Yahoo Tuning Groups Ultimate Backup tuning-math A potentially informative property of tunings

Take a generator of 260.76 cents and a period of 1206.55 cents. This
defines a linear tuning which belongs to a family of related linear
temperaments. The simplest mapping is the "beep" mapping, which distributes
the 27;25 interval:

[(1, 0), (2, -2), (3, -3)]

but after 6 iterations of the generator, there's a better 5:1 at (1, 6),
about 15 cents flat (compared with the 51 cent sharp "beep" version of the
interval). That means this particular tuning is consistent with "beep"
temperament only up to a range of 5 generators -- or to coin a phrase, its
"consistency range" with respect to "beep" is 5. In comparison, top
meantone has a "consistency range" of 34: its (17, -35) version of 5:1 is
only 2 cents flat, compared with the 4-cent sharp (4, -4). Quarter-comma
meantone has a "consistency range" of 29, since it has a better 3:1 at
(-11, 30).

First of all, I don't like the term "consistency range", but I couldn't
think of anything better. I'd appreciate ideas for what to call this
property. Secondly, the fact that this property varies from one particular
tuning of a temperament to another implies that there's a particular tuning
with the maximum "consistency range" for any given temperament. This seems
like it would be a useful property to maximize.

Once a tuning exceeds its "consistency range" limit for a specific
temperament, it gives way to (mutates into?) a related temperament that
shares part of the mapping. So "beep" mutates into "superpelog" at 6
iterations, and "superpelog" in turn mutates into a new temperament at 31
iterations:

[(1, 0), (2, -2), (3, -3)] beep
[(1, 0), (2, -2), (1, 6)] superpelog
[(1, 0), (2, -2), (9, -31)] unnamed temperament

The same thing happens with 7-limit versions of top meantone:

[(1, 0), (2, -1), (4, -4), (2, 2)]
[(1, 0), (2, -1), (4, -4), (-1, 9)]
[(1, 0), (2, -1), (4, -4), (7, -10)]

The mutation sequence for quarter-comma meantone skips over (-1, 9) and
goes straight to (7, -10), followed by (-6, 21).

Obviously, the mutation sequence varies from one tuning of the generator
and period to another; a 442.2 cent tuning of "father" mutates to
semisixths, but "top father" mutates to a bizarre temperament with mapping
[(-2, 8), (5, -9), (5, -7)]. On the other hand, knowing the mutation
sequence and the consistency range limit could be important when trying to
get a feel for the usefulness of a new tuning.

--
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

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> [(1, 0), (2, -2), (1, 6)] superpelog

Why are you calling this "superpelog"? It's a Dave Keenan style
duplicated pelog, isn't it?

> [(1, 0), (2, -2), (9, -31)] unnamed temperament

This can't be what you want; it gives a complex "comma" of 588 cents.

> The same thing happens with 7-limit versions of top meantone:
>
> [(1, 0), (2, -1), (4, -4), (2, 2)]
> [(1, 0), (2, -1), (4, -4), (-1, 9)]
> [(1, 0), (2, -1), (4, -4), (7, -10)]
>
> The mutation sequence for quarter-comma meantone skips over (-1, 9) and
> goes straight to (7, -10), followed by (-6, 21).

The latter isn't going to win any awards, as the error is almost
identical. They differ by 31 generators; going up by 205 generators
to (-79, 195) or down 174 generators to (80, -184) cuts the error
almost in half, but at the cost of absurd complexity. If you are going
to do that you may as well go whole-hog to (-164, 400) or (166,-389)
giving you two microtemperaments (error less than 1/5 cent) for
{2,5,7}-JI bizarrely attached to a meantone fifth. I don't think this
really works as a way of analyzing meantone.

🔗Paul Erlich <perlich@aya.yale.edu>

Gotcha. Yup, it was essentially this 'consistency range' deal which
caused us originally to reject certain temperaments (you might search
for 'funky' on this list) which, as I recall, had a consistency range
of 0.

> First of all, I don't like the term "consistency range", but I
couldn't
> think of anything better.

