loglog!

/tuning-math/files/Paul/et5loglog.gif

Ok, easy! No moat needed, at least for ETs. Just draw a
circle around the origin and grow the radius until it would
include something that exceeds a single bound -- a "TOP
notes per 1200 cents" bound. For ETs at least. Choose a
bound according to sensibilities in the 5-limit, round it
to the nearest ten, and use it for all limits.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> /tuning-math/files/Paul/et5loglog.gif
>
> Ok, easy! No moat needed, at least for ETs. Just draw a
> circle around the origin and grow the radius until it would
> include something that exceeds a single bound -- a "TOP
> notes per 1200 cents" bound. For ETs at least. Choose a
> bound according to sensibilities in the 5-limit, round it
> to the nearest ten, and use it for all limits.

That's great, Carl, but in loglog land the origin is arbitrary.

>> /tuning-math/files/Paul/et5loglog.gif
>>
>> Ok, easy! No moat needed, at least for ETs. Just draw a
>> circle around the origin and grow the radius until it would
>> include something that exceeds a single bound -- a "TOP
>> notes per 1200 cents" bound. For ETs at least. Choose a
>> bound according to sensibilities in the 5-limit, round it
>> to the nearest ten, and use it for all limits.
>
>That's great, Carl, but in loglog land the origin is arbitrary.

Lessee,

log(TOP notes per 1200 cents)
log2(2)= 1
log2(1)= 0
stop!

...and if you take 1 cent as being the Reinhard cutoff, you get
another zero for log(error). Poof! Instant origin.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> /tuning-math/files/Paul/et5loglog.gif
>
> Ok, easy! No moat needed, at least for ETs. Just draw a
> circle around the origin and grow the radius until it would
> include something that exceeds a single bound -- a "TOP
> notes per 1200 cents" bound. For ETs at least. Choose a
> bound according to sensibilities in the 5-limit, round it
> to the nearest ten, and use it for all limits.
>
> -Carl

Something like that may be worth looking at, except for the fact that
the origin (as in the point 0,0) does not appear anywhere on a log log
graph, as Paul has been at pains to point out several times over the
past few days.

But I agree that one could use a circle to take the bite out of the
side that I talked about elsewhere. At least we agree on that general
shape!

One can freely choose not only the radius, but also the center point.
And it would still be wise to try to fit it into a moat, for reasons
already given.

Whether the same circle can be generalised in such a simple way to
other limits remains to be seen, but seems unlikely.

And it still seems simpler to me to draw a straight line on a linear
plot than a circle on a log plot. And the linear axes themselves are
closer to representing "pain" than log ones.

>One can freely choose not only the radius, but also the center point.
>And it would still be wise to try to fit it into a moat, for reasons
>already given.
>
>Whether the same circle can be generalised in such a simple way to
>other limits remains to be seen, but seems unlikely.

Why do you say that?

>And it still seems simpler to me to draw a straight line on a linear
>plot than a circle on a log plot. And the linear axes themselves are
>closer to representing "pain" than log ones.

Didn't Gen ask for a linear plot?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> /tuning-math/files/Paul/et5loglog.gif
>
> Ok, easy! No moat needed, at least for ETs. Just draw a
> circle around the origin

Where's the origin, Carl? I don't see it.

> and grow the radius until it would
> include something that exceeds a single bound -- a "TOP
> notes per 1200 cents" bound.

I'm not following.

> For ETs at least. Choose a
> bound according to sensibilities in the 5-limit, round it
> to the nearest ten, and use it for all limits.

The complexity measures cannot be compared across different
dimensionalities, any more than lengths can be compared with areas
can be compared with volumes.

>> For ETs at least. Choose a
>> bound according to sensibilities in the 5-limit, round it
>> to the nearest ten, and use it for all limits.
>
>The complexity measures cannot be compared across different
>dimensionalities, any more than lengths can be compared with areas
>can be compared with volumes.

Not if it's number of notes, I guess. I've suggested in the
past adjusting for it, crudely, by dividing by pi(lim). But
I think by fiddling with the variables in my scheme procedure
one could do better.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> For ETs at least. Choose a
> >> bound according to sensibilities in the 5-limit, round it
> >> to the nearest ten, and use it for all limits.
> >
> >The complexity measures cannot be compared across different
> >dimensionalities, any more than lengths can be compared with areas
> >can be compared with volumes.
>
> Not if it's number of notes, I guess.

What's number of notes??

> I've suggested in the
> past adjusting for it, crudely, by dividing by pi(lim).

Huh? What's that?

