A Christmas present for George

Here is something you might find to be a more expeditious way of
exploring these scales/temperaments; this example has a circle of
fifths from C to F given in terms of the brats for B and D, called b
and d.

[4/3645*(-64*d+5504*b*d+1341*b)/(4*d+1)/b,
4*(16+477*b+2048*b*d)/(-64*d+5504*b*d+1341*b),
(16*d+3037*b*d+768*b)/(16+477*b+2048*b*d),
2*(16*d+2269*b*d+576*b)/(16*d+3037*b*d+768*b),
16*(4*d+1)*(-1+54*b)/(16*d+2269*b*d+576*b),
80*b/(-1+54*b), 3/2, 3/2, 3/2, 3/2, 3/2, 3/2]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> [4/3645*(-64*d+5504*b*d+1341*b)/(4*d+1)/b,
> 4*(16+477*b+2048*b*d)/(-64*d+5504*b*d+1341*b),
> (16*d+3037*b*d+768*b)/(16+477*b+2048*b*d),
> 2*(16*d+2269*b*d+576*b)/(16*d+3037*b*d+768*b),
> 16*(4*d+1)*(-1+54*b)/(16*d+2269*b*d+576*b),
> 80*b/(-1+54*b), 3/2, 3/2, 3/2, 3/2, 3/2, 3/2]

I'd better give the brats which go with this:

4, 4, d, 2, 16*(4*d+1)/(d*(-16+35*b)), b,
3/2, 3/2, 3/2, 3/2, 3/2, 3/2

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

The same family of temperaments can also be given in terms of the
brats for d and e:

[4/3645*(5504*d+1341*d*e+1341)/(d*e+4*d+1),
4*(512*d*e+2048*d+477)/(5504*d+1341*d*e+1341),
(3037*d+768*d*e+768)/(512*d*e+2048*d+477),
2*(2269*d+576*d*e+576)/(3037*d+768*d*e+768),
(829*d*e+3456*d+864)/(2269*d+576*d*e+576),
1280*(d*e+4*d+1)/(829*d*e+3456*d+864),
3/2, 3/2, 3/2, 3/2, 3/2, 3/2]

brats: 4, 4, d, 2, e, 16/35*(d*e+4*d+1)/(d*e),
3/2, 3/2, 3/2, 3/2, 3/2, 3/2

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

I haven't heard from George on whether he would find these things
useful, but I suspect so, and am giving another one:

Circle of fifths, C to F:

[4*(8192*f*d-69*f-2048*d+3456)/(21809*f*d-5376*d-69*f+9216),
(12288*f*d-207*f-3072*d+5248)/(8192*f*d-69*f-2048*d+3456),
5*(3645*f*d-896*d+1536)/(12288*f*d-207*f-3072*d+5248),
2/5*(13617*f*d-3328*d+5760)/(3645*f*d-896*d+1536),
5120/3*(12*f*d-3*d+5)/(13617*f*d-3328*d+5760),
3/2, 3/2, 3/2, 3/2, 3/2, 3/2,
1/1215*(21809*f*d-5376*d-69*f+9216)/(12*f*d-3*d+5)]

Brats, C to F:
[4, 3, d, 2, 64/3*(12*f*d-3*d+5)/(207*f-128)/d,
3/2, 3/2, 3/2, 3/2, 3/2, 3/2, f]

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
>
> I haven't heard from George

Sorry. I've been away over the holiday weekend and didn't see the
previous 3 messages till Tuesday, then spent some time trying to
figure out how useful they would be.

> on whether he would find these things
> useful, but I suspect so,

Sorry again, but it looks as if they would be only marginally useful
to me. (But I do appreciate the thought and effort that went into
your timely gift. :-)

I shouldn't have been so hasty in submitting that last batch of
requests, because I've found all sorts of other brat combinations --
literally dozens of possibilities involving various trade-offs.
After doing more extensive comparative listening to a lot of these,
I'm drawing some new conclusions, e.g.: a D major brat of 2.75 is
better than 2.5, because the M3:5th beat ratio (2 in the former vs.
2.5 in the latter) seems to be the most important factor to unify the
beating.

So I still haven't settled on the 2 or 3 "best" well-temperaments --
more time needed to listen, compare, and see if any more ideas
materialize.

Anyway, thanks for your efforts and especially your patience, Gene.
I'll try to be less flaky before asking for any more rationalizing.

Hoping you had a nice Christmas,

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> >
> > I haven't heard from George
> > on whether he would find these things
> > useful, but I suspect so,
>
> Sorry again, but it looks as if they would be only marginally
useful
> to me. (But I do appreciate the thought and effort that went into
> your timely gift. :-)
>
> I shouldn't have been so hasty in submitting that last batch of
> requests, because ...

Once again, I shouldn't have been so hasty in my conclusion about the
usefulness of these formulas -- once I realized that I didn't have to
go so far as to put the ratios in the form n/d in order to explore
their possibilities. They proved to be very useful!

