Topic: well temperaments : just won't die

9 scales

Thread (27 messages)

From: Carl Lumma (2007-02-18)
Subject: well temperaments : just won't die

So!

After all my playing around with the idea of '21st-century
circulating temperaments', I decided to have a go at more
historically-appropriate fare.  I'll still be improving on
history, of course.  Just not as much.  :)

I started with Werckmeister's pure octaves.  This assumption
can easily be changed later if desired.

Then I decided there would only be two kinds of 5th.
Using 3 or more will get you more shades of contrast, but
no new colors (basic ways of making the compromise).  I
don't think it's worth it for the complexity it adds.

To get enough contrast from ET without going too close
to meantone, 1/5- and 1/6-comma systems are basically the
only games in town.  But how to distribute these on the
chain of fifths?

Here I used two constraints: 1. The scale must have at
least one 3rd audibly better than ET. and 2. The scale
must not have any 3rds > 404 cents.  You might not think
that Pythagorean thirds sound like ass, but I do.

I used these constraints to get the number circular
permutations down to something manageable.  Then I looked
at the 3rds in Scala.

Not all patterns produced unique patterns of thirds.  In
these cases I chose scales having their best 7:4 on keys
with better thirds.

I used Scala to rotate the scales into historical 'good
keys' position.  Not all scales had good transitions
along the chain of fifths, and those were discarded.

The following four scales is the result:

! 12_lumma_5thcomma[246].scl
!
 1/5-comma SLSSLLLSLLLS temperament.
 12
!
 99.609
 199.218
 298.827
 393.744
 502.737
 597.654
 697.263
 796.872
 896.481
 1000.782
 1095.699
 2/1
!
! 2 x 394-cent 3rds on F C.
! 4 x 398-cent 3rds on Eb Bb G D.
! 6 x 403-cent 3rds elsewhere.

! 12_lumma_5thcomma[327].scl
!
 1/5-comma SSSLSLLLSLLL temperament.
 12
!
 94.917
 194.526
 294.135
 393.744
 498.045
 592.962
 697.263
 796.872
 891.789
 996.090
 1091.007
 2/1
!
! 3 x 394-cent 3rds on F C G.
! 2 x 398-cent 3rds on Bb D.
! 7 x 403-cent 3rds elsewhere.

! 12_lumma_6thcomma[1335].scl
!
 1/6-comma SSSLSLLSLLLS temperament.
 12
!
 98.045
 196.090
 298.045
 396.090
 501.955
 596.090
 698.045
 796.090
 894.135
 1000.000
 1094.135
 2/1
!
! 1 x 392-cent 3rd on F.
! 3 x 396-cent 3rds on Bb C G.
! 3 x 400-cent 3rds on Eb D E.
! 5 x 404-cent 3rds elsewhere.

! 12_lumma_6thcomma[2226].scl
!
 1/6-comma SSSSLLLSLLLS temperament.
 12
!
 98.045
 196.090
 298.045
 392.180
 501.955
 596.090
 698.045
 796.090
 894.135
 1000.000
 1094.135
 2/1
!
! 2 x 392-cent 3rds on F C.
! 2 x 396-cent 3rds on Bb G.
! 2 x 400-cent 3rds on Eb D.
! 6 x 404-cent 3rds elsewhere.

I'm not aware of anything like this having been done
before (though I'd guess it has), but I feel pretty good
about it.

I'm also not aware of any historical temperaments using
these patterns.  It seems like they couldn't get past the
idea of bunching the fifths at either end of the chain.
But maybe there are Victorian temperaments like this.

Comments sought,

-Carl

From: Carl Lumma (2007-02-18)
Subject: Re: well temperaments : just won't die

[I wrote...]
> To get enough contrast from ET without going too close
> to meantone, 1/5- and 1/6-comma systems are basically the
> only games in town.  But how to distribute these on the
> chain of fifths?

I forgot the part about how one of the fifths is pure,
to avoid harmonic waste and make tuning easier.

-Carl

From: Tom Dent (2007-02-18)
Subject: Re: well temperaments : just won't die

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> So!
> 
> (...)
> I decided there would only be two kinds of 5th.
> Using 3 or more will get you more shades of contrast, but
> no new colors (basic ways of making the compromise).  I
> don't think it's worth it for the complexity it adds.
> 

The temperament-calculators Neidhardt and Sorge clearly did, though.

> I'm also not aware of any historical temperaments using
> these patterns.  It seems like they couldn't get past the
> idea of bunching the fifths at either end of the chain.
> But maybe there are Victorian temperaments like this.
> 
> Comments sought,
> 
> -Carl


The two guys I mentioned above brought out a large clutch of such
temperament schemes in the 1720s onwards. They used, by habit, both
1/6 and 1/12 comma chunks. Also in the main they were rather kinder to
A major - which Carl seems to be relegated to the status of a rare and
remote key! ... 

For example: 
so-called 'Neidhardt I' (Poletti's favourite unobtrusive WT ;)

C-E 392 
G-B 394
F-A and D-F# 396
A-C# and Bb-D 400
Eb-G 402
all others 404

pattern of fifths 222211001100

where 2 -> 1/6 comma, 1 -> 1/12 comma. The apparent favouring of
sharps is somewhat counteracted by better fifths on the flat side. 

Sorge 1744 Harpsichord tuning 

F-A, C-E, G-B 396
D-F#, Bb-F 398
A-C#, E-G#, Eb-G 400
rest 404

pattern of fifths: 222021100110


... I don't quite see the point of the 1/5 comma tunings, if you don't
use the possibility of putting four tempered fifths in a row. If
you're interested in fifths and major thirds, it seems most logical to
decide how narrow your 'best' major third should be, then divide it
into four regular fifths. 

In this respect the 1/6 comma tunings are preferable. The first one
has the slight flaw that A-C# is 'worse' than E-G#. The second is a
perfectly decent arrangement - though I would transpose it to
SSSSSLLLSLLL for most purposes.