I like it fine, and in multi-period-per-octave tunings, I would
multiply by the number of periods per octave, so that the right
number of tones is involved. But which intervals are you testing? You
would need to know this too to determine the "maximum consistency"
version of any temperament.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> The mutation sequence for quarter-comma meantone skips over (-1, 9) and
> goes straight to (7, -10), followed by (-6, 21).

If you want to go out this far to map 7, it probably makes more sense
to use a sharper fifth and (12, -22) for the map. This is a
temperament for meantones in the neighborhood of 67-et, and since
55-et falls in there we could be evil and call it "moztone", but given
the nature of Mozart's use of meantone that hardly makes sense. In any
case, its badness is so high because of the high complexity to get to
7s harmony it hardly makes sense to use this for an alternative to
septimal meantone. Jon might be happy with its TOP tuning, since
octaves are only a smidgen flat.

🔗Carl Lumma <ekin@lumma.org>

>> The mutation sequence for quarter-comma meantone skips over (-1, 9) and
>> goes straight to (7, -10), followed by (-6, 21).
>
>If you want to go out this far to map 7, it probably makes more sense
>to use a sharper fifth and (12, -22) for the map. This is a
>temperament for meantones in the neighborhood of 67-et, and since
>55-et falls in there we could be evil and call it "moztone", but given
>the nature of Mozart's use of meantone that hardly makes sense. In any
>case, its badness is so high because of the high complexity to get to
>7s harmony it hardly makes sense to use this for an alternative to
>septimal meantone. Jon might be happy with its TOP tuning, since
>octaves are only a smidgen flat.

I wonder if this has anything to do with the attached?

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

>> The mutation sequence for quarter-comma meantone skips over (-1, 9) and
>> goes straight to (7, -10), followed by (-6, 21).
//
>> > I wonder if this has anything to do with the attached?
>>
>> Which got repeated to me via e-mail, but I'm glad I checked
>> the web site, which sucks.
>>
>> Here is the file...
>>
>> http://lumma.org/tuning/01.gif
>>
>> -Carl
>
>I see two searchlights, nothing else.

Oh, bother. This would have been much better:

http://lumma.org/tuning/05.gif

-Ca.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Take a generator of 260.76 cents and a period of 1206.55 cents. This
> defines a linear tuning which belongs to a family of related linear
> temperaments. The simplest mapping is the "beep" mapping, which
distributes
> the 27;25 interval:
>
> [(1, 0), (2, -2), (3, -3)]
>
> but after 6 iterations of the generator, there's a better 5:1 at
(1, 6),
> about 15 cents flat (compared with the 51 cent sharp "beep" version
of the
> interval). That means this particular tuning is consistent
with "beep"
> temperament only up to a range of 5 generators -- or to coin a
phrase, its
> "consistency range" with respect to "beep" is 5. In comparison, top
> meantone has a "consistency range" of 34: its (17, -35) version of
5:1 is
> only 2 cents flat, compared with the 4-cent sharp (4, -4). Quarter-
comma
> meantone has a "consistency range" of 29, since it has a better 3:1
at
> (-11, 30).
>
> First of all, I don't like the term "consistency range", but I
couldn't
> think of anything better. I'd appreciate ideas for what to call this
> property.

Since you're describing a relationship or comparison between two
temperaments, I would suggest "compatibility range". The
term "consistency" is usually used to describe only relationships
within a single temperament.

--George

🔗Herman Miller <hmiller@IO.COM>

On Mon, 19 Jan 2004 00:52:19 -0000, "Gene Ward Smith" <gwsmith@svpal.org>
wrote:

>--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
>> [(1, 0), (2, -2), (1, 6)] superpelog
>
>Why are you calling this "superpelog"? It's a Dave Keenan style
>duplicated pelog, isn't it?

It approximates two different versions of pelog, which turned up in the
Scala archive as pelog_pa.scl and pelog_pb.scl, credited to "von
Hornbostel" (presumably Erich von Hornbostel). The "type a" pelog is what
we're calling pelogic, but I'm actually more interested right now in the
"type b" pelog, which sounds to my ears a bit more authentic. In any rate,
there's so much variability among pelog scales that both versions are
useful.

