Yahoo Tuning Groups Ultimate Backup tuning-math finding a moat in 7-limit commas a bit tougher . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > /tuning-math/files/Erlich/planar.gif
> >
> > Paul,
> >
> > Please do another one of these without the labels, so we have a
> chance
> > of eyeballing the moats.
>
> /tuning-math/files/Erlich/planar0.gif

Thanks Paul. Fascinating to look at, isn't it. So organic. Some order,
some randomness.

I think that planar temperaments are inherently less useful than
linear (which are less useful than equal). This is mostly due to the
melodic dimension, which Herman mentions all the time, but we are
completely ignoring (except in so far as harmonic complexity implies
melodic complexity). We are not measuring things like evenness and
transposability when deciding what is in and what is out. And that's
OK. We have to learn to crawl before we can walk.

But because planar are inherently less even and less transposable than
linear I think there are only a very few interesting or useful 7-limit
planars.

Since you favour linear moats, I suggest
50/49
49/48
64/63
81/80
126/125
225/224
245/243

🔗Paul Erlich <perlich@aya.yale.edu>

I completely agree if you replace "less useful" with "more complex".

> This is mostly due to the
> melodic dimension, which Herman mentions all the time, but we are
> completely ignoring (except in so far as harmonic complexity implies
> melodic complexity).

I disagree that it's about an ignored melodic dimension. Instead,
it's as I said before, these complexity values are not directly
comparable, because what's the length of an area? What's the area of
a volume.

> We are not measuring things like evenness and
> transposability when deciding what is in and what is out. And that's
> OK. We have to learn to crawl before we can walk.

Well, we're definitely agreed that a 7-limit planar temperament based
on a particular comma is quite a bit more complex than a 5-limit
linear temperament based on that same comma.

> But because planar are inherently less even and less transposable
than
> linear I think there are only a very few interesting or useful 7-
limit
> planars.

Sure. I kind of figured the ragismic planar deserved to be in there,
but I wouldn't insist on it.

> Since you favour linear moats,

Where did you get that idea? Curved is fine too.

> I suggest
> 50/49
> 49/48
> 64/63
> 81/80
> 126/125
> 225/224
> 245/243

I definitely wouldn't want to throw out 28/27, 36/35 . . .

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > /tuning-math/files/Erlich/planar.gif
> >
> > Paul,
> >
> > Please do another one of these without the labels, so we have a
> chance
> > of eyeballing the moats.
>
> My eyeballs are telling me the same thing as when the labels were
> there:

Maybe so, but it's also conceivable that one could be prejudiced by
the slope on the labels, so it's better not to have them when
moat-spotting. Although we still need the labelled one to refer to.

> /tuning-math/files/Erlich/myemoat.gif

Woah! 42 7-limit planars, when we only have 17 or 18 5-limit linears.
No way. I can see several linear moats less inclusive than that one
that are much wider (especially when the width is counted as the
percentage of the distance to the origin).

It would be good to draw two parallel lines showing the two sides of
the moat (and even fill with colour between them) so we can easily see
the moat width.

🔗Paul Erlich <perlich@aya.yale.edu>

I agree it's probably too large a number for the purposes of a paper,
but I'm not sure there have to be fewer than the number of 5-limit
linears. There is simply a greater variety in the 7-limit planar
world, which works against the fact that we'd use a considerably
lower bound on the most complex comma in 7-limit planar vs. 5-limit
linear.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > > wrote:
> > > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
> <perlich@a...>
> > > wrote:
> > > > >
> /tuning-math/files/Erlich/planar.gif
> > > >
> > > > Paul,
> > > >
> > > > Please do another one of these without the labels, so we have a
> > > chance
> > > > of eyeballing the moats.
> > >
> > >
> /tuning-math/files/Erlich/planar0.gif
> >
> > Thanks Paul. Fascinating to look at, isn't it. So organic. Some
> order,
> > some randomness.
> >
> > I think that planar temperaments are inherently less useful than
> > linear (which are less useful than equal).
>
> I completely agree if you replace "less useful" with "more complex".
>
> > This is mostly due to the
> > melodic dimension, which Herman mentions all the time, but we are
> > completely ignoring (except in so far as harmonic complexity implies
> > melodic complexity).
>
> I disagree that it's about an ignored melodic dimension. Instead,
> it's as I said before, these complexity values are not directly
> comparable, because what's the length of an area? What's the area of
> a volume.
>
> > We are not measuring things like evenness and
> > transposability when deciding what is in and what is out. And that's
> > OK. We have to learn to crawl before we can walk.
>
> Well, we're definitely agreed that a 7-limit planar temperament based
> on a particular comma is quite a bit more complex than a 5-limit
> linear temperament based on that same comma.
>
> > But because planar are inherently less even and less transposable
> than
> > linear I think there are only a very few interesting or useful 7-
> limit
> > planars.
>
> Sure. I kind of figured the ragismic planar deserved to be in there,
> but I wouldn't insist on it.
>
> > Since you favour linear moats,
>
> Where did you get that idea? Curved is fine too.

