I've added an important new lattice to my "meantone"

Dictionary entry, at the bottom:

http://www.ixpres.com/interval/dict/meantone.htm

This attempts to show visually how a meantone "wraps"

the lattice into a cylinder, thus closing one of the

theoretically infinite dimensions of the JI lattice.

Does anyone know the math that will apply sines and

cosines, to warp the lattice-lines and points so that

they actually *look* like they're sitting on the

curved face of a cylinder?

-monz

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

>

> I've added an important new lattice to my "meantone"

> Dictionary entry, at the bottom:

>

> http://www.ixpres.com/interval/dict/meantone.htm

>

>

> This attempts to show visually how a meantone "wraps"

> the lattice into a cylinder, thus closing one of the

> theoretically infinite dimensions of the JI lattice.

>

>

> Does anyone know the math that will apply sines and

> cosines, to warp the lattice-lines and points so that

> they actually *look* like they're sitting on the

> curved face of a cylinder?

Well, assuming you're looking at the cylinder in a direction

perpendicular to its axis, it's pretty easy -- first of all, only use

a slice of the lattice running perpendicular to, and with width of

one, syntonic comma . . . then divide that width into 360

degrees . . . then, for each point, you can use either the sine or

the cosine of that angle to determine the final horizontal position,

while the vertical position remains the same! Of course, that will

have no "perspective" effect, nor will it impart any sense

of "translucency" or "opacity" to the surface of the cylinder . . .

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

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

> >

> > I've added an important new lattice to my "meantone"

> > Dictionary entry, at the bottom:

> >

> > http://www.ixpres.com/interval/dict/meantone.htm

I still have a problem with the way you're plotting the "meantone

chain", i.e.,

"each meantone chain itself . . ."

What is the meaning of it? And what is it for meantones like 31-tET

or 69-tET or LucyTuning? Does it still exist?

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Tuesday, January 29, 2002 1:39 PM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

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

> > >

> > > I've added an important new lattice to my "meantone"

> > > Dictionary entry, at the bottom:

> > >

> > > http://www.ixpres.com/interval/dict/meantone.htm

>

> I still have a problem with the way you're plotting the "meantone

> chain", i.e.,

>

> "each meantone chain itself . . ."

>

> What is the meaning of it?

I'm trying to show that, even acknowledging that the meantone

cylinder is *always the same* (which I mention on the webpage),

the way each fraction-of-a-comma meantone "cuts" across the

cylinder is different.

I don't know if there's any special meaning to that, and I

know that you've argued that there isn't. But the actual

amount of tempering in a meantone is an acoustical reality

which can be modeled as a mathematical property that's easy

to see on this lattice as the angle of spiral which the

meantone chain produces across the face of the cylinder.

Would you like me to create a 1/6-comma example for contrast?

Using the view that's on my webpage, the 1/4-comma chain

has an angle something like this:

-._

'-._

'-._

Whereas 1/6-comma is nearly vertical.

> And what is it for meantones like 31-tET

> or 69-tET or LucyTuning? Does it still exist?

Well ... the lattice I'm using here is "8ve-equivalent",

so I can't put any EDOs on them. But my formula can easily

include 2 as a prime-factor, in which case the whole lattice

is stretched vertically, and EDOs simply form vertical chains.

But I'm not sure how well it works, because in my lattice

formula, 2 is the smallest step in ratio-space, and so

the entire 69-EDO, for example, would be crammed into a

space smaller than that which separates 1:1 from 3:2,

making it hard to give any real visual relevance.

But of course I could easily change the "step sizes" of

my lattice metrics too. Could even reverse it, and make

2 the *largest* step. Hmmm.... I'm thinking that that

might be a really useful idea... then it'd be easy to show

EDOs.

As for LucyTuning, I don't know ... other than the fact

that it's audibly indistinguishable from 3/10-comma MT,

which I could easily plot. But as far as actually putting

pi on my lattice, I don't have a clue.

-monz

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

> > And what is it for meantones like 31-tET

> > or 69-tET or LucyTuning? Does it still exist?

>

>

> Well ... the lattice I'm using here is "8ve-equivalent",

> so I can't put any EDOs on them.

31-tET or 69-tET or LucyTuning are no more or less "8ve-equivalent"

than fraction-of-a-comma meantones. So I'm not sure what you meant my

this.

> But I'm not sure how well it works, because in my lattice

> formula, 2 is the smallest step in ratio-space, and so

> the entire 69-EDO, for example, would be crammed into a

> space smaller than that which separates 1:1 from 3:2,

> making it hard to give any real visual relevance.

If this whole business really had any acoustical meaning, wouldn't

the 69-tET line simply be somewhere between the 1/3-comma meantone

line and the 2/7-comma meantone line?

>

> But of course I could easily change the "step sizes" of

> my lattice metrics too. Could even reverse it, and make

> 2 the *largest* step. Hmmm.... I'm thinking that that

> might be a really useful idea... then it'd be easy to show

> EDOs.

>

>

> As for LucyTuning, I don't know ... other than the fact

> that it's audibly indistinguishable from 3/10-comma MT,

> which I could easily plot. But as far as actually putting

> pi on my lattice, I don't have a clue.

I'm hoping this sort of consideration will give you a clue that what

you're plotting isn't actually meaningful.

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Tuesday, January 29, 2002 2:30 PM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

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

>

> > > And what is it for meantones like 31-tET

> > > or 69-tET or LucyTuning? Does it still exist?

> >

> >

> > Well ... the lattice I'm using here is "8ve-equivalent",

> > so I can't put any EDOs on them.

>

> 31-tET or 69-tET or LucyTuning are no more or less "8ve-equivalent"

> than fraction-of-a-comma meantones. So I'm not sure what you meant my

> this.

I simply meant that I don't include 2 as part of the calculation

on these lattices, thus I can't graph EDOs (= 2^x).

> > But I'm not sure how well it works, because in my lattice

> > formula, 2 is the smallest step in ratio-space, and so

> > the entire 69-EDO, for example, would be crammed into a

> > space smaller than that which separates 1:1 from 3:2,

> > making it hard to give any real visual relevance.

>

> If this whole business really had any acoustical meaning, wouldn't

> the 69-tET line simply be somewhere between the 1/3-comma meantone

> line and the 2/7-comma meantone line?

>

> >

> > As for LucyTuning, I don't know ... other than the fact

> > that it's audibly indistinguishable from 3/10-comma MT,

> > which I could easily plot. But as far as actually putting

> > pi on my lattice, I don't have a clue.

>

> I'm hoping this sort of consideration will give you a clue that what

> you're plotting isn't actually meaningful.

Well ... *I'm* hoping that I can continue to find ways to

"warp" my lattices so that these equivalent mathematics

*do* correspond visually! Keep helping me! :)

-monz

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In my last post, I wrote:

> Well ... *I'm* hoping that I can continue to find ways to

> "warp" my lattices so that these equivalent mathematics

> *do* correspond visually! Keep helping me! :)

I should have said "acoustically equivalent mathematics".

-monz

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

>

> > From: paulerlich <paul@s...>

> > To: <tuning-math@y...>

> > Sent: Tuesday, January 29, 2002 2:30 PM

> > Subject: [tuning-math] Re: new cylindrical meantone lattice

> >

> >

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

> >

> > > > And what is it for meantones like 31-tET

> > > > or 69-tET or LucyTuning? Does it still exist?

> > >

> > >

> > > Well ... the lattice I'm using here is "8ve-equivalent",

> > > so I can't put any EDOs on them.

> >

> > 31-tET or 69-tET or LucyTuning are no more or less "8ve-

equivalent"

> > than fraction-of-a-comma meantones. So I'm not sure what you

meant my

> > this.

>

>

> I simply meant that I don't include 2 as part of the calculation

> on these lattices, thus I can't graph EDOs (= 2^x).

It sounds to me like you're just plugging things into the numbers

without really understanding what they mean.

> Well ... *I'm* hoping that I can continue to find ways to

> "warp" my lattices so that these equivalent mathematics

> *do* correspond visually! Keep helping me! :)

It's absolutely wonderful that you're going to be producing real

cylindrical lattices. I'm hoping that, at least on some of your

webpages, you won't confuse the reader with your fractional lattice

points -- there is so much valuable information there already without

them -- and I still don't know what they're supposed to mean,

_especially_ once you've wrapped the lattice into a cylinder.

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Tuesday, January 29, 2002 4:24 PM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

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

>

> > I simply meant that I don't include 2 as part of the calculation

> > on these lattices, thus I can't graph EDOs (= 2^x).

>

> It sounds to me like you're just plugging things into the numbers

> without really understanding what they mean.

Huh?

Most of my lattices don't use 2 as a factor, but it is certainly

possible to include 2, and I have done so on occasion. For example,

to produce lattices which portray ancient Greek systems, I've

occassionaly included 2, because the Greeks really weren't

thinking in terms of "8ve"-equivalence, but rather more in

terms of tetrachordal equivalence (or similarity, anyway).

By including 2 in my formula, and perhaps reversing the

prime-lengths so that 2 is the longest (as I wrote in another

post), I can show EDOs as well as all the usual JI ratios, and

fraction-of-a-comma meantones too.

> It's absolutely wonderful that you're going to be producing real

> cylindrical lattices. I'm hoping that, at least on some of your

> webpages, you won't confuse the reader with your fractional lattice

> points -- there is so much valuable information there already without

> them -- and I still don't know what they're supposed to mean,

> _especially_ once you've wrapped the lattice into a cylinder.

Well, now that I finally have a firm understanding of the

cylindrical meantone lattice, I can see at least a few of

the objections you've been leveling at me:

- The lattice of "rational implications" is indeed identical

for all meantones. Tempering out the syntonic comma

causes the JI lattices to wrap into identical cylinders.

- Something I found most interesting: the distance of all

meantone pitches as represented by the circumferential

lines around the cylinder (the lines which represent

the syntonic comma), perpendicular to the cylinder itself,

is *also* identical for all meantones.

This was a surprise when I first realized it, but upon

further reflection, it's an obvious result of the above.

*But* ... those agreements noted, I don't understand why

you still object to my representation of the various meantone

spirals around the cylinder.

The different meantone systems are tuned in different ways,

and if the difference between any two systems is large enough,

it's audible. So what's wrong with showing that visually,

by having the meantones slice the cylinder in their own

particular way according to the math involved?

And I *still* don't understand how a note that I factor as,

for example, 3^(2/3) * 5^(1/3) (ignoring 2), which is the

1/6-comma meantone "whole tone", can be represented as

anything else. There is no other combination of exponents

for 3 and 5 which will plot that point in exactly that spot.

I would really appreciate much more help with this. I've

been trying to understand it for a couple of years, to no avail.

And remember ... the whole purpose in making these lattices

is to eventually include this capability in my JustMusic software.

