Re: Another BP linear temperament?

In-Reply-To: <9k4o1h+jqn@eGroups.com>
Dan Stearns wrote:

> > The BP linear temperament was so obvious that I could hardly call it
> > much of a contribution... it's an observation at best!

Well, that's all I'm cataloguing.

I've written a little script to go with my temperament finder for this.
Your BP mapping is the best in the "7-limit" with a tritave as the
interval of equivalence.

Here's the printout:

7/30, 279.0 cent generator

basis:
(1.0, 0.23246075055125442)

mapping by period and generator:
[(1, 0), (-1, 7), (1, 2), (2, -1)]

mapping by steps:
[(17, 13), (11, 8), (25, 19), (30, 23)]

unison vectors:
[[9, 2, -7, 0], [-13, 1, 0, 7]]

highest interval width: 8
complexity measure: 8 (9 for smallest MOS)
highest error: 0.003704 (4.445 cents)
unique

The top 10 list should be at <http://x31eq.com/tritave.txt> and
the script at <http://x31eq.com/tritave.py>.

Graham

--- In tuning-math@y..., graham@m... wrote:
> Here's the printout:
>
> 7/30, 279.0 cent generator

Oops! Your generator should be 442.1 cents. You've multiplied by cents
per octave when it should have been cents per tritave.

I get 442.6 cents as the 7-limit optimum. I notice you put "7 limit"
in scare quotes, and rightly so. I'd call what you've given "7-limit
without 2s".

By analogy:
For an octave repeating scale, "7 limit" means ratios of all (whole
number) non-multiples of 2 up to 7, i.e {1,3,5,7} (and their octave
equivalents).

So for a tritave repeating scale I think "7 limit" should mean ratios
of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.

-- Dave Keenan

--- In tuning-math@y..., "Dave
Keenan" <D.KEENAN@U...> wrote:

> I get 442.6 cents as the 7-limit optimum. I notice you put "7 limit"
> in scare quotes, and rightly so. I'd call what you've given "7-limit
> without 2s".
>
> By analogy:
> For an octave repeating scale, "7 limit" means ratios of all (whole
> number) non-multiples of 2 up to 7, i.e {1,3,5,7} (and their octave
> equivalents).
>
> So for a tritave repeating scale I think "7 limit" should mean ratios
> of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.

Since the "orthodox" BP
consonances are ratios of {3,5,7},
and the triple-BP consonances
are ratios of {3,5,7,11,13}, and
their tritave-equivalents, and
these have been referred to as
"7-limit" and "13-limit",
respectively, I think this could
lead to confusion . . . let's
abandon the term "limit" when
leaving the octave-equivalent
world and adopt more explicit
mathematical terminology.

(Remember, Pierce's idea for the
scale that became known as BP
was that if the even partials are
absent, the 3:1 would take on the
role of the octave, and ratios
involving even numbers would no
longer be "consonances" . . .
whether or not the idea holds up
in practice, it's conceptually neat
enough that it makes sense to at
least honor it on equal footing
with other conceptions . . .
Graham, or anyone else who can
"hear" 3:1 equivalence . . . does it
help to eliminate even partials
from the timbres?)

In-Reply-To: <9kafes+8qo4@eGroups.com>
Paul wrote:

> Graham, or anyone else who can
> "hear" 3:1 equivalence . . . does it
> help to eliminate even partials
> from the timbres?)

It's all in the timbre to me. Tritaves with odd-number partials have a
stronger tritave equivalence than octaves with sawtooth timbres. The same
is true of phi with a phi timbre. I'm guessing it would also be true of
octaves with only even partials, but haven't tried yet.

Graham

In-Reply-To: <9kabno+v09n@eGroups.com>
Dave Keenan wrote:

> > Here's the printout:
> >
> > 7/30, 279.0 cent generator
>
> Oops! Your generator should be 442.1 cents. You've multiplied by cents
> per octave when it should have been cents per tritave.

Yes, I mentioned this in a later post.