>> >> For ETs at least. Choose a
>> >> bound according to sensibilities in the 5-limit, round it
>> >> to the nearest ten, and use it for all limits.
>> >
>> >The complexity measures cannot be compared across different
>> >dimensionalities, any more than lengths can be compared with areas
>> >can be compared with volumes.
>>
>> Not if it's number of notes, I guess.
>
>What's number of notes??

Complexity units.

>> I've suggested in the
>> past adjusting for it, crudely, by dividing by pi(lim).
>
>Huh? What's that?

If we're counting dyads, I suppose higher limits ought to do
better with constant notes. If we're counting complete chords,
they ought to do worse. Yes/no?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> For ETs at least. Choose a
> >> >> bound according to sensibilities in the 5-limit, round it
> >> >> to the nearest ten, and use it for all limits.
> >> >
> >> >The complexity measures cannot be compared across different
> >> >dimensionalities, any more than lengths can be compared with
areas
> >> >can be compared with volumes.
> >>
> >> Not if it's number of notes, I guess.
> >
> >What's number of notes??
>
> Complexity units.

It's only that (or very nearly that) in the ET cases. So it the below
still relevant?

> >> I've suggested in the
> >> past adjusting for it, crudely, by dividing by pi(lim).
> >
> >Huh? What's that?
>
> If we're counting dyads, I suppose higher limits ought to do
> better with constant notes.
> If we're counting complete chords,
> they ought to do worse. Yes/no?

>> >> >The complexity measures cannot be compared across different
>> >> >dimensionalities, any more than lengths can be compared with
>> >> >areas can be compared with volumes.
>> >>
>> >> Not if it's number of notes, I guess.
>> >
>> >What's number of notes??
>>
>> Complexity units.
>
>It's only that (or very nearly that) in the ET cases.

Your creepy complexity is giving notes, clearly.

>So it the below
>still relevant?

Yes! It's a fundamental question about how to view complexity.
I'd be most interested in your answer.

>> >> I've suggested in the
>> >> past adjusting for it, crudely, by dividing by pi(lim).
>> >
>> >Huh? What's that?
>>
>> If we're counting dyads, I suppose higher limits ought to do
>> better with constant notes.
>> If we're counting complete chords,
>> they ought to do worse. Yes/no?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >The complexity measures cannot be compared across different
> >> >> >dimensionalities, any more than lengths can be compared with
> >> >> >areas can be compared with volumes.
> >> >>
> >> >> Not if it's number of notes, I guess.
> >> >
> >> >What's number of notes??
> >>
> >> Complexity units.
> >
> >It's only that (or very nearly that) in the ET cases.
>
> Your creepy complexity is giving notes, clearly.

Hmm . . .And what do you propose to use for the 5-limit linear and 7-
limit planar cases?

> >So it the below
> >still relevant?
>
> Yes! It's a fundamental question about how to view complexity.
> I'd be most interested in your answer.

Again, I view complexity as a measure of length, area, volume . . .
in the Tenney lattice with taxicab metric. We're measuring the size
of the finite dimensions of the periodicity slice, periodicity tube,
periodicity block . . .

> >> >> I've suggested in the
> >> >> past adjusting for it, crudely, by dividing by pi(lim).
> >> >
> >> >Huh? What's that?
> >>
> >> If we're counting dyads, I suppose higher limits ought to do
> >> better with constant notes.
> >> If we're counting complete chords,
> >> they ought to do worse. Yes/no?

Still have no idea how to approach this questioning, or what the
thinking behind it is . . .

>> >> >> >The complexity measures cannot be compared across different
>> >> >> >dimensionalities, any more than lengths can be compared with
>> >> >> >areas can be compared with volumes.
>> >> >>
>> >> >> Not if it's number of notes, I guess.
>> >> >
>> >> >What's number of notes??
>> >>
>> >> Complexity units.
>> >
>> >It's only that (or very nearly that) in the ET cases.
>>
>> Your creepy complexity is giving notes, clearly.
>
>Hmm . . .And what do you propose to use for the 5-limit linear and
>7-limit planar cases?

First of all, what is the real name for creepy complexity? L1?

Second of all, since I do not know how to calculate it (as far as
I can tell the details are lost in a myriad of messages between
you and Gene), I was unaware it was undefined for these cases.

>> >So it the below
>> >still relevant?
>>
>> Yes! It's a fundamental question about how to view complexity.
>> I'd be most interested in your answer.
>
>Again, I view complexity as a measure of length, area, volume . . .
>in the Tenney lattice with taxicab metric. We're measuring the size
>of the finite dimensions of the periodicity slice, periodicity tube,
>periodicity block . . .

The units in all cases should be notes. We just need a way to
compare a slice with a tube, etc.