The point is to be enable the user to create beautiful diagrams

which s/he can then manipulate to create beautiful music.

(Well, OK ... whatever kind of music s/he wants. Some people

really don't like beautiful ...)

-monz

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

> > From: paulerlich <paul@s...>

> > To: <tuning-math@y...>

> > Sent: Tuesday, January 29, 2002 4:24 PM

> > Subject: [tuning-math] Re: new cylindrical meantone lattice

> >

> >

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

> >

> > > I simply meant that I don't include 2 as part of the calculation

> > > on these lattices, thus I can't graph EDOs (= 2^x).

> >

> > It sounds to me like you're just plugging things into the numbers

> > without really understanding what they mean.

>

>

> Huh?

>

> Most of my lattices don't use 2 as a factor, but it is certainly

> possible to include 2, and I have done so on occasion. For example,

> to produce lattices which portray ancient Greek systems, I've

> occassionaly included 2, because the Greeks really weren't

> thinking in terms of "8ve"-equivalence, but rather more in

> terms of tetrachordal equivalence (or similarity, anyway).

Right . . . but when you assume octave equivalence, you ignore 2. But

what are you really plotting on the lattice, and why are you

representing it mathematically the way you are? That's the kind of

thinking that I'd encourage you to do more of . . . though that

should carry no more weight than anyone else's opinion . . .

> By including 2 in my formula, and perhaps reversing the

> prime-lengths so that 2 is the longest (as I wrote in another

> post), I can show EDOs as well as all the usual JI ratios, and

> fraction-of-a-comma meantones too.

But what happens to the cylinder? It seems that, without necessarily

ignoring all the instances of the number 2, you would want to make

these diagrams octave-invariant, wouldn't you?

>

> Well, now that I finally have a firm understanding of the

> cylindrical meantone lattice, I can see at least a few of

> the objections you've been leveling at me:

>

>

> - Something I found most interesting: the distance of all

> meantone pitches as represented by the circumferential

> lines around the cylinder (the lines which represent

> the syntonic comma), perpendicular to the cylinder itself,

> is *also* identical for all meantones.

>

> This was a surprise when I first realized it, but upon

> further reflection, it's an obvious result of the above.

I could imagine a formula where these distances would be *very

slightly* different from meantone to meantone. The idea is simply

that you'll allow mistuning of consonant intervals to be reflected as

small changes in length.

But feel free to ignore that.

> *But* ... those agreements noted, I don't understand why

> you still object to my representation of the various meantone

> spirals around the cylinder.

What do those spirals represent? And what does it say about

LucyTuning and ETs that you can't construct such spirals from _these_

meantones?

> The different meantone systems are tuned in different ways,

> and if the difference between any two systems is large enough,

> it's audible. So what's wrong with showing that visually,

> by having the meantones slice the cylinder in their own

> particular way according to the math involved?

(I mentioned another way of showing that visually above. A slight

acoustical difference merits at most a slight visual difference, IMO).

>

> And I *still* don't understand how a note that I factor as,

> for example, 3^(2/3) * 5^(1/3) (ignoring 2), which is the

> 1/6-comma meantone "whole tone", can be represented as

> anything else. There is no other combination of exponents

> for 3 and 5 which will plot that point in exactly that spot.

Sure there are -- they'd just be irrational exponents. But when some

meantones, like LucyTuning, will require irrational exponents anyway

in order to get a "spiral" happening for them, that implies to me

that irrational exponents are just as meaningful as rational

ones. . . .

If you insist on having the spirals, then you _need_ to find a way to

get them to work for LucyTuning and Golden meantone and ET meantones,

etc. Otherwise I can't imagine how they could _possibly_ be

meaningful. (P.S. Congratulations on discovering them -- they may be

an original Monzo contribution!)

> And remember ... the whole purpose in making these lattices

> is to eventually include this capability in my JustMusic software.

Well . . . meantones aren't "Just" by any of the definitions

proposed. So maybe a new name is in order?

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Tuesday, January 29, 2002 9:24 PM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

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

>

> > By including 2 in my formula, and perhaps reversing the

> > prime-lengths so that 2 is the longest (as I wrote in another

> > post), I can show EDOs as well as all the usual JI ratios, and

> > fraction-of-a-comma meantones too.

>

> But what happens to the cylinder? It seems that, without necessarily

> ignoring all the instances of the number 2, you would want to make

> these diagrams octave-invariant, wouldn't you?

I'll have to try to answer this question and the previous

(which I snipped) more fully another time. But I have one here:

what the heck is the difference between "octave equivalent" and

"octave invariant"? Is there a difference?

> > *But* ... those agreements noted, I don't understand why

> > you still object to my representation of the various meantone

> > spirals around the cylinder.

>

> What do those spirals represent?

The actual mathematical tuning of the fraction-of-a-comma meantones.

> And what does it say about LucyTuning and ETs that you can't

> construct such spirals from _these_ meantones?

I says that while LucyTuning and meantone-like ETs are audibly

indistinguishable from certain fraction-of-a-comma meantones,

they are mathematically entirely different.

Again, I refer you to my (very vague but seemingly always

getting clearer) ideas on finity. Xenharmonic Bridges in

effect here.

> > The different meantone systems are tuned in different ways,

> > and if the difference between any two systems is large enough,

> > it's audible. So what's wrong with showing that visually,

> > by having the meantones slice the cylinder in their own

> > particular way according to the math involved?

>

> (I mentioned another way of showing that visually above. A slight

> acoustical difference merits at most a slight visual difference, IMO).

OK, Paul, I can buy that! As I've said before many times, I'd

love to enlist your help and for the two of us to work together

to create some really killer lattice formulae.

You know that I'm very fond of my particular formula, but

I'm open-minded and willing to revise it, or better, to create

new kinds of lattices from scratch.

> > And I *still* don't understand how a note that I factor as,

> > for example, 3^(2/3) * 5^(1/3) (ignoring 2), which is the

> > 1/6-comma meantone "whole tone", can be represented as

> > anything else. There is no other combination of exponents

> > for 3 and 5 which will plot that point in exactly that spot.

>

> Sure there are -- they'd just be irrational exponents. But when some

> meantones, like LucyTuning, will require irrational exponents anyway

> in order to get a "spiral" happening for them, that implies to me

> that irrational exponents are just as meaningful as rational

> ones. . . .

>

> If you insist on having the spirals, then you _need_ to find a way to

> get them to work for LucyTuning and Golden meantone and ET meantones,

> etc. Otherwise I can't imagine how they could _possibly_ be

> meaningful. (P.S. Congratulations on discovering them -- they may be

> an original Monzo contribution!)

Hmmm ... yes, I'm thinking that the meantone-spiral thing really

might be a useful new contribution to tuning theory. Thanks for

the acknowledgement!

I'd definitely like to see more about what you say here.

Hope this sparks an interest in several of you, so that a

real discussion might ensue. I'm still pretty confused about

the "non-uniqueness" business myself.

> > And remember ... the whole purpose in making these lattices

> > is to eventually include this capability in my JustMusic software.

>

> Well . . . meantones aren't "Just" by any of the definitions

> proposed. So maybe a new name is in order?

Yeah, I've been sweating over that name ever since having been

on the tuning list for about a year.

But then again ... one of the things I seek to show, which I

interpret as being along the same lines as your Hypothesis, is

that all of our various tuning systems (in other words, the

way "finity" works) have some ultimate basis in the manipulation

of the prime series, which is at the heart of my whole

JustMusic theory and software design.

Maybe "PrimeMusic" would be better? But it doesn't have the

same ring, the logo wouldn't look as cool, and ... JustMusic

happens to have the same initials as JoeMonzo, which makes

abbreviations in emails very convenient for me! :)

-monz

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

>

> > From: paulerlich <paul@s...>

> > To: <tuning-math@y...>

> > Sent: Tuesday, January 29, 2002 9:24 PM

> > Subject: [tuning-math] Re: new cylindrical meantone lattice

> >

> >

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

> >

> > > By including 2 in my formula, and perhaps reversing the

> > > prime-lengths so that 2 is the longest (as I wrote in another

> > > post), I can show EDOs as well as all the usual JI ratios, and

> > > fraction-of-a-comma meantones too.

> >

> > But what happens to the cylinder? It seems that, without

necessarily

> > ignoring all the instances of the number 2, you would want to

make

> > these diagrams octave-invariant, wouldn't you?

>

>

> I'll have to try to answer this question and the previous

> (which I snipped) more fully another time. But I have one here:

> what the heck is the difference between "octave equivalent" and

> "octave invariant"? Is there a difference?

Not that I can think of right now.

>

>

> > > *But* ... those agreements noted, I don't understand why

> > > you still object to my representation of the various meantone

> > > spirals around the cylinder.

> >

> > What do those spirals represent?

>

> The actual mathematical tuning of the fraction-of-a-comma meantones.

Hmm . . .

> > And what does it say about LucyTuning and ETs that you can't

> > construct such spirals from _these_ meantones?

>

>

> I says that while LucyTuning and meantone-like ETs are audibly

> indistinguishable from certain fraction-of-a-comma meantones,

> they are mathematically entirely different.

Not really.

> Again, I refer you to my (very vague but seemingly always

> getting clearer) ideas on finity. Xenharmonic Bridges in

> effect here.

Can you elaborate, please?

> > > The different meantone systems are tuned in different ways,

> > > and if the difference between any two systems is large enough,

> > > it's audible. So what's wrong with showing that visually,

> > > by having the meantones slice the cylinder in their own

> > > particular way according to the math involved?

> >

> > (I mentioned another way of showing that visually above. A slight

> > acoustical difference merits at most a slight visual difference,

IMO).

>

> OK, Paul, I can buy that! As I've said before many times, I'd

> love to enlist your help and for the two of us to work together

> to create some really killer lattice formulae.

>

> You know that I'm very fond of my particular formula, but

> I'm open-minded and willing to revise it, or better, to create

> new kinds of lattices from scratch.

Somewhere a long time ago, perhaps in the Mills times, I posted a

proposed formula for these lengths. But I'm not too picky about it.

Why not show mistuning as a tiny "break" in the consonant connections?

>

> > > And I *still* don't understand how a note that I factor as,

> > > for example, 3^(2/3) * 5^(1/3) (ignoring 2), which is the

> > > 1/6-comma meantone "whole tone", can be represented as

> > > anything else. There is no other combination of exponents

> > > for 3 and 5 which will plot that point in exactly that spot.

> >

> > Sure there are -- they'd just be irrational exponents. But when

some

> > meantones, like LucyTuning, will require irrational exponents

anyway

> > in order to get a "spiral" happening for them, that implies to me

> > that irrational exponents are just as meaningful as rational

> > ones. . . .

