The grooviest linear temperaments for 7-limit music

Gene, who's way, way ahead of any other theorist on this list (and
possibly anywhere) has (like Dave Keenan and Graham Breed before him)
completed a comprehensive search for linear temperaments for 7-limit
music. He proposed a 'badness' measure defined as

step^3 cent

where step is a measure of the typical number of notes in a scale for
this temperament (given any desired degree of harmonic depth), and
cent is a measure of the deviation from JI 'consonances' in cents.
He then ranked his 505 temperaments by 'goodness'. The familiar ones
don't come in until later, so bear with me . . .

Topping off Gene's list are some very funky simple temperaments, with
errors larger than 12-tET but nonetheless sure to be of interest to
many, especially those working in adaptive tuning and adaptive
timbring. They also should interest those looking for planar
temperaments, and all should interest those looking for very simple
JI scales without wolves. For these, I quote the simplest pair of
unison vectors:

> (1) <21/20,27/25>
> (2) <8/7,15/14>
> (3) <9/8,15/14>
> (4) <25/24,49/48>
> (5) <15/14,25/24>
> (6) <21/20,25/24>
> (7) <15/14,35/32>
> (8) <7/6,16/15>
> (9) <16/15,21/20>

Then we have a shadow of 12-tET that maybe Gene can describe more
fully -- it would seem to be related to tritone substitution (real
jazzy, dude).

> (10) <36/35,50/49>

Then we have Gene's own creation, without a sign of the complex comma
he discovered and named it after:

> (11) <2401/2400,4375/4374> Ennealimmal

Then more funky ones:

> (12) <16/15,49/48>
> (13) <10/9,16/15>

Then the one that I think is Dave Keenan's chain-of-minor-thirds as
in this 11-note scale:
http://www.uq.net.au/~zzdkeena/Music/ChainOfMinor3rds.htm

> (14) <49/48,126/125>

Another funky one:

> (15) <28/27,49/48>

And then:

> (16) <225/224,1029/1024> Miracle

> (17) <50/49,64/63> Paultone

More funk:

> (18) <16/15,50/49>

This one which I think came up as number 1 on Graham's ranking of
these:

> (19) <126/125,1728/1715>

And then another 12-tET-shadow system I'll let Gene talk about, which
is what you get when you assume the meantone dominant seventh is
4:5:6:7:

> (20) <36/35,64/63>

Gene only posted the top 20. I'm quite curious where we'll find
the "conventional" 7-limit meantone.

I've also asked Gene to look at the 5-limit. 11-limit is around the
corner I'm sure.

Jam on!

Thanks for this summary Paul, but ...

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> He proposed a 'badness' measure defined as
>
> step^3 cent
>
> where step is a measure of the typical number of notes in a scale
for
> this temperament (given any desired degree of harmonic depth),

What the heck does that mean? What is this number for 7-limit
meantone, paultone, miracle?

How does he justify cubing it?

> and
> cent is a measure of the deviation from JI 'consonances' in cents.

Yes but which measure of deviation? minimum maximum absolute or
minimum root mean squared or something else?

How does he justify not applying a human sensory correction to this?

--- In tuning@y..., "dkeenanuqnetau" <D.KEENAN@U...> wrote:
> Thanks for this summary Paul, but ...

You mean you haven't been on tuning-math@yahoogroups.com ? Get thee
hence :)

> > He proposed a 'badness' measure defined as
> >
> > step^3 cent
> >
> > where step is a measure of the typical number of notes in a scale
> for
> > this temperament (given any desired degree of harmonic depth),
>
> What the heck does that mean?

step is the RMS of the numbers of generators required to get to each
ratio of the tonality diamond from the 1/1, I think.

> What is this number for 7-limit
> meantone, paultone, miracle?

As I recall, it was about 11 for paultone, 30-40 for miracle . ..

> How does he justify cubing it?

Just squaring it seems to yield an infinite list of more and more
complex temperaments entering the top of the ranking the farther out
you look. But I'm trying to convince him that a different kind of
criterion might be better . . . and failing to convince myself . . .
>
> > and
> > cent is a measure of the deviation from JI 'consonances' in cents.
>
> Yes but which measure of deviation? minimum maximum absolute or
> minimum root mean squared or something else?

RMS

> How does he justify not applying a human sensory correction to this?

A human sensory correction?