Then again you might object that this was too sharp-favouring... well,
that's why Neidhardt and Sorge used their little 1/12 comma chunks, to
even things out.

In practice it might be relatively easy to start here (assuming you
can tune 1/6 comma by ear!)

222220002000

Then notice that Eb-G is a bit sharp for a relatively 'central' key -
tweak Eb up a bit :

222220001100

 - now B-D# is also a bit oversharp - tweak B up a bit... 

222211001100

- whoops! you just got Neidhardt I again. Hours of fun.

~~~T~~~

From: Carl Lumma (2007-02-18)
Subject: Re: well temperaments : just won't die

> > So!
> > 
> > (...)
> > I decided there would only be two kinds of 5th.
> > Using 3 or more will get you more shades of contrast, but
> > no new colors (basic ways of making the compromise).  I
> > don't think it's worth it for the complexity it adds.
> 
> The temperament-calculators Neidhardt and Sorge clearly did, though.

Did they, or is that an assumption on your part from the
(apparent) fact that wrote down such scales?

> > I'm also not aware of any historical temperaments using
> > these patterns.  It seems like they couldn't get past the
> > idea of bunching the fifths at either end of the chain.
> > But maybe there are Victorian temperaments like this.
> > 
> > Comments sought,
> > 
> > -Carl
>
> The two guys I mentioned above brought out a large clutch of such
> temperament schemes in the 1720s onwards.

Did either of them systematically check cyclic permutations
against criteria?

> They used, by habit, both
> 1/6 and 1/12 comma chunks.

I suspect most of what they did was by habit, i.e. not
fully generalized i.e. not all assumptions examined.

> Also in the main they were rather kinder to
> A major - which Carl seems to be relegated to the
> status of a rare and remote key! ...

My '21st-century temperaments' all aim for A and E.
I understand these historical things aim for F and C.
'zthat true?  If I could get something reasonable
on F C G I accepted the pattern, and let other criteria
choose between patterns.  I got the fit as good as
I could otherwise.

> For example: 
> so-called 'Neidhardt I' (Poletti's favourite unobtrusive WT ;)
> 
> C-E 392 
> G-B 394
> F-A and D-F# 396
> A-C# and Bb-D 400
> Eb-G 402
> all others 404

Ok, so this is centered on C not F.  Also I like the
lack of pythag. 3rds.

> Sorge 1744 Harpsichord tuning 
> 
> F-A, C-E, G-B 396
> D-F#, Bb-F 398
> A-C#, E-G#, Eb-G 400
> rest 404
> 
> pattern of fifths: 222021100110

And again.

> ... I don't quite see the point of the 1/5 comma tunings, if
> you don't use the possibility of putting four tempered fifths
> in a row.

I was just about to post on this point.  The alternative is
1/7-comma.  And this seems to be the single pattern of
interest:

! 12_lumma_7thcomma[2343].scl
!
 1/7-comma SSSSSLSLLSLL temperament.
 12
!
 96.927
 197.207
 297.486
 394.414
 501.396
 594.972
 698.603
 798.882
 895.810
 999.441
 1096.369
 2/1
!
! 2 x 394-cent 3rds on F C.
! 3 x 398-cent 3rds on Bb G D.
! 4 x 401-cent 3rds on Ab Eb A B.
! 3 x 404-cent 3rds elsewhere.

However we can see that there is arguably a different
compromise going on with the 1/5-comma systems I posted,
even though their best possible thirds don't occur.

> If you're interested in fifths and major thirds, it seems most
> logical to decide how narrow your 'best' major third should be,
> then divide it into four regular fifths.

How narrow it should be is 386 cents.  It seems more logical
to put an upper bound on it.

> In this respect the 1/6 comma tunings are preferable. The first one
> has the slight flaw that A-C# is 'worse' than E-G#. The second is a
> perfectly decent arrangement - though I would transpose it to
> SSSSSLLLSLLL for most purposes.

That's how I had it originally, but I moved it to get F
better.  Somewhere I picked up that F is supposed to be best.
I assumed this was for accompaniment, which I think is a
miserable use for a keyboard anyway but I was trying to
play by the rules.

> Then again you might object that this was too sharp-favouring...
> well, that's why Neidhardt and Sorge used their little 1/12
> comma chunks, to even things out.

Thanks!  Knowing this, I'll move them back!

-Carl

From: Tom Dent (2007-02-18)
Subject: Re: well temperaments : just won't die

I can't say much about Neidhardt / Sorge's motivation, so far as
choosing this or that exact pattern of fifths was concerned.

They were both fans of ET, so *perhaps* they thought of it as a basic
background of 1/12 comma with a bit of 1/6 comma or Pythagorean
flavouring thrown in.

Systematic combinatorics probably wasn't involved: more like
progressive adjustments, until one got an apparently decent, or even
elegant, result.

They did add one further criterion: no pythagorean *minor* thirds.
This rules out a lot of possibilities, unless you have 1/12 comma or
similarly small bits to play with.

With 1/7 comma you do get something decent, certainly... since you
tolerate 32:27, I would also add

SSSSSLLSLLLS

which is a bit more unequal: 3 at 394, 2 at 398, 3 at 401, 4 at 404.

John Barnes went even further in dilution and (in an appendix to his
1979 article, where he said the tuning had already been used in a Bach
recital) gave the following pattern of 1/8 comma fifths: SSSSSSLLSLLS.

A question below.


--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> Not all scales had good transitions
> along the chain of fifths, and those were discarded.
> 

... What could be a bad transition, if you allow yourself only a pure
or narrow tempered fifth?

~~~T~~~

From: Carl Lumma (2007-02-18)
Subject: Re: well temperaments : just won't die

> They did add one further criterion: no pythagorean *minor* thirds.