>> >> >> I've suggested in the
>> >> >> past adjusting for it, crudely, by dividing by pi(lim).
>> >> >
>> >> >Huh? What's that?
>> >>
>> >> If we're counting dyads, I suppose higher limits ought to do
>> >> better with constant notes.
>> >> If we're counting complete chords,
>> >> they ought to do worse. Yes/no?
>
>Still have no idea how to approach this questioning, or what the
>thinking behind it is . . .

Think scales. What relations, if any, do we expect, for n
notes, as lim goes up:

() More dyads or fewer dyads?
() More complete chords or less complete chords?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >> >The complexity measures cannot be compared across
different
> >> >> >> >dimensionalities, any more than lengths can be compared
with
> >> >> >> >areas can be compared with volumes.
> >> >> >>
> >> >> >> Not if it's number of notes, I guess.
> >> >> >
> >> >> >What's number of notes??
> >> >>
> >> >> Complexity units.
> >> >
> >> >It's only that (or very nearly that) in the ET cases.
> >>
> >> Your creepy complexity is giving notes, clearly.
> >
> >Hmm . . .And what do you propose to use for the 5-limit linear and
> >7-limit planar cases?
>
> First of all, what is the real name for creepy complexity? L1?

Yes, it's the L1 norm of the *monzo-wedgie* in the Tenney lattice. In
other words, it's the 'taxicab area' of the (nontorsional) vanishing
bivector, something which seems to give three times the number of
notes in the 5-limit ET case.

> Second of all, since I do not know how to calculate it (as far as
> I can tell the details are lost in a myriad of messages between
> you and Gene), I was unaware it was undefined for these cases.

Well, I tried applying it to 5-limit linear, and posted a few
results, but even fewer looked 'creepy', unfortunately.

> >> >So it the below
> >> >still relevant?
> >>
> >> Yes! It's a fundamental question about how to view complexity.
> >> I'd be most interested in your answer.
> >
> >Again, I view complexity as a measure of length, area,
volume . . .
> >in the Tenney lattice with taxicab metric. We're measuring the
size
> >of the finite dimensions of the periodicity slice, periodicity
tube,
> >periodicity block . . .
>
> The units in all cases should be notes.

I disagree, since I feel the Tenney lattice is much more appropriate
than the symmetrical cubic lattice.

> We just need a way to
> compare a slice with a tube, etc.

Doesn't seem possible to do this in a fair way.

> >> >> >> I've suggested in the
> >> >> >> past adjusting for it, crudely, by dividing by pi(lim).
> >> >> >
> >> >> >Huh? What's that?
> >> >>
> >> >> If we're counting dyads, I suppose higher limits ought to do
> >> >> better with constant notes.
> >> >> If we're counting complete chords,
> >> >> they ought to do worse. Yes/no?
> >
> >Still have no idea how to approach this questioning, or what the
> >thinking behind it is . . .
>
> Think scales.

Well that's different. What kind of scales? ET? DE? JI? Other?

> What relations, if any, do we expect, for n
> notes, as lim goes up:

For a given scale? Then this is even more different . . .

> () More dyads or fewer dyads?

Certainly not fewer.

> () More complete chords or less complete chords?

Certainly not more.

>> First of all, what is the real name for creepy complexity? L1?
>
>Yes, it's the L1 norm of the *monzo-wedgie* in the Tenney lattice. In
>other words, it's the 'taxicab area' of the (nontorsional) vanishing
>bivector, something which seems to give three times the number of
>notes in the 5-limit ET case.

Ok.

>> >Again, I view complexity as a measure of length, area,
>> >volume . . . in the Tenney lattice with taxicab metric. We're
>> >measuring the size of the finite dimensions of the periodicity
>> >slice, periodicity tube, periodicity block . . .
>>
>> The units in all cases should be notes.
>
>I disagree, since I feel the Tenney lattice is much more appropriate
>than the symmetrical cubic lattice.

Why would that make any difference?

>> We just need a way to
>> compare a slice with a tube, etc.
>
>Doesn't seem possible to do this in a fair way.

Well, sure, but after a coupla margaritas... :)

>> >> >> >> I've suggested in the
>> >> >> >> past adjusting for it, crudely, by dividing by pi(lim).
>> >> >> >
>> >> >> >Huh? What's that?
>> >> >>
>> >> >> If we're counting dyads, I suppose higher limits ought to do
>> >> >> better with constant notes.
>> >> >> If we're counting complete chords,
>> >> >> they ought to do worse. Yes/no?
>> >
>> >Still have no idea how to approach this questioning, or what the
>> >thinking behind it is . . .
>>
>> Think scales.
>
>Well that's different. What kind of scales? ET? DE? JI? Other?