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

> Just squaring it seems to yield an infinite list of more and more
> complex temperaments entering the top of the ranking the farther
out
> you look.

An order of growth estimate shows there should be an infinite list
for step^2, but not neccesarily for anything higher, and looking far
out makes it clear step^3 gives a finite list. What this means, of
course, is that in some sense step^2 is the right way to measure
goodness. Step^3 weighs the small systems more heavily, and that is
why we see so many of them to start with.

I wrote,

> I'm quite curious where we'll find
> the "conventional" 7-limit meantone.

#21 is the <2401/2400, 3136/3125> 68+31 system

#22 is the <49/48,225/224> 10+9 system -- I have a paper by John
Negri with a 19-tET keyboard having 10 whites and 9 flats -- I guess
that makes this the Negri system?

#23 is conventional meantone.

#24 is a funky 7-tET shadow.

#25 is a funky 3-chain 33-tET-like thing.

#26 is Gene's Orwell tuning.

#27 is a decent-smelling 6+9 system . . . this deserves its own
webpage or article (as does 68+31 and Graham's #1)

#28 is the <2401/2400, 65625/65536> 140+31 system.

#29 is <50/49, 875/864> -- I have no idea what that is.

#30 is Gene's 15+12=27 system.

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

> completed a comprehensive search for linear temperaments for 7-
limit
> music.

I'm afraid I haven't completed it; I have a preliminary list of 505
temperaments, however, and probably at least 300 of these should be
tossed.

> Then we have a shadow of 12-tET that maybe Gene can describe more
> fully -- it would seem to be related to tritone substitution (real
> jazzy, dude).
>
> > (10) <36/35,50/49>

The least-squares generators are 1/4 and 5.000375 / 28, so we may as
well equate this with the 28-et system we discussed before, which you
said at the time was jazzy.

> Then we have Gene's own creation, without a sign of the complex
comma
> he discovered and named it after:
>
> > (11) <2401/2400,4375/4374> Ennealimmal

I think Graham had found this already. The ennealimma is important in
this system, since it means you always divide the octave into nine
equal parts as part of it.

> And then another 12-tET-shadow system I'll let Gene talk about,
which
> is what you get when you assume the meantone dominant seventh is
> 4:5:6:7:
>
> > (20) <36/35,64/63>

It's hyper-Pythagorean meantone, with *sharp* fifths, which have the
effect of making the 7-limit harmony better at the serious expense of
the 3 and 5 limits. Perfect for Setharization, and the sort of thing
Margo likes even without any. You might say it's what you get from
the 12-et if you take its 7-limit approximations seriously, so it
could be used as a 12-note temperament.

> Gene only posted the top 20. I'm quite curious where we'll find
> the "conventional" 7-limit meantone.

Number 24.

> I've also asked Gene to look at the 5-limit. 11-limit is around the
> corner I'm sure.

Now that I've got wedge products to help, I'm sure it is quite doable.

--- In tuning@y..., "ideaofgod" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > completed a comprehensive search for linear temperaments for 7-
> limit
> > music.
>
> I'm afraid I haven't completed it; I have a preliminary list of 505
> temperaments, however, and probably at least 300 of these should be
> tossed.
>
> > Then we have a shadow of 12-tET that maybe Gene can describe more
> > fully -- it would seem to be related to tritone substitution
(real
> > jazzy, dude).
> >
> > > (10) <36/35,50/49>

> The least-squares generators are 1/4 and 5.000375 / 28, so we may
as
> well equate this with the 28-et system we discussed before, which
you
> said at the time was jazzy.

Right -- I just realized this includes the diminished or octatonic
scale right before you posted this. This makes me more confident of
my old dream of getting six fingerboards for the ETs from 26 to 31.

> > Then we have Gene's own creation, without a sign of the complex
> comma
> > he discovered and named it after:
> >
> > > (11) <2401/2400,4375/4374> Ennealimmal
>
> I think Graham had found this already.

I didn't think so. Graham?

> The ennealimma is important in
> this system, since it means you always divide the octave into nine
> equal parts as part of it.

Wicked. So perhaps we should get Joseph to try 27-out-of-72? How does
it go . . . I'd like to make a lattice.