Lame.  I wonder why.  They're perfectly good.
 
> John Barnes went even further in dilution and (in an appendix to
> his 1979 article, where he said the tuning had already been used
> in a Bach recital) gave the following pattern of 1/8 comma
> fifths: SSSSSSLLSLLS.

Bah!
 
> A question below.
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> >
> > Not all scales had good transitions
> > along the chain of fifths, and those were discarded.
> > 
> 
> ... What could be a bad transition, if you allow yourself
> only a pure or narrow tempered fifth?

Some patterns produced non-sine-ish circulation.  Does that
make sense?

-Carl

From: Tom Dent (2007-02-18)
Subject: Re: well temperaments : just won't die

> > > Not all scales had good transitions
> > > along the chain of fifths, and those were discarded.
> > > 
> > 
> > ... What could be a bad transition, if you allow yourself
> > only a pure or narrow tempered fifth?
> 
> Some patterns produced non-sine-ish circulation.  Does that
> make sense?
> 
> -Carl
>

Yes... so you rule out double-centred things like SSSLLLSSSLLL. 

Personally I would prefer to be able to adjust things continuously
rather than working to a grid. Otherwise it's not clear who's making
the musical decisions, you or the grid.
~~~T~~~

From: Gene Ward Smith (2007-02-19)
Subject: Re: well temperaments : just won't die

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> To get enough contrast from ET without going too close
> to meantone, 1/5- and 1/6-comma systems are basically the
> only games in town.  But how to distribute these on the
> chain of fifths?

Here's something you can try: if you use six fifths of 256/171 (about 
1/6 comma), one fifth of  102889341747/68719476736 = 3^7 19^6 2^(-36),
(about 1/7 comma) and five pure fifths, the circle will close. If 
anywhere in the circle of fifths you have two 3/2s and one 256/171 in 
any order, you get a 19/16 minor third. Can you get a Lumma-style 
circulating temperament out of these which gives you as many 19/16 
minor thirds as you can contrive? Bonus points for 1-19/16-3/2 JI 
chords in there.

From: Carl Lumma (2007-02-19)
Subject: Re: well temperaments : just won't die

> > To get enough contrast from ET without going too close
> > to meantone, 1/5- and 1/6-comma systems are basically the
> > only games in town.  But how to distribute these on the
> > chain of fifths?
> 
> Here's something you can try: if you use six fifths of
> 256/171 (about 1/6 comma), one fifth of
> 102889341747/68719476736 = 3^7 19^6 2^(-36),
> (about 1/7 comma) and five pure fifths, the circle will
> close. If anywhere in the circle of fifths you have two
> 3/2s and one 256/171 in any order, you get a 19/16 minor
> third. Can you get a Lumma-style circulating temperament
> out of these which gives you as many 19/16 minor thirds
> as you can contrive? Bonus points for 1-19/16-3/2 JI 
> chords in there.

I'm going to submit moh-ha-ha for this.

!
 Rational well temperament.
 12
!
 19/18
 323/288
 19/16
 323/256
 171/128
 361/256
 551/368
 19/12
 323/192
 57/32
 513/272
 2/1
!

-Carl

From: Gordon Collins (2007-02-19)
Subject: Re: well temperaments : just won't die

Carl wrote:

> The scale
> must not have any 3rds > 404 cents.  You might not think
> that Pythagorean thirds sound like ass, but I do.

You have plenty of company among 18th-century theorists.  In particular, 404 is very close to Sorge's tolerance limit of  9/11 SC wide.  (He actually wrote, "5/12 of a Lesser Diesis", which is a little bit smaller [17.1 cents], but this is the largest major third he used in his temperaments.)



> ! 12_lumma_6thcomma[1335].scl
> !
> 1/6-comma SSSLSLLSLLLS temperament.

If you modify this to SSSSLSLLLSLL, you get a temperament that satisfies all the following criteria:

1) The worst major third is within Sorge's limit.
2) The best major third is C-E, and is as good as possible given 1).
3) The best minor third is A-C.
4) The error graph for the major thirds has only a single peak or plateau.
5) The error graph for the minor thirds has only a single peak or plateau.

! 12_Sorgean_6th-comma.scl
!
Sorgean 1/6-Pyth-comma temperament SSSSLSLLLSLL.
 12
!
 94.135
 196.090
 298.045
 392.180
 498.045
 592.180
 698.045
 796.090
 894.135
 996.090
 1094.135
 2/1


SSSLSSLLLSLL also works, but its best major third is not as good.

If you allow 1/12-PC fifths as well, as the historical authors did, there are many more possibilities, having smoother progressions of thirds.

Gordon

From: Carl Lumma (2007-02-19)
Subject: Re: well temperaments : just won't die

> > The scale
> > must not have any 3rds > 404 cents.  You might not think
> > that Pythagorean thirds sound like ass, but I do.
> 
> You have plenty of company among 18th-century theorists.

Good to know.

> > ! 12_lumma_6thcomma[1335].scl
> > !
> > 1/6-comma SSSLSLLSLLLS temperament.
> 
> If you modify this to SSSSLSLLLSLL, you get a temperament that
> satisfies all the following criteria:
> 
> 1) The worst major third is within Sorge's limit.
> 2) The best major third is C-E, and is as good as possible given 1).
> 3) The best minor third is A-C.
> 4) The error graph for the major thirds has only a single peak or
> plateau.
> 5) The error graph for the minor thirds has only a single peak or
> plateau.

Why are these last two desirable?

-Carl

From: Gordon Collins (2007-02-19)
Subject: Re: well temperaments : just won't die

Carl wrote:

>> 4) The error graph for the major thirds has only a single peak or
>> plateau.
>> 5) The error graph for the minor thirds has only a single peak or
>> plateau.
>
>Why are these last two desirable?

They mean that there is a consistent, coherent progression of dissonance around the circle of fifths.  Stated in more detail: there exists a note such that, as you move around the circle of fifths in either direction from C to this note, the major thirds on the notes stay the same or get progressively worse.  Similarly for the minor thirds.

I'm pretty sure that the criteria were mentioned in some form by someone in the 18th-century, but at the moment I can't pin down who did.

You expressed the desirability yourself in your response to Tom Dent:
"
>> ... What could be a bad transition, if you allow yourself
>> only a pure or narrow tempered fifth?
>
>Some patterns produced non-sine-ish circulation.
"

Gordon

From: Tom Dent (2007-02-19)
Subject: Re: well temperaments : just won't die

--- In tuning@yahoogroups.com, "Gordon Collins" <clavier@...> wrote:
>
> 404 is very close to Sorge's tolerance limit of  9/11 SC wide.  (He
actually wrote, "5/12 of a Lesser Diesis", which is a little bit
smaller [17.1 cents], but this is the largest major third he used in
his temperaments.)
> 

I've never seen the actual text of his works ... it would be
interesting to see if he said anything about how to put them into
practice.

Didn't he have a limit for minor thirds too? At least, none of his
published formulas have 3 pure fifths in succession.


> If you modify this to SSSSLSLLLSLL, you get a temperament that
satisfies all the following criteria:
> 
>(...)

> 5) The error graph for the minor thirds has only a single peak or
plateau.

... I claim this peak is outside Sorge's limits.