> >

> > If you insist on having the spirals, then you _need_ to find a

way to

> > get them to work for LucyTuning and Golden meantone and ET

meantones,

> > etc. Otherwise I can't imagine how they could _possibly_ be

> > meaningful. (P.S. Congratulations on discovering them -- they may

be

> > an original Monzo contribution!)

>

>

> Hmmm ... yes, I'm thinking that the meantone-spiral thing really

> might be a useful new contribution to tuning theory. Thanks for

> the acknowledgement!

You got it, though I'm not banking on the "useful" part :)

Particularly irksome is that you must choose a 1/1, such as C, which

flies in the face of the true nature of meantone as a transposible

system.

----- Original Message -----

From: paulerlich <paul@stretch-music.com>

To: <tuning-math@yahoogroups.com>

Sent: Wednesday, January 30, 2002 11:10 AM

Subject: [tuning-math] Re: new cylindrical meantone lattice

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

> >

> > > From: paulerlich <paul@s...>

> > > To: <tuning-math@y...>

> > > Sent: Tuesday, January 29, 2002 9:24 PM

> > > Subject: [tuning-math] Re: new cylindrical meantone lattice

> > >

> > >

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

> > >

> > what the heck is the difference between "octave equivalent" and

> > "octave invariant"? Is there a difference?

>

> Not that I can think of right now.

I just posted something else about this a few minutes ago.

I'm totally confused, in studying this list's archives,

about the differences between "octave-invariant" and

"octave-equivalent", and don't understand what differences

there are between these and unison-vectors and periods.

> >

> >

> > > > *But* ... those agreements noted, I don't understand why

> > > > you still object to my representation of the various meantone

> > > > spirals around the cylinder.

> > >

> > > What do those spirals represent?

> >

> > The actual mathematical tuning of the fraction-of-a-comma meantones.

>

> Hmm . . .

>

> > > And what does it say about LucyTuning and ETs that you can't

> > > construct such spirals from _these_ meantones?

> >

> >

> > I says that while LucyTuning and meantone-like ETs are audibly

> > indistinguishable from certain fraction-of-a-comma meantones,

> > they are mathematically entirely different.

>

> Not really.

Huh? Take a look at the "ratios" I post below.

2^(2/10) * 3^(-2/10) * 5^(3/10) is *mathematically*

totally different from 2^(2pi+1)/4pi, even tho they're

acoustically identical.

> > Again, I refer you to my (very vague but seemingly always

> > getting clearer) ideas on finity. Xenharmonic Bridges in

> > effect here.

>

> Can you elaborate, please?

Sure.

(Well, I'll be cornswoggled -- that doesn't mean anything ...

I just realized I had explained this on an updated definition

I made for "LucyTuning", but I never uploaded it until now!)

It's all in the Dictionary now at

http://www.ixpres.com/interval/dict/lucy.htm

But I'll post it anyway:

>> This [LucyTuning] generator or "5th" is composed of three

>> Large (3L) plus one small note (s), i.e. (3L+s)

>> = (~190.986*3) + (~122.535) = ~695.493 cents or ratio of

>>

>> 2^(3/2pi) * ( 2/(2^(5/2pi)) )^(1/2)

>>

>> = 2^( (2pi + 1) / 4pi )

>>

>> = 2^( 1/2 + 1/4pi )

>>

>> = ~1.494412.

>>

>>

>> This generator is audibly indistinguishable from that

>> of 3/10-comma quasi-meantone:

>>

>> 2^(2/10) * 3^(-2/10) * 5^(3/10) 3/10-comma quasi-meantone

"5th"

>> - 2^(2pi+1)/4pi Lucytuning "5th"

>> ------------------------------------------

>> 2^(-12pi-10)/40pi * 3^(-2/10) * 5^(3/10) = ~0.010148131 cent = ~1/99

cent

That last number is the "ratio" of the tiny xenharmonic bridge

I'm talking about.

If I could find some way to represent LucyTuning on the flat

lattice (which means finding some way to represent pi in a

universe where everything is factored by 3 and 5), then bend

the lattice into a meantone cylinder, then warp the cylinder

so that the LucyTuning "5th" occupies the same point as the

3/10-comma meantone "5th", I'd have it.

The fact that pi is transcendental, irrational, whatever,

makes it hard for me to figure out how to do this.

> Somewhere a long time ago, perhaps in the Mills times, I posted a

> proposed formula for these lengths. But I'm not too picky about it.

> Why not show mistuning as a tiny "break" in the consonant connections?

Hmm ... sounds interesting. You mean like on the recent Blackjack

lattice you posted? Please elaborate.

> > > > And I *still* don't understand how a note that I factor as,

> > > > for example, 3^(2/3) * 5^(1/3) (ignoring 2), which is the

> > > > 1/6-comma meantone "whole tone", can be represented as

> > > > anything else. There is no other combination of exponents

> > > > for 3 and 5 which will plot that point in exactly that spot.

> > >

> > > Sure there are -- they'd just be irrational exponents. But when

> > > some meantones, like LucyTuning, will require irrational exponents

> > > anyway in order to get a "spiral" happening for them, that

> > > implies to me that irrational exponents are just as meaningful

> > > as rational ones. . . .

I'm still not getting this. It seems to me that you're thinking

in terms of something other than the 3x5 plane within which I'm working.

But I can certainly buy what you're saying about irrational exponents.

As I said, if I could figure out *how* to lattice them, I would.

(See the bit above about the xenharmonic bridge.)

> > Hmmm ... yes, I'm thinking that the meantone-spiral thing really

> > might be a useful new contribution to tuning theory. Thanks for

> > the acknowledgement!

>

> You got it, though I'm not banking on the "useful" part :)

> Particularly irksome is that you must choose a 1/1, such as C, which

> flies in the face of the true nature of meantone as a transposible

> system.

Ah! ... Paul, this is where you'd understand my ideas a little

better if you finally succumb to my always begging you to join

my justmusic group! </justmusic>

The idea is that the user could:

1) Choose meantone as the type of tuning desired: JustMusic

then draws the syntonic-comma based cylinder on the lattice.

2) Define which meantone by fraction-of-a-comma (or Lucy or Golden,

if I ever figure out how): JustMusic draws the spiral around

the cylinder.

3) Use the mouse to roll the cylinder around on the lattice

to get whichever key-center is desired, or simply input the

key and let JustMusic do the rolling.

That's just a brief outline.

Sure do hope to see you over there! (and anyone else on this

list who is intrigued but hasn't signed up yet)

-monz

Do You Yahoo!?

Get your free @yahoo.com address at http://mail.yahoo.com

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

> > Not really.

>

>

> Huh? Take a look at the "ratios" I post below.

> 2^(2/10) * 3^(-2/10) * 5^(3/10) is *mathematically*

> totally different from 2^(2pi+1)/4pi, even tho they're

> acoustically identical.

Again, Monz, you're looking too closely at the way the math looks,

and not thinking about what it means.

I could write pi in fifteen different ways that look totally

mathematically different to you, but they're all be the same.

> > > Again, I refer you to my (very vague but seemingly always

> > > getting clearer) ideas on finity. Xenharmonic Bridges in

> > > effect here.

> >

> > Can you elaborate, please?

>

>

> Sure.

>

> (Well, I'll be cornswoggled -- that doesn't mean anything ...

> I just realized I had explained this on an updated definition

> I made for "LucyTuning", but I never uploaded it until now!)

>

>

> It's all in the Dictionary now at

> http://www.ixpres.com/interval/dict/lucy.htm

>

> But I'll post it anyway:

>

> >> This [LucyTuning] generator or "5th" is composed of three

> >> Large (3L) plus one small note (s), i.e. (3L+s)

> >> = (~190.986*3) + (~122.535) = ~695.493 cents or ratio of

> >>

> >> 2^(3/2pi) * ( 2/(2^(5/2pi)) )^(1/2)

> >>

> >> = 2^( (2pi + 1) / 4pi )

> >>

> >> = 2^( 1/2 + 1/4pi )

> >>

> >> = ~1.494412.

> >>

> >>

> >> This generator is audibly indistinguishable from that

> >> of 3/10-comma quasi-meantone:

> >>

> >> 2^(2/10) * 3^(-2/10) * 5^(3/10) 3/10-comma quasi-

meantone

> "5th"

> >> - 2^(2pi+1)/4pi Lucytuning "5th"

> >> ------------------------------------------

> >> 2^(-12pi-10)/40pi * 3^(-2/10) * 5^(3/10) = ~0.010148131

cent = ~1/99

> cent

>

>

> That last number is the "ratio" of the tiny xenharmonic bridge

> I'm talking about.

That's no kind of unison vector, Monz. All that is is a tiny

difference between two functionally identical (as meantones in 5-

limit) tuning systems. A real unison vector expresses a relationship

within _one_ tuning system that allows it to take a pitch in more

than one sense (and thus lead you on the road to finity). This

interval you're talking about does not lead you in that direction at

all (as you claimed above, in case you're wondering what I'm rambling

on about).

> If I could find some way to represent LucyTuning on the flat

> lattice (which means finding some way to represent pi in a

> universe where everything is factored by 3 and 5), then bend

> the lattice into a meantone cylinder, then warp the cylinder

> so that the LucyTuning "5th" occupies the same point as the

> 3/10-comma meantone "5th", I'd have it.

>

> The fact that pi is transcendental, irrational, whatever,

> makes it hard for me to figure out how to do this.

If you want people to help you figure this out, you'll have to

determine what the above construction is all about. What does it

mean? Why do we want to see it on our perfectly good lattices-wrapped-

onto-cylinders?

P.S. There are so many wonderful varieties of cylindrical tunings

your JustMusic software could be helping people visualize -- they'll

all look different -- why not let the meantones look pretty much the

same -- compositionally, they pretty much are, wouldn't you say?

>

>

> > Somewhere a long time ago, perhaps in the Mills times, I posted a

> > proposed formula for these lengths. But I'm not too picky about

it.

> > Why not show mistuning as a tiny "break" in the consonant

connections?

>

>

> Hmm ... sounds interesting. You mean like on the recent Blackjack

> lattice you posted? Please elaborate.

>

>

> > > > > And I *still* don't understand how a note that I factor as,

> > > > > for example, 3^(2/3) * 5^(1/3) (ignoring 2), which is the

> > > > > 1/6-comma meantone "whole tone", can be represented as

> > > > > anything else. There is no other combination of exponents

> > > > > for 3 and 5 which will plot that point in exactly that spot.

> > > >

> > > > Sure there are -- they'd just be irrational exponents. But

when

> > > > some meantones, like LucyTuning, will require irrational

exponents

> > > > anyway in order to get a "spiral" happening for them, that

> > > > implies to me that irrational exponents are just as meaningful

> > > > as rational ones. . . .

>

>

> I'm still not getting this. It seems to me that you're thinking

> in terms of something other than the 3x5 plane within which I'm

working.

>

> But I can certainly buy what you're saying about irrational

exponents.