The nine-note MOS of this tuning is (maybe coincidentally) similar to the
tuning of the giant bamboo flutes on the CD _Music of the Gambuh Theater_,
which isn't exactly 9-ET.

This tuning also has some useful resources in its own right, beyond the
pelog approximations, such as a pretty good 3:5:7:9:11 chord.

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

Herman,

I quite agree that it would be very useful to know, for any n-limit
temperament, the max number of generators before we obtain a better
approximation for some n-limit consonance, than that given by the
temperament's mapping.

I'd love to see this figure for all our old favourites. The only thing
that bothers me is that I assume it will vary according to which
particular optimum generator we use, and if so, then it isn't entirely
a property of the temperament (i.e. the map).

But otherwise, it works fine for me to say that temperament X has a
_consistency_limit_ of Y generators.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
> wrote:
> > Take a generator of 260.76 cents and a period of 1206.55 cents.
This
> > defines a linear tuning which belongs to a family of related
linear
> > temperaments. The simplest mapping is the "beep" mapping, which
> distributes
> > the 27;25 interval:
> >
> > [(1, 0), (2, -2), (3, -3)]
> >
> > but after 6 iterations of the generator, there's a better 5:1 at
> (1, 6),
> > about 15 cents flat (compared with the 51 cent sharp "beep"
version
> of the
> > interval). That means this particular tuning is consistent
> with "beep"
> > temperament only up to a range of 5 generators -- or to coin a
> phrase, its
> > "consistency range" with respect to "beep" is 5. In comparison,
top
> > meantone has a "consistency range" of 34: its (17, -35) version
of
> 5:1 is
> > only 2 cents flat, compared with the 4-cent sharp (4, -4).
Quarter-
> comma
> > meantone has a "consistency range" of 29, since it has a better
3:1
> at
> > (-11, 30).
> >
> > First of all, I don't like the term "consistency range", but I
> couldn't
> > think of anything better. I'd appreciate ideas for what to call
this
> > property.
>
> Since you're describing a relationship or comparison between two
> temperaments, I would suggest "compatibility range". The
> term "consistency" is usually used to describe only relationships
> within a single temperament.

I don't see any comparison between two temperaments in what Herman is
proposing! It all looks "within temperament" to me.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> Herman,
>
> I quite agree that it would be very useful to know, for any n-limit
> temperament, the max number of generators before we obtain a better
> approximation for some n-limit consonance, than that given by the
> temperament's mapping.
>
> I'd love to see this figure for all our old favourites. The only
thing
> that bothers me is that I assume it will vary according to which
> particular optimum generator we use, and if so, then it isn't
entirely
> a property of the temperament (i.e. the map).
>
> But otherwise, it works fine for me to say that temperament X has a
> _consistency_limit_ of Y generators.

I'd avoid the term 'limit' since it's already so loaded. But I agree
with Dave, and disagree with George, about the appropriateness
of 'consistency' here.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
> > --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> > > Take a generator of 260.76 cents and a period of 1206.55 cents.
This
> > > defines a linear tuning which belongs to a family of related
linear
> > > temperaments. The simplest mapping is the "beep" mapping, which
distributes
> > > the 27;25 interval:
> > >
> > > [(1, 0), (2, -2), (3, -3)]
> > >
> > > but after 6 iterations of the generator, there's a better 5:1
at (1, 6),
> > > about 15 cents flat (compared with the 51 cent sharp "beep"
version of the
> > > interval). That means this particular tuning is consistent
with "beep"
> > > temperament only up to a range of 5 generators -- or to coin a
phrase, its
> > > "consistency range" with respect to "beep" is 5. In comparison,
top
> > > meantone has a "consistency range" of 34: its (17, -35) version
of 5:1 is
> > > only 2 cents flat, compared with the 4-cent sharp (4, -4).
Quarter-comma
> > > meantone has a "consistency range" of 29, since it has a better
3:1 at
> > > (-11, 30).
> > >
> > > First of all, I don't like the term "consistency range", but I
couldn't
> > > think of anything better. I'd appreciate ideas for what to call
this
> > > property.
> >
> > Since you're describing a relationship or comparison between two
> > temperaments, I would suggest "compatibility range". The
> > term "consistency" is usually used to describe only relationships
> > within a single temperament.
>
> I don't see any comparison between two temperaments in what Herman
is
> proposing! It all looks "within temperament" to me.

