Topic: A Christmas present for George

2 scales

Thread (14 messages)

From: Gene Ward Smith (2005-12-25)
Subject: 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]

From: Gene Ward Smith (2005-12-25)
Subject: Re: A Christmas present for George

--- 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

From: Gene Ward Smith (2005-12-25)
Subject: Re: A Christmas present for George

--- 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

From: Gene Ward Smith (2005-12-27)
Subject: Re: A Christmas present for George

--- 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]

From: George D. Secor (2005-12-28)
Subject: Re: A Christmas present for George

> --- 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

From: George D. Secor (2006-01-03)
Subject: the simplest pan-proportionally beating 12-tone temperament (was: Xmas present)

--- 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:
http://groups.yahoo.com/group/tuning/messages/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

From: Carl Lumma (2006-01-03)
Subject: Re: [tuning-math] the simplest pan-proportionally beating  12-tone temperament

>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:
>http://groups.yahoo.com/group/tuning/messages/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

From: George D. Secor (2006-01-03)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- 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:
> >http://groups.yahoo.com/group/tuning/messages/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

From: Carl Lumma (2006-01-03)
Subject: Re: [tuning-math] Re: the simplest pan-proportionally beating  12-tone temperament

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

From: wallyesterpaulrus (2006-01-03)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- 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 . . .

From: George D. Secor (2006-01-05)
Subject: More brat versatility? (was: A Christmas present for George)

> --- 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

From: George D. Secor (2006-01-05)
Subject: Re: More brat versatility? (was: A Christmas present for 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

From: Carl Lumma (2006-05-11)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament (was: Xmas present)

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

From: George D. Secor (2006-05-12)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament (was: Xmas present)

--- 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