> I get 442.6 cents as the 7-limit optimum. I notice you put "7 limit"
> in scare quotes, and rightly so. I'd call what you've given "7-limit
> without 2s".

Oh, it does include the 2s, but not enough of them. It uses a 2-5-7
basis, but the consonance limit is the octave-equivalent 7-limit matrix.
So it won't include 7:4 or 5:4.

> By analogy:
> For an octave repeating scale, "7 limit" means ratios of all (whole
> number) non-multiples of 2 up to 7, i.e {1,3,5,7} (and their octave
> equivalents).
>
> So for a tritave repeating scale I think "7 limit" should mean ratios
> of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.

Sounds good, but it's a case of typing in the matrices. This was a quick
experiment.

Graham

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave
> Keenan" <D.KEENAN@U...> wrote:
> > So for a tritave repeating scale I think "7 limit" should mean
ratios
> > of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.
>
> Since the "orthodox" BP
> consonances are ratios of {3,5,7},

So isn't 7:9 included, or do you really mean ratios of {1,5,7} and
their tritave equivalents?

> and the triple-BP consonances
> are ratios of {3,5,7,11,13}, and
> their tritave-equivalents,

You mean {1,5,7,11,13} and their tritave-equivalents?

> these have been referred to as
> "7-limit" and "13-limit",
> respectively, I think this could
> lead to confusion . . . let's
> abandon the term "limit" when
> leaving the octave-equivalent
> world and adopt more explicit
> mathematical terminology.

Yes please!

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9kabno+v09n@e...>
> Dave Keenan wrote:
> > So for a tritave repeating scale I think "7 limit" should mean
ratios
> > of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.
>
> Sounds good, but it's a case of typing in the matrices. This was a
quick
> experiment.

Duh! I included 6 above (which is of course tritave equivalent to 2
and therefore redundant). This stuff is tricky. It should have been:

So for a tritave repeating scale I think "7 limit" should mean
ratios of all non-multiples of 3 up to 7, i.e. {1,2,4,5,7} (and
of course their tritave equivalents).

But I agree with Paul, just spell it out, like after the "i.e." above.

Regards,
-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "Dave
> > Keenan" <D.KEENAN@U...> wrote:
> > > So for a tritave repeating scale I think "7 limit" should mean
> ratios
> > > of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.
> >
> > Since the "orthodox" BP
> > consonances are ratios of {3,5,7},
>
> So isn't 7:9 included, or do you really mean ratios of {1,5,7} and
> their tritave equivalents?

The latter.
>
> > and the triple-BP consonances
> > are ratios of {3,5,7,11,13}, and
> > their tritave-equivalents,
>
> You mean {1,5,7,11,13} and their tritave-equivalents?

Same thing -- right?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > > and the triple-BP consonances
> > > are ratios of {3,5,7,11,13}, and
> > > their tritave-equivalents,
> >
> > You mean {1,5,7,11,13} and their tritave-equivalents?
>
> Same thing -- right?

Yes, but not in lowest terms, so could be confusing.

You wouldn't write "ratios of {2,3,5,7,9} and their octave
equivalents", would you? And there will be times when folks will not
want to always write "and their tritave equivalents" because they feel
it is given by the context. In this case "3" or "9" in place of "1"
would definitely suggest tritave-specific.

Just being my usual pedantic self. Sorry.

-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> You wouldn't write "ratios of {2,3,5,7,9} and their octave
> equivalents", would you?

Why not? I might even use 4 instead of 2.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > You wouldn't write "ratios of {2,3,5,7,9} and their octave
> > equivalents", would you?
>
> Why not? I might even use 4 instead of 2.

I thought I answered "Why not?" in the same message you quote from.

Why would you?

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > > You wouldn't write "ratios of {2,3,5,7,9} and their octave
> > > equivalents", would you?
> >
> > Why not? I might even use 4 instead of 2.
>
> I thought I answered "Why not?" in the same message you quote from.
>
> Why would you?

So that the resulting ratios would tend to be the familar voicings
within one octave: 3:4, 4:5, 4:7 . . .