For starters, check out these two well-temperaments that I stumbled
across:

! WTPB-24a.scl
!
George Secor's 24-triad proportional-beating well-temperament (24a)
12
!
256/243
272/243
32/27
304/243
4/3
1024/729
364/243
128/81
2032/1215
16/9
152/81
2/1

The major-triad brats are all exact simple ratios (starting on C): 4,
4, 3, 2, 1.5, 16/9, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5

The minor-triad brats are also all exact simple ratios (starting on
C): 1.125, 1.5, 15/7, 4.2, 1, 1.5, 1, 1, 1, 1, 1, 1

It's also musically useful in that it has a reasonable progression of
key colors around the circle of fifths (best triad on C, worst on B,
F#, C#, and G#).

I'm therefore submitting it as my answer to Aaron Johnson's question:
/tuning/topicId_59689.html#61731
> what is the simplest possible 12-note temperament where all 24
major and minor
> triads have rationally proportional beating? here 'simplest' means
that the
> brats (beat ratios for the un-initiated) are the lowest numbers in
the
> numerator and denominator that they can be.....

I also found another solution with brats that are almost as simple:

! WTPB-24b.scl
!
George Secor's 24-triad proportional-beating well-temperament (24b)
12
!
256/243
4076/3645
32/27
169/135
4/3
1024/729
202/135
128/81
2033/1215
16/9
6832/3645
2/1

The major-triad brats are (starting on C): 4, 4, 2.25, 2, 16/9, 1.6,
1.5, 1.5, 1.5, 1.5, 1.5, 1.5

A couple of the minor brats (on G and E) are not as simple as in 24a,
but 24b is IMO more useful as a well-temperament.

--George

>For starters, check out these two well-temperaments that I stumbled
>across:
>
>! WTPB-24a.scl
>!
>George Secor's 24-triad proportional-beating well-temperament (24a)
> 12
>!
> 256/243
> 272/243
> 32/27
> 304/243
> 4/3
> 1024/729
> 364/243
> 128/81
> 2032/1215
> 16/9
> 152/81
> 2/1
>
>The major-triad brats are all exact simple ratios (starting on C): 4,
>4, 3, 2, 1.5, 16/9, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5
>
>The minor-triad brats are also all exact simple ratios (starting on
>C): 1.125, 1.5, 15/7, 4.2, 1, 1.5, 1, 1, 1, 1, 1, 1
>
>It's also musically useful in that it has a reasonable progression of
>key colors around the circle of fifths (best triad on C, worst on B,
>F#, C#, and G#).
>
>I'm therefore submitting it as my answer to Aaron Johnson's question:
>/tuning/topicId_59689.html#61731
>> what is the simplest possible 12-note temperament where all 24
>> major and minor triads have rationally proportional beating? here
>> 'simplest' means that the brats (beat ratios for the un-initiated)
>> are the lowest numbers in the numerator and denominator that they
>> can be.....

Is this simpler than Wendell's scale?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >! WTPB-24a.scl
> >!
> >George Secor's 24-triad proportional-beating well-temperament (24a)
> > 12
> >!
> > 256/243
> > 272/243
> > 32/27
> > 304/243
> > 4/3
> > 1024/729
> > 364/243
> > 128/81
> > 2032/1215
> > 16/9
> > 152/81
> > 2/1
> >
> >The major-triad brats are all exact simple ratios (starting on C):
4,
> >4, 3, 2, 1.5, 16/9, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5
> >
> >The minor-triad brats are also all exact simple ratios (starting
on
> >C): 1.125, 1.5, 15/7, 4.2, 1, 1.5, 1, 1, 1, 1, 1, 1
> >
> >It's also musically useful in that it has a reasonable progression
of
> >key colors around the circle of fifths (best triad on C, worst on
B,
> >F#, C#, and G#).
> >
> >I'm therefore submitting it as my answer to Aaron Johnson's
question:
> >/tuning/topicId_59689.html#61731
> >> what is the simplest possible 12-note temperament where all 24
> >> major and minor triads have rationally proportional beating? here
> >> 'simplest' means that the brats (beat ratios for the un-
initiated)
> >> are the lowest numbers in the numerator and denominator that they
> >> can be.....
>
> Is this simpler than Wendell's scale?
>
> -Carl

Yes, because:

1) Wendell's rational version (per GWS) has one major-triad brat
approximating 2 (C major, ~1.9795), whereas all of the major-triad
brats in 24a are *exact* simple ratios. However, if that one brat in
Wendell's temperament were exact, then his major brats would be
simpler, so on this point it could be considered a close call.

2) Half of Wendell's minor-triad brats aren't simple ratios, and most
of those don't even approximate simple ratios, whereas all of the
minor-triad brats in 24a are *exact* simple ratios; on this point,
it's not even close.

Apart from that, the two temperaments are in different categories;
24a has a much higher key contrast.