> > And then another 12-tET-shadow system I'll let Gene talk about,
> which
> > is what you get when you assume the meantone dominant seventh is
> > 4:5:6:7:
> >
> > > (20) <36/35,64/63>
>
> It's hyper-Pythagorean meantone, with *sharp* fifths, which have
the
> effect of making the 7-limit harmony better at the serious expense
of
> the 3 and 5 limits. Perfect for Setharization, and the sort of
thing
> Margo likes even without any. You might say it's what you get from
> the 12-et if you take its 7-limit approximations seriously, so it
> could be used as a 12-note temperament.

I posted, a couple of years ago (it might have been back on the Mills
list), the optimal generator for this system. 704.something?
>
> > Gene only posted the top 20. I'm quite curious where we'll find
> > the "conventional" 7-limit meantone.
>
> Number 24.

I thought you said number 23.

I wrote,

> Then the one that I think is Dave Keenan's chain-of-minor-thirds as
> in this 11-note scale:
> http://www.uq.net.au/~zzdkeena/Music/ChainOfMinor3rds.htm
>
> > (14) <49/48,126/125>

That isn't it, but I think Dave mentions this one too. It's basically
a chain of 31-tET minor thirds.

> This one which I think came up as number 1 on Graham's ranking of
> these:
>
> > (19) <126/125,1728/1715>

Nope, this one is Dave Keenan's chain-of-minor-thirds. You gain a lot
of accuracy, and don't lose much simplicity, by using 31-tET minor
thirds instead of 19-tET minor thirds.

I wrote,

> #21 is the <2401/2400, 3136/3125> 68+31 system

This is Graham's #1. If Graham had found ennealimmal, wouldn't it
have been higher than this?

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> I wrote,
>
> > #21 is the <2401/2400, 3136/3125> 68+31 system
>
> This is Graham's #1. If Graham had found ennealimmal, wouldn't it
> have been higher than this?

It should have been, unless he tossed it as being beyond his upper
limit.

I wrote,

> --- In tuning@y..., "dkeenanuqnetau" <D.KEENAN@U...> wrote:
>
> > > He proposed a 'badness' measure defined as
> > >
> > > step^3 cent
> > >
> > > where step is a measure of the typical number of notes in a
scale
> > for
> > > this temperament (given any desired degree of harmonic depth),
> >
> > What the heck does that mean?
>
> step is the RMS of the numbers of generators required to get to
each
> ratio of the tonality diamond from the 1/1, I think.

I'd like Gene to verify the tonality diamond thing . . . maybe it's
weighted in some way . . . the numbers seem a bit high (but may still
represent some degree of harmonic depth, and the proportion between
them may remain the same for any other degree of harmonic depth).
>
> > What is this number for 7-limit
> > meantone, paultone, miracle?
>
> As I recall, it was about 11 for paultone, 30-40 for miracle . ..

21.1 for conventional meantone, 17.5 for paultone, 36.6 for
miracle . . .

--- In tuning@y..., "ideaofgod" <genewardsmith@j...> wrote:

> > > (20) <36/35,64/63>
>
> It's hyper-Pythagorean meantone, with *sharp* fifths, which have
the
> effect of making the 7-limit harmony better at the serious expense
of
> the 3 and 5 limits. Perfect for Setharization, and the sort of
thing
> Margo likes even without any.

Well . . . she actually likes to takes her 7s from much farther out
in the chains of sharp fifths she uses, since she likes them to be
much more accurate than this (right, Margo)?

> You might say it's what you get from
> the 12-et if you take its 7-limit approximations seriously, so it
> could be used as a 12-note temperament.

Well, you could also say that about the octatonic temperament which
came in higher, so this is not a unique characterization.
Saying "hyper-Pythagorean meantone" or something like that, along
with a mention that the regular dominant seventh chords act as
4:5:6:7, would get the point across, though.

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

> I'd like Gene to verify the tonality diamond thing . . . maybe it's
> weighted in some way . . . the numbers seem a bit high

It's unweighted--7/5 counts equally with 3. That could be changed, of
course, but this way has some advantages. What do you mean by the
numbers being high?

> 21.1 for conventional meantone, 17.5 for paultone, 36.6 for
> miracle . . .

These I think must be old numbers. I have

steps steps^2 steps^3
Miracle 12.45 94.7 720
Meantone 23.32 148.4 945
Paultone 46.26 196.3 833

If you remember I changed my system because of your suggestion.