What range of exponents are acceptable to you? Isn't 1 near the
(geometric) middle of them?

>
> > I suggest
> > 50/49
> > 49/48
> > 64/63
> > 81/80
> > 126/125
> > 225/224
> > 245/243
>
> I definitely wouldn't want to throw out 28/27, 36/35 . . .

Gene, I hope you're happy I'm using slashes here. I agree there isn't
likely to be any confusion in this discussion since we're not talking
about individual pitches at all.

Why not. I have enough trouble wondering why anyone would use a
5-limit _linear_ temperament that was non-unique, 7-limit planar
stretches my credibility even further. Can you propose a scale or
finite tuning in these that you think might be useful as an
approximation of 7-limit JI?

Moat-wise, I can see my way to adding 36/35 and 128/125. That probably
gives the biggest moat possible (percentage-wise) particularly if you
use an exponent greater than 1. Unless you were to have one with an
exponent less than 1 (which I don't like) and go all the way up to
include 21/20 (which seems lidicrous to me).

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> > I definitely wouldn't want to throw out 28/27, 36/35 . . .
>
> Gene, I hope you're happy I'm using slashes here. I agree there
isn't
> likely to be any confusion in this discussion since we're not
talking
> about individual pitches at all.
>
> Why not. I have enough trouble wondering why anyone would use a
> 5-limit _linear_ temperament that was non-unique, 7-limit planar
> stretches my credibility even further. Can you propose a scale or
> finite tuning in these that you think might be useful as an
> approximation of 7-limit JI?

Not right now, must jet soon . . . This is Herman's department, or
maybe Gene's . . .

> Moat-wise, I can see my way to adding 36/35 and 128/125. That
probably
> gives the biggest moat possible (percentage-wise) particularly if
you
> use an exponent greater than 1. Unless you were to have one with an
> exponent less than 1 (which I don't like)

Maybe you'll reconsider when you look at the ET graphs I just posted.

> and go all the way up to
> include 21/20 (which seems lidicrous to me).

It doesn't seem that lidicrous :) to me . . . Seriously, I think all
kinds of novel effects could be obtained if 21/20 vanished, and if
you used full 1:2:3:4:5:6:7:8:9:10 chords, there would certainly be
no confusion over what the chords were 'representing' -- you might
simply have to use the kinds of timbres that George and I were
talking about . . . Maybe Herman would like to entertain us with some
sort of example . . .

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > I definitely wouldn't want to throw out 28/27, 36/35 . . .
> >
> > Gene, I hope you're happy I'm using slashes here. I agree there
> isn't
> > likely to be any confusion in this discussion since we're not
> talking
> > about individual pitches at all.
> >
> > Why not. I have enough trouble wondering why anyone would use a
> > 5-limit _linear_ temperament that was non-unique, 7-limit planar
> > stretches my credibility even further. Can you propose a scale or
> > finite tuning in these that you think might be useful as an
> > approximation of 7-limit JI?
>
> Not right now, must jet soon . . . This is Herman's department, or
> maybe Gene's . . .
>
> > Moat-wise, I can see my way to adding 36/35 and 128/125. That
> probably
> > gives the biggest moat possible (percentage-wise) particularly if
> you
> > use an exponent greater than 1. Unless you were to have one with an
> > exponent less than 1 (which I don't like)
>
> Maybe you'll reconsider when you look at the ET graphs I just posted.
>
> > and go all the way up to
> > include 21/20 (which seems lidicrous to me).
>
> It doesn't seem that lidicrous :) to me . . .