Any scale that is a manifestation of the given temperament.

>> What relations, if any, do we expect, for n
>> notes, as lim goes up:
>
>For a given scale? Then this is even more different . . .

Ultimately if we can't show a relation to notes in scales
we've gone off the deep end.

With so-called Graham complexity, as you grow the scale
you get nothing, nothing, nothing, then boom, 2 complete
chords. Then every note you add after than gives you
another pair of chords. Thus, Graham complexity seems
important.

Likewise, Herman was onto something with his consintency
range thing.

>> () More dyads or fewer dyads?
>
>Certainly not fewer.
>
>> () More complete chords or less complete chords?
>
>Certainly not more.

Thank you.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> First of all, what is the real name for creepy complexity? L1?
> >
> >Yes, it's the L1 norm of the *monzo-wedgie* in the Tenney lattice.
In
> >other words, it's the 'taxicab area' of the (nontorsional)
vanishing
> >bivector, something which seems to give three times the number of
> >notes in the 5-limit ET case.
>
> Ok.
>
> >> >Again, I view complexity as a measure of length, area,
> >> >volume . . . in the Tenney lattice with taxicab metric. We're
> >> >measuring the size of the finite dimensions of the periodicity
> >> >slice, periodicity tube, periodicity block . . .
> >>
> >> The units in all cases should be notes.
> >
> >I disagree, since I feel the Tenney lattice is much more
appropriate
> >than the symmetrical cubic lattice.
>
> Why would that make any difference?

Tenney makes the lower primes simpler than the higher primes.

> >> >> >> >> I've suggested in the
> >> >> >> >> past adjusting for it, crudely, by dividing by pi(lim).
> >> >> >> >
> >> >> >> >Huh? What's that?
> >> >> >>
> >> >> >> If we're counting dyads, I suppose higher limits ought to
do
> >> >> >> better with constant notes.
> >> >> >> If we're counting complete chords,
> >> >> >> they ought to do worse. Yes/no?
> >> >
> >> >Still have no idea how to approach this questioning, or what
the
> >> >thinking behind it is . . .
> >>
> >> Think scales.
> >
> >Well that's different. What kind of scales? ET? DE? JI? Other?
>
> Any scale that is a manifestation of the given temperament.
>
> >> What relations, if any, do we expect, for n
> >> notes, as lim goes up:
> >
> >For a given scale? Then this is even more different . . .
>
> Ultimately if we can't show a relation to notes in scales
> we've gone off the deep end.

That's a separate issue. I mean are you comparing a given scale
across *different* temperaments? It seems like you are, but above you
say "Any scale that is a manifestation of the given temperament", so
I'm not sure what you're getting from what.

>> >> >Again, I view complexity as a measure of length, area,
>> >> >volume . . . in the Tenney lattice with taxicab metric. We're
>> >> >measuring the size of the finite dimensions of the periodicity
>> >> >slice, periodicity tube, periodicity block . . .
>> >>
>> >> The units in all cases should be notes.
>> >
>> >I disagree, since I feel the Tenney lattice is much more
>> >appropriate than the symmetrical cubic lattice.
>>
>> Why would that make any difference?
>
>Tenney makes the lower primes simpler than the higher primes.

You're still enclosing notes.

>> >> >> >> >> I've suggested in the
>> >> >> >> >> past adjusting for it, crudely, by dividing by pi(lim).
>> >> >> >> >
>> >> >> >> >Huh? What's that?
>> >> >> >>
>> >> >> >> If we're counting dyads, I suppose higher limits ought to
>do
>> >> >> >> better with constant notes.
>> >> >> >> If we're counting complete chords,
>> >> >> >> they ought to do worse. Yes/no?
>> >> >
>> >> >Still have no idea how to approach this questioning, or what
>the
>> >> >thinking behind it is . . .
>> >>
>> >> Think scales.
>> >
>> >Well that's different. What kind of scales? ET? DE? JI? Other?
>>
>> Any scale that is a manifestation of the given temperament.
>>
>> >> What relations, if any, do we expect, for n
>> >> notes, as lim goes up:
>> >
>> >For a given scale? Then this is even more different . . .
>>
>> Ultimately if we can't show a relation to notes in scales
>> we've gone off the deep end.
>
>That's a separate issue. I mean are you comparing a given scale
>across *different* temperaments?