~~~T~~~

From: Gordon Collins (2007-02-19)
Subject: Re: well temperaments : just won't die

Tom Dent wrote:

> I've never seen the actual text of [Sorge's] works ... it would be
> interesting to see if he said anything about how to put them into
> practice.

Do you mean practice as in actual tuning, or practice as in designing temperaments?


> Didn't he have a limit for minor thirds too? At least, none of his
> published formulas have 3 pure fifths in succession.

Not that I know of, though I haven't seen his _Anweisung zur Stimmung_ yet.  In his _Gespraech_, I find several mentions of the limit on major thirds, but no limit on minor thirds.  His published temps only have 3-4 pure fifths all together, so the lack of three in a row doesn't seem to be telling.

Gordon

From: Carl Lumma (2007-02-19)
Subject: Re: well temperaments : just won't die

--- In tuning@yahoogroups.com, "Gordon Collins" <clavier@...> wrote:
>
> Carl wrote:
> 
> >> 4) The error graph for the major thirds has only a single peak or
> >> plateau.
> >> 5) The error graph for the minor thirds has only a single peak or
> >> plateau.
> >
> >Why are these last two desirable?
> 
> They mean that there is a consistent, coherent progression of
> dissonance around the circle of fifths.

Oh, I see what you meant now.

> You expressed the desirability yourself in your response to
> Tom Dent:
> "
> >> ... What could be a bad transition, if you allow yourself
> >> only a pure or narrow tempered fifth?
> >
> >Some patterns produced non-sine-ish circulation.
> "
> 
> Gordon

Yes.  But where I differ is on the importance of the minor
thirds.

-Carl

From: Aaron Krister Johnson (2007-02-19)
Subject: Re: well temperaments : just won't die

Carl,

This is an interesting post...my comment would be that I think a damn
good well-temperament was devised by myself, with slight modification
by George Secor. It's in the .scl archives as 'johnson-secor_rwt.scl'
It contains five pure 24/19s, which are the largest thirds, which keep
the parameter you have set to stay at or below ~404cents (a 24/19 is
404.442 cents). The thirds are:

1 361/288           391.116 cents  
1 4864/3879         391.750 cents  
1 240/191           395.354 cents  
1 2413/1920         395.666 cents  
1 160/127           399.892 cents  
1 3629/2880         400.204 cents  
1 1293/1024         403.808 cents  
5 24/19             404.442 cents

That's 5 fifths that are smaller than 12-eq, which is on par with
Young, except it beats Young because it's smallest third is samller
than Young, and it's largest third is smaller than the 81/64s you see
in Young (6 of them--yuk!)

And the fifths are very fine too:

1 190/127           697.405 cents  
1 1143/764          697.417 cents  
1 431/288           697.943 cents  
1 29032/19395       698.351 cents  
2 256/171           698.577 cents  
6 3/2               701.955 cents  perfect fifth

With 6 perfect fifths, and no Pythagorean thirds, this is a great
temperament...and, it's rational, and, equal-beating (if you like that
as a feature) to boot!