Definition of "lidicrous": so ludicrous that you can't type correctly. ;-)

> Seriously, I think all
> kinds of novel effects could be obtained if 21/20 vanished,

"All kinds of novel effects" is one thing and "approximating 7-limit
JI" is another.

> and if
> you used full 1:2:3:4:5:6:7:8:9:10 chords, there would certainly be
> no confusion over what the chords were 'representing' -- you might
> simply have to use the kinds of timbres that George and I were
> talking about . . . Maybe Herman would like to entertain us with some
> sort of example . . .

It's the lack of counterexamples I'm more worried about.

I understand you claim that 12-ET is an approximation of JI for all
limits.

If the obtaining of relative consonance by using timbres of poorly
defined pitch in massive otonalities is a sufficient criterion for
temperament-hood (JI approximation) then please give me a non-trivial
planar tuning that _doesn't_ work like that. Otherwise we have a
reductio ad absurdum.

By the way, the TOP tuning of the 21/20 planar temperament has the
following errors in the primes (to the nearest cent).
2 +10 c
3 -15 c
5 +23 c
7 -27 c

So we have the following large errors in certain intervals
2:3 -25 c
7:10 +60 c

3:4 +35 c
5:7 -50 c

3:5 +38 c
4:7 -47 c

The approximations of 3:4 and 5:7 are the same interval, so are the
approximations of 3:5 and 4:7, and 2:3 is the same as 7:10.

🔗Herman Miller <hmiller@IO.COM>

On Tue, 03 Feb 2004 01:58:33 -0000, "Paul Erlich" <perlich@aya.yale.edu>
wrote:

>> Why not. I have enough trouble wondering why anyone would use a
>> 5-limit _linear_ temperament that was non-unique, 7-limit planar
>> stretches my credibility even further. Can you propose a scale or
>> finite tuning in these that you think might be useful as an
>> approximation of 7-limit JI?
>
>Not right now, must jet soon . . . This is Herman's department, or
>maybe Gene's . . .

Tempering 36/35 gives you a perfectly symmetrical 4:5:6:7:9 chord, which
could be of some use; the 6/5 is the same as the 7/6. Looks similar to
12-ET; a 12-note subset might be appropriate.

TOP tuning: [1195.264647, 1894.449645, 2797.308862, 3382.119722]

The difference between this tuning and top dominant seventh is VERY minor.

TOP 36/35 TOP Dominant Seventh
9/8 203.105 203.47
5/4 406.78 406.94
3/2 699.185 699.35
5/3 902.859 902.82
7/4 991.59 991.76
2/1 1195.265 1195.23

Tempering out 28/27 gives you something close to 5-ET, but unevenly spaced;
a series of three fifths gives you a 7/2. Adding a just 5/1 gives you sharp
thirds like those of 15-ET.

TOP tuning: [1193.415676, 1912.390908, 2786.313714, 3350.341372]

TOP 28/27 TOP Blackwood
3/2 718.975 717.54
7/4 963.51 956.72
2/1 1193.416 1195.9

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > wrote:
> >
> > > > I definitely wouldn't want to throw out 28/27, 36/35 . . .
> > >
> > > Gene, I hope you're happy I'm using slashes here. I agree there
> > isn't
> > > likely to be any confusion in this discussion since we're not
> > talking
> > > about individual pitches at all.
> > >
> > > Why not. I have enough trouble wondering why anyone would use a
> > > 5-limit _linear_ temperament that was non-unique, 7-limit planar
> > > stretches my credibility even further. Can you propose a scale
or
> > > finite tuning in these that you think might be useful as an
> > > approximation of 7-limit JI?
> >
> > Not right now, must jet soon . . . This is Herman's department,
or
> > maybe Gene's . . .
> >
> > > Moat-wise, I can see my way to adding 36/35 and 128/125. That
> > probably
> > > gives the biggest moat possible (percentage-wise) particularly
if
> > you
> > > use an exponent greater than 1. Unless you were to have one
with an
> > > exponent less than 1 (which I don't like)
> >
> > Maybe you'll reconsider when you look at the ET graphs I just
posted.
> >
> > > and go all the way up to
> > > include 21/20 (which seems lidicrous to me).
> >
> > It doesn't seem that lidicrous :) to me . . .
>
> Definition of "lidicrous": so ludicrous that you can't type
correctly. ;-)
>
> > Seriously, I think all
> > kinds of novel effects could be obtained if 21/20 vanished,
>
> "All kinds of novel effects" is one thing and "approximating 7-limit
> JI" is another.
>
> > and if
> > you used full 1:2:3:4:5:6:7:8:9:10 chords, there would certainly
be
> > no confusion over what the chords were 'representing' -- you
might
> > simply have to use the kinds of timbres that George and I were
> > talking about . . . Maybe Herman would like to entertain us with
some
> > sort of example . . .
>
> It's the lack of counterexamples I'm more worried about.
>
> I understand you claim that 12-ET is an approximation of JI for all
> limits.

