of NOT tuning in this
manner on
> > > MMM; however, it seems to me it has the best claim.
> >
> > I most certainly disagree. One of the important features of TOP
you
> > have thrown out is that TOP is still TOP even if you only look at
the
> > intervals narrower than X (where X can be an octave or whatever).
I
> > hope Graham chimes in too.
>
> The same is true of NOT.

No it isn't. Sometimes with NOT, the worst interval is, say 5:1 (and
its powers). Ignore everything wider than an octave, and 5:1 doesn't
get considered, so the tuning changes. This doesn't happen with TOP.

> You merely ignore anything other than ratios
> of odd integers, and you are in business *exactly* like TOP.

Not quite.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...>
wrote:
>
> > In that case it's "like TOP but without octaves" not "like TOP
but with
> > pure octaves" as the subject line says.
>
> That would be completely wrong, since the tuning *does* include
octaves.
> Youy make it sound like some kind of Bohlen-Pierce deal; it's an
> entirely practical alternative.
>
> You can hide 3, or any other
> > prime, behind the sofa and TOP will still work with what you have
left.
> > Why have a special name for throwing away the most important
one?
>
> Because lots and lots and lots of people insist on pure octaves, and
> it does make life simpler in some theoretical respects.

What we've been calling "Kees" is what Graham and I are advocating in
this regard, as the appropriate modification of TOP to the pure-
octaves case. Octave-equivalence is built in in a nicer, more
reasonable way, that IMO reflects more of the desirable properties
TOP had to begin with.

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

> No it isn't. Sometimes with NOT, the worst interval is, say 5:1 (and
> its powers). Ignore everything wider than an octave, and 5:1 doesn't
> get considered, so the tuning changes. This doesn't happen with TOP.

Who said you are supposed to ignore everything larger than an octave?
That would be crazy.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
>
> > No it isn't. Sometimes with NOT, the worst interval is, say 5:1
(and
> > its powers). Ignore everything wider than an octave, and 5:1
doesn't
> > get considered, so the tuning changes. This doesn't happen with TOP.
>
> Who said you are supposed to ignore everything larger than an octave?

You're obviously not reading my latest posts, or my paper, very
carefully.

> That would be crazy.

An octave, or any other interval you choose. Not crazy at all. We know
that most measures of dissonance show a lot more "flatness" (less depth
in the local minima) as one moves to intervals of wide SPAN. So the
idea is to check what happens if we reduce the weight on, or eliminate
from consideration entirely, the really wide intervals. What happens
with TOP? Nothing -- the optimal tuning stays exactly the same! This is
good and tells us we haven't erred by weighting the wide intervals too
highly. With NOT, however, one wide interval can dictate the tuning;
eliminate wide intervals from consideration, and the NOT tuning
changes. It's not stable under different possible treatments of SPAN.

There's more to it, but this is just one angle on why I don't like NOT.

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

> What we've been calling "Kees" is what Graham and I are advocating in
> this regard, as the appropriate modification of TOP to the pure-
> octaves case. Octave-equivalence is built in in a nicer, more
> reasonable way, that IMO reflects more of the desirable properties
> TOP had to begin with.

Yes, but its theoretical relationship to TOP is much more complicated,
despite how often it turns out to be simply a stretched version of
TOP. In more practical terms, NOT weights the lower limits more, which
you may or may not want.

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

> You're obviously not reading my latest posts, or my paper, very
> carefully.

I'll return the complement--your remarks on NOT seem to indicate
you've simply ignored what I've said about it.

What happens
> with TOP? Nothing -- the optimal tuning stays exactly the same!

Same with NOT.

Gene Ward Smith wrote:

> Yes, but its theoretical relationship to TOP is much more complicated,
> despite how often it turns out to be simply a stretched version of
> TOP. In more practical terms, NOT weights the lower limits more, which
> you may or may not want.

The version I now prefer is always a stretched version of TOP because it preserves the quantity

[max(w)-min(w)]/[max(w)+min(w)]

where w are the weighted primes. This is the easiest way of defining the TOP error for an equal temperament, and looks like a simple theoretical relationship to me. If it happens to give the same result as a Kees metric so much the better.

As you describe it, NOT's bogus for ratios that do involve a 2. It gives 15:8 and 5:3 equal weight. That's wrong. It's why we ditched prime-based measures in favor of odd limits in the first place. You could use it as a rough gess, but I can't see the point because max({w,1})-min({w,1}) is better, and hardly complicated (until you come to optimize it, but that's a different matter).

Graham

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
>
> > What we've been calling "Kees" is what Graham and I are advocating
in
> > this regard, as the appropriate modification of TOP to the pure-
> > octaves case. Octave-equivalence is built in in a nicer, more
> > reasonable way, that IMO reflects more of the desirable properties
> > TOP had to begin with.
>
> Yes, but its theoretical relationship to TOP is much more complicated,
> despite how often it turns out to be simply a stretched version of
> TOP.