> As I said, if I could figure out *how* to lattice them, I would.

> (See the bit above about the xenharmonic bridge.)

>

>

>

> > > Hmmm ... yes, I'm thinking that the meantone-spiral thing really

> > > might be a useful new contribution to tuning theory. Thanks for

> > > the acknowledgement!

> >

> > You got it, though I'm not banking on the "useful" part :)

> > Particularly irksome is that you must choose a 1/1, such as C,

which

> > flies in the face of the true nature of meantone as a transposible

> > system.

>

>

> Ah! ... Paul, this is where you'd understand my ideas a little

> better if you finally succumb to my always begging you to join

> my justmusic group! </justmusic>

>

>

> The idea is that the user could:

>

> 1) Choose meantone as the type of tuning desired: JustMusic

> then draws the syntonic-comma based cylinder on the lattice.

>

> 2) Define which meantone by fraction-of-a-comma (or Lucy or Golden,

> if I ever figure out how): JustMusic draws the spiral around

> the cylinder.

>

> 3) Use the mouse to roll the cylinder around on the lattice

> to get whichever key-center is desired, or simply input the

> key and let JustMusic do the rolling.

>

>

> That's just a brief outline.

>

> Sure do hope to see you over there! (and anyone else on this

> list who is intrigued but hasn't signed up yet)

>

>

>

> -monz

>

>

>

>

>

> _________________________________________________________

> Do You Yahoo!?

> Get your free @yahoo.com address at http://mail.yahoo.com

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

> > Somewhere a long time ago, perhaps in the Mills times, I posted a

> > proposed formula for these lengths. But I'm not too picky about

it.

> > Why not show mistuning as a tiny "break" in the consonant

connections?

>

>

> Hmm ... sounds interesting. You mean like on the recent Blackjack

> lattice you posted?

Well, sort of. Except that all the consonant intervals would be

broken.

>Please elaborate.

Well, for example, you could do it like in Hall's hexagonal lattices,

where he puts a little number representing the mistuning of each

interval, but you could do it visually, like representing each

consonant interval as a twig and putting a physical "break", the size

of the mistuning, into it.

> > > > > And I *still* don't understand how a note that I factor as,

> > > > > for example, 3^(2/3) * 5^(1/3) (ignoring 2), which is the

> > > > > 1/6-comma meantone "whole tone", can be represented as

> > > > > anything else. There is no other combination of exponents

> > > > > for 3 and 5 which will plot that point in exactly that spot.

> > > >

> > > > Sure there are -- they'd just be irrational exponents. But

when

> > > > some meantones, like LucyTuning, will require irrational

exponents

> > > > anyway in order to get a "spiral" happening for them, that

> > > > implies to me that irrational exponents are just as meaningful

> > > > as rational ones. . . .

>

>

> I'm still not getting this. It seems to me that you're thinking

> in terms of something other than the 3x5 plane within which I'm

working.

No I'm not :)

> But I can certainly buy what you're saying about irrational

exponents.

> As I said, if I could figure out *how* to lattice them, I would.

> (See the bit above about the xenharmonic bridge.)

You'll have to determine what the "spiral" means for some

hypothetical musician who might want to see it, if anything. Once you

define the problem, it can be solved.

> > > Hmmm ... yes, I'm thinking that the meantone-spiral thing really

> > > might be a useful new contribution to tuning theory. Thanks for

> > > the acknowledgement!

> >

> > You got it, though I'm not banking on the "useful" part :)

> > Particularly irksome is that you must choose a 1/1, such as C,

which

> > flies in the face of the true nature of meantone as a transposible

> > system.

>

>

> Ah! ... Paul, this is where you'd understand my ideas a little

> better if you finally succumb to my always begging you to join

> my justmusic group! </justmusic>

>

>

> The idea is that the user could:

>

> 1) Choose meantone as the type of tuning desired: JustMusic

> then draws the syntonic-comma based cylinder on the lattice.

Right . . .

> 2) Define which meantone by fraction-of-a-comma (or Lucy or Golden,

> if I ever figure out how): JustMusic draws the spiral around

> the cylinder.

And the spiral helps me, as a musician, do . . . ?

> 3) Use the mouse to roll the cylinder around on the lattice

> to get whichever key-center is desired, or simply input the

> key and let JustMusic do the rolling.

Well this I was expecting anyway.

> That's just a brief outline.

I'm afraid you haven't told me anything new :) :)

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Wednesday, January 30, 2002 8:17 PM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

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

>

> > > Not really.

> >

> >

> > Huh? Take a look at the "ratios" I post below.

> > 2^(2/10) * 3^(-2/10) * 5^(3/10) is *mathematically*

> > totally different from 2^(2pi+1)/4pi, even tho they're

> > acoustically identical.

>

> Again, Monz, you're looking too closely at the way the math looks,

> and not thinking about what it means.

>

> I could write pi in fifteen different ways that look totally

> mathematically different to you, but they're all be the same.

OK, I've admitted many times that I'm math-challenged.

I'll ask more about this below ...

(I've rewritten the following quote for clearer presentation)

> > >> This generator is audibly indistinguishable from that

> > >> of 3/10-comma quasi-meantone:

> > >>

> > >>

> > >> 3/10-comma quasi-meantone "5th"

> > >> "-" Lucytuning "5th"

> > >> ---------------------------------

> > >> ~0.010148131 cent = ~1/99 cent

> > >>

> > >> =

> > >>

> > >> 2^(2/10) * 3^(-2/10) * 5^(3/10)

> > >> - 2^(2pi+1)/4pi

> > >> ------------------------------------------

> > >> 2^(-12pi-10)/40pi * 3^(-2/10) * 5^(3/10)

> >

> >

> > That last number is the "ratio" of the tiny xenharmonic bridge

> > I'm talking about.

>

> That's no kind of unison vector, Monz. All that is is a tiny

> difference between two functionally identical (as meantones in 5-

> limit) tuning systems. A real unison vector expresses a relationship

> within _one_ tuning system that allows it to take a pitch in more

> than one sense (and thus lead you on the road to finity). This

> interval you're talking about does not lead you in that direction at

> all (as you claimed above, in case you're wondering what I'm rambling

> on about).

OK, in case there are other subtle distinctions between xenharmonic

bridges and unison-vectors, I'm going to stick with my terminology.

I'm saying that these two tunings *are* different (altho the auditory

system can't hear the difference), so the xenharmonic bridge here

*is* allowing the listener to accept 2^(2/10) * 3^(-2/10) * 5^(3/10)

to be the same as 2^(2pi+1)/4pi .

Am I missing your point simply because pi is a number that cannot

be finitely quantized?

> > If I could find some way to represent LucyTuning on the flat

> > lattice (which means finding some way to represent pi in a

> > universe where everything is factored by 3 and 5), then bend

> > the lattice into a meantone cylinder, then warp the cylinder

> > so that the LucyTuning "5th" occupies the same point as the

> > 3/10-comma meantone "5th", I'd have it.

> >

> > The fact that pi is transcendental, irrational, whatever,

> > makes it hard for me to figure out how to do this.

>

> If you want people to help you figure this out, you'll have to

> determine what the above construction is all about. What does it

> mean? Why do we want to see it on our perfectly good lattices-wrapped-

> onto-cylinders?

Sheesh, I don't know! I was hoping *you* could help me figure

that out! (OK, good questions from you *do* do that.)

> P.S. There are so many wonderful varieties of cylindrical tunings

> your JustMusic software could be helping people visualize -- they'll

> all look different --

Wow, Paul, I'm amazed that you wrote this just now.

Another project I've been working on for a couple of months

is a MIDI-file of Beethoven's "Moonlight" Sonata, tuned in

Kirnberger III well-temperament. I chose Kirnberger simply

because it's quite likely to have been a tuning that Beethoven's

piano tuner might have used, and because it has a simplicity

and elegance (in terms of portraying it on my lattice) that

make it easy to lattice, where other WTs are more complicated.

Anyway, one of the things that I found fascinating about

Kirnberger III (and this probably applies to many other WTs

as well ... haven't looked yet) is that the ends of the

tuning chain invoke the skhisma, which means in effect

(assuming the ~2-cent difference falls outside the capability

of a human tuner in Beethoven's day), that it's a closed tuning.

Thus, a nice cylinder, perpendicular to the skhisma, which is

*extremely* different from the meantone cylinder.

> ... -- why not let the meantones look pretty much the

> same -- compositionally, they pretty much are, wouldn't you say?

Yes, I think I can agree with that.

But then again, why would a composer choose one particular

meantone over any other? There must be *some* reason/s

... so what's wrong with making a visual model of those choices

too?

Perhaps there are deeper things about the differences between

different meantones that we haven't noticed yet. Having nice

pictures of them would make it easier to find those as-yet

undiscovered aspects, I think.

-monz

Do You Yahoo!?

Get your free @yahoo.com address at http://mail.yahoo.com

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Wednesday, January 30, 2002 8:24 PM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

> > > Why not show mistuning as a tiny "break" in the consonant

> > > connections?

> >

> >

> > Hmm ... sounds interesting. You mean like on the recent Blackjack

> > lattice you posted?

>

> Well, sort of. Except that all the consonant intervals would be

> broken.

>

> > Please elaborate.

>

> Well, for example, you could do it like in Hall's hexagonal lattices,

> where he puts a little number representing the mistuning of each

> interval, but you could do it visually, like representing each

> consonant interval as a twig and putting a physical "break", the size

> of the mistuning, into it.

Hmmm ... this sounds *really* interesting! I'd like to see

more on this from you and the others who understand how to do it.

> > [me, monz]

> > The idea is that the user could:

> >

> > 1) Choose meantone as the type of tuning desired: JustMusic

> > then draws the syntonic-comma based cylinder on the lattice.

>

> Right . . .

>

> > 2) Define which meantone by fraction-of-a-comma (or Lucy or Golden,

> > if I ever figure out how): JustMusic draws the spiral around

> > the cylinder.

>

> And the spiral helps me, as a musician, do . . . ?

Sheesh ... I don't know. I'm just groping in the dark with this

right now. If I was able to actually *implement* it into my

software, I could play around with it and maybe offer some ideas.

But the project has been stalled for a while now.

>

> > 3) Use the mouse to roll the cylinder around on the lattice

> > to get whichever key-center is desired, or simply input the

> > key and let JustMusic do the rolling.

>

> Well this I was expecting anyway.

Really? Hmmm ... if you were expecting it, then why'd you ask

me the bit about choosing "a 1/1, such as C, which flies in the

face of the true nature of meantone as a transposible system" ?

If the user can transpose/roll the cylinder at will, what

difference does it make which note is chosen as 1/1 ?

> > That's just a brief outline.

>

> I'm afraid you haven't told me anything new :) :)

OK, sorry ... but I'd still like to have you on the justmusic list.