here's the .scl file:
http://www.akjmusic.com/johnson-secor_rwt.scl

-A.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> So!
> 
> After all my playing around with the idea of '21st-century
> circulating temperaments', I decided to have a go at more
> historically-appropriate fare.  I'll still be improving on
> history, of course.  Just not as much.  :)
> 
> I started with Werckmeister's pure octaves.  This assumption
> can easily be changed later if desired.
> 
> Then I decided there would only be two kinds of 5th.
> Using 3 or more will get you more shades of contrast, but
> no new colors (basic ways of making the compromise).  I
> don't think it's worth it for the complexity it adds.
> 
> To get enough contrast from ET without going too close
> to meantone, 1/5- and 1/6-comma systems are basically the
> only games in town.  But how to distribute these on the
> chain of fifths?
> 
> Here I used two constraints: 1. The scale must have at
> least one 3rd audibly better than ET. and 2. The scale
> must not have any 3rds > 404 cents.  You might not think
> that Pythagorean thirds sound like ass, but I do.
> 
> I used these constraints to get the number circular
> permutations down to something manageable.  Then I looked
> at the 3rds in Scala.
> 
> Not all patterns produced unique patterns of thirds.  In
> these cases I chose scales having their best 7:4 on keys
> with better thirds.
> 
> I used Scala to rotate the scales into historical 'good
> keys' position.  Not all scales had good transitions
> along the chain of fifths, and those were discarded.
> 
> The following four scales is the result:
> 
> ! 12_lumma_5thcomma[246].scl
> !
>  1/5-comma SLSSLLLSLLLS temperament.
>  12
> !
>  99.609
>  199.218
>  298.827
>  393.744
>  502.737
>  597.654
>  697.263
>  796.872
>  896.481
>  1000.782
>  1095.699
>  2/1
> !
> ! 2 x 394-cent 3rds on F C.
> ! 4 x 398-cent 3rds on Eb Bb G D.
> ! 6 x 403-cent 3rds elsewhere.
> 
> ! 12_lumma_5thcomma[327].scl
> !
>  1/5-comma SSSLSLLLSLLL temperament.
>  12
> !
>  94.917
>  194.526
>  294.135
>  393.744
>  498.045
>  592.962
>  697.263
>  796.872
>  891.789
>  996.090
>  1091.007
>  2/1
> !
> ! 3 x 394-cent 3rds on F C G.
> ! 2 x 398-cent 3rds on Bb D.
> ! 7 x 403-cent 3rds elsewhere.
> 
> ! 12_lumma_6thcomma[1335].scl
> !
>  1/6-comma SSSLSLLSLLLS temperament.
>  12
> !
>  98.045
>  196.090
>  298.045
>  396.090
>  501.955
>  596.090
>  698.045
>  796.090
>  894.135
>  1000.000
>  1094.135
>  2/1
> !
> ! 1 x 392-cent 3rd on F.
> ! 3 x 396-cent 3rds on Bb C G.
> ! 3 x 400-cent 3rds on Eb D E.
> ! 5 x 404-cent 3rds elsewhere.
> 
> ! 12_lumma_6thcomma[2226].scl
> !
>  1/6-comma SSSSLLLSLLLS temperament.
>  12
> !
>  98.045
>  196.090
>  298.045
>  392.180
>  501.955
>  596.090
>  698.045
>  796.090
>  894.135
>  1000.000
>  1094.135
>  2/1
> !
> ! 2 x 392-cent 3rds on F C.
> ! 2 x 396-cent 3rds on Bb G.
> ! 2 x 400-cent 3rds on Eb D.
> ! 6 x 404-cent 3rds elsewhere.
> 
> I'm not aware of anything like this having been done
> before (though I'd guess it has), but I feel pretty good
> about it.
> 
> I'm also not aware of any historical temperaments using
> these patterns.  It seems like they couldn't get past the
> idea of bunching the fifths at either end of the chain.
> But maybe there are Victorian temperaments like this.
> 
> Comments sought,
> 
> -Carl
>

From: Cameron Bobro (2007-02-20)
Subject: Re: well temperaments : just won't die

Hahaha! Well, Carl, you guessed right if you were thinking that
I'd go for things like 24/19 in a 12 WT, here's one of a number of
Nineteensy WTs I did over the last couple of years...

Bobrova Cheerful 12wt
  0:          1/1               0.000 unison, perfect prime
  1:         19/18             93.603 undevicesimal semitone
  2:         19/17            192.558 quasi-meantone
  3:         19/16            297.513 19th harmonic
  4:        361/288           391.116
  5:          4/3             498.045 perfect fourth
  6:        361/256           595.026 two (19th harmonic)
  7:        323/216           696.603
  8:         19/12            795.558 undevicesimal minor sixth
  9:         57/34            894.513
 10:         57/32            999.468
 11:        361/192          1093.071
 12:          2/1            1200.000 octave

thirds like this...

 4:   1 361/288             391.116 cents  
  4:   1 64/51               393.090 cents  
  4:   2 171/136             396.468 cents  
  4:   2 34/27               399.090 cents  septendecimal major third
  4:   1 323/256             402.468 cents  
  4:   5 24/19               404.442 cents  smaller undevicesimal 

and fifths -5, -6, 0, -5, 0, 0, -3, 0, 0, 0, -3, 0 to use a Tom Dent 
stylee notation :-)

It doesn't suck and it does sound pretty cheerful, I think. It's 
also very similar to a bunch of WTs as you can see, although I don't 
know if anyone else started from a 19(x19) EDL then tinkered with it.

Two thirds concretely softer than 12-tET. A bunch of those 64/51 
thirds would be even nicer, wouldn't it.