But its theoretical grounding on its own terms is simpler than that of
NOT. There are no special constraints for "Kees" -- pure octaves fall
naturally out of the very desiderata that determine all the other
intervals.

> In more practical terms, NOT weights the lower limits more, which
> you may or may not want.

I don't quite see it that way, though I might be missing something.
Instead, I see that it seems to weight the primes more, at the expense
of other consonant intervals in the same odd limit.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
>
> > You're obviously not reading my latest posts, or my paper, very
> > carefully.
>
> I'll return the complement--your remarks on NOT seem to indicate
> you've simply ignored what I've said about it.

Really? I did my best. What did I seem to ignore?

>> What happens
> > with TOP? Nothing -- the optimal tuning stays exactly the same!
>
> Same with NOT.

Maybe I did something wrong. What's the NOT meantone tuning, and which
intervals carry the maximum damage in this tuning?

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> Gene Ward Smith wrote:
>
> > Yes, but its theoretical relationship to TOP is much more
complicated,
> > despite how often it turns out to be simply a stretched version of
> > TOP. In more practical terms, NOT weights the lower limits more,
which
> > you may or may not want.
>
> The version I now prefer is always a stretched version of TOP

Really? You don't want to ditch L_inf in favor of L_2?

> because it
> preserves the quantity
>
> [max(w)-min(w)]/[max(w)+min(w)]
>
> where w are the weighted primes.

But sometimes the tuning that acheives this is not unique. Should the
choice of a unique tuning then be the same in the "unstretched"
(tempered octaves) and "stretched" (pure octaves) cases?

> This is the easiest way of defining
> the TOP error for an equal temperament,

But what about higher-dimensional temperaments?

> and looks like a simple
> theoretical relationship to me.

But what's the justification for it in general?

> If it happens to give the same result
> as a Kees metric so much the better.
>
> As you describe it, NOT's bogus for ratios that do involve a 2. It
> gives 15:8 and 5:3 equal weight. That's wrong.

I agree.

> It's why we ditched
> prime-based measures in favor of odd limits in the first place.
You
> could use it as a rough gess, but I can't see the point because
> max({w,1})-min({w,1}) is better, and hardly complicated (until you
come
> to optimize it, but that's a different matter).
>
>
> Graham
>

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

> > I'll return the complement--your remarks on NOT seem to indicate
> > you've simply ignored what I've said about it.
>
> Really? I did my best. What did I seem to ignore?

That NOT works the same way as TOP.

> Maybe I did something wrong. What's the NOT meantone tuning, and which
> intervals carry the maximum damage in this tuning?

5-limit: 698.02
7-limit: 697.65
11-limit 31&43: 697.65
11-limit 31&50: 696.90

wallyesterpaulrus wrote:

> Really? You don't want to ditch L_inf in favor of L_2?

Yes, but then it wouldn't be like TOP, and we were talking about like TOP.

> But sometimes the tuning that acheives this is not unique. Should the > choice of a unique tuning then be the same in the "unstretched" > (tempered octaves) and "stretched" (pure octaves) cases?

It helps if they're the same, but I don't really care as long as the error's the same.

>>This is the easiest way of defining >>the TOP error for an equal temperament,
> > But what about higher-dimensional temperaments?

Too difficult by any method. But calculating it means finding the weighted primes, and then it's the same as an ET. And if it comes to a numeric algorithm it should be faster if it has to optimize for one less variable.

>>and looks like a simple >>theoretical relationship to me.
> > But what's the justification for it in general?

(max(w)-min(w))/(max(w)+min(w))

It looks like the minimax of ratios of the form p:q, where p and q are both prime, instead of p:1. Such intervals are generally more musically useful. It means a bias towards smaller intervals, like with an odd limit.

It put me in mind of a metric where, instead of removing all factors of 2, you add or remove them until the number of prime factors in the numerator and denominator match. I'm not sure if that's what's being optimized here, though. I can't see how I'd tell it to make 11:4 simpler than 11:2. Feed in 11:2 instead of 11 as the prime, maybe?

Anyway, here's the way it treats 7-prime 16-numerator limited intervals. Note the way it tends to favour small intervals.

16:15 weighted as 15:4 -
15:14 weighted as is
10:9 weighted as is
9:8 weighted as 9:4 -
8:7 weighted as 7:2 -
7:6 weighted as 7:3 -
6:5 weighted as 5:3 -
5:4 weighted as 5:2 -
4:3 weighted as 3:2 -
7:5 weighted as is
10:7 weighted as 7:5 -
3:2 weighted as is
8:5 weighted as 5:2 -
5:3 weighted as is
12:7 weighted as 7:3 -
7:4 weighted as 7:2 -
16:9 weighted as 9:4 -
9:5 weighted as 10:9 +
15:8 weighted as 15:4 -
2:1 always perfect
15:7 weighted as 15:14+
9:4 weighted as is
16:7 weighted as 7:4 -
7:3 weighted as is
12:5 weighted as 5:3 -
5:2 weighted as is
8:3 weighted as 3:2 -
14:5 weighted as 7:5 -
3:1 weighted as 3:2 +
16:5 weighted as 5:2 -
10:3 weighted as 5:3 -
7:2 weighted as is
15:4 weighted as is
4:1 always perfect
9:2 weighted as 9:4 +
14:3 weighted as 7:3 -
5:1 weighted as 5:2 +
16:3 weighted as 3:2 -
6:1 weighted as 3:2
7:1 weighted as 7:2 +
15:2 weighted as 15:4+
8:1 always perfect
9:1 weighted as 9:4 +
10:1 weighted as 5:2
12:1 weighted as 3:2
14:1 weighted as 7:2
15:1 weighted as 15:4+
16:1 always perfect

