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A new rational welltemperament
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Hi,
Spurred on by my recent Python code for rational approximations, and
wanting for some time to develop a welltemperament with 24/19 instead
of 81/64 as a widethird 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 EG# and AbC be 24/19, and
CE is of course the octave residue of that. Other than that, I tried
to use the smallest rational approximations I could while preserving
traditional welltemperament 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 welltemperament 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 0 Attachment
 In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
> Tune it up and play...I would love some comments, and I hope I might
Great! This scale is epimorphic in more than one way, so it's a nice
> inspire others to take this work further, or improve it!
example among other talents. It's also an authentic welltemperament,
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 welltemperament should suit nineteenth century music pretty
well. 0 Attachment
 In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:>
C#> 19/18 == (256/243)*(513/512)
> ! johnson_ratwell.scl
> !
> a rational welltemperament with five 24/19's
> 12
> !
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 noneepomoric!
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 justpure 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 epimoricdeviating too?
A.S. 0 Attachment
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 welltemperament with 24/19 instead
> of 81/64 as a widethird 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 EG# and AbC be 24/19, and
> CE 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 welltemperament 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 welltemperament 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 welltemperament well enough to
use one. (Oh, OK, I do use 12EDO 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

No virus found in this outgoing message.
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On 6/1/06, Yahya AbdalAziz <yahya@...> wrote:> With G# =Ab ?
Of course; that's what makes it a well temperament. Unequal but closed.
 0 Attachment
 In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:>
Scala adds "no fifth greater than 3/2" to the definition; otherwise, I
> On 6/1/06, Yahya AbdalAziz <yahya@...> wrote:
> > With G# =Ab ?
>
> Of course; that's what makes it a well temperament. Unequal but closed.
suppose, it is extraordinaire. 0 Attachment
 In tuning@yahoogroups.com, "a_sparschuh" <a_sparschuh@...>
wrote:>
wrote:
>  In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@>
> >
"A".
> > ! johnson_ratwell.scl
> > !
> > a rational welltemperament 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 noneepomoric!
> 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 justpure 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
> Question: Why became that "A" not epimoricdeviating too?
Hi,
> A.S.
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. 0 Attachment
> Hi,
Insired by this, I came up with:
>
> Spurred on by my recent Python code for rational approximations,
> and wanting for some time to develop a welltemperament with
> 24/19 instead of 81/64 as a widethird 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.
! 12_mohhaha.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 pureoctaves scale based on direct approximations
to 12tET 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
19/16based 'diminished 7th' chords rooted on adjacent 3:2
fifths.
And don't forget strangeion...
! 12_strangeion.scl
!
19limit "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
! / \ /
! DD#E
! / \ / \ /
! BCC#
! / \ / \ /
! G#AA#
! /
! F
!
!  = 17/16
! / = 19/16
I'd love to hear anybody's reactions to playing with these.
Carl 0 Attachment
 In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
> My calculations indicate that we could change the 'A' to 191/114 and
It's fine by me, though personally I find the 139limit quite higher
> preserve that property entirely....any comments, Gene, or George?
enough without going all the way to the 191 limit. 0 Attachment
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 welltemperament with
> > 24/19 instead of 81/64 as a widethird 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_mohhaha.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 pureoctaves scale based on direct approximations
> to 12tET 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
> 19/16based 'diminished 7th' chords rooted on adjacent 3:2
> fifths.
>
> And don't forget strangeion...
>
> ! 12_strangeion.scl
> !
> 19limit "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
> ! / \ /
> ! DD#E
> ! / \ / \ /
> ! BCC#
> ! / \ / \ /
> ! G#AA#
> ! /
> ! F
> !
> !  = 17/16
> ! / = 19/16
>
> I'd love to hear anybody's reactions to playing with these.
>
> Carl
> 0 Attachment
 In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:>
wrote:
>  In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@>
>
and
> > My calculations indicate that we could change the 'A' to 191/114
> > preserve that property entirely....any comments, Gene, or
George?
>
higher
> It's fine by me, though personally I find the 139limit quite
> 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 sonicallyalthough with trying it, I
suspect it's hard to notice) would the 'A' missing a superparticular
cofactor be?
Are there any ways to improve the scale I posted that would:
1) satisfy superparticular cofactor fetishes?
2) satisfy being lower than 139limit?
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. 0 Attachment
 In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
hi!
> ! 12_mohhaha.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/xenharmonicbridge.aspx
" Eratosthenes 3==19 bridge, so it skips 5 primes in between"
That epimoric riddleplay 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. 0 Attachment
> > ! 12_mohhaha.scl
Interesting. Thanks, A.S.!
> > !
> > 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/xenharmonicbridge.aspx
> " Eratosthenes 3==19 bridge, so it skips 5 primes in between"
>
> That epimoric riddleplay 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.
Carl 0 Attachment
 In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
> > It's fine by me, though personally I find the 139limit quite
If I were to choose, yes.
> 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 sonicallyalthough with trying it, I
No importance whatever. But keeping the prime limit low only has the
> suspect it's hard to notice) would the 'A' missing a superparticular
> cofactor be?
effect for me that when I run the "show data" command with Scala, it
can keep its enthusiasm within better bounds. 0 Attachment
 In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:>
instead
> Hi,
>
> Spurred on by my recent Python code for rational approximations, and
> wanting for some time to develop a welltemperament with 24/19
> of 81/64 as a widethird 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.
tried
>
> The idea is to have the backbone thirds EG# and AbC be 24/19, and
> CE is of course the octave residue of that. Other than that, I
> to use the smallest rational approximations I could while preserving
Aaron, sorry I've taken so long to reply.
> traditional welltemperament 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 welltemperament 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
This is really intriguing in that it:
1) produces 8 simple proportionalbeating 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 welltemperament.)
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):
! AKJGDSRWT.scl
!
A.K. Johnson/G. Secor proportionalbeating rational welltemperament
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 (SecorVRWT.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 0 Attachment
 In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:> A.K. Johnson/G. Secor proportionalbeating rational welltemperament
A.S.
> 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
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