Eventually I hope to characterize temperaments by the kinds of
scales they can manifest. But before that, things like Graham
complexity and consistency range need to be cultured to fix
a relationship between a single temperament and its scales.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >Again, I view complexity as a measure of length, area,
> >> >> >volume . . . in the Tenney lattice with taxicab metric. We're
> >> >> >measuring the size of the finite dimensions of the
periodicity
> >> >> >slice, periodicity tube, periodicity block . . .
> >> >>
> >> >> The units in all cases should be notes.
> >> >
> >> >I disagree, since I feel the Tenney lattice is much more
> >> >appropriate than the symmetrical cubic lattice.
> >>
> >> Why would that make any difference?
> >
> >Tenney makes the lower primes simpler than the higher primes.
>
> You're still enclosing notes.

An infinite number of them, except for the ET case, where (and only
where) cubic and Tenney will agree -- the cells are shaped
differently, but there are always the same number of them.

> >> >> >> >> >> I've suggested in the
> >> >> >> >> >> past adjusting for it, crudely, by dividing by pi
(lim).
> >> >> >> >> >
> >> >> >> >> >Huh? What's that?
> >> >> >> >>
> >> >> >> >> If we're counting dyads, I suppose higher limits ought
to
> >do
> >> >> >> >> better with constant notes.
> >> >> >> >> If we're counting complete chords,
> >> >> >> >> they ought to do worse. Yes/no?
> >> >> >
> >> >> >Still have no idea how to approach this questioning, or what
> >the
> >> >> >thinking behind it is . . .
> >> >>
> >> >> Think scales.
> >> >
> >> >Well that's different. What kind of scales? ET? DE? JI? Other?
> >>
> >> Any scale that is a manifestation of the given temperament.
> >>
> >> >> What relations, if any, do we expect, for n
> >> >> notes, as lim goes up:
> >> >
> >> >For a given scale? Then this is even more different . . .
> >>
> >> Ultimately if we can't show a relation to notes in scales
> >> we've gone off the deep end.
> >
> >That's a separate issue. I mean are you comparing a given scale
> >across *different* temperaments?
>
> Eventually I hope to characterize temperaments by the kinds of
> scales they can manifest. But before that, things like Graham
> complexity and consistency range need to be cultured to fix
> a relationship between a single temperament and its scales.

You're only talking about *linear* temperaments here, right?

>>>>>>>We're measuring the size of the finite dimensions of the
>>>>>>>periodicity slice, periodicity tube, periodicity block . . .
>>>>>>
>>>>>>The units in all cases should be notes.
>>>>>
>>>>>I disagree, since I feel the Tenney lattice is much more
>>>>>appropriate than the symmetrical cubic lattice.
>>>>
>>>>Why would that make any difference?
>>>
>>>Tenney makes the lower primes simpler than the higher primes.
>>
>>You're still enclosing notes.
>
>An infinite number of them, except for the ET case, where (and
>only where) cubic and Tenney will agree -- the cells are shaped
>differently, but there are always the same number of them.

Yes but units are still notes.

>> >> >> >> >> >> I've suggested in the
>> >> >> >> >> >> past adjusting for it, crudely, by dividing by
>> >> >> >> >> >> pi(lim).
>> >> >> >> >> >
>> >> >> >> >> >Huh? What's that?
>> >> >> >> >>
>> >> >> >> >> If we're counting dyads, I suppose higher limits
>> >> >> >> >> ought to do
>> >> >> >> >> better with constant notes.
>> >> >> >> >> If we're counting complete chords,
>> >> >> >> >> they ought to do worse. Yes/no?
//
>> >> >> What relations, if any, do we expect, for n
>> >> >> notes, as lim goes up:
//
>> >> Ultimately if we can't show a relation to notes in scales
>> >> we've gone off the deep end.
>> >
>> >That's a separate issue. I mean are you comparing a given scale
>> >across *different* temperaments?
>>
>> Eventually I hope to characterize temperaments by the kinds of
>> scales they can manifest. But before that, things like Graham
>> complexity and consistency range need to be cultured to fix
>> a relationship between a single temperament and its scales.
>
>You're only talking about *linear* temperaments here, right?

No, I hope for a single complexity measure for all temperaments.
I've thought about PTs in terms of a lattice of generators, and
wracked my brain against the issue of scales coming from that.

-Carl

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

> No, I hope for a single complexity measure for all temperaments.

And how about JI? Or do you just arbitrarily leave that series off
your Pascal's triangle? If it works for all temperaments of all
dimensions, it certainly has to work for JI.

>> No, I hope for a single complexity measure for all temperaments.
>
>And how about JI? Or do you just arbitrarily leave that series off
>your Pascal's triangle? If it works for all temperaments of all
>dimensions, it certainly has to work for JI.

Good point.

-Carl