--- In tuning@y..., "ideaofgod" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > I'd like Gene to verify the tonality diamond thing . . . maybe
it's
> > weighted in some way . . . the numbers seem a bit high
>
> It's unweighted--7/5 counts equally with 3. That could be changed,
of
> course, but this way has some advantages. What do you mean by the
> numbers being high?

Well they're wrong anyway.
>
> > 21.1 for conventional meantone, 17.5 for paultone, 36.6 for
> > miracle . . .
>
> These I think must be old numbers. I have
>
> steps steps^2 steps^3
> Miracle 12.45 94.7 720
> Meantone 23.32 148.4 945
> Paultone 46.26 196.3 833
>
> If you remember I changed my system because of your suggestion.

Gene, this makes no sense. 46.26 for Paultone? And if I square this
number, I get 2140 -- not 196.3 !

> > steps steps^2 steps^3
> > Miracle 12.45 94.7 720
> > Meantone 23.32 148.4 945
> > Paultone 46.26 196.3 833

Oh -- perhaps you're already multiplying by cents error! Assuming
that, I see that in fact the numbers are

step
Miracle 7.6064
Meantone 6.3636
Paultone 4.2434

So I would multiply by roughly e to get the size of the typical,
harmonically full scale . . .

Let's move this back to tuning-math@yahoogroups.com . . . the natives
are getting restless . . .

Hello, there, Paul, and this is a quick reply to your accurate comment
in a message </tuning/topicId_31054.html#31084>.

While I certainly appreciate the special attractions of 22-tET and
other tunings where the 63:64 or septimal comma is dispersed, you're
right that I often lean toward tunings with 7-based intervals derived
either from longer chains of regular fifths tempered rather gently in
the wide direction, or from two chains of fifths at some convenient
distance from each other.

In fact, the two-keyboards-in-9-tET scheme is a brilliant reversal of
the last pattern, since here each keyboard has near-pure 6:7 thirds
(2/9 octave), but we mix notes from the two keyboards to get pure
fifths -- the opposite of a usual situation with two keyboards each
providing a chain of just or near-just fifths, with the 7-based
intervals produced by mixing their notes.

Returning to the solution of a longer chain of regular fifths, there
are two regions of special interest if we're going after a single
chain of fifths tuning, or something very close to it. The Pythagorean
region around 702 cents is one obvious choice, with various ways of
handling the 3-7 schisma. My own chosen solution is to take a chain of
23 fifths or 24 notes with the middle fifth "virtually tempered" by
the full schisma (about 3.80 cents), and all the others pure. Graham
Breed has advocated 135-tET as an ideal regular schisma temperament.

Since the region around 704 cents was mentioned in this thread, I
would note that around 704.6 cents we get a virtually just 4:7 from 15
fifths up, with the other 2-3-7-9 ratios within about 5 cents of just.

If one is seeking near-pure 5-limit ratios in the area of 704 cents,
then the "Noble Fifth" tuning suggested by Erv Wilson's Scale Tree and
described by Keenan Pepper, with fifths at around 704.10 cents, has an
interval of 21 fifths up with a size of something like 386 cents.
While it's an exotic touch at a few locations from a 24-note
neo-Gothic viewpoint, someone looking for a 5-limit tuning and ready
to use lots of notes per octave might find it of interest.

However, I wouldn't describe the "Noble Fifth" tuning as very accurate
for ratios of 2-3-7-9, and in my view it takes much of its charm in
this department from the "ambiguity" of ratios such a large major
third (13 fourths up) at around 447 cents, or 17:22 if one wants an
integer approximation. This may be closer to the "near-10:13" quality
of 24-tET or 29-tET than to a pure or near-pure 7:9.

This suggests to me that someone looking for a balance between 5-limit
and 7-limit ratios in this part of the spectrum might favor something
around 46-tET, often discussed in connection with the srutar, for
example.

From a neo-Gothic viewpoint, of course, the complex ratios formed by
shorter chains of fifths in this region are a very major attraction.
However, as this discussion suggests, there are many ways of looking
at a tuning.