Graham

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> wrote:
>
> > > I'll return the complement--your remarks on NOT seem to indicate
> > > you've simply ignored what I've said about it.
> >
> > Really? I did my best. What did I seem to ignore?
>
> That NOT works the same way as TOP.

I'm not ignoring it, I'm questioning it (see below).

> > Maybe I did something wrong. What's the NOT meantone tuning, and
which
> > intervals carry the maximum damage in this tuning?
>
> 5-limit: 698.02
> 7-limit: 697.65
> 11-limit 31&43: 697.65
> 11-limit 31&50: 696.90

You didn't answer my final question. Would you? Let's stick to 5-
limit for that, OK?

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

> You didn't answer my final question. Would you? Let's stick to 5-
> limit for that, OK?

I presume the question was rhetorical.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
>
> > You didn't answer my final question. Would you? Let's stick to 5-
> > limit for that, OK?
>
> I presume the question was rhetorical.

As usual, when you assumed this, you assumed incorrectly. Now let me
ask you for the *third* time to answer it.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> wallyesterpaulrus wrote:
>
> > Really? You don't want to ditch L_inf in favor of L_2?
>
> Yes, but then it wouldn't be like TOP, and we were talking about
>like TOP.

It would be like TOP in certain ways, as you yourself took pains to
point out. But I thought your (snipped) statement was meant more
generally than just within some sense of "like TOP".

> > But sometimes the tuning that acheives this is not unique. Should
the
> > choice of a unique tuning then be the same in the "unstretched"
> > (tempered octaves) and "stretched" (pure octaves) cases?
>
> It helps if they're the same,

Helps? You seemed to be giving a specific proposal: stretch the TOP
tuning. But it's really not that specific . . .

> but I don't really care as long as the
> error's the same.

Hmm . . .

> >>This is the easiest way of defining
> >>the TOP error for an equal temperament,
> >
> > But what about higher-dimensional temperaments?
>
> Too difficult by any method.

Well, it's extremely simple when the codimension is 1, at least.

> But calculating it means finding the
> weighted primes, and then it's the same as an ET. And if it comes
to a
> numeric algorithm it should be faster if it has to optimize for one
less
> variable.
>
> >>and looks like a simple
> >>theoretical relationship to me.
> >
> > But what's the justification for it in general?
>
> (max(w)-min(w))/(max(w)+min(w))
>
> It looks like the minimax of ratios of the form p:q, where p and q
are
> both prime, instead of p:1. Such intervals are generally more
musically
> useful. It means a bias towards smaller intervals, like with an
odd limit.
>
> It put me in mind of a metric where, instead of removing all
factors of
> 2, you add or remove them until the number

"Weighted number", perhaps?

> of prime factors in the
> numerator and denominator match. I'm not sure if that's what's
being
> optimized here, though. I can't see how I'd tell it to make 11:4
> simpler than 11:2. Feed in 11:2 instead of 11 as the prime, maybe?
>
> Anyway, here's the way it treats 7-prime 16-numerator limited
intervals.
> Note the way it tends to favour small intervals.
>
> 16:15 weighted as 15:4 -
> 15:14 weighted as is
> 10:9 weighted as is
> 9:8 weighted as 9:4 -
> 8:7 weighted as 7:2 -
> 7:6 weighted as 7:3 -
> 6:5 weighted as 5:3 -
> 5:4 weighted as 5:2 -
> 4:3 weighted as 3:2 -
> 7:5 weighted as is
> 10:7 weighted as 7:5 -
> 3:2 weighted as is
> 8:5 weighted as 5:2 -
> 5:3 weighted as is
> 12:7 weighted as 7:3 -
> 7:4 weighted as 7:2 -
> 16:9 weighted as 9:4 -
> 9:5 weighted as 10:9 +
> 15:8 weighted as 15:4 -
> 2:1 always perfect
> 15:7 weighted as 15:14+
> 9:4 weighted as is
> 16:7 weighted as 7:4 -
> 7:3 weighted as is
> 12:5 weighted as 5:3 -
> 5:2 weighted as is
> 8:3 weighted as 3:2 -
> 14:5 weighted as 7:5 -
> 3:1 weighted as 3:2 +
> 16:5 weighted as 5:2 -
> 10:3 weighted as 5:3 -
> 7:2 weighted as is
> 15:4 weighted as is
> 4:1 always perfect
> 9:2 weighted as 9:4 +
> 14:3 weighted as 7:3 -
> 5:1 weighted as 5:2 +
> 16:3 weighted as 3:2 -
> 6:1 weighted as 3:2
> 7:1 weighted as 7:2 +
> 15:2 weighted as 15:4+
> 8:1 always perfect
> 9:1 weighted as 9:4 +
> 10:1 weighted as 5:2
> 12:1 weighted as 3:2
> 14:1 weighted as 7:2
> 15:1 weighted as 15:4+
> 16:1 always perfect
>
>
> Graham

I'm not sure I understand this table . . .