-Cameron Bobro 


--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> 
wrote:
>
> Carl,
> 
> This is an interesting post...my comment would be that I think a 
damn
> good well-temperament was devised by myself, with slight 
modification
> by George Secor. It's in the .scl archives as 'johnson-
secor_rwt.scl'
> It contains five pure 24/19s, which are the largest thirds, which 
keep
> the parameter you have set to stay at or below ~404cents (a 24/19 
is
> 404.442 cents). The thirds are:
> 
> 1 361/288           391.116 cents  
> 1 4864/3879         391.750 cents  
> 1 240/191           395.354 cents  
> 1 2413/1920         395.666 cents  
> 1 160/127           399.892 cents  
> 1 3629/2880         400.204 cents  
> 1 1293/1024         403.808 cents  
> 5 24/19             404.442 cents
> 
> That's 5 fifths that are smaller than 12-eq, which is on par with
> Young, except it beats Young because it's smallest third is samller
> than Young, and it's largest third is smaller than the 81/64s you 
see
> in Young (6 of them--yuk!)
> 
> And the fifths are very fine too:
> 
> 1 190/127           697.405 cents  
> 1 1143/764          697.417 cents  
> 1 431/288           697.943 cents  
> 1 29032/19395       698.351 cents  
> 2 256/171           698.577 cents  
> 6 3/2               701.955 cents  perfect fifth
> 
> With 6 perfect fifths, and no Pythagorean thirds, this is a great
> temperament...and, it's rational, and, equal-beating (if you like 
that
> as a feature) to boot!
> 
> here's the .scl file:
> http://www.akjmusic.com/johnson-secor_rwt.scl
> 
> -A.
> 
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> >
> > So!
> > 
> > After all my playing around with the idea of '21st-century
> > circulating temperaments', I decided to have a go at more
> > historically-appropriate fare.  I'll still be improving on
> > history, of course.  Just not as much.  :)
> > 
> > I started with Werckmeister's pure octaves.  This assumption
> > can easily be changed later if desired.
> > 
> > Then I decided there would only be two kinds of 5th.
> > Using 3 or more will get you more shades of contrast, but
> > no new colors (basic ways of making the compromise).  I
> > don't think it's worth it for the complexity it adds.
> > 
> > To get enough contrast from ET without going too close
> > to meantone, 1/5- and 1/6-comma systems are basically the
> > only games in town.  But how to distribute these on the
> > chain of fifths?
> > 
> > Here I used two constraints: 1. The scale must have at
> > least one 3rd audibly better than ET. and 2. The scale
> > must not have any 3rds > 404 cents.  You might not think
> > that Pythagorean thirds sound like ass, but I do.
> > 
> > I used these constraints to get the number circular
> > permutations down to something manageable.  Then I looked
> > at the 3rds in Scala.
> > 
> > Not all patterns produced unique patterns of thirds.  In
> > these cases I chose scales having their best 7:4 on keys
> > with better thirds.
> > 
> > I used Scala to rotate the scales into historical 'good
> > keys' position.  Not all scales had good transitions
> > along the chain of fifths, and those were discarded.
> > 
> > The following four scales is the result:
> > 
> > ! 12_lumma_5thcomma[246].scl
> > !
> >  1/5-comma SLSSLLLSLLLS temperament.
> >  12
> > !
> >  99.609
> >  199.218
> >  298.827
> >  393.744
> >  502.737
> >  597.654
> >  697.263
> >  796.872
> >  896.481
> >  1000.782
> >  1095.699
> >  2/1
> > !
> > ! 2 x 394-cent 3rds on F C.
> > ! 4 x 398-cent 3rds on Eb Bb G D.
> > ! 6 x 403-cent 3rds elsewhere.
> > 
> > ! 12_lumma_5thcomma[327].scl
> > !
> >  1/5-comma SSSLSLLLSLLL temperament.
> >  12
> > !
> >  94.917
> >  194.526
> >  294.135
> >  393.744
> >  498.045
> >  592.962
> >  697.263
> >  796.872
> >  891.789
> >  996.090
> >  1091.007
> >  2/1
> > !
> > ! 3 x 394-cent 3rds on F C G.
> > ! 2 x 398-cent 3rds on Bb D.
> > ! 7 x 403-cent 3rds elsewhere.
> > 
> > ! 12_lumma_6thcomma[1335].scl
> > !
> >  1/6-comma SSSLSLLSLLLS temperament.
> >  12
> > !
> >  98.045
> >  196.090
> >  298.045
> >  396.090
> >  501.955
> >  596.090
> >  698.045
> >  796.090
> >  894.135
> >  1000.000
> >  1094.135
> >  2/1
> > !
> > ! 1 x 392-cent 3rd on F.
> > ! 3 x 396-cent 3rds on Bb C G.
> > ! 3 x 400-cent 3rds on Eb D E.
> > ! 5 x 404-cent 3rds elsewhere.
> > 
> > ! 12_lumma_6thcomma[2226].scl
> > !
> >  1/6-comma SSSSLLLSLLLS temperament.
> >  12
> > !
> >  98.045
> >  196.090
> >  298.045
> >  392.180
> >  501.955
> >  596.090
> >  698.045
> >  796.090
> >  894.135
> >  1000.000
> >  1094.135
> >  2/1
> > !
> > ! 2 x 392-cent 3rds on F C.
> > ! 2 x 396-cent 3rds on Bb G.
> > ! 2 x 400-cent 3rds on Eb D.
> > ! 6 x 404-cent 3rds elsewhere.
> > 
> > I'm not aware of anything like this having been done
> > before (though I'd guess it has), but I feel pretty good
> > about it.
> > 
> > I'm also not aware of any historical temperaments using
> > these patterns.  It seems like they couldn't get past the
> > idea of bunching the fifths at either end of the chain.
> > But maybe there are Victorian temperaments like this.
> > 
> > Comments sought,
> > 
> > -Carl
> >
>

From: Carl Lumma (2007-02-20)
Subject: Re: well temperaments : just won't die

Hi Aaron,

Johnson-Secor-RWT is in my "WellTemperamentComparator"
spreadsheet, and I also mention it in this thread, because
it's one of my favorite temperaments.

-Carl

From: Carl Lumma (2007-02-20)
Subject: Re: well temperaments : just won't die

> Hahaha! Well, Carl, you guessed right if you were thinking that
> I'd go for things like 24/19 in a 12 WT, here's one of a number of
> Nineteensy WTs I did over the last couple of years...
> 
> Bobrova Cheerful 12wt
>   0:          1/1               0.000 unison, perfect prime
>   1:         19/18             93.603 undevicesimal semitone
>   2:         19/17            192.558 quasi-meantone
>   3:         19/16            297.513 19th harmonic
>   4:        361/288           391.116
>   5:          4/3             498.045 perfect fourth
>   6:        361/256           595.026 two (19th harmonic)
>   7:        323/216           696.603
>   8:         19/12            795.558 undevicesimal minor sixth
>   9:         57/34            894.513
>  10:         57/32            999.468
>  11:        361/192          1093.071
>  12:          2/1            1200.000 octave

Cool.  I don't recognize the use of 19/17 in these... interesting.
I wish you'd give a .scl file instead of Scala's "show" output...

-Carl

From: Cameron Bobro (2007-02-20)
Subject: Re: well temperaments : just won't die

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
> Cool.  I don't recognize the use of 19/17 in these... interesting.
> I wish you'd give a .scl file instead of Scala's "show" output...