Things have been extremely slow there lately, so there's no

concern over dealing with the volume.

-monz

Do You Yahoo!?

Get your free @yahoo.com address at http://mail.yahoo.com

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

>

> OK, in case there are other subtle distinctions between xenharmonic

> bridges and unison-vectors, I'm going to stick with my terminology.

You're depriving it of much-needed meaning. You'll have to revise

your definition, then.

> I'm saying that these two tunings *are* different (altho the

auditory

> system can't hear the difference), so the xenharmonic bridge here

> *is* allowing the listener to accept 2^(2/10) * 3^(-2/10) * 5^(3/10)

> to be the same as 2^(2pi+1)/4pi .

Now you're claiming that our auditory system cares about these

irrational ratios???

> Am I missing your point simply because pi is a number that cannot

> be finitely quantized?

No.

> > > If I could find some way to represent LucyTuning on the flat

> > > lattice (which means finding some way to represent pi in a

> > > universe where everything is factored by 3 and 5), then bend

> > > the lattice into a meantone cylinder, then warp the cylinder

> > > so that the LucyTuning "5th" occupies the same point as the

> > > 3/10-comma meantone "5th", I'd have it.

> > >

> > > The fact that pi is transcendental, irrational, whatever,

> > > makes it hard for me to figure out how to do this.

> >

> > If you want people to help you figure this out, you'll have to

> > determine what the above construction is all about. What does it

> > mean? Why do we want to see it on our perfectly good lattices-

wrapped-

> > onto-cylinders?

>

>

> Sheesh, I don't know! I was hoping *you* could help me figure

> that out! (OK, good questions from you *do* do that.)

>

>

> > P.S. There are so many wonderful varieties of cylindrical tunings

> > your JustMusic software could be helping people visualize --

they'll

> > all look different --

>

> Wow, Paul, I'm amazed that you wrote this just now.

>

> Another project I've been working on for a couple of months

> is a MIDI-file of Beethoven's "Moonlight" Sonata, tuned in

> Kirnberger III well-temperament. I chose Kirnberger simply

> because it's quite likely to have been a tuning that Beethoven's

> piano tuner might have used, and because it has a simplicity

> and elegance (in terms of portraying it on my lattice) that

> make it easy to lattice, where other WTs are more complicated.

>

> Anyway, one of the things that I found fascinating about

> Kirnberger III (and this probably applies to many other WTs

> as well ... haven't looked yet) is that the ends of the

> tuning chain invoke the skhisma, which means in effect

> (assuming the ~2-cent difference falls outside the capability

> of a human tuner in Beethoven's day), that it's a closed tuning.

> Thus, a nice cylinder, perpendicular to the skhisma, which is

> *extremely* different from the meantone cylinder.

Schismic tuning, gets lots of discussion, usually with reference to

17-tone, 29-tone, 41-tone, 53-tone, and similar MOSs of often 1/8- to

1/9-schisma tempered fifths. But once you've created a finite 12-tone

periodicity block, such as a well-temperament, you can no longer

attribute much importance to the schisma, which is one of only an

infinite number of unison vectors which, in pairs, can produce the 12-

tone periodicity block (or a torsional "multiple" if you're not

careful).

> > ... -- why not let the meantones look pretty much the

> > same -- compositionally, they pretty much are, wouldn't you say?

>

>

> Yes, I think I can agree with that.

>

> But then again, why would a composer choose one particular

> meantone over any other?

It would depend on the criteria they chose to determine their

meantone. Are they concentrating on the fifths? Do they care about

the thirds only? Do they minimize maximum error, or total error? See

my table in your meantone definition page for examples of what

meantones different desiderata can lead to.

> There must be *some* reason/s

> ... so what's wrong with making a visual model of those choices

> too?

>

> Perhaps there are deeper things about the differences between

> different meantones that we haven't noticed yet. Having nice

> pictures of them would make it easier to find those as-yet

> undiscovered aspects, I think.

I could come up with lots of nice mathematical formulae for

differentiating them, but if none of them have any meaning that we

can understand, how can we choose one over another?

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

> > From: paulerlich <paul@s...>

> > To: <tuning-math@y...>

> > Sent: Wednesday, January 30, 2002 8:24 PM

> > Subject: [tuning-math] Re: new cylindrical meantone lattice

> >

> >

> > > > Why not show mistuning as a tiny "break" in the consonant

> > > > connections?

> > >

> > >

> > > Hmm ... sounds interesting. You mean like on the recent

Blackjack

> > > lattice you posted?

> >

> > Well, sort of. Except that all the consonant intervals would be

> > broken.

> >

> > > Please elaborate.

> >

> > Well, for example, you could do it like in Hall's hexagonal

lattices,

> > where he puts a little number representing the mistuning of each

> > interval, but you could do it visually, like representing each

> > consonant interval as a twig and putting a physical "break", the

size

> > of the mistuning, into it.

>

>

> Hmmm ... this sounds *really* interesting! I'd like to see

> more on this from you and the others who understand how to do it.

Seems simple enough. What more do you want? Did Hall do the hexagonal

ones in the article I sent you? Do I need to send you another Hall

article?

> > > 3) Use the mouse to roll the cylinder around on the lattice

> > > to get whichever key-center is desired, or simply input the

> > > key and let JustMusic do the rolling.

> >

> > Well this I was expecting anyway.

>

>

> Really? Hmmm ... if you were expecting it, then why'd you ask

> me the bit about choosing "a 1/1, such as C, which flies in the

> face of the true nature of meantone as a transposible system" ?

>

> If the user can transpose/roll the cylinder at will, what

> difference does it make which note is chosen as 1/1 ?

Because the *spiral* will be *pinned* to 1/1 no matter how we roll

the cylinder, correct?

----- Original Message -----

From: paulerlich <paul@stretch-music.com>

To: <tuning-math@yahoogroups.com>

Sent: Wednesday, January 30, 2002 9:33 PM

Subject: [tuning-math] Re: new cylindrical meantone lattice

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

> >

> > OK, in case there are other subtle distinctions between xenharmonic

> > bridges and unison-vectors, I'm going to stick with my terminology.

>

> You're depriving it of much-needed meaning. You'll have to revise

> your definition, then.

How so? You were the one who pointed out to me that my xenharmonic

bridge concept, while very similar, was not identical to Fokker's

unison-vector concept.

So since I know in my mind -- even if I'm not expressing it entirely

clearly to others, or in the Dictionary -- what a xenharmonic bridge

is, then I'll use that term when I mean that concept. Keep the criticism

and questions coming ... it should help me to hone the definition.

> > I'm saying that these two tunings *are* different (altho

> > the auditory system can't hear the difference), so the

> > xenharmonic bridge here *is* allowing the listener to accept

> > 2^(2/10) * 3^(-2/10) * 5^(3/10) to be the same as 2^(2pi+1)/4pi .

>

> Now you're claiming that our auditory system cares about these

> irrational ratios???

Huh? No, I'm not claiming that. As always, I believe that our

auditory system cares about fairly-low-integer/prime ratios.

I'm stating that LucyTuning is not the same as 3/10-comma

meantone. Sure, it sounds the same. But then why didn't Harrison

and Lucy simply write about 3/10-comma instead?, which I think would

be a lot easier to understand mathematically.

There is a difference, and there's a little tiny xenharmonic bridge

in effect which blurs that difference and allows us to accept them

as being exactly the same. *This* kind of fudging and blurring

is what I originally was trying to express with the xenharmonic bridge

concept.

> > Am I missing your point simply because pi is a number that cannot

> > be finitely quantized?

>

> No.

So then please try to elaborate more ... I still don't get

what you're saying.

> > > ... -- why not let the meantones look pretty much the

> > > same -- compositionally, they pretty much are, wouldn't you say?

> >

> >

> > Yes, I think I can agree with that.

> >

> > But then again, why would a composer choose one particular

> > meantone over any other?

>

> It would depend on the criteria they chose to determine their

> meantone. Are they concentrating on the fifths? Do they care about

> the thirds only? Do they minimize maximum error, or total error? See

> my table in your meantone definition page for examples of what

> meantones different desiderata can lead to.

Well there you go: there's the answer as to why the spirals

are useful!

By seeing how a particular meantone spirals around the

cylindrical lattice -- which still has the JI points and

distances marked on it, albeit warped a bit so as to fit around

the circle -- one can see which JI intervals it favors.

Thus, it can be seen from my 1/4-comma lattice at the bottom of

http://www.ixpres.com/interval/dict/meantone.htm

that 1/4-comma meantone gives all the "major 3rds" and "minor 6ths"

exactly.

If you refer to the dotted lines on these graphs,

http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm

which show the fraction-of-a-comma tempering of each meantone note,

and imagine that these lattices are wrapped around a cylinder,

then for the "tonic" chord:

- it could be seen from the spiral of 1/3-comma meantone that

it gives the "minor 3rd" and "major 6th" exactly;

- it could be seen from the 2/7-comma spiral that both the "major"

and "minor" "3rd" and "6th" all have an equal amount of error.

- 1/5-comma favors the "5ths/4ths" and "major 3rd / minor 6th"

(which have the same error) over the "minor 3rd / major 6th";

Etc.

-monz

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> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Wednesday, January 30, 2002 9:35 PM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

> > > Well, for example, you could do it like in Hall's hexagonal

> > > lattices, where he puts a little number representing the

> > > mistuning of each interval, but you could do it visually,

> > > like representing each consonant interval as a twig and

> > > putting a physical "break", the size of the mistuning, into it.

> >

> >

> > Hmmm ... this sounds *really* interesting! I'd like to see

> > more on this from you and the others who understand how to do it.

>

> Seems simple enough. What more do you want? Did Hall do the hexagonal

> ones in the article I sent you? Do I need to send you another Hall

> article?

Yes, he did the hexagonal ones.

Does he have more articles that I'd find of interest?

What I meant was: perhaps you could make a quick-and-dirty graphic

illustrating that "twig" idea. That was what I found interesting.

> > > > 3) Use the mouse to roll the cylinder around on the lattice

> > > > to get whichever key-center is desired, or simply input the

> > > > key and let JustMusic do the rolling.

> > >

> > > Well this I was expecting anyway.

> >

> >

> > Really? Hmmm ... if you were expecting it, then why'd you ask

> > me the bit about choosing "a 1/1, such as C, which flies in the

> > face of the true nature of meantone as a transposible system" ?

> >

> > If the user can transpose/roll the cylinder at will, what

> > difference does it make which note is chosen as 1/1 ?

>

> Because the *spiral* will be *pinned* to 1/1 no matter how we roll

> the cylinder, correct?

Doesn't have to be. It could be shifted along either axis by any

fraction of 3 or 5 or both that you'd like. It doesn't affect the

*relative* relationships between the meantone and the JI, only the

specific relationships between specific pairs of notes.

