Topic: A new rational well-temperament

3 scales

Thread (16 messages)

From: Aaron Krister Johnson (2006-06-01)
Subject: 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

From: Gene Ward Smith (2006-06-01)
Subject: Re: A new rational well-temperament

--- In [email protected], "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.

From: a_sparschuh (2006-06-01)
Subject: Re: A new rational well-temperament

--- In [email protected], "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.

From: Yahya Abdal-Aziz (2006-06-02)
Subject: RE: A new rational well-temperament

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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From: Keenan Pepper (2006-06-02)
Subject: Re: [tuning] RE: A new rational well-temperament

On 6/1/06, Yahya Abdal-Aziz <[email protected]> wrote:
> With G# =Ab ?

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

From: Gene Ward Smith (2006-06-02)
Subject: Re: A new rational well-temperament

--- In [email protected], "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.

From: Aaron Krister Johnson (2006-06-02)
Subject: Re: A new rational well-temperament

--- In [email protected], "a_sparschuh" <a_sparschuh@...> 
wrote: 
> 
> --- In [email protected], "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.

From: Carl Lumma (2006-06-02)
Subject: Re: 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. 

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

From: Gene Ward Smith (2006-06-02)
Subject: Re: A new rational well-temperament

--- In [email protected], "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.

From: Aaron Krister Johnson (2006-06-02)
Subject: Re: A new rational well-temperament

Cool! I'll have to check these out........ 
 
-Aaron. 
 
 
--- In [email protected], "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 
>

From: Aaron Krister Johnson (2006-06-02)
Subject: Re: A new rational well-temperament

--- In [email protected], "Gene Ward Smith" <genewardsmith@...>  
wrote:  
>  
> --- In [email protected], "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.

From: a_sparschuh (2006-06-02)
Subject: Re: A new rational well-temperament

--- In [email protected], "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.

From: Carl Lumma (2006-06-02)
Subject: Re: A new rational well-temperament

> > ! 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

From: Gene Ward Smith (2006-06-02)
Subject: Re: A new rational well-temperament

--- In [email protected], "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.

From: George D. Secor (2006-06-05)
Subject: Re: A new rational well-temperament

--- In [email protected], "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

From: a_sparschuh (2006-06-06)
Subject: Re: A new rational well-temperament

--- In [email protected], "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.