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> > wrote:
> >
> > > You didn't answer my final question. Would you? Let's stick to 5-
> > > limit for that, OK?
> >
> > I presume the question was rhetorical.
>
> As usual, when you assumed this, you assumed incorrectly. Now let me
> ask you for the *third* time to answer it.

For 5-limit meantone, everything shaper than 1/4-comma has the biggest
tuning damage on 6/5, less than that and it's on 3/2. I presume you
knew that, so I presumed the question was rhetorical.

wallyesterpaulrus wrote:

> Helps? You seemed to be giving a specific proposal: stretch the TOP > tuning. But it's really not that specific . . .

I'm proposing that any pure-octave TOP should keep the error as

(max(w)-min(w))/(max(w)+min(w))

where w are the weighted primes. The easiest way to do that is to stretch the real TOP. And the same works for other weighted-prime measures. For the RMS, the expression is

1 - <w>2/<w2>

or

std(w)/rms(w)

I suppose these are values in projective space, right?

Now to this:

>>16:15 weighted as 15:4 -
>>15:14 weighted as is
>>10:9 weighted as is
>> 9:8 weighted as 9:4 -
>> 8:7 weighted as 7:2 -
>> 7:6 weighted as 7:3 -
>> 6:5 weighted as 5:3 -
>> 5:4 weighted as 5:2 -
>> 4:3 weighted as 3:2 -
>> 7:5 weighted as is
>>10:7 weighted as 7:5 -
>> 3:2 weighted as is
>> 8:5 weighted as 5:2 -
>> 5:3 weighted as is
>>12:7 weighted as 7:3 -
>> 7:4 weighted as 7:2 -
>>16:9 weighted as 9:4 -
>> 9:5 weighted as 10:9 +
>>15:8 weighted as 15:4 -
>> 2:1 always perfect
>>15:7 weighted as 15:14+
>> 9:4 weighted as is
>>16:7 weighted as 7:4 -
>> 7:3 weighted as is
>>12:5 weighted as 5:3 -
>> 5:2 weighted as is
>> 8:3 weighted as 3:2 -
>>14:5 weighted as 7:5 -
>> 3:1 weighted as 3:2 +
>>16:5 weighted as 5:2 -
>>10:3 weighted as 5:3 -
>> 7:2 weighted as is
>>15:4 weighted as is
>> 4:1 always perfect
>> 9:2 weighted as 9:4 +
>>14:3 weighted as 7:3 -
>> 5:1 weighted as 5:2 +
>>16:3 weighted as 3:2 -
>> 6:1 weighted as 3:2
>> 7:1 weighted as 7:2 +
>>15:2 weighted as 15:4+
>> 8:1 always perfect
>> 9:1 weighted as 9:4 +
>>10:1 weighted as 5:2
>>12:1 weighted as 3:2
>>14:1 weighted as 7:2
>>15:1 weighted as 15:4+
>>16:1 always perfect

> I'm not sure I understand this table . . .

It's saying what weight an interval gets in my proposed metric. If you simply threw away the primes, 15:8 would be weighted the same as 15:1. Here, 15:1 is weighted as 15:4, but 6:5 and 5:3 are both weighted as 5:3. That means 5:3 is sill a stronger consonance than 15:8, as with the odd limit. The stretched-TOP may be a strict minimax with a metric something like this. I'm not sure.

The + and - at the end show whether an interval is better or worse using this metric compared to the octave-specific Tenney one.

Graham

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > >
> > > --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> > > wrote:
> > >
> > > > You didn't answer my final question. Would you? Let's stick
to 5-
> > > > limit for that, OK?
> > >
> > > I presume the question was rhetorical.
> >
> > As usual, when you assumed this, you assumed incorrectly. Now let
me
> > ask you for the *third* time to answer it.
>
> For 5-limit meantone, everything shaper than 1/4-comma has the
biggest
> tuning damage on 6/5, less than that and it's on 3/2.

But this assumes octave-equivalence, which isn't assumed by the TOP
error criterion that NOT uses for most intervals . . .

> I presume you
> knew that, so I presumed the question was rhetorical.

I wasn't asking this question about meantones in general but about
your 5-limit "NOT" meantone in particular. And the answer shouldn't
be the kind of octave-equivalent one that you gave above, right?

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

> But this assumes octave-equivalence, which isn't assumed by the TOP
> error criterion that NOT uses for most intervals . . .