!
Bobrova Cheerful 12 WT based on *19 EDL
 12
!
 19/18
 19/17
 19/16
 361/288
 4/3
 361/256
 323/216
 19/12
 57/34
 57/32
 361/192
 2/1


There you go... I wonder how it does in your spreadsheet? I had 
almost forgotten about this one, it sounds better than I had 
remembered. The real question for historical performance would be
where the thirds lie as far as keys.

From: Carl Lumma (2007-02-20)
Subject: Re: well temperaments : just won't die

> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> > Cool.  I don't recognize the use of 19/17 in these... interesting.
> > I wish you'd give a .scl file instead of Scala's "show" output...
>
> !
> Bobrova Cheerful 12 WT based on *19 EDL
>  12
> !
>  19/18
>  19/17
>  19/16
>  361/288
>  4/3
>  361/256
>  323/216
>  19/12
>  57/34
>  57/32
>  361/192
>  2/1
> 
> There you go... I wonder how it does in your spreadsheet? I had 
> almost forgotten about this one, it sounds better than I had 
> remembered. The real question for historical performance would be
> where the thirds lie as far as keys.

It's like 3 2/9-comma 5ths, 2 1/6-comma 5ths, and the rest pure
fifths.  Very similar to johnson-secor, with a lot of the
same intervals (including the 5 24/19s), but some differences
too.  Interesting!

You're welcome to try to add it to my spreadsheet, but it's
not the most extensible thing in the world, despite my efforts
to make it so.

-Carl

From: Aaron Krister Johnson (2007-02-20)
Subject: Re: well temperaments : just won't die

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> Hi Aaron,
> 
> Johnson-Secor-RWT is in my "WellTemperamentComparator"
> spreadsheet, and I also mention it in this thread, because
> it's one of my favorite temperaments.

That's right--I just forgot...

You should also consider some modified meantones, although I think as
long as you are at or below 1/6-comma, you get thirds that are larger
than your upper bound.

-A.

From: Cameron Bobro (2007-02-20)
Subject: Re: well temperaments : just won't die

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> Then I decided there would only be two kinds of 5th.
> Using 3 or more will get you more shades of contrast, but
> no new colors (basic ways of making the compromise).  I
> don't think it's worth it for the complexity it adds.

The open strings of the string instruments are important,
in my opinion. For example, the first four fifths ascending
in size seems natural; diminishing or a real hodge-podge of sizes 
would be the worst. My "Cheerful" WT is ho-hum in this respect,
I think. 

> To get enough contrast from ET without going too close
> to meantone, 1/5- and 1/6-comma systems are basically the
> only games in town.  But how to distribute these on the
> chain of fifths?

Well that's why I think 3-4 sizes of fifths is better, but
you're setting yourself a stringent goal. 

> Here I used two constraints: 1. The scale must have at
> least one 3rd audibly better than ET. and 2. The scale
> must not have any 3rds > 404 cents.  You might not think
> that Pythagorean thirds sound like ass, but I do.

I like Pythagorean thirds, but in the case of a general 
purpose 12WT, I think they'd have to be in the context of
a very embossed tuning, which means you'd have to have
an equal amount of truly 5/4ish thirds to balance them.

However, at a certain point going for "highly embossed", 
a person needs to stop piddling with 12 and just stick in some 
split keys, in my opinion. So I'll go along with the 404c 
limit, especially as it's not a problem to do. :-)

By the way, I believe that contrasting seconds are very 
important, not just thirds- there's a great deal of
step-wise motion in Bach for example. 
 
> Not all patterns produced unique patterns of thirds.  In
> these cases I chose scales having their best 7:4 on keys
> with better thirds.

Makes sense, the soft keys will be even softer.
 
> The following four scales is the result:

>  1/5-comma SLSSLLLSLLLS temperament.

See open strings comment above...
 
> ! 12_lumma_5thcomma[327].scl
> !
>  1/5-comma SSSLSLLLSLLL temperament.

That would be alright...

>  12
> !
>  94.917
>  194.526
>  294.135
>  393.744
>  498.045
>  592.962
>  697.263
>  796.872
>  891.789
>  996.090
>  1091.007
>  2/1
> !
> ! 3 x 394-cent 3rds on F C G.
> ! 2 x 398-cent 3rds on Bb D.
> ! 7 x 403-cent 3rds elsewhere.

Looks like this one's good to go, doesn't it?
> 
> ! 12_lumma_6thcomma[1335].scl
> !
>  1/6-comma SSSLSLLSLLLS temperament.
>  12
> !
>  98.045
>  196.090
>  298.045
>  396.090
>  501.955
>  596.090
>  698.045
>  796.090
>  894.135
>  1000.000
>  1094.135
>  2/1
> !
> ! 1 x 392-cent 3rd on F.
> ! 3 x 396-cent 3rds on Bb C G.
> ! 3 x 400-cent 3rds on Eb D E.
> ! 5 x 404-cent 3rds elsewhere.

Pretty 12-tETish I bet but you'd have to listen.
> 
> ! 12_lumma_6thcomma[2226].scl
> !
>  1/6-comma SSSSLLLSLLLS temperament.
>  12
> !
>  98.045
>  196.090
>  298.045
>  392.180
>  501.955
>  596.090
>  698.045
>  796.090
>  894.135
>  1000.000
>  1094.135
>  2/1
> !
> ! 2 x 392-cent 3rds on F C.
> ! 2 x 396-cent 3rds on Bb G.
> ! 2 x 400-cent 3rds on Eb D.
> ! 6 x 404-cent 3rds elsewhere.

I'll bet this one sounds quite good.
 
> I'm also not aware of any historical temperaments using
> these patterns.  It seems like they couldn't get past the
> idea of bunching the fifths at either end of the chain.

Maybe because at least at the beginning, the first keys would
want to be meantone-y? and as I mentioned 
above, maybe the open strings as well.