--George

At 02:42 PM 1/3/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>>
>> >! WTPB-24a.scl
>> >!
>> >George Secor's 24-triad proportional-beating well-temperament (24a)
>> > 12
>> >!
>> > 256/243
>> > 272/243
>> > 32/27
>> > 304/243
>> > 4/3
>> > 1024/729
>> > 364/243
>> > 128/81
>> > 2032/1215
>> > 16/9
>> > 152/81
>> > 2/1
>> >
>> >The major-triad brats are all exact simple ratios (starting on C):
>> >4, 4, 3, 2, 1.5, 16/9, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5
>> >
>> >The minor-triad brats are also all exact simple ratios
>> >(starting on C):
>> > 1.125, 1.5, 15/7, 4.2, 1, 1.5, 1, 1, 1, 1, 1, 1
//
>> Is this simpler than Wendell's scale?
//
>Yes, because:
//
>2) Half of Wendell's minor-triad brats aren't simple ratios, and most
>of those don't even approximate simple ratios, whereas all of the
>minor-triad brats in 24a are *exact* simple ratios; on this point,
>it's not even close.

Hey, you're right. I thought Bob's claim was that all brats were
1, 2, or 1.5, major and minor. But according to Scala, the minor
brats are...

-0.859416
-1.000000
-0.721053
-1.000000
-1.000000
-1.000000
-0.688679
-0.781893
-1.004566
-0.644769
-0.815758
-1.000000
-0.859416

Hrm...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Hey, you're right. I thought Bob's claim was that all brats were
> 1, 2, or 1.5, major and minor.

You have the beat ratios that involve the minor thirds in the major
triad, and then you have the beat ratios in the minor triad. I'm pretty
sure Bob was talking about the former and not the latter . . .

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
>
> I haven't heard from George on whether he would find these things
> useful, but I suspect so, and am giving another one:
>
> Circle of fifths, C to F:
> ...
> Brats, C to F:
> [4, 3, d, 2, 64/3*(12*f*d-3*d+5)/(207*f-128)/d,
> 3/2, 3/2, 3/2, 3/2, 3/2, 3/2, f]

Gene, I found that this one was not as helpful. For C and G, [4, 4],
[3, 4], or even [3, 3] would have been much more useful. (But the
first two were great -- thanks!) Would it be very difficult for you
to give me formulas for the fifth-ratios with a greater number of
independent brat variables, e.g., [c, g, d, a, e, b, 1.5, 1.5, 1.5,
1.5, 1.5, f] in versions having f, e, or b as the dependent (or
unassigned) variable? I expect that the formulas would be quite
complicated, but I can't imagine that it would be more time-consuming
than having to make up many sets of formulas with differing
combinations of constants, and I believe I could make good use of
them in a spreadsheet.

BTW, the "best" (IMO) high-contrast well-temperament I've been able
to come up with so far is one that I found apart from (and prior to)
using your formulas, namely this:

! GS5_23WT.scl
!
George Secor's rational 5/23-comma proportional-beating well-
temperament
12
!
5175/4912
15801/14122
46575/39296
70725/56488
419175/314368
1725/1228
42251/28244
15525/9824
47265/28244
139725/78592
575/307
2/1

It has 11 exact major-triad brats [4, 4, 2.5, 2, 23/12, 1.5, 1.5,
1.5, 1.5, 1.5, ~1,5] (starting on C), with the unassigned brat for F
being very close to 1.5. There are also 8 exact minor-triad brats,
with the remaining 4 approximating reasonably simple ratios: [~9/7,
~1.4, ~19/11, 2.5, 8/3, 1, 1, 1, 1, 1, 1, ~1].

What makes this the "best" one (so far) is that it excels on several
points: 1) good major and 2) minor brats, 3) a superb progression of
key color (such as is desirable in a well-temperament), and 4) no
fifth tempered by an excessive amount (max. is -5.0473 cents, for
D:A).

The approximate 1.5 (major) and 1 (minor) brats on F (1.51569 and
1.003426, respectively) are associated with a fifth tempered narrow
by only 0.0730 cents, so the amount of actual pitch error involved
with these approximated brats is quite small, which leads me to
believe that exactness of brats probably has more significance
as "eye candy" than as something that will appeal to the ear. In any
case, I'm already convinced that points 3) and 4) above are much more
important than 1) and 2) -- but if you can have all of them in a
single temperament, then so much the better.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> ...
> BTW, the "best" (IMO) high-contrast well-temperament I've been able
> to come up with so far is one that I found apart from (and prior to)
> using your formulas, namely this:
> ...
> It has 11 exact major-triad brats [4, 4, 2.5, 2, 23/12, 1.5, 1.5,
> 1.5, 1.5, 1.5, ~1,5] (starting on C), with the unassigned brat for F
> being very close to 1.5. ...

Correction: the major-triad brats should be:

[4, 4, 2.5, 2, 23/12, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5, ~1,5]>

--George

Hiya George,

How is 24b less contrasty than 24a? They both have the
same worst maj 3rd of 408 cents, but 24b actually has one
more of them.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@...> wrote:
>
> Hiya George,
>
> How is 24b less contrasty than 24a? They both have the
> same worst maj 3rd of 408 cents, but 24b actually has one
> more of them.
>
> -Carl

They both have three pythagorean major 3rds, which are the worst ones.
The big difference is on the consonant side of the circle, where the C
major triad has total absolute error of 7.6c in 24a, but 13.7c in 24b.
Try these in Scala and you'll hear that the difference is *HUGE*!

--George