I'm not sure what this means, but you only apply TOP methods to ratios
of odd numbers, such as 5/3.

> I wasn't asking this question about meantones in general but about
> your 5-limit "NOT" meantone in particular. And the answer shouldn't
> be the kind of octave-equivalent one that you gave above, right?

TOP is octave equivalent, so why not?

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> It's saying what weight an interval gets in my proposed metric. If
you
> simply threw away the primes, 15:8 would be weighted the same as
15:1.
> Here, 15:1 is weighted as 15:4,

How do you arrive at this? Can you flesh it out starting with your
proposed metric? I know, I'm brain-dead (that darn sushi . . .)

< but 6:5 and 5:3 are both weighted as
> 5:3. That means 5:3 is sill a stronger consonance than 15:8, as with
> the odd limit. The stretched-TOP may be a strict minimax with a
metric
> something like this. I'm not sure.
>
> The + and - at the end show whether an interval is better or worse
using
> this metriccompared to the octave-specific Tenney one.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
>
> > But this assumes octave-equivalence, which isn't assumed by the TOP
> > error criterion that NOT uses for most intervals . . .
>
> I'm not sure what this means, but you only apply TOP methods to ratios
> of odd numbers, such as 5/3.

Really? I thought that for "NOT" you were minimizing TOP damage over
*all* intervals subject to the constraint that the octaves are pure.
Many of your statements implied this interpretation.

> > I wasn't asking this question about meantones in general but about
> > your 5-limit "NOT" meantone in particular. And the answer shouldn't
> > be the kind of octave-equivalent one that you gave above, right?
>
> TOP is octave equivalent, so why not?

TOP is most certainly not octave-equivalent in any sense I can think
of. For one example of one aspect of this, note in TOP meantone, while
5:1 takes on the maximum damage, 5:4 and 8:5 don't; while 5:3 takes on
the maximum damage, 6:5 doesn't; and while 3:2 and 3:1 take on the
maximum damage, 6:1, 12:1, 24:1, and 48:1 don't. So even if my question
were intended in the general sense in which you took it (it wasn't),
your answer assumed way too much octave-equivalence.

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

> Really? I thought that for "NOT" you were minimizing TOP damage over
> *all* intervals subject to the constraint that the octaves are pure.
> Many of your statements implied this interpretation.

What I meant was that it is the exact same minimax computation, with 2
removed from the picture.

> > TOP is octave equivalent, so why not?
>
> TOP is most certainly not octave-equivalent in any sense I can think
> of.

Sorry, I meant NOT. No Octave Tempering.

At 05:57 PM 1/10/2006, you wrote:
>--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
>wrote:
>>
>> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
>> wrote:
>>
>> > But this assumes octave-equivalence, which isn't assumed by the TOP
>> > error criterion that NOT uses for most intervals . . .
>>
>> I'm not sure what this means, but you only apply TOP methods to ratios
>> of odd numbers, such as 5/3.
>
>Really? I thought that for "NOT" you were minimizing TOP damage over
>*all* intervals subject to the constraint that the octaves are pure.
>Many of your statements implied this interpretation.
>
>> > I wasn't asking this question about meantones in general but about
>> > your 5-limit "NOT" meantone in particular. And the answer shouldn't
>> > be the kind of octave-equivalent one that you gave above, right?
>>
>> TOP is octave equivalent, so why not?
>
>TOP is most certainly

Gene meant to say NOT there.

-Carl

wallyesterpaulrus wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> >>It's saying what weight an interval gets in my proposed metric. If > you >>simply threw away the primes, 15:8 would be weighted the same as > 15:1. >>Here, 15:1 is weighted as 15:4,
> > How do you arrive at this? Can you flesh it out starting with your > proposed metric? I know, I'm brain-dead (that darn sushi . . .)

It's a variant on the naive octave-equivalent prime metric Gene seems to be using. So you take an interval, throw away all factors of two and take the Tenney complexity of the resulting interval. In that case, you start with 15:8, throw away the 8 to get 15:1 and the complexity is log(15).

What I'm proposing is, instead of throwing away all twos, adjust the twos so that there are the same number of prime factors in the numberator and denominator. So you take 15:8 and factorize it

15:8 = 5*3:2*2*2

There are three factors in the numerator and only two in the denominator. The octave-equivalent value is then

15:4 = 5*2:2*2

and the complexity is log(60).

This metric shares the property of the naive prime-based one that you have a true group containing those intervals where the octave-equivalent metric agrees with the octave-specific one. But it's closer to the Kees metric because it differentiates 15:8 (complexity log(60)) from 6:5 (complexity log(15)).

Fish is good for your brain.

Graham

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> wrote:
>
> > Really? I thought that for "NOT" you were minimizing TOP damage
over
> > *all* intervals subject to the constraint that the octaves are
pure.
> > Many of your statements implied this interpretation.
>
> What I meant was that it is the exact same minimax computation,
with 2
> removed from the picture.

Perhaps the two interpretations are equivalent?