Well, I don't know about 13-limit, but in 11-limit for example, there
appear to be instances where it works, and 19-equal can work with two
different mappings (and thus two different "stretch" factors) in the
11-limit . . .

> If the obtaining of relative consonance by using timbres of poorly
> defined pitch in massive otonalities is a sufficient criterion for
> temperament-hood (JI approximation) then please give me a non-
trivial
> planar tuning that _doesn't_ work like that.

Since the consensus seems to be that father (16:15) doesn't work that
way, then the 16:15 and 15:14 planar temperaments probably won't work
that way either . . .

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > If the obtaining of relative consonance by using timbres of poorly
> > defined pitch in massive otonalities is a sufficient criterion for
> > temperament-hood (JI approximation) then please give me a non-
> trivial
> > planar tuning that _doesn't_ work like that.
>
> Since the consensus seems to be that father (16:15) doesn't work that
> way, then the 16:15 and 15:14 planar temperaments probably won't work
> that way either . . .

There is a consensus that "father" isn't an approximation of 5-limit
JI, but I don't think that is because anyone tried it with large
otonalities using inharmonic timbres. It may well work in that way.

Perhaps we should limit such tests to otonalities having at most one
note per prime (or odd) in the limit. e.g. If you can't make a
convincing major triad then it aint 5-limit. And you can't use
scale-spectrum timbres although you can use inharmonics that have no
relation to the scale.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> > > If the obtaining of relative consonance by using timbres of
poorly
> > > defined pitch in massive otonalities is a sufficient criterion
for
> > > temperament-hood (JI approximation) then please give me a non-
> > trivial
> > > planar tuning that _doesn't_ work like that.
> >
> > Since the consensus seems to be that father (16:15) doesn't work
that
> > way, then the 16:15 and 15:14 planar temperaments probably won't
work
> > that way either . . .
>
> There is a consensus that "father" isn't an approximation of 5-limit
> JI, but I don't think that is because anyone tried it with large
> otonalities using inharmonic timbres. It may well work in that way.
>
> Perhaps we should limit such tests to otonalities having at most one
> note per prime (or odd) in the limit. e.g. If you can't make a
> convincing major triad then it aint 5-limit. And you can't use
> scale-spectrum timbres although you can use inharmonics that have no
> relation to the scale.

yes, mastuuuhhhhh . . . =(

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > Perhaps we should limit such tests to otonalities having at most one
> > note per prime (or odd) in the limit. e.g. If you can't make a
> > convincing major triad then it aint 5-limit. And you can't use
> > scale-spectrum timbres although you can use inharmonics that have no
> > relation to the scale.
>
> yes, mastuuuhhhhh . . . =(

It was just a suggestion. I wrote "perhaps we should" and "e.g.".

What does "=(" mean?

I'm guessing you think it's a bad idea.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> > > Perhaps we should limit such tests to otonalities having at
most one
> > > note per prime (or odd) in the limit. e.g. If you can't make a
> > > convincing major triad then it aint 5-limit. And you can't use
> > > scale-spectrum timbres although you can use inharmonics that
have no
> > > relation to the scale.
> >
> > yes, mastuuuhhhhh . . . =(
>
> It was just a suggestion. I wrote "perhaps we should" and "e.g.".
>
> What does "=(" mean?
>
> I'm guessing you think it's a bad idea.

It's a picture of me succumbing to your authority.

🔗Dave Keenan <d.keenan@bigpond.net.au>

I don't have any, and I'd rather you succumbed to the good sense of my
arguments. :-)