A new rational well-temperament

Hi,

Spurred on by my recent Python code for rational approximations, and
wanting for some time to develop a well-temperament with 24/19 instead
of 81/64 as a wide-third basis, and inspired by George Secor and Gene
Ward Smith's work in the area of rational temperament, I came up with
the following yesterday.

The idea is to have the backbone thirds E-G# and Ab-C be 24/19, and
C-E is of course the octave residue of that. Other than that, I tried
to use the smallest rational approximations I could while preserving
traditional well-temperament qualities.

Tune it up and play...I would love some comments, and I hope I might
inspire others to take this work further, or improve it!

! johnson_ratwell.scl
!
a rational well-temperament with five 24/19's
12
!
19/18
103/92
32/27
361/288
4/3
38/27
208/139
19/12
129/77
16/9
152/81
2/1

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:

> Tune it up and play...I would love some comments, and I hope I might
> inspire others to take this work further, or improve it!

Great! This scale is epimorphic in more than one way, so it's a nice
example among other talents. It's also an authentic well-temperament,
with no fifth wider than 3/2. Scala tells me that this is similar to
Herman Miller's "Arrow" temperaments, but searching did turn those up,
so I hope Herman can explain.

This mild well-temperament should suit nineteenth century music pretty
well.

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
>
> ! johnson_ratwell.scl
> !
> a rational well-temperament with five 24/19's
> 12
> !
C#> 19/18 == (256/243)*(513/512)
D > 103/92 = (9/8)*(206/207)
Eb> 32/27 == (6/5)*(81/80) pyth. minor 3rd
E > 361/288= (5/4)*(361/360)
F > 4/3
F#> 38/27 == (1024/729)*(513/512)
G > 208/139= (3/2)*(416/417)
G#> 19/12 == (128/81)*(513/512)
A > 129/77 = (5/3)*(387/385)=(27/16)*(688/693)both none-epomoric!
Bb> 16/9
b > 152/81 = (15/8)*(1216/1215)
C'> 2/1

As far as i'm able to see:
All -but except yours "A"- deviate only from just-pure merely
about an small epimoric cofactor in order to yield the tempering.

Hence i can't understand:
Why did you took the special "A" in a different way from its
superparticular neighbourhood, unlike yours other 11 ratios?
Please -be so kind to- explain me yours extraordinary choice on "A".
Question: Why became that "A" not epimoric-deviating too?
A.S.

Hi all,

On Thu Jun 1, 2006, Aaron Krister Johnson wrote:
>
> Hi,
>
> Spurred on by my recent Python code for rational approximations, and
> wanting for some time to develop a well-temperament with 24/19 instead
> of 81/64 as a wide-third basis, and inspired by George Secor and Gene
> Ward Smith's work in the area of rational temperament, I came up with
> the following yesterday.
>
> The idea is to have the backbone thirds E-G# and Ab-C be 24/19, and
> C-E is of course the octave residue of that. ...

With G# =Ab ?

> ... Other than that, I tried
> to use the smallest rational approximations I could while preserving
> traditional well-temperament qualities.
>
> Tune it up and play...I would love some comments, and I hope I might
> inspire others to take this work further, or improve it!
>
> ! johnson_ratwell.scl

Great name! At first I thought, "I know who Johnson
is, but who is Ratwell?!" ;-)

> !
> a rational well-temperament with five 24/19's
> 12
> !
> 19/18
> 103/92
> 32/27
> 361/288
> 4/3
> 38/27
> 208/139
> 19/12
> 129/77
> 16/9
> 152/81
> 2/1

Well, Aaron, I hope some day to understand the
virtues of a well-temperament well enough to
use one. (Oh, OK, I do use 12-EDO for jazzy
stuff, and for first audition of JI stuff.) But
since most of my music doesn't require extensive
key modulation, I don't expect I can be much use
to you at present with this temperament - anything
I wrote using it would almost certainly not exploit
its potential particularly well.

Still, I've never knowingly used the 19 limit, and
it might be fun to try!

Regards,
Yahya

--
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On 6/1/06, Yahya Abdal-Aziz <yahya@melbpc.org.au> wrote:
> With G# =Ab ?

Of course; that's what makes it a well temperament. Unequal but closed.