> > > TOP is octave equivalent, so why not?
> >
> > TOP is most certainly not octave-equivalent in any sense I can
think
> > of.
>
> Sorry, I meant NOT. No Octave Tempering.

OK. Now I await your answer(s) to my question(s) . . .

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> wallyesterpaulrus wrote:
> > --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...>
wrote:
> >
> >>It's saying what weight an interval gets in my proposed metric.
If
> > you
> >>simply threw away the primes, 15:8 would be weighted the same as
> > 15:1.
> >>Here, 15:1 is weighted as 15:4,
> >
> > How do you arrive at this? Can you flesh it out starting with
your
> > proposed metric? I know, I'm brain-dead (that darn sushi . . .)
>
> It's a variant on the naive octave-equivalent prime metric Gene
seems to
> be using. So you take an interval, throw away all factors of two
and
> take the Tenney complexity of the resulting interval. In that
case, you
> start with 15:8, throw away the 8 to get 15:1 and the complexity is
log(15).
>
> What I'm proposing is, instead of throwing away all twos, adjust
the
> twos so that there are the same number of prime factors in the
> numberator and denominator.

Oh, weird. Why should "number of prime factors" mean anything?

> Fish is good for your brain.

Tuna especially makes me tired. Perhaps it's the mercury.

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

> > > Really? I thought that for "NOT" you were minimizing TOP damage
> over
> > > *all* intervals subject to the constraint that the octaves are
> pure.

That doesn't make sense, though. How can you ignore octaves if you are
using them to find Tenney height with?

> > > Many of your statements implied this interpretation.

To you, maybe. I was talking about the minimax computation, which as
I've said is exactly the same, except 2 is constrained to be exact.

> > What I meant was that it is the exact same minimax computation,
> with 2
> > removed from the picture.
>
> Perhaps the two interpretations are equivalent?

What I've been saying is that doing the exact same minimax
computation, subject to the constraint octaves are pure, is the same
as doing the TOP thing only using ratios of odd integers. This should
be clear if you think about it a bit. If you get rid of 2 and live in
the Bohlen-Pierce universe, what does TOP become?

wallyesterpaulrus wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

>>What I'm proposing is, instead of throwing away all twos, adjust > the >>twos so that there are the same number of prime factors in the >>numberator and denominator.
> > Oh, weird. Why should "number of prime factors" mean anything?

Because the calculation's over all ratios where the numerator and denominator both have a single prime factor. But I'm sure there's something more subtle going on.

The reason it works is that it favours small intervals over n:1 ratios. Another metric would be to take the smallest octave equivalent value, so that the numerator and denominator are approximately the same. That's obviously going to be similar to the Kees metric.

Graham

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> wrote:
>
> > > > Really? I thought that for "NOT" you were minimizing TOP
damage
> > over
> > > > *all* intervals subject to the constraint that the octaves
are
> > pure.
>
>
> That doesn't make sense, though.

Sure it does.

> How can you ignore octaves if you are
> using them to find Tenney height with?

I don't know what you mean, "using them to find Tenney height with."
Tenney height is calculated as usual; it's just that for ratios with
no primes other than 2, it becomes irrelevant.

> > > > Many of your statements implied this interpretation.
>
> To you, maybe. I was talking about the minimax computation, which as
> I've said is exactly the same, except 2 is constrained to be exact.

Huh? This looks exactly like the intepretation I was talking about
(see above). So why do you say, "To you, maybe"?

> > > What I meant was that it is the exact same minimax computation,
> > with 2
> > > removed from the picture.
> >
> > Perhaps the two interpretations are equivalent?
>
> What I've been saying is that doing the exact same minimax
> computation, subject to the constraint octaves are pure, is the same
> as doing the TOP thing only using ratios of odd integers. This
should
> be clear if you think about it a bit.

OK, so you're saying the two interpretations are equivalent. Yet
somehow, one of the interpretations isn't even equivalent to itself
(see above).

> If you get rid of 2 and live in
> the Bohlen-Pierce universe, what does TOP become?

I'd say it remains TOP as defined for that universe ({3,5,7}).
There's no such thing as meantone in this universe, so I'm not sure
how to answer "what does TOP become".

Meanwhile, my non-rhetorical questions remain unanswered.

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

> I'd say it remains TOP as defined for that universe ({3,5,7}).
> There's no such thing as meantone in this universe, so I'm not sure
> how to answer "what does TOP become".

Well, you're right. That was a terrible way to try to explain it.

> Meanwhile, my non-rhetorical questions remain unanswered.

Which was? If you want to know how to compute NOT, you find the
minimum over the subspace of tuning maps <1 x3 x5 ... xp| which temper
out all the commas of the temperament, of the maximum of |xp/log2(p) -
1| for all the primes p. TOP is exactly the same, except now we start
from
<x2 x3 x5 ... xp| and add |x2-1| to the things being minimaxed.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> wallyesterpaulrus wrote:
> > --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...>
wrote:
>
> >>What I'm proposing is, instead of throwing away all twos, adjust
> > the
> >>twos so that there are the same number of prime factors in the
> >>numberator and denominator.
> >
> > Oh, weird. Why should "number of prime factors" mean anything?
>
> Because the calculation's over all ratios where the numerator and
> denominator both have a single prime factor.