-monz

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> From: monz <joemonz@yahoo.com>

> To: <tuning-math@yahoogroups.com>

> Cc: <justmusic@yahoogroups.com>; Ken Fasano <kenfasano@hotmail.com>

> Sent: Thursday, January 31, 2002 1:30 AM

> Subject: Re: [tuning-math] Re: new cylindrical meantone lattice

>

>

> > > [me, monz]

> > > If the user can transpose/roll the cylinder at will, what

> > > difference does it make which note is chosen as 1/1 ?

> >

> > [Paul]

> > Because the *spiral* will be *pinned* to 1/1 no matter how we roll

> > the cylinder, correct?

>

> [monz]

> Doesn't have to be. It could be shifted along either axis by any

> fraction of 3 or 5 or both that you'd like. It doesn't affect the

> *relative* relationships between the meantone and the JI, only the

> specific relationships between specific pairs of notes.

There's a flat-lattice example of this at the bottom of

http://www.ixpres.com/interval/monzo/meantone/lattices/PB-MT.htm

The last two pictures on the page shows a Duodene JI

periodicity-block with 2/9-comma meantone latticed within it.

The first picture has the bounding unison-vectors and the

meantone both centered on 1/1. But the Duodene ends up having

three pitches lying on the right (i.e., positive-3) boundary,

one of which is at a corner. So all three of them may be

exchanged for pitches a comma away, which would fall on

the other boundary, and either of the two corner pitches

may be exchanged for pitches a diesis away, which would

fall on the opposite corners.

So I slid things around a bit.

About the last picture, I quote from the page:

>> If one keeps the JI pitches in place, and moves the bounding

>> unison-vectors and the meantone 1/2-step to the right along

>> the 3-axis, so that the boundaries enclose only 12

>> pitches (the minimal set for this pair of unison-vectors)

>> and the meantone is exactly centered and symmetrical to those

>> 12 pitches, the entire system is centered and symmetrical

>> around the ratio 3^(1/2), or in terms of "8ve"-equivalent

>> ratios, the square-root of 3/2.

>>

>> In fact this structure more accurately portrays what the

>> meantone really represents: because of the disappearance

>> of the syntonic comma unison-vector, this flat lattice

>> should be imagined to wrap around as a cylinder, so that

>> the right and left edges connect. Thus the centered

>> meantone may imply either of any pair of pitches which

>> would be separated by a comma on the flat lattice, and

>> each of those pairs of points on the flat lattice map to

>> the same point on a cylinder.

-monz

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> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Wednesday, January 30, 2002 8:17 PM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

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

>

> > > Not really.

> >

> >

> > Huh? Take a look at the "ratios" I post below.

> > 2^(2/10) * 3^(-2/10) * 5^(3/10) is *mathematically*

> > totally different from 2^(2pi+1)/4pi, even tho they're

> > acoustically identical.

>

> Again, Monz, you're looking too closely at the way the math looks,

> and not thinking about what it means.

>

> I could write pi in fifteen different ways that look totally

> mathematically different to you, but they're all be the same.

>

>

> ...

>

> > >> 3/10-comma quasi-meantone "5th"

> > >> "-" Lucytuning "5th"

> > >> ---------------------------------

> > >> ~0.010148131 cent = ~1/99 cent

> > >>

> > >> =

> > >>

> > >> 2^(2/10) * 3^(-2/10) * 5^(3/10)

> > >> - 2^(2pi+1)/4pi

> > >> ------------------------------------------

> > >> 2^(-12pi-10)/40pi * 3^(-2/10) * 5^(3/10)

> >

> >

> > That last number is the "ratio" of the tiny xenharmonic bridge

> > I'm talking about.

>

> That's no kind of unison vector, Monz. All that is is a tiny

> difference between two functionally identical (as meantones in 5-

> limit) tuning systems. A real unison vector expresses a relationship

> within _one_ tuning system that allows it to take a pitch in more

> than one sense (and thus lead you on the road to finity). This

> interval you're talking about does not lead you in that direction at

> all (as you claimed above, in case you're wondering what I'm rambling

> on about).

>

> > If I could find some way to represent LucyTuning on the flat

> > lattice (which means finding some way to represent pi in a

> > universe where everything is factored by 3 and 5), then bend

> > the lattice into a meantone cylinder, then warp the cylinder

> > so that the LucyTuning "5th" occupies the same point as the

> > 3/10-comma meantone "5th", I'd have it.

> >

> > The fact that pi is transcendental, irrational, whatever,

> > makes it hard for me to figure out how to do this.

>

> If you want people to help you figure this out, you'll have to

> determine what the above construction is all about. What does it

> mean? Why do we want to see it on our perfectly good lattices-wrapped-

> onto-cylinders?

Paul, *you* were the one who kept nagging me about "... but how

are you going to lattice LucyTuning or Golden Meantone on this

kind of lattice?".

So if your point is that LucyTuning and 3/10-comma meantone

are "functionally identical", then I can simply lattice

LucyTuning *as* 3/10-comma meantone and call it a day.

Now, about that bit where I wrote: "... finding some way to

represent pi in a universe where everything is factored by

3 and 5", I had an idea for a simpler beginning approach.

Let's start with meantone-like EDOs instead. As an example,

we want to lattice 1/3-comma meantone alongside 19-EDO.

Can you devise some formula that would find the fractional

powers of 3 and 5 that would be needed to plot 19-EDO on a

trajectory that would follow closely alongside the 1/3-comma

trajectory? Now, I think *that* would be a meaningful lattice!

-monz

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

> > You're depriving it of much-needed meaning. You'll have to revise

> > your definition, then.

>

>

> How so? You were the one who pointed out to me that my xenharmonic

> bridge concept, while very similar, was not identical to Fokker's

> unison-vector concept.

Yes, but at least before, you could still use xenharmonic bridges to

imply finity.

> > > I'm saying that these two tunings *are* different (altho

> > > the auditory system can't hear the difference), so the

> > > xenharmonic bridge here *is* allowing the listener to accept

> > > 2^(2/10) * 3^(-2/10) * 5^(3/10) to be the same as 2^

(2pi+1)/4pi .

> >

> > Now you're claiming that our auditory system cares about these

> > irrational ratios???

>

>

> Huh? No, I'm not claiming that. As always, I believe that our

> auditory system cares about fairly-low-integer/prime ratios.

So what kind of "acceptance" are you talking about above???

> I'm stating that LucyTuning is not the same as 3/10-comma

> meantone. Sure, it sounds the same.

Well, it's _slightly_ different.

But then why didn't Harrison

> and Lucy simply write about 3/10-comma instead?, which I think would

> be a lot easier to understand mathematically.

Because they liked pi!

> There is a difference, and there's a little tiny xenharmonic bridge

> in effect which blurs that difference and allows us to accept them

> as being exactly the same. *This* kind of fudging and blurring

> is what I originally was trying to express with the xenharmonic

bridge

> concept.

But it doesn't induce *finity* unless the fudging is between two

different constructions from the same set of basis intervals (usually

primes).

> > > Am I missing your point simply because pi is a number that

cannot

> > > be finitely quantized?

> >

> > No.

>

>

> So then please try to elaborate more ... I still don't get

> what you're saying.

You claimed this whole xenharmonic bridge stuff was relevant to

finity, and I'm arguing that here, it's not.

> > > > ... -- why not let the meantones look pretty much the

> > > > same -- compositionally, they pretty much are, wouldn't you

say?

> > >

> > >

> > > Yes, I think I can agree with that.

> > >

> > > But then again, why would a composer choose one particular

> > > meantone over any other?

> >

> > It would depend on the criteria they chose to determine their

> > meantone. Are they concentrating on the fifths? Do they care

about

> > the thirds only? Do they minimize maximum error, or total error?

See

> > my table in your meantone definition page for examples of what

> > meantones different desiderata can lead to.

>

>

> Well there you go: there's the answer as to why the spirals

> are useful!

>

> By seeing how a particular meantone spirals around the

> cylindrical lattice -- which still has the JI points and

> distances marked on it, albeit warped a bit so as to fit around

> the circle -- one can see which JI intervals it favors.

>

>

> Thus, it can be seen from my 1/4-comma lattice at the bottom of

> http://www.ixpres.com/interval/dict/meantone.htm

>

> that 1/4-comma meantone gives all the "major 3rds" and "minor 6ths"

> exactly.

>

>

> If you refer to the dotted lines on these graphs,

> http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm

>

> which show the fraction-of-a-comma tempering of each meantone note,

> and imagine that these lattices are wrapped around a cylinder,

> then for the "tonic" chord:

>

> - it could be seen from the spiral of 1/3-comma meantone that

> it gives the "minor 3rd" and "major 6th" exactly;

>

> - it could be seen from the 2/7-comma spiral that both the "major"

> and "minor" "3rd" and "6th" all have an equal amount of error.

>

> - 1/5-comma favors the "5ths/4ths" and "major 3rd / minor 6th"

> (which have the same error) over the "minor 3rd / major 6th";

>

> Etc.

So the _angle_ is meaningful.

But I still have problems with

(a) the density of points along the line, which doesn't appear to be

meaningful;

(b) the fact that you have to pin the spiral to a particular "1/1"

origin, which ruins the rotational symmetry of the cylindrical

meantone lattice

(c) the fact that you can't yet plot non-rational-fraction-of-a-comma

meantones this way, though the _angle_ part should be just as

meaningful for those.

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

> Yes, he did the hexagonal ones.

Oh, good.

> Does he have more articles that I'd find of interest?

Sure -- also his book, _Musical Acoustics_.

> What I meant was: perhaps you could make a quick-and-dirty graphic

> illustrating that "twig" idea. That was what I found interesting.

Use your imagination!

> > Because the *spiral* will be *pinned* to 1/1 no matter how we

roll

> > the cylinder, correct?

>

>

> Doesn't have to be. It could be shifted along either axis by any

> fraction of 3 or 5 or both that you'd like. It doesn't affect the

> *relative* relationships between the meantone and the JI,

Well, then, I think it should be *optional*, because for me, it's

more relevant to see the relationships between the notes on the

rolled lattice, without any differentiation.

> only the

> specific relationships between specific pairs of notes.

How do you mean?

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

> >> If one keeps the JI pitches in place, and moves the bounding

> >> unison-vectors and the meantone 1/2-step to the right along

> >> the 3-axis, so that the boundaries enclose only 12

> >> pitches (the minimal set for this pair of unison-vectors)

> >> and the meantone is exactly centered and symmetrical to those

> >> 12 pitches, the entire system is centered and symmetrical

> >> around the ratio 3^(1/2), or in terms of "8ve"-equivalent

> >> ratios, the square-root of 3/2.

That's great.

> >> In fact this structure more accurately portrays what the

> >> meantone really represents:

No way, Jose.

