Re: Another BP linear temperament

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Dave,
>
> The way I look at it, what I did was simply remap the unison vectors
> so that the 2D comma is 118098/117649

Ok <goes and figures> that's (2^1 * 3^10 * 7^-6) = 6.7 cents

> and the primary chroma is 343/324.

That's (2^-2 * 3^-4 * 7^3) = 99 cents. Is this for the 9-note per
tritave MOS?

> This means that the dimensions are 9/7 and 7/6 with a period
> of 3/1 as opposed to the usual 3/2 and 5/4 with a period of 2/1.

Ok. But for this to be a "generalised meantone" in any sense other
than the one which is much better described by the term "linear
temperament", then a chain of _four_ 9/7's would have to be
tritave-equivalent to a 7/6. This is not the case. It needs a chain of
five according to your comma.

> The 1/6 comma meantone generalization is exactly analogous to
standard
> 1/5 comma meantone in terms of fractionalizing a comma and altering
> the size of the generator in a 1D chain.

I suspect you mean standard 1/4 comma meantone.

But this is just a linear temperament. How is it more specifically a
"generalised meantone"? Is it merely because it seeks to approximate
only two primes (other than the interval-of-equivalence), in this case
2 and 7? That certainly isn't enough for _me_ to consider it a
"generalised meantone". What do others think?

So this BP temperament still has an approximate 7:9 generator but it's
typically a narrow one, where the previous BP temperament has a wide
one. The mapping from primes to numbers of generators is:

Prime No. generators
----- --------------
2 -6
3 0
5 (don't care, but 2 is best)
7 -1

The MA optimum {1,2,7} generator is 434.0 cents giving a maximum error
of 1.2 cents. This generator is, as you said, a 7:9 narrowed by 1/6 of
the above comma.

If 5's are included, but not 4's, i.e. {1,2,5.7}, the optimum
generator is 436.0 cents (1/7 comma wide) with errors of 12.3 cents.

The optimum for the full 7-limit, i.e. {1,2,4,5,7}, is 435.15 cents
(in between the previous two) but with errors of 14.1 cents. It has
MOS cardinalities of 4 (5) 9 13 (22) 35 (48), (improper in
parenthesis).

-- Dave Keenan

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> <<But this is just a linear temperament. How is it more specifically
a
> "generalised meantone"?>>
>
> Again, it's a generalization of meantone because the "tone" and the
> "mean" are generalized! The process of determining a comma and
> fractionalizing and distributing it are also all exactly analogous
if
> one strips away the particulars.

Ok. Can you tell me what intervals are analogous to the major and
minor tone in this temperament. I'm concerned that you're generalising
"tone" to mean "an interval of any size", which I think would be a bad
move.

> It's the process, not the specific tuning -- besides, if the octave
> divided into 12 and 13 equidistant parts can rightfully be called
> "equal temperament" in either instance, then I see no reason why
what
> I'm doing here can't be called "meantone"!
>
> That said, I never was happy with "generalized meantone" as a
blanket
> term for this sort of thing, and I did post several times a while
back
> trolling for better, less likely to cause confusion, types of terms.
> Unfortunately this never really went anywhere. So, as I haven't
> thought of anything better or less confusing myself, I've just stuck
> with my initial intuitive feeling as that seems obvious and simple
> enough for communication purposes...

So it's just a particular way of arriving at a linear temperament? Why
not call the process "comma distribution"?

> Does any of this make any better sense now? If not, well, then I'm
> convinced that Graham was actually right, and I am quite incapable
of
> communicating these things.

I don't think he said (or meant) that exactly. Yes, it makes better
sense now and we are quite capable of communicating (it takes two).

Regards,
-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> But this is just a linear temperament. How is it more specifically
a
> "generalised meantone"? Is it merely because it seeks to
approximate
> only two primes (other than the interval-of-equivalence), in this
case
> 2 and 7? That certainly isn't enough for _me_ to consider it a
> "generalised meantone". What do others think?

I'd have to agree with you Dave -- I think "meantone" would have to
mean something a bit more specific, even if you "generalize" it. But
that's just terminology.
>
> So this BP temperament still has an approximate 7:9 generator but
it's
> typically a narrow one, where the previous BP temperament has a
wide
> one. The mapping from primes to numbers of generators is:
>
> Prime No. generators
> ----- --------------
> 2 -6
> 3 0
> 5 (don't care, but 2 is best)
> 7 -1
>
> The MA optimum {1,2,7} generator is 434.0 cents giving a maximum
error
> of 1.2 cents. This generator is, as you said, a 7:9 narrowed by 1/6
of
> the above comma.
>
> If 5's are included, but not 4's, i.e. {1,2,5.7}, the optimum
> generator is 436.0 cents (1/7 comma wide) with errors of 12.3 cents.
>
> The optimum for the full 7-limit, i.e. {1,2,4,5,7}, is 435.15 cents
> (in between the previous two) but with errors of 14.1 cents.

So I take it Dan's suggestion of which commas temper out doesn't look
as good? I'm a little lost here.

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Dave,
>
> <<Ok. But for this to be a "generalised meantone" in any sense other
> than the one which is much better described by the term "linear
> temperament", then a chain of _four_ 9/7's would have to be
> tritave-equivalent to a 7/6. It needs a chain of five according to
> your comma.>>
>
> I disagree, and I think your looking at this in much too narrow a
way.
> The process of determining a comma and fractionalizing and
> distributing it is what's being generalized...

But no one ever called schismic temperament, which seems to have
predated meantone, a "generalized meantone" . . .
>
> Again, it's a generalization of meantone because the "tone" and the
> "mean" are generalized!

Ah . . . this may be what Dave and I are missing. What are the
JI "tones" here?

Anyway, let's not let squabbles over terminology (Margo suggested
some nice ideas for this) blind us to the fact that all of the others
on this list have great pools of insight from which they are drawing.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > The MA optimum {1,2,7} generator is 434.0 cents giving a maximum
> error
> > of 1.2 cents. This generator is, as you said, a 7:9 narrowed by
1/6
> of
> > the above comma.
> >
> > If 5's are included, but not 4's, i.e. {1,2,5.7}, the optimum
> > generator is 436.0 cents (1/7 comma wide) with errors of 12.3
cents.
> >
> > The optimum for the full 7-limit, i.e. {1,2,4,5,7}, is 435.15
cents
> > (in between the previous two) but with errors of 14.1 cents.
>
> So I take it Dan's suggestion of which commas temper out doesn't
look
> as good? I'm a little lost here.

No. Dan's statements were spot-on. If you only want to approximate
ratios of {1,2,7}*3^n then a generator which is a 7:9 reduced by 1/6
of the comma he gave (2^1 * 3^10 * 7^-6 = 6.7 cents), is superb (max
1.2c errors and only 6 generators per triad). But if you wanted to
include any other non-multiples of 3 (e.g. 4 or 5) then it's very
ordinary.

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> No. Dan's statements were spot-on. If you only want to approximate
> ratios of {1,2,7}*3^n then a generator which is a 7:9 reduced by
1/6
> of the comma he gave (2^1 * 3^10 * 7^-6 = 6.7 cents), is superb
(max
> 1.2c errors and only 6 generators per triad). But if you wanted to
> include any other non-multiples of 3 (e.g. 4 or 5) then it's very
> ordinary.

I see. So if you've only got two dimensions in the lattice, Dan's the
man!!