Incidentally, if I were describing a "hyper-Pythagorean" tuning of the
22-tET variety, I might express the ambiguity as one between 35:36 and
27:28, with the same interval of 1/22 octave representing, for example,
the difference either between a near-4:5 and a near-7:9 third (7 vs. 8
steps), or between a near-7:9 third and a near-3:4 fourth (8 vs. 9 steps).
However, I'm not sure if that's the same kind of concept with these ratios
as the one notated in this thread.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> Incidentally, if I were describing a "hyper-Pythagorean" tuning of
the
> 22-tET variety, I might express the ambiguity as one between 35:36
and
> 27:28, with the same interval of 1/22 octave representing, for
example,
> the difference either between a near-4:5 and a near-7:9 third (7
vs. 8
> steps), or between a near-7:9 third and a near-3:4 fourth (8 vs. 9
steps).
> However, I'm not sure if that's the same kind of concept with these
ratios
> as the one notated in this thread.

Hi Margo,

You've found two just intervals which come out as the 1-step interval
of 22-tone equal temperament: 35:36 and 27:28. Therefore, dividing
these ratios by one another gives you a unison vector of 22-tone
equal temperament:

(27:28)/(35:36) = 243:245

Since we are talking about ratios involving the three primes 3, 5,
and 7, a three-dimensional just lattice is the space in which they
live, and three independent unison vectors are required to define a
closed temperament, such as 22-tET. Besides the one we found above,
it's easy to find two others, by dividing other pairs of just
intervals that come out the same in 22-tET:

dividing two ratios for the 9-step interval:
(9:16)/(4:7) = 63:64

dividing two ratios for the 11-step interval:
(7:10)/(5:7) = 49:50

Together, the three unison vectors
<49:50, 63:64, 243:245>

define 22-tone equal temperament -- no other temperament has these
three unison vectors.

What you saw in this thread was _pairs_ of unison vectors like
<80:81, 125:126>

A pair of unison vectors won't collapse the three-dimensional just
lattice into a closed tuning; instead, they will collapse it into a
linear temperament. A linear temperament has a potentially unlimited
number of notes, but it's constructed very simply. There is a single
generator, an interval which is iterated over and over again to get
more and more notes from the first. And there is a period of
repetition, with which every note is transposed indefinitely upwards
and downwards. The period of repetition is usually an octave (or
whatever interval of equivalence is chosen), but often turns out to
be a fraction 1/N of an octave, where N is an integer greater than 1.

Hopefully this clarifies some of the things that may have been
puzzling you about this thread . . . if not, I would be more than
happy to attempt to answer further questions.

Cheers,
Paul

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> JI scales without wolves. For these, I quote the simplest pair of
> unison vectors:
>
> > (1) <21/20,27/25>
> > (2) <8/7,15/14>
> > (3) <9/8,15/14>
> > (4) <25/24,49/48>
> > (5) <15/14,25/24>
> > (6) <21/20,25/24>
> > (7) <15/14,35/32>
> > (8) <7/6,16/15>
> > (9) <16/15,21/20>
>
> Then we have a shadow of 12-tET that maybe Gene can describe more
> fully -- it would seem to be related to tritone substitution (real
> jazzy, dude).
>
> > (10) <36/35,50/49>
... etc

The descriptive comments are useful, but why bother posting these
lists of unison vector pairs to the tuning list. They don't even mean
anything to _me_, with a math background.

I think most people recognise linear temperaments by the approximate
size of the generator in cents (or as an approximate ratio) and the
number of periods per octave.

Also this particular ranking of them appears to be seriously out of
touch with reality.

--- In tuning@y..., "dkeenanuqnetau" <D.KEENAN@U...> wrote:

> The descriptive comments are useful, but why bother posting these
> lists of unison vector pairs to the tuning list. They don't even
mean
> anything to _me_, with a math background.

Well you're right, but now that I've begun explaining them to Margo,
perhaps that will change.

> I think most people recognise linear temperaments by the
approximate
> size of the generator in cents (or as an approximate ratio) and the
> number of periods per octave.

Me too, but unfortunately Gene didn't initially give that data. The
follow-ups are on

tuning-math@yahoogroups.com

> Also this particular ranking of them appears to be seriously out of
> touch with reality.

I find it to be very appealing. Ennealimmal is a hugely complex
system, but having it up there in the ranking is very useful so that

(a) someone who is obsessed with eliminating the last vestiges of
beating in their music and doesn't shy away from scales with around
45 notes (sound familiar?) might have something useful to take away
from all this;
(b) one has a sense of exactly what level of complexity _is_ required
if one wants to get fantastic accuracy -- I think it's a useful
landmark in this terrain.