I haven't really been following this discussion, so I may be
misinterpreting something here, but it sure looks to me like Herman
is comparing or relating "this particular tuning" with "'beep'
temperament" in the first sentence of the following excerpt (which I
am requoting from above):

> > > ... That means this particular tuning is consistent with "beep"
> > > temperament only up to a range of 5 generators -- or to coin a
phrase, its
> > > "consistency range" with respect to "beep" is 5.

So if "this particular tuning" and "'beep' temperament" are both
temperaments, is he not comparing two temperaments?

--George

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "George D. Secor"
> <gdsecor@y...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, Herman Miller
<hmiller@I...>
> wrote:
> > > > Take a generator of 260.76 cents and a period of 1206.55
cents.
> This
> > > > defines a linear tuning which belongs to a family of related
> linear
> > > > temperaments. The simplest mapping is the "beep" mapping,
which
> distributes
> > > > the 27;25 interval:
> > > >
> > > > [(1, 0), (2, -2), (3, -3)]
> > > >
> > > > but after 6 iterations of the generator, there's a better 5:1
> at (1, 6),
> > > > about 15 cents flat (compared with the 51 cent sharp "beep"
> version of the
> > > > interval). That means this particular tuning is consistent
> with "beep"
> > > > temperament only up to a range of 5 generators -- or to coin
a
> phrase, its
> > > > "consistency range" with respect to "beep" is 5. In
comparison,
> top
> > > > meantone has a "consistency range" of 34: its (17, -35)
version
> of 5:1 is
> > > > only 2 cents flat, compared with the 4-cent sharp (4, -4).
> Quarter-comma
> > > > meantone has a "consistency range" of 29, since it has a
better
> 3:1 at
> > > > (-11, 30).
> > > >
> > > > First of all, I don't like the term "consistency range", but
I
> couldn't
> > > > think of anything better. I'd appreciate ideas for what to
call
> this
> > > > property.
> > >
> > > Since you're describing a relationship or comparison between
two
> > > temperaments, I would suggest "compatibility range". The
> > > term "consistency" is usually used to describe only
relationships
> > > within a single temperament.
> >
> > I don't see any comparison between two temperaments in what
Herman
> is
> > proposing! It all looks "within temperament" to me.
>
> I haven't really been following this discussion, so I may be
> misinterpreting something here, but it sure looks to me like Herman
> is comparing or relating "this particular tuning" with "'beep'
> temperament" in the first sentence of the following excerpt (which
I
> am requoting from above):
>
> > > > ... That means this particular tuning is consistent
with "beep"
> > > > temperament only up to a range of 5 generators -- or to coin
a
> phrase, its
> > > > "consistency range" with respect to "beep" is 5.
>
> So if "this particular tuning" and "'beep' temperament" are both
> temperaments, is he not comparing two temperaments?

George, as explained at the top of this message, "this particular
tuning" does result from applying "'beep' temperament", so no.

🔗Paul Erlich <perlich@aya.yale.edu>

Note that Herman also says, above, 'Quarter-comma meantone has
a "consistency range" of 29, since it has a better 3:1 at (-11, 30)'.
This is an example which concerns a temperament you already
understand, and clearly only one, not two.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> > ...
> > So if "this particular tuning" and "'beep' temperament" are both
> > temperaments, is he not comparing two temperaments?
>
> George, as explained at the top of this message, "this particular
> tuning" does result from applying "'beep' temperament", so no.
>
> Note that Herman also says, above, 'Quarter-comma meantone has
> a "consistency range" of 29, since it has a better 3:1 at (-11,
30)'.
> This is an example which concerns a temperament you already
> understand, and clearly only one, not two.

Okay, I get it. Thanks.

--George