It is? How so? And don't some of the ratios have 1 in the numerator
or denominator?

> But I'm sure there's
> something more subtle going on.

Please elaborate.

> The reason it works is that it favours small intervals over n:1
>ratios.

Many possibilities for how much to favor them seem possible.

> Another metric would be to take the smallest octave equivalent
value,
> so that the numerator and denominator are approximately the same.
> That's obviously going to be similar to the Kees metric.

Is there something less "subtle going on" in these alternatives?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
>
> > I'd say it remains TOP as defined for that universe ({3,5,7}).
> > There's no such thing as meantone in this universe, so I'm not sure
> > how to answer "what does TOP become".
>
> Well, you're right. That was a terrible way to try to explain it.
>
> > Meanwhile, my non-rhetorical questions remain unanswered.
>
> Which was?

One of them was, which set or group of intervals carry the maximum
damage in NOT meantone?

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

> One of them was, which set or group of intervals carry the maximum
> damage in NOT meantone?

Your question is not well-defined. What is the limit? What are the
intervals--a tonality diamond?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
>
> > One of them was, which set or group of intervals carry the maximum
> > damage in NOT meantone?
>
> Your question is not well-defined. What is the limit?

5-limit, as the last few times I asked.

> What are the
> intervals--a tonality diamond?

*All* intervals are to be considered -- that's why I asked "which set
or group of intervals".

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
wrote:
>
> > Your question is not well-defined. What is the limit?
>
> 5-limit, as the last few times I asked.
>
> > What are the
> > intervals--a tonality diamond?
>
> *All* intervals are to be considered -- that's why I asked "which set
> or group of intervals".

Then you need a weighting or the question doesn't make sense.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
> >
> > > Your question is not well-defined. What is the limit?
> >
> > 5-limit, as the last few times I asked.
> >
> > > What are the
> > > intervals--a tonality diamond?
> >
> > *All* intervals are to be considered -- that's why I asked "which
set
> > or group of intervals".
>
> Then you need a weighting or the question doesn't make sense.

Read the subject line. Or remember back one post that we're talking
about NOT. It seems like every time you're about to answer my question,
you snip out a part of it, and then tell me I haven't informed you of
that part!

wallyesterpaulrus wrote:

>>>Oh, weird. Why should "number of prime factors" mean anything?
>>
>>Because the calculation's over all ratios where the numerator and >>denominator both have a single prime factor.
> > It is? How so? And don't some of the ratios have 1 in the numerator > or denominator?

You have a set of Tenney-weighted approximate prime intervals, which I call w. Then you take (max(w)-min(w))/(max(w)+min(w)) as an error measure. The numerator is the highest value for the difference between any two weighted primes. The denominator is the highest weight that an interval between those two primes might have. So this is really a formula for the highest Tenney-weighted error in all intervals p:q where p and q are both within the prime limit.

>>But I'm sure there's >>something more subtle going on.
> > Please elaborate.

I don't know what metric that error implies. But it must have something to so with favoring ratios of the form p:q over p:q.

>>The reason it works is that it favours small intervals over n:1 >>ratios. > > Many possibilities for how much to favor them seem possible.

Yes, this one has the advantage that it gives identical results to TOP when you optimally temper the octaves.

>> Another metric would be to take the smallest octave equivalent > > value, > >>so that the numerator and denominator are approximately the same. >>That's obviously going to be similar to the Kees metric.
> > Is there something less "subtle going on" in these alternatives?

I don't know what's going on, but it might imply any of these metrics.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> wallyesterpaulrus wrote:
>
> >>>Oh, weird. Why should "number of prime factors" mean anything?
> >>
> >>Because the calculation's over all ratios where the numerator and
> >>denominator both have a single prime factor.
> >
> > It is? How so? And don't some of the ratios have 1 in the
numerator
> > or denominator?
>
> You have a set of Tenney-weighted approximate prime intervals,
which I
> call w. Then you take (max(w)-min(w))/(max(w)+min(w)) as an error
> measure. The numerator is the highest value for the difference
between
> any two weighted primes. The denominator is the highest weight
that an
> interval between those two primes might have. So this is really a
> formula for the highest Tenney-weighted error in all intervals p:q
where
> p and q are both within the prime limit.
>
> >>But I'm sure there's
> >>something more subtle going on.
> >
> > Please elaborate.
>
> I don't know what metric that error implies. But it must have
something
> to so with favoring ratios of the form p:q over p:q.

Huh?

> >>The reason it works is that it favours small intervals over n:1
> >>ratios.
> >
> > Many possibilities for how much to favor them seem possible.
>
> Yes, this one has the advantage that it gives identical results to
TOP
> when you optimally temper the octaves.

Really? Can you show an example?