--- In tuning@y..., "dkeenanuqnetau" <D.KEENAN@U...> wrote:

> The descriptive comments are useful, but why bother posting these
> lists of unison vector pairs to the tuning list. They don't even
mean
> anything to _me_, with a math background.

Dave, I sent you a copy of _The Forms Of Tonality_. Doesn't it mean
something to, after reading that paper, that the pair of unison
vectors 50:49 and 64:63 is going to define the "paultone" system?
Even with all the pretty pictures?

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <D.KEENAN@U...> wrote:
> > Also this particular ranking of them appears to be seriously out
of
> > touch with reality.
>
> I find it to be very appealing. Ennealimmal is a hugely complex
> system, but having it up there in the ranking is very useful so that
>
> (a) someone who is obsessed with eliminating the last vestiges of
> beating in their music and doesn't shy away from scales with around
> 45 notes (sound familiar?) might have something useful to take away
> from all this;
> (b) one has a sense of exactly what level of complexity _is_
required
> if one wants to get fantastic accuracy -- I think it's a useful
> landmark in this terrain.

I suspect even these folks would find some things about that ranking
rather silly. I suspect they would be better served by a small value
of the parameter k (say 2 cents) in

step^2 * exp((cents/k)^2)

See my post to tuning-math. Of course this parameter k is intimately
related to your parameter s for harmonic entropy.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <D.KEENAN@U...> wrote:
>
> > The descriptive comments are useful, but why bother posting these
> > lists of unison vector pairs to the tuning list. They don't even
> mean
> > anything to _me_, with a math background.
>
> Dave, I sent you a copy of _The Forms Of Tonality_. Doesn't it mean
> something to, after reading that paper, that the pair of unison
> vectors 50:49 and 64:63 is going to define the "paultone" system?
> Even with all the pretty pictures?

Sure but it takes a considerable amount of work to unpack the meaning
of a new pair of unfamiliar unison vectors, in order to say "Oh I know
that one, that's two chains of slightly wide 6:7's a half octave
apart" or whatever. In fact I realise I have no idea how to unpack
that at all. I haven't been following tuning-math.

--- In tuning@y..., "dkeenanuqnetau" <D.KEENAN@U...> wrote:

> I suspect even these folks would find some things about that
ranking
> rather silly. I suspect they would be better served by a small
value
> of the parameter k (say 2 cents) in
>
> step^2 * exp((cents/k)^2)

But what if we want to reach _all_ folks? We (perhaps you're still in
this, perhaps not) want to write a paper that will introduce some
general concepts, and then give say 30 or 40 specific examples, so
that the greatest number of people are likely to find something
useful to them.

--- In tuning@y..., "dkeenanuqnetau" <D.KEENAN@U...> wrote:

> Sure but it takes a considerable amount of work to unpack the
meaning
> of a new pair of unfamiliar unison vectors, in order to say "Oh I
know
> that one, that's two chains of slightly wide 6:7's a half octave
> apart" or whatever.

Me too! Therefore, I did what I did out of laziness.

> In fact I realise I have no idea how to unpack
> that at all. I haven't been following tuning-math.

Gene has something that we now call the "wedgie" (technically, "wedge
invariant") that apparantly helps in this unpacking endeavor. So far,
I think he's the only one who really understands it. Follows to

tuning-math@yahoogroups.com

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

/tuning/topicId_31054.html#31067

>
> Wicked. So perhaps we should get Joseph to try 27-out-of-72? How
does it go . . . I'd like to make a lattice.
>

Great. I will always like to try new things... However, I must
confess I'm having some trouble with this discussion. I see various
commas associated with, apparently, certain scales, but that's about
as far as I'm getting with it. :(

Maybe I can get the "rundown" later, when more has been decided abou
it...

JP

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

/tuning/topicId_31054.html#31067
>

> Wicked. So perhaps we should get Joseph to try 27-out-of-72? How
does it go . . . I'd like to make a lattice.
>

Oh... suddenly it dawned on me that each of these scales has two
pairs of unison vectors... going in two different dimensions like a
meantone... so these are all, apparently, meantone-like "thingies..."
(I guess I should have seen that in "linear temperaments..." :) )

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> Maybe I can get the "rundown" later, when more has been decided
abou
> it...
>
> JP

Yup . . . we're going to present all this as a paper and the 27-tone
(really though you'd use 45) system in question will be in there . . .