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> On 6/1/06, Yahya Abdal-Aziz <yahya@...> wrote:
> > With G# =Ab ?
>
> Of course; that's what makes it a well temperament. Unequal but closed.

Scala adds "no fifth greater than 3/2" to the definition; otherwise, I
suppose, it is extraordinaire.

--- In tuning@yahoogroups.com, "a_sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@>
wrote:
> >
> > ! johnson_ratwell.scl
> > !
> > a rational well-temperament with five 24/19's
> > 12
> > !
> C#> 19/18 == (256/243)*(513/512)
> D > 103/92 = (9/8)*(206/207)
> Eb> 32/27 == (6/5)*(81/80) pyth. minor 3rd
> E > 361/288= (5/4)*(361/360)
> F > 4/3
> F#> 38/27 == (1024/729)*(513/512)
> G > 208/139= (3/2)*(416/417)
> G#> 19/12 == (128/81)*(513/512)
> A > 129/77 = (5/3)*(387/385)=(27/16)*(688/693)both none-epomoric!
> Bb> 16/9
> b > 152/81 = (15/8)*(1216/1215)
> C'> 2/1
>
> As far as i'm able to see:
> All -but except yours "A"- deviate only from just-pure merely
> about an small epimoric cofactor in order to yield the tempering.
>
> Hence i can't understand:
> Why did you took the special "A" in a different way from its
> superparticular neighbourhood, unlike yours other 11 ratios?
> Please -be so kind to- explain me yours extraordinary choice on
"A".
> Question: Why became that "A" not epimoric-deviating too?
> A.S.

Hi,

Well, I hadn't thought about it that way until you pointed it
out....
:)
My calculations indicate that we could change the 'A' to 191/114 and
preserve that property entirely....any comments, Gene, or George?

It's possible for 'D' to be 19/17 or 28/25, too, but I don't like
the step sizes that result as much, so I traded them for higher
ratios.

-Aaron.

> Hi,
>
> Spurred on by my recent Python code for rational approximations,
> and wanting for some time to develop a well-temperament with
> 24/19 instead of 81/64 as a wide-third basis, and inspired by
> George Secor and Gene Ward Smith's work in the area of rational
> temperament, I came up with the following yesterday.

Insired by this, I came up with:

! 12_moh-ha-ha.scl
!
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
!

and

! 12_fun.scl
!
Rational well temperament based on 577/289, 3/2, and 19/16.
12
!
19/18
18464/16473
19/16
361/288
1154/867
361/256
73856/49419
10963/6936
9232/5491
4616/2601
208297/110976
577/289
!

The first is a pure-octaves scale based on direct approximations
to 12-tET with 'simple' ratios. It's similar to Aaron's, but
swaps two of his '24/19' thirds for one '81/80' third on C#.

The second uses flat octaves, and is built from three
19/16-based 'diminished 7th' chords rooted on adjacent 3:2
fifths.

And don't forget strangeion...

! 12_strangeion.scl
!
19-limit "dodekaphonic" scale.
12
!
17/16 !.......C#
19/17 !........D
19/16 !.......D#
323/256 !......E
8192/6137 !....F
361/256 !.....F#
6137/4096 !....G
512/323 !.....G#
32/19 !........A
34/19 !.......A#
32/17 !........B
2/1 !..........C
!
! F#--G
! / \ /
! D---D#--E
! / \ / \ /
! B---C---C#
! / \ / \ /
! G#--A---A#
! /
! F
!
! --- = 17/16
! / = 19/16

I'd love to hear anybody's reactions to playing with these.

-Carl

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:

> My calculations indicate that we could change the 'A' to 191/114 and
> preserve that property entirely....any comments, Gene, or George?

It's fine by me, though personally I find the 139-limit quite higher
enough without going all the way to the 191 limit.

Cool! I'll have to check these out........