> >> because of the disappearance

> >> of the syntonic comma unison-vector, this flat lattice

> >> should be imagined to wrap around as a cylinder, so that

> >> the right and left edges connect. Thus the centered

> >> meantone may imply either of any pair of pitches which

> >> would be separated by a comma on the flat lattice, and

> >> each of those pairs of points on the flat lattice map to

> >> the same point on a cylinder.

This is true also for an infinite number of pitches on the flat

lattice that are even further from the line. So the centering of the

meantone line between 1/1 and 3/2 is irrelevant. So is the use of 2/9-

comma meantone as opposed to any other meantone, as far as I can tell.

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

>

> Paul, *you* were the one who kept nagging me about "... but how

> are you going to lattice LucyTuning or Golden Meantone on this

> kind of lattice?".

>

> So if your point is that LucyTuning and 3/10-comma meantone

> are "functionally identical", then I can simply lattice

> LucyTuning *as* 3/10-comma meantone and call it a day.

Not quite -- it would have to be *very slightly different*.

>

>

>

> Now, about that bit where I wrote: "... finding some way to

> represent pi in a universe where everything is factored by

> 3 and 5", I had an idea for a simpler beginning approach.

>

> Let's start with meantone-like EDOs instead. As an example,

> we want to lattice 1/3-comma meantone alongside 19-EDO.

>

> Can you devise some formula that would find the fractional

> powers of 3 and 5 that would be needed to plot 19-EDO on a

> trajectory that would follow closely alongside the 1/3-comma

> trajectory? Now, I think *that* would be a meaningful lattice!

OK, I'm in favor of thinking in this direction, as it addresses at

least one of the three objections I raised in a post to you a few

minutes ago.

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Thursday, January 31, 2002 1:05 PM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

> > >> because of the disappearance

> > >> of the syntonic comma unison-vector, this flat lattice

> > >> should be imagined to wrap around as a cylinder, so that

> > >> the right and left edges connect. Thus the centered

> > >> meantone may imply either of any pair of pitches which

> > >> would be separated by a comma on the flat lattice, and

> > >> each of those pairs of points on the flat lattice map to

> > >> the same point on a cylinder.

>

> This is true also for an infinite number of pitches on the

> flat lattice that are even further from the line. So the

> centering of the meantone line between 1/1 and 3/2 is

> irrelevant. So is the use of 2/9-comma meantone as opposed

> to any other meantone, as far as I can tell.

I don't know about that, Paul. My intuition tells me that

if composers choose particular flavors of meantone based on

criteria such as the amount-of-error-from-JI relationships

among the basic consonant intervals (M3/m6, m3/M6, p4/p5),

then they *do* intend to emphasize/deemphasize specific

JI intervals, because the meantone they choose will do that.

The centering of the meantone line -- which I prefer to call

a spiral since it belongs on a cylindrical lattice -- within

the PB, distributes the amount of error from JI as evenly as

possible among the intervals closest to the 1/1. That seems

to me to be something with an actual musical application.

-monz

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----- Original Message -----

From: paulerlich <paul@stretch-music.com>

To: <tuning-math@yahoogroups.com>

Sent: Thursday, January 31, 2002 1:12 PM

Subject: [tuning-math] Re: new cylindrical meantone lattice

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

> >

> > Paul, *you* were the one who kept nagging me about "... but how

> > are you going to lattice LucyTuning or Golden Meantone on this

> > kind of lattice?".

> >

> > So if your point is that LucyTuning and 3/10-comma meantone

> > are "functionally identical", then I can simply lattice

> > LucyTuning *as* 3/10-comma meantone and call it a day.

>

> Not quite -- it would have to be *very slightly different*.

OK, I think we're getting somewhere with this.

> > Now, about that bit where I wrote: "... finding some way to

> > represent pi in a universe where everything is factored by

> > 3 and 5", I had an idea for a simpler beginning approach.

> >

> > Let's start with meantone-like EDOs instead. As an example,

> > we want to lattice 1/3-comma meantone alongside 19-EDO.

> >

> > Can you devise some formula that would find the fractional

> > powers of 3 and 5 that would be needed to plot 19-EDO on a

> > trajectory that would follow closely alongside the 1/3-comma

> > trajectory? Now, I think *that* would be a meaningful lattice!

>

> OK, I'm in favor of thinking in this direction, as it addresses at

> least one of the three objections I raised in a post to you a few

> minutes ago.

And that objection would be this one, correct? :

> (c) the fact that you can't yet plot

> non-rational-fraction-of-a-comma meantones this way,

> though the _angle_ part should be just as meaningful

> for those.

So, how about a formula that plots 19-EDO as, literally,

a close cousin to 1/3-comma meantone spiral? How does

take something that's roots of 2, and change it into

"8ve"-equivalent fractional powers of 3 and 5?

Now about your other two objections:

> (a) the density of points along the line, which doesn't

> appear to be meaningful;

I'm hoping that the post I just sent before this one,

about composer choosing particular flavors of meantone,

addresses this one.

> (b) the fact that you have to pin the spiral to a particular

> "1/1" origin, which ruins the rotational symmetry of the

> cylindrical meantone lattice

I've already said elsewhere that the spiral doesn't have

to be pinned to anything. It can float anywhere the user

wants it. What's important is the angle of the spiral,

as you've noted.

-monz

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

> I don't know about that, Paul. My intuition tells me that

> if composers choose particular flavors of meantone based on

> criteria such as the amount-of-error-from-JI relationships

> among the basic consonant intervals (M3/m6, m3/M6, p4/p5),

> then they *do* intend to emphasize/deemphasize specific

> JI intervals, because the meantone they choose will do that.

>

> The centering of the meantone line -- which I prefer to call

> a spiral since it belongs on a cylindrical lattice -- within

> the PB, distributes the amount of error from JI as evenly as

> possible among the intervals closest to the 1/1. That seems

> to me to be something with an actual musical application.

So you're deriving 2/9-comma meantone as an optimal meantone? What

weights are you using to do so? I showed the meantones that I could

easily derive from various weightings of the basic consonant

intervals (M3/m6, m3/M6, p4/p5), and 2/9-comma wasn't one of

them . . .

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

> > OK, I'm in favor of thinking in this direction, as it addresses

at

> > least one of the three objections I raised in a post to you a few

> > minutes ago.

>

>

> And that objection would be this one, correct? :

>

> > (c) the fact that you can't yet plot

> > non-rational-fraction-of-a-comma meantones this way,

> > though the _angle_ part should be just as meaningful

> > for those.

>

>

> So, how about a formula that plots 19-EDO as, literally,

> a close cousin to 1/3-comma meantone spiral? How does

> take something that's roots of 2, and change it into

> "8ve"-equivalent fractional powers of 3 and 5?

Well, maybe there's another way to get the right spiral.

> Now about your other two objections:

>

>

> > (a) the density of points along the line, which doesn't

> > appear to be meaningful;

>

>

> I'm hoping that the post I just sent before this one,

> about composer choosing particular flavors of meantone,

> addresses this one.

Not at all -- I was referring to the fact that, for example, in 5/18-

comma meantone, the points on the spiral are rather far apart from

one another -- that doesn't seem particularly meaningful.

> > (b) the fact that you have to pin the spiral to a particular

> > "1/1" origin, which ruins the rotational symmetry of the

> > cylindrical meantone lattice

>

>

> I've already said elsewhere that the spiral doesn't have

> to be pinned to anything. It can float anywhere the user

> wants it. What's important is the angle of the spiral,

> as you've noted.

So maybe a set of arrows (say from every _true_ lattice point)

pointing at that angle would be preferable to a spiral.

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Friday, February 01, 2002 1:03 AM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

>

> So you're deriving 2/9-comma meantone as an optimal meantone?

Optimal for the *Duodene*, specifically.

And I'm not really sure if it's an optimal, just guessing.

> What weights are you using to do so? I showed the meantones

> that I could easily derive from various weightings of the basic

> consonant intervals (M3/m6, m3/M6, p4/p5), and 2/9-comma wasn't

> one of them . . .

Didn't do it that way at all. Simply looked at the lattice

of the shifted-boundary Duodene PB and saw that 2/9-comma

slashed right across the middle of it. Since the angle

of the meantone line on the flat lattice (and of the spiral

on the cylindrical) graphically shows the tempering of the

meantone in relation to the nearest JI pitches, I moved it

around until it was centered perfectly within the shifted PB,

and it seemed to distribute the error the most evenly.

So I suppose it's only optimal for the intervals within

the Duodene PB. If that's the case, then it says to me

that 2/9-comma meantone will tend to imply JI tonal

structures which derive from the Duodene.

-monz

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

>

> > From: paulerlich <paul@s...>

> > To: <tuning-math@y...>

> > Sent: Friday, February 01, 2002 1:03 AM

> > Subject: [tuning-math] Re: new cylindrical meantone lattice

> >

> >

> >

> > So you're deriving 2/9-comma meantone as an optimal meantone?

>

>

> Optimal for the *Duodene*, specifically.

> And I'm not really sure if it's an optimal, just guessing.

>

>

> > What weights are you using to do so? I showed the meantones

> > that I could easily derive from various weightings of the basic

> > consonant intervals (M3/m6, m3/M6, p4/p5), and 2/9-comma wasn't

> > one of them . . .

>

>

> Didn't do it that way at all. Simply looked at the lattice

> of the shifted-boundary Duodene PB and saw that 2/9-comma

> slashed right across the middle of it.

Hmm . . . so you're not using only the consonant intervals, as you

said you were.

Anyway, can you show me how it slashes right down the middle, which

some other meantone doesn't?

> Since the angle

> of the meantone line on the flat lattice (and of the spiral

> on the cylindrical) graphically shows the tempering of the

> meantone in relation to the nearest JI pitches, I moved it

> around until it was centered perfectly within the shifted PB,

> and it seemed to distribute the error the most evenly.

Couldn't any other meantone do exactly the same thing?

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Friday, February 01, 2002 1:06 AM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

> > So, how about a formula that plots 19-EDO as, literally,

> > a close cousin to 1/3-comma meantone spiral? How does

> > take something that's roots of 2, and change it into

> > "8ve"-equivalent fractional powers of 3 and 5?

>

> Well, maybe there's another way to get the right spiral.

I'm all ears.

>

> > Now about your other two objections:

> >

> >

> > > (a) the density of points along the line, which doesn't

> > > appear to be meaningful;

> >

> >

> > I'm hoping that the post I just sent before this one,

> > about composer choosing particular flavors of meantone,

> > addresses this one.

>

> Not at all -- I was referring to the fact that, for example,

> in 5/18-comma meantone, the points on the spiral are rather

> far apart from one another -- that doesn't seem particularly

> meaningful.

