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.