> >> Another metric would be to take the smallest octave equivalent
> >
> > value,
> >
> >>so that the numerator and denominator are approximately the same.
> >>That's obviously going to be similar to the Kees metric.
> >
> > Is there something less "subtle going on" in these alternatives?
>
> I don't know what's going on, but it might imply any of these
metrics.

Hmm . . .

wallyesterpaulrus wrote:

>>I don't know what metric that error implies. But it must have > something >>to so with favoring ratios of the form p:q over p:q.
> > Huh?

p:q over p:1, sorry.

>>Yes, this one has the advantage that it gives identical results to > TOP >>when you optimally temper the octaves.
> > Really? Can you show an example?

Any equal temperament will do. So take 12-equal in the 7-limit. The weighted mapping is

<12/log2(2), 19/log2(3), 28/log2(5), 34/log2(7)]/12

= <1.0, 0.99897, 1.00491, 1.00925]

The highest number here is 1.00925 and the lowest is 0.99897.

(1.00925-0.99897)/(1.00925+0.99897) = 0.01028/2.00822 = 0.005119

I hope that's the TOP damage, with a rounding error in the last digit. To convert it to cents/octave, multiply it by 1200 to get 6.143.

You said you'd noticed this before when I originally mentioned it.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> wallyesterpaulrus wrote:
>
> >>I don't know what metric that error implies. But it must have
> > something
> >>to so with favoring ratios of the form p:q over p:q.
> >
> > Huh?
>
> p:q over p:1, sorry.

Aha.

> >>Yes, this one has the advantage that it gives identical results
to
> > TOP
> >>when you optimally temper the octaves.
> >
> > Really? Can you show an example?
>
> Any equal temperament will do. So take 12-equal in the 7-limit.
The
> weighted mapping is
>
> <12/log2(2), 19/log2(3), 28/log2(5), 34/log2(7)]/12
>
> = <1.0, 0.99897, 1.00491, 1.00925]
>
> The highest number here is 1.00925 and the lowest is 0.99897.
>
> (1.00925-0.99897)/(1.00925+0.99897) = 0.01028/2.00822 = 0.005119

I have no idea what this has to do with the weighting formula we were
talking about here, which you said has something to do with the
number of primes in the numerator and denominator wanting to be the
same, or whatever it is you said.

> I hope that's the TOP damage, with a rounding error in the last
digit.
> To convert it to cents/octave, multiply it by 1200 to get 6.143.

Yes, I get 6.14368381323901.

> You said you'd noticed this before when I originally mentioned it.

I thought I was the one who showed something like this to you as a
way of calculating the TOP damage for ETs, after you said it was too
hard; you then cleverly compressed the calculation into fewer steps.
But here, I thought you were talking about a different weighting
scheme, and don't see how the above illustrates any such weighting
scheme.

Me:
>>>>Yes, this one has the advantage that it gives identical results > to >>>TOP >>>
>>>>when you optimally temper the octaves.

Paul:
>>>Really? Can you show an example?
<snip>
> I have no idea what this has to do with the weighting formula we were > talking about here, which you said has something to do with the > number of primes in the numerator and denominator wanting to be the > same, or whatever it is you said.

Yes, so probably an example wasn't what you wanted. I was talking about two formulae on the weighted primes, w. On is the max error of the primes

max(|1-w|)

the other is

(max(w)-min(w))/(max(w)+min(w))

Both of them give the TOP error when w is for the optimal tuning. The second one gives the TOP error regardless of the octave stretch applied to w.

The point about that second formula is that the number of intervals is balanced on the top and bottom. So it directly calculates the worst error of all intervals of the form p:q where p and q are prime, whereas the first formula is the worst error of intervals p:1. All this assuming Tenney weighting.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Me:
> >>>>Yes, this one has the advantage that it gives identical results
> > to
> >>>TOP
> >>>
> >>>>when you optimally temper the octaves.
>
> Paul:
> >>>Really? Can you show an example?
> <snip>
> > I have no idea what this has to do with the weighting formula we
were
> > talking about here, which you said has something to do with the
> > number of primes in the numerator and denominator wanting to be
the
> > same, or whatever it is you said.
>
> Yes, so probably an example wasn't what you wanted.

:)
How about a demonstration?

> I was talking about
> two formulae on the weighted primes, w. On is the max error of the
primes
>
> max(|1-w|)
>
> the other is
>
> (max(w)-min(w))/(max(w)+min(w))
>
> Both of them give the TOP error when w is for the optimal tuning.
The
> second one gives the TOP error regardless of the octave stretch
applied
> to w.
>
> The point about that second formula is that the number of intervals
is
> balanced on the top and bottom. So it directly calculates the
worst
> error of all intervals of the form p:q where p and q are prime,
whereas
> the first formula is the worst error of intervals p:1. All this
> assuming Tenney weighting.

I'm still not seeing how this supports what you said about another
weird weighting scheme where the most relevant intervals, or whatever
you said, are the ones with the same number of prime factors in the
numerator and denominator.