OK, the only way I can respond to this properly is to go ahead

and create a 5/18-comma lattice and examine it. That's not going

to happen until tomorrow. But try to remember this if I don't

respond soon, because I do actually want to make that lattice

and have a look, and try to answer you.

>

> > > (b) the fact that you have to pin the spiral to a particular

> > > "1/1" origin, which ruins the rotational symmetry of the

> > > cylindrical meantone lattice

> >

> >

> > I've already said elsewhere that the spiral doesn't have

> > to be pinned to anything. It can float anywhere the user

> > wants it. What's important is the angle of the spiral,

> > as you've noted.

>

> So maybe a set of arrows (say from every _true_ lattice point)

> pointing at that angle would be preferable to a spiral.

Well, I think arrows are a good idea, sure. But again,

I'd leave the choice of spirals or arrows up to the user.

The whole idea behind JustMusic is to give people a visual

way to create music, by manipulating graphs of the sonic math.

In effect, the computer screen becomes a lattice instrument.

So I want it to be as flexible as possible, and want to

leave as many decisions as possible to the user.

-monz

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

>

> > From: paulerlich <paul@s...>

> > To: <tuning-math@y...>

> > Sent: Friday, February 01, 2002 1:06 AM

> > Subject: [tuning-math] Re: new cylindrical meantone lattice

> >

> >

> > > So, how about a formula that plots 19-EDO as, literally,

> > > a close cousin to 1/3-comma meantone spiral? How does

> > > take something that's roots of 2, and change it into

> > > "8ve"-equivalent fractional powers of 3 and 5?

> >

> > Well, maybe there's another way to get the right spiral.

>

>

> I'm all ears.

Maybe Gene can help.

>

>

> >

> > > Now about your other two objections:

> > >

> > >

> > > > (a) the density of points along the line, which doesn't

> > > > appear to be meaningful;

> > >

> > >

> > > I'm hoping that the post I just sent before this one,

> > > about composer choosing particular flavors of meantone,

> > > addresses this one.

> >

> > Not at all -- I was referring to the fact that, for example,

> > in 5/18-comma meantone, the points on the spiral are rather

> > far apart from one another -- that doesn't seem particularly

> > meaningful.

>

>

> OK, the only way I can respond to this properly is to go ahead

> and create a 5/18-comma lattice and examine it. That's not going

> to happen until tomorrow.

It's already on your meantone webpage applet!!

> >

> > > > (b) the fact that you have to pin the spiral to a particular

> > > > "1/1" origin, which ruins the rotational symmetry of the

> > > > cylindrical meantone lattice

> > >

> > >

> > > I've already said elsewhere that the spiral doesn't have

> > > to be pinned to anything. It can float anywhere the user

> > > wants it. What's important is the angle of the spiral,

> > > as you've noted.

> >

> > So maybe a set of arrows (say from every _true_ lattice point)

> > pointing at that angle would be preferable to a spiral.

>

>

> Well, I think arrows are a good idea, sure. But again,

> I'd leave the choice of spirals or arrows up to the user.

Cool!

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Friday, February 01, 2002 1:15 AM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

> > [me, monz]

> > Didn't do it that way at all. Simply looked at the lattice

> > of the shifted-boundary Duodene PB and saw that 2/9-comma

> > slashed right across the middle of it.

>

> Hmm . . . so you're not using only the consonant intervals, as you

> said you were.

Did I say that?!

I suppose what I'm really doing is basing the position of the

meantone on the position of the defining unison-vectors.

>

> Anyway, can you show me how it slashes right down the middle, which

> some other meantone doesn't?

>

> > Since the angle

> > of the meantone line on the flat lattice (and of the spiral

> > on the cylindrical) graphically shows the tempering of the

> > meantone in relation to the nearest JI pitches, I moved it

> > around until it was centered perfectly within the shifted PB,

> > and it seemed to distribute the error the most evenly.

>

> Couldn't any other meantone do exactly the same thing?

Yeah, actually, I think you're right about that. Too tired

to see it now ... I'll have to make several of these and compare

them. So most likely I'll just keep adding more to that

Duodene webpage, and you can give me feedback from that

when you see them.

-monz

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

>

> > From: paulerlich <paul@s...>

> > To: <tuning-math@y...>

> > Sent: Friday, February 01, 2002 1:15 AM

> > Subject: [tuning-math] Re: new cylindrical meantone lattice

> >

> >

> > > [me, monz]

> > > Didn't do it that way at all. Simply looked at the lattice

> > > of the shifted-boundary Duodene PB and saw that 2/9-comma

> > > slashed right across the middle of it.

> >

> > Hmm . . . so you're not using only the consonant intervals, as

you

> > said you were.

>

>

> Did I say that?!

>

> I suppose what I'm really doing is basing the position of the

> meantone on the position of the defining unison-vectors.

OK . . . then it doesn't depend on the intervals _or_ on the pitches

in the scale. For there are other ways to "capture" the Duodene than

with the parallelogram of the two unison vectors you've chosen.

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Friday, February 01, 2002 1:27 AM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

> > I suppose what I'm really doing is basing the position of the

> > meantone on the position of the defining unison-vectors.

>

> OK . . . then it doesn't depend on the intervals _or_ on the pitches

> in the scale. For there are other ways to "capture" the Duodene than

> with the parallelogram of the two unison vectors you've chosen.

Really?! Do tell!

I had a hunch that there might be some hexagonal PBs that

define the Duodene as well, and upon looking now at

http://www.ixpres.com/interval/td/erlich/intropblockex.htm

I can see how the hexagonal PB in the bottom graphic could

be shifted slightly to enclose the Duodene.

But are there any other unison-vectors that will enclose it?

Or is [4 -1],[0 -3],[-4 -2] the only set from which any two

will create the Duodene?

-monz

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

> Really?! Do tell!

>

> I had a hunch that there might be some hexagonal PBs that

> define the Duodene as well, and upon looking now at

> http://www.ixpres.com/interval/td/erlich/intropblockex.htm

> I can see how the hexagonal PB in the bottom graphic could

> be shifted slightly to enclose the Duodene.

There you go!

> But are there any other unison-vectors that will enclose it?

> Or is [4 -1],[0 -3],[-4 -2] the only set from which any two

> will create the Duodene?

Probably -- but you only used two of those. And anyway, we may have

no reason for talking about this, because you seem to concur that

perhaps any meantone would do what 2/9-comma did even for the block

in question.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> you seem to concur that

> perhaps any meantone would do what 2/9-comma did even for the block

> in question.

I mean even for the parallelogram in question.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> Maybe Gene can help.

I haven't been thinking about the spirals, because I don't know what they are for. I have a suggestion for JustMusic lattices, which is to take lattices of higher dimensions, project them onto a plane and give a way of rotating the lattice to look at different projections--some sort of "knobs" you can "turn", giving you orthogonal matricies.

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Friday, February 01, 2002 1:42 AM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> >

> > you seem to concur that perhaps any meantone would do

> > what 2/9-comma did even for the block

> > in question.

>

> I mean even for the parallelogram in question.

Well, it makes sense. But as I said, I think I'd like

to make some different lattices and compare them.

Still, even if you feel that this "choice of meantone"

thing is a moot point, I still think it's fascinating

to lattice the meantones this way and *see* their

deviation from JI.

Another thing I like is that since 12-EDO is ~= 1/11-comma

meantone, it too can be plotted and examined in this way.

That's something I'd like to study for a bit.

-monz

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

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > Maybe Gene can help.

>

> I haven't been thinking about the spirals, because I don't know

>what they are for. I have a suggestion for JustMusic lattices, which

>is to take lattices of higher dimensions, project them onto a plane

>and give a way of rotating the lattice to look at different

>projections--some sort of "knobs" you can "turn",

Note the 5 "knobs" you can "turn" in Dave Keenan's 4-dimensional

Tumbling Dekany:

http://www.uq.net.au/~zzdkeena/Music/StereoDekany.xls

So you mean like this?

> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Friday, February 01, 2002 2:05 AM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > Maybe Gene can help.

>

> I haven't been thinking about the spirals, because I don't

> know what they are for. I have a suggestion for JustMusic

> lattices, which is to take lattices of higher dimensions,

> project them onto a plane and give a way of rotating the

> lattice to look at different projections--some sort of

> "knobs" you can "turn", giving you orthogonal matricies.

Thanks, Gene. This has been part of the desiderata ever

since I figured out how to lattice >3 dimensions, c. 1998.

-monz

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> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Friday, February 01, 2002 1:21 AM

> Subject: [tuning-math] Re: new cylindrical meantone lattice

>

>

> > > > Now about your other two objections:

> > > >

> > > >

> > > > > (a) the density of points along the line, which doesn't

> > > > > appear to be meaningful;

> > > >

> > > >

> > > > I'm hoping that the post I just sent before this one,

> > > > about composer choosing particular flavors of meantone,

> > > > addresses this one.

> > >

> > > Not at all -- I was referring to the fact that, for example,

> > > in 5/18-comma meantone, the points on the spiral are rather

> > > far apart from one another -- that doesn't seem particularly

> > > meaningful.

> >

> >

> > OK, the only way I can respond to this properly is to go ahead

> > and create a 5/18-comma lattice and examine it. That's not going

> > to happen until tomorrow.

>

> It's already on your meantone webpage applet!!

Ah, OK ... I really need to clean this page up.

You're confusing an older idea I had, which I should probably

clarify or maybe even delete, with what I'm talking about now.

That applet was an idea that I had, against which you argued

quite strongly, and I pretty much agree that there's not a

whole lot of meaning in it.

The significant thing is *does* show is the *angle* of the

meantone, which corresponds to the angle of spiral on the cylinder.

The *insignificant* part of it is the distribution of points

along those lines. Those are merely the chains of JI pitches

that the meantones give exactly, which have no significance at

all on a flat lattice like this ... other than that one plotted

step in each direction (positive and negative) along the meantone

axis shows the single pair of complementary JI intervals which

are given exactly and really *are* audibly significant in the

meantone.

The distribution of points that really matters is that of the

chain of meantone generators, which can be seen on the cylindrical

lattice at the bottom of the "meantone" webpage, and here:

http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm

And those don't vary a lot ... only a tiny bit of variation in

lattice taxicab step-size from one meantone to the next, within

a fairly narrow overall range.

Since the meantone cylinder is exactly perpendicular to the

syntonic-comma metric, the circles which ring the cylinder

and represent those commas for each pair of JI pitches are

equidistant.

The meantone generators are all plotted at some point along

each of these circles, the exact point being determined by

the intersection of the meantone spiral with the syntonic-comma

circle.

Thus, the only difference in the plotting of the chains of

generators (along the spirals) between various meantones is

the slight change in spacing of those points due to the

varying angle of the meantone spiral.

So in fact, on my newer meantone cylinder lattices, the density

of points along the spiral really doesn't change all that much.

Does that clear up your first objection, and thus all three?

-monz

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