> But maybe there are Victorian temperaments like this.

I wouldn't be surprised if there were two distinct varieties in
the Victorian age-  very ETish tunings for the pros doing 
chromaticism, and some popular tunings for the home piano boom 
which leaned toward open-sounding white key music.

From: Carl Lumma (2007-02-20)
Subject: Re: well temperaments : just won't die

> > Here I used two constraints: 1. The scale must have at
> > least one 3rd audibly better than ET. and 2. The scale
> > must not have any 3rds > 404 cents.  You might not think
> > that Pythagorean thirds sound like ass, but I do.
> 
> I like Pythagorean thirds, but in the case of a general 
> purpose 12WT, I think they'd have to be in the context of
> a very embossed tuning, which means you'd have to have
> an equal amount of truly 5/4ish thirds to balance them.
> 
> However, at a certain point going for "highly embossed", 
> a person needs to stop piddling with 12 and just stick in some 
> split keys, in my opinion.

Or go with a generalized keyboard, in mine.  But this whole
exercise is really one of, 'what should I be putting my
halberstadt hardware in (and using my halberstadt skills with)?'.

> >  12
> > !
> >  94.917
> >  194.526
> >  294.135
> >  393.744
> >  498.045
> >  592.962
> >  697.263
> >  796.872
> >  891.789
> >  996.090
> >  1091.007
> >  2/1
> > !
> > ! 3 x 394-cent 3rds on F C G.
> > ! 2 x 398-cent 3rds on Bb D.
> > ! 7 x 403-cent 3rds elsewhere.
> 
> Looks like this one's good to go, doesn't it?

In the end I didn't like the majority of 3rds being
worse than equal.

> > ! 12_lumma_6thcomma[2226].scl
> > !
> >  1/6-comma SSSSLLLSLLLS temperament.
> >  12
> > !
> >  98.045
> >  196.090
> >  298.045
> >  392.180
> >  501.955
> >  596.090
> >  698.045
> >  796.090
> >  894.135
> >  1000.000
> >  1094.135
> >  2/1
> > !
> > ! 2 x 392-cent 3rds on F C.
> > ! 2 x 396-cent 3rds on Bb G.
> > ! 2 x 400-cent 3rds on Eb D.
> > ! 6 x 404-cent 3rds elsewhere.
> 
> I'll bet this one sounds quite good.

This is one of the three that made it into
the final rounds for listening tests (that I'll
do at a keyboard, this weekend, God willing).

-Carl

From: George D. Secor (2007-02-20)
Subject: Re: well temperaments : just won't die

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> 
wrote:
>
> Carl,
> 
> This is an interesting post...my comment would be that I think a damn
> good well-temperament was devised by myself, with slight modification
> by George Secor. It's in the .scl archives as 'johnson-secor_rwt.scl'
> It contains five pure 24/19s, which are the largest thirds, which keep
> the parameter you have set to stay at or below ~404cents (a 24/19 is
> 404.442 cents). ...

Aaron, I don't know if you saw it way back when, but within a few days 
of coming up with the modification, I suggested changing the ratio for 
D from 3629/3240 to 1093/976, which results in:

1) Smaller numbers in the ratio,
2) Less variation in the sizes of the 5ths,
3) Slightly better harmonic balance, and
4) Almost exact integers in the ratios between the 5th & m3rd beat 
rates for both the G and D major triads.

I doubt that you'll be able to hear any difference, but why not have a 
look?

--George

From: Carl Lumma (2007-02-23)
Subject: Re: well temperaments : just won't die

I've solved my 1/7-comma '3rds quality not being like a
sine wave around the circle of fifths' problem, with the
fifths pattern SSSSSLLSSLLL.

So here are my final entries into the pure-octaves WT
gamut:

! 12_lumma_5thcomma[246].scl
!
 1/5-comma SLSSLLLSLLLS temperament.
 12
!
 94.917
 194.526
 294.135
 393.744
 498.045
 592.962
 697.263
 796.872
 896.481
 996.090
 1091.007
 2/1
!
! 2 x 394-cent 3rds on C G.
! 4 x 398-cent 3rds on Bb F D A.
! 6 x 403-cent 3rds elsewhere.

! 12_lumma_6thcomma[2226].scl
!
 1/6-comma SSSSLLLSLLLS temperament.
 12
!
 94.135
 196.090
 294.135
 392.180
 498.045
 592.180
 698.045
 796.090
 894.135
 996.090
 1090.225
 2/1
!
! 2 x 392-cent 3rds on C G.
! 2 x 396-cent 3rds on F D.
! 2 x 400-cent 3rds on Bb A.
! 6 x 404-cent 3rds elsewhere.

! 12_lumma_7thcomma[2262].scl
!
 1/7-comma SSSSLLSSLLLS temperament.
 12
!
 96.929
 197.208
 297.488
 394.415
 501.396
 598.325
 698.604
 795.533
 895.811
 999.442
 1096.370
 2/1
!
! 2 x 394-cent 3rds on F C.
! 2 x 398-cent 3rds on Bb G.
! 6 x 401-cent 3rds on Eb D A E B F#.
! 2 x 405-cent 3rds elsewhere.

-Carl

From: martinsj013 (2009-02-28)
Subject: Re: well temperaments : just won't die

Hello, I am a new joiner here and have been searching the message
archive for topics of interest.  That is my excuse for resurrecting an
old topic from Feb 2007, to defend Thomas Young:

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
> <snip>
> That's 5 fifths that are smaller than 12-eq, which is on par with
> Young, except it beats Young because it's smallest third is samller
> than Young, and it's largest third is smaller than the 81/64s you see
> in Young (6 of them--yuk!)
> <snip>

Surely not!  Six pure 5ths in Young 2, resulting in three 81/64s; and
only four pure 5ths in Young 1, resulting in one 81/64.

Kind regards,
Steve M.