-Aaron.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Hi,
> >
> > Spurred on by my recent Python code for rational approximations,
> > and wanting for some time to develop a well-temperament with
> > 24/19 instead of 81/64 as a wide-third basis, and inspired by
> > George Secor and Gene Ward Smith's work in the area of rational
> > temperament, I came up with the following yesterday.
>
> Insired by this, I came up with:
>
> ! 12_moh-ha-ha.scl
> !
> 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
> !
>
> and
>
> ! 12_fun.scl
> !
> Rational well temperament based on 577/289, 3/2, and 19/16.
> 12
> !
> 19/18
> 18464/16473
> 19/16
> 361/288
> 1154/867
> 361/256
> 73856/49419
> 10963/6936
> 9232/5491
> 4616/2601
> 208297/110976
> 577/289
> !
>
> The first is a pure-octaves scale based on direct approximations
> to 12-tET with 'simple' ratios. It's similar to Aaron's, but
> swaps two of his '24/19' thirds for one '81/80' third on C#.
>
> The second uses flat octaves, and is built from three
> 19/16-based 'diminished 7th' chords rooted on adjacent 3:2
> fifths.
>
> And don't forget strangeion...
>
> ! 12_strangeion.scl
> !
> 19-limit "dodekaphonic" scale.
> 12
> !
> 17/16 !.......C#
> 19/17 !........D
> 19/16 !.......D#
> 323/256 !......E
> 8192/6137 !....F
> 361/256 !.....F#
> 6137/4096 !....G
> 512/323 !.....G#
> 32/19 !........A
> 34/19 !.......A#
> 32/17 !........B
> 2/1 !..........C
> !
> ! F#--G
> ! / \ /
> ! D---D#--E
> ! / \ / \ /
> ! B---C---C#
> ! / \ / \ /
> ! G#--A---A#
> ! /
> ! F
> !
> ! --- = 17/16
> ! / = 19/16
>
> I'd love to hear anybody's reactions to playing with these.
>
> -Carl
>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@>
wrote:
>
> > My calculations indicate that we could change the 'A' to 191/114
and
> > preserve that property entirely....any comments, Gene, or
George?
>
> It's fine by me, though personally I find the 139-limit quite
higher
> enough without going all the way to the 191 limit.

So does that mean you would prefer the first version? How important
to you theoretically (or even sonically--although with trying it, I
suspect it's hard to notice) would the 'A' missing a superparticular
co-factor be?

Are there any ways to improve the scale I posted that would:
1) satisfy superparticular co-factor fetishes?
2) satisfy being lower than 139-limit?
3) keep the fifths from C to E sounding smooth and perceptibly
similar in size?

I can't see any right now........am I missing something?

-Aaron.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
hi!
> ! 12_moh-ha-ha.scl
> !
> Rational well temperament.
> 12
> !
> 19/18 ! = (256/243)(513/512)
> 323/288!= (9/8)(323/324) = (10/9)(323/320)
> 19/16 ! = (32/27)(513/512)
> 323/256!= (81/64)(323/324) = (5/4)(323/320)
> 171/128!= (4/3)(513/512)
> 361/256!= (45/32)(361/360)
> 551/368!= (3/2)(551/552)
> 19/12 ! = (128/81)(513/512)
> 323/192!= (27/16)(323/324) = (5/3)(323/320)
> 57/32 ! = (16/9)(513/512)
> 513/272!= (32/17)(513/512) = (15/8)(171/170)
> 2
> !
Hence it looks i.m.o. nearer to pythagorean than to syntonic,
basing mostly on:
http://tonalsoft.com/enc/x/xenharmonic-bridge.aspx
" Eratosthenes 3==19 bridge, so it skips 5 primes in between"

That epimoric riddle-play makes real fun.
I think above defactorized superparticular decompositions
tell more about how the tempering of the intervals is done,
than merely only the original bare(scl-)ratios alone.
A.S.

> > ! 12_moh-ha-ha.scl
> > !
> > Rational well temperament.
> > 12
> > !
> > 19/18 ! = (256/243)(513/512)
> > 323/288!= (9/8)(323/324) = (10/9)(323/320)
> > 19/16 ! = (32/27)(513/512)
> > 323/256!= (81/64)(323/324) = (5/4)(323/320)
> > 171/128!= (4/3)(513/512)
> > 361/256!= (45/32)(361/360)
> > 551/368!= (3/2)(551/552)
> > 19/12 ! = (128/81)(513/512)
> > 323/192!= (27/16)(323/324) = (5/3)(323/320)
> > 57/32 ! = (16/9)(513/512)
> > 513/272!= (32/17)(513/512) = (15/8)(171/170)
> > 2
> > !
> Hence it looks i.m.o. nearer to pythagorean than to syntonic,
> basing mostly on:
> http://tonalsoft.com/enc/x/xenharmonic-bridge.aspx
> " Eratosthenes 3==19 bridge, so it skips 5 primes in between"
>
> That epimoric riddle-play makes real fun.
> I think above defactorized superparticular decompositions
> tell more about how the tempering of the intervals is done,
> than merely only the original bare(scl-)ratios alone.
> A.S.

Interesting. Thanks, A.S.!

-Carl

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:

> > It's fine by me, though personally I find the 139-limit quite
> higher
> > enough without going all the way to the 191 limit.
>
> So does that mean you would prefer the first version?

If I were to choose, yes.

How important
> to you theoretically (or even sonically--although with trying it, I
> suspect it's hard to notice) would the 'A' missing a superparticular
> co-factor be?

No importance whatever. But keeping the prime limit low only has the
effect for me that when I run the "show data" command with Scala, it
can keep its enthusiasm within better bounds.

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:
>
> Hi,
>
> Spurred on by my recent Python code for rational approximations, and
> wanting for some time to develop a well-temperament with 24/19
instead
> of 81/64 as a wide-third basis, and inspired by George Secor and
Gene
> Ward Smith's work in the area of rational temperament, I came up
with
> the following yesterday.
>
> The idea is to have the backbone thirds E-G# and Ab-C be 24/19, and
> C-E is of course the octave residue of that. Other than that, I
tried
> to use the smallest rational approximations I could while preserving
> traditional well-temperament qualities.
>
> Tune it up and play...I would love some comments, and I hope I might
> inspire others to take this work further, or improve it!
>
> ! johnson_ratwell.scl
> !
> a rational well-temperament with five 24/19's
> 12
> !
> 19/18
> 103/92
> 32/27
> 361/288
> 4/3
> 38/27
> 208/139
> 19/12
> 129/77
> 16/9
> 152/81
> 2/1

Aaron, sorry I've taken so long to reply.

This is really intriguing in that it:
1) produces 8 simple proportional-beating major triads (on all of the
most dissonant ones), while
2) keeping the max error for the major 3rd around 18 cents.

I was able to accomplish each of these things in separate well-
temperaments, but not both at once. (And as Gene noted, it's an
excellent well-temperament.)

Unfortunately, the major brats on C, G, D, and A are not simple, so I
couldn't resist seeing if those could be improved. By changing the
ratios for G, D, and A I was able to get simpler brats: 2.75 for C,
2.25 for D, and 2 for A, with a leftover of ~2.491803 for G (pretty
close to 2.5):

! AKJ-GDS-RWT.scl
!
A.K. Johnson/G. Secor proportional-beating rational well-temperament
with five 24/19's
12
!
19/18
3629/3240
32/27
361/288
4/3
38/27
431/288
19/12
2413/1440
16/9
152/81
2/1

Half of the minor brats are exactly 1, and the others are not all
that bad, considering that most of those are approximations of
reasonably simple brats. I tried it in Scala, and I think it sounds
pretty good! And the 6 just fifths should make it reasonably easy to
tune by ear.

I've had a couple of days to decide whether or not I prefer this to
my rationalized Ellis #2 (Secor-VRWT.scl). It's not an easy call,
but I think I would have to go with the VRWT because of:
1) its higher key contrast (more consonant C major triad), and
2) my personal preference for slightly tempered (vs. just) fifths on
the worst triads -- which is to say, I prefer to have the total error
of the fifths of the worst triads distributed more or less equally,
as opposed to putting all of that error on 1 or 2 of the fifths.

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
> A.K. Johnson/G. Secor proportional-beating rational well-temperament
> with five 24/19's
> 12
> !
> 19/18 ! = = (256/243)(513/512)
> 3629/3240! =(9/8)(3629/3780) = (10/9)(3629/3600)
> 32/27 ! = = (6/5)(80/81)
> 361/288 ! = (5/4)((361/360)
> 4/3 ! = = = (11/8)(32/33)
> 38/27 ! = = (7/5)(190/189)
> 431/288 ! = (3/2)(431/432)
> 19/12 ! = = (25/16)(76/75)
> 2413/1440 !=(5/3)(2413/2400)
> 16/9 ! = = =(7/4)(64/63)
> 152/81 ! = =(243/128)(513/